Variations on a theorem of Tate
Stefan Patrikis
Princeton University, Department of Mathematics, Fine Hall, Washington Road, Princeton,
NJ 08544
Current address: Department of Mathematics, The University of Utah, Salt Lake City, UT
84112
E-mail address: patrikis@math.utah.edu
URL: http://www.math.utah.edu/˜patrikis/

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2000 Mathematics Subject Classification. Primary 11R39, 11F80, 14C15
Key words and phrases. Galois representations, algebraic automorphic representations, motives
for motivated cycles, monodromy, Kuga-Satake construction, hyperkähler varieties

Partially supported by an NDSEG fellowship and NSF grant DMS-1303928.
Abstract. Let F be a number field. These notes explore Galois-theoretic, automorphic, and motivic analogues and refinements of Tate’s basic result that continuous projective representations
Gal(F/F) → PGLn (C) lift to GLn (C). We take special interest in the interaction of this result with algebraicity (for automorphic representations) and geometricity (in the sense of Fontaine-Mazur). On
the motivic side, we study refinements and generalizations of the classical Kuga-Satake construction.
Some auxiliary results touch on: possible infinity-types of algebraic automorphic representations;comparison of the automorphic and Galois “Tannakian formalisms”; monodromy (independenceof-`) questions for abstract Galois representations.

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Version 4 - Submitted to MEMOContents
Chapter 1. Introduction
1.1. Introduction
1.2. What is assumed of the reader: background references
1.2.1. Automorphic representations
1.2.2. `-adic Galois representations
1.2.3. Motives
1.2.4. Connecting the dots
1.3. Acknowledgments
1.4. Notation

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Chapter 2. Foundations & examples
2.1. Review of lifting results
2.2. `-adic Hodge theory preliminaries
2.2.1. Basics
2.2.2. Labeled Hodge-Tate-Sen weights
2.2.3. Induction and `-adic Hodge theory
2.3. GL1
2.3.1. The automorphic side
2.3.2. Galois GL1
2.4. Coefficients: generalizing Weil’s CM descent of type A Hecke characters
2.4.1. Coefficients in Hodge theory
2.4.2. CM descent2.5. W-algebraic representations

2.6. Further examples: the Hilbert modular case and GL2 × GL2 →− GL4

2.6.1. General results on the GL2 × GL2 →− GL4 functorial transfer
2.6.2. The Hilbert modular case
2.6.3. A couple of questions
2.7. Galois lifting: Hilbert modular case
2.7.1. Outline
2.7.2. GL2 (Q` ) → PGL2 (Q` )2.8. Spin examples

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Chapter 3. Galois and automorphic lifting
3.1. Lifting W-algebraic representations
3.1.1. Notation and central character calculation
e F)
3.1.2. Generalities on lifting from G(AF ) to G(A3.1.3. Algebraicity of lifts: the ideal case

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3.2. Galois lifting: the general case
3.3. Applications: comparing the automorphic and Galois formalisms
3.3.1. Notions of automorphy
3.3.2. Automorphy of projective representations
3.4. Monodromy of abstract Galois representations
3.4.1. A general decomposition
3.4.2. Lie-multiplicity-free representations

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Chapter 4. Motivic lifting
4.1. Motivated cycles: generalities
4.1.1. Lifting Hodge structures
4.1.2. Motives for homological equivalence and the Tannakian formalism
4.1.2.1. Neutral Tannakian categories
4.1.2.2. Homological motives
4.1.3. Motivated cycles
4.1.4. Motives with coefficients
4.1.5. Hodge symmetry in MF,E
4.1.6. Motivic lifting: the potentially CM case
4.2. Motivic lifting: the hyperkähler case
4.2.1. Setup
4.2.2. The Kuga-Satake construction
4.2.3. A simple case
4.2.4. Arithmetic descent: preliminary reduction
4.2.5. Arithmetic descent: the generic case
4.2.6. Non-generic cases: dim(T Q ) odd
4.2.7. Non-generic cases: dim T Q even
4.3. Towards a generalized Kuga-Satake theory
4.3.1. A conjecture
4.3.2. Motivic lifting: abelian varieties
4.3.3. Coda

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Index of symbols

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Index of terms and concepts

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Bibliography

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Version 4 - Submitted to MEMOCHAPTER 1

Introduction
1.1. Introduction
Let F be a number field, with ΓF = Gal(F/F) its Galois group relative to a choice of algebraic
closure. Tate’s theorem that H 2 (ΓF , Q/Z) vanishes (see [Ser77, Theorem 4] or Theorem 2.1.1 below) encodes one of the basic features of the representation theory of ΓF : since coker(µ∞ (C) → C× )is uniquely divisible, H 2 (ΓF , C× ) vanishes as well, and therefore all obstructions to lifting projective representations ΓF → PGLn (C) vanish. More generally, as explained in [Con11] (Lemmae  H of linear algebraic groups over Q` with central torus
2.1.4 below), for any surjection He ` ). The simplicity of Tate’s
kernel, any homomorphism ΓF → H(Q` ) lifts (continuously) to H(Qtheorem is striking: replacing Q/Z by some finite group Z/m of coefficients, the vanishing result
breaks down.
Two other groups, more or less fanciful, extend ΓF and conjecturally encode the key structural features of, respectively, automorphic representations and (pure) motives. The more remote
automorphic Langlands group, LF , at present exists primarily as heuristic, whereas the motivic
Galois group GF would take precise form granted Grothendieck’s Standard Conjectures, and in
the meantime can be approximated by certain unconditional substitutes, for instance via Deligne’s
theory of absolute Hodge cycles, or André’s theory of motivated cycles (see §4.1.3). In either case,however, we can ask for an analogue of Tate’s lifting theorem and attempt to prove unconditional
results. The basic project of this book is to pursue these analogies wherever they may lead; thefundamental importance of Tate’s theorem is affirmed by the fruitfulness of these investigations,which lead us to reconsider and reinvigorate classic works of Weil ([Wei56]) and Kuga and Satake([KS67]).
We begin by translating the heuristic lifting problem for the (conjectural) automorphic Langlands group into a concrete and well-defined question about automorphic representations. Lete be an inclusion of connected reductive F-groups having the same derived group; basic
G ⊂ G
cases to keep in mind are SLn ⊂ GLn or Sp2n ⊂ GSp2n . For simplicity, we will throughout assume
these groups are split, although many of the results described continue to hold in greater generality.
e∨ denote the respective Langlands dual groups, which are related by a surjective
We let G∨ and Ge∨ → G∨ whose kernel is a central torus in Ge∨ (in the SLn ⊂ GLn case, thishomomorphism G
dual homomorphism is the usual surjection GLn (C) → PGLn (C)); thus on the dual side we havea setup analogous to the one in Tate’s lifting theorem, and indeed the whole point of the Langlands group LF is that, loosely, automorphic representations of G(AF ) should correspond to maps
LF → G∨ . The automorphic analogue of Tate’s theorem is then the assertion that automorphic
e F ), i.e. the fibers of therepresentations of G(AF ) extend to automorphic representations of G(A∨∨efunctorial transfer associated to G → G are non-empty. We show (Proposition 3.1.4) that cuspe F ). This result is elementary, but our analogies
idal representations of G(AF ) do indeed lift to G(Ainsist that we investigate it further. For instance, the analogous lifting result is false for GF , and1

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Version 4 - Submitted to MEMOa priori there is no reason to believe that ‘lifting problems’ for the three groups ΓF , LF , and GF
should qualitatively admit the same answer: a general automorphic representation has seemingly
no connection with either `-adic representations or motives; and a general `-adic representation has
no apparent connection with classical automorphic representations or motives. Two quite distinct
kinds of transcendence–one complex, one `-adic–prevent these overlapping theories from being
reduced to one another. A key problem, therefore, is to identify the overlap and ask how the lifting
problem behaves when restricted to this (at least conjectural) common ground; it is only in posing
this refined form of the lifting problem that the relevant structures emerge.
What automorphic representations, then, do we expect to correspond to `-adic Galois representations or motives? Putting off for the time being precise definitions, we now informally state thefundamental conjectures, due to Fontaine-Mazur, Langlands, Tate, Serre, and Grothendieck, thatlay out the basic expectations.
Conjecture 1.1.1 (Preliminary form: see Conjecture 1.2.1 for the precise version). There is abijection, characterized by local compatibilities, between• “Algebraic” cuspidal automorphic representations of GLn (AF ).
• Irreducible “geometric” Galois representations ΓF → GLn (Q` ).
• Irreducible pure motives over F (with Q-coefficients).
(The only direction of this correspondence whose construction is known is from motives to
Galois representations: simply take `-adic cohomology.) This book will focus on studying theanalogues of Tate’s lifting theorem at this conjectural intersection of the three subjects of automorphic forms, motives, and Galois representations. The automorphic and Galois-theoretic aspects
will be more or less fully treated, and we will find that the motivic aspect is a deep and largely
unconsidered problem; but our Galois-theoretic and automorphic investigations will allow us toframe a precise conjectural response to the motivic lifting question.
To continue, then, we must discuss notions of algebraicity for automorphic representations;then we can see whether the automorphic lifting analogous to Tate’s theorem can be made to respect
this ‘algebraicity.’ Weil laid the foundation for this discussion in his paper [Wei56], by showingthat for Hecke characters (automorphic representations of GL1 (AF )), an integrality condition on
the archimedean component suffices to imply algebraicity of the coefficients of the corresponding
L-series (i.e. algebraicity of its Satake parameters). Waldschmidt ([Wal82]) later proved necessityof this condition. Weil, Serre, and others also showed that a subset, the ‘type A0 ’ Hecke charac×
ters (Definition 2.3.2), moreover give rise to compatible systems of Galois characters ΓF → Q`
(and, via the theory of CM abelian varieties, motives underlying these compatible systems). The
general feeling since has been that the most obvious analogue of Weil’s type A0 condition should
govern the existence of associated `-adic representations. To be precise, let π be an automorphic
representation of our F-group G, and let T be a maximal torus of G. We will use the terminology‘L-algebraic’ of [BG11] as the general analogue of type A0 characters. That is, fixing at each v|∞∼
an isomorphism ιv : F v →− C, we can write (in Langlands’ normalization of [Lan89]) the restriction
to WF v of its L-parameter asrecv (πv ) : z 7→ ιv (z)µιv ῑv (z)νιv ∈ T ∨ (C).
with µιv , νιv ∈ X • (T )C and µιv − νιv ∈ X • (T ) (here and throughout, ῑv denotes the complex conjugateof ιv ). Unless there is risk of confusion, we will omit reference to the embedding ιv , writingµv = µιv , etc.
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Version 4 - Submitted to MEMODefinition 1.1.2. The automorphic representation π is L-algebraic if for all v|∞, µv and νv lie
in X • (T ).
The naı̈ve reason for focusing on this condition is that, for G = GLn , it lets us see the Hodge
numbers of the (hoped-for) corresponding motive. Clozel ([Clo90]), noticing that cohomologicalrepresentations need not satisfy this condition, but have parameters µv , νv ∈ ρ + X • (T ), where ρis the half-sum of the positive roots,1 studied this alternative integrality condition for G = GLn .
Buzzard and Gee ([BG11]) have recently elaborated on this latter condition (‘C-algebraic’) and its
relation with L-algebraicity. For GL1 , the two notions C and L-algebraic coincide, and in generalthey are both useful and distinct generalizations of Weil’s type A0 condition.
But Weil’s ‘integrality condition,’ which he calls ‘type A’ (Definition 2.3.2), is more generalthan the type A0 condition, and its analogues in higher rank seem largely to have been neglected.2
It is of independent interest to resuscitate this condition, but we moreover find it essential forunderstanding the interaction of descent problems and algebraicity, including our original lifting
question. By this we mean the following problem: given connected reductive (say quasi-split)
F-groups H and G with a morphism of L-groups L H → LG, and given a cuspidal L-algebraic
representation π of G(AF ) which is known to be in the image of the associated functorial transfer,
is π the transfer of an L-algebraic representation? The answer is certainly ‘no,’ even for GL1 , but
it fails to be ‘yes’ in a tightly constrained way. I believe the most useful general notion is thefollowing:
Definition 1.1.3. We say that π is W-algebraic if for all v|∞, µv and νv lie in 21 X • (T ).
This is more restrictive than Weil’s type A condition (which allows twists |·|r for r ∈ Q as well),and it is easy to concoct examples of automorphic representations that have algebraic Satake parameters but are not even twists of W-algebraic representations. It is one of our guiding principles,suggested by the (archimedean) Ramanujan conjecture, that all such examples are degenerate, andup to unwinding these degeneracies, all representations with algebraic Satake parameters should
be built up from W-algebraic pieces.
We can now describe the algebraic refinement for automorphic representations of the aforementioned (Proposition 3.1.4) lifting result. The answer depends on whether the field F has realembeddings, so we state it in two parts:
Proposition 1.1.4 (See Proposition 3.1.12). Let F be a CM field, and for simplicity let G be ae be a split F-group containing G as its derived group. Let π
split semi-simple F-group, and let G
be a cuspidal representation of G(AF ). Assume π∞ is tempered.
(1) If π is L-algebraic, then there exists an L-algebraic lift to a cuspidal automorphic repree F ).
sentation π̃ of G(A(2) If π is W-algebraic, then there exists a W-algebraic lift π̃.
In §2.4 we develop a conjectural framework that allows this result to be extended to all totally
imaginary F. See page 10 of this introduction. In contrast, over totally real fields, we have thefollowing:
1

For some choice of Borel containing T . Such a choice was implicit in defining a dual group, with its Borel B∨
containing a maximal torus T ∨ , and the above archimedean L-parameters.
2
One substantial use of type A but not A0 Hecke characters since the early work of Weil and Shimura occurs
in the paper [BR93], in which Blasius-Rogawski associate motives to certain tensor products of Hilbert modular
representations.
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Proposition 1.1.5 (See Proposition 3.1.14). Now suppose F is totally real, with π as before.
e for the center of G,e and Ze0 for its connected compoContinue to assume π∞ is tempered. Write Z3nent.
(1) If π is L-algebraic, then it admits a W-algebraic lift π̃.
(2) For the ‘only if’ direction of the following statement assume Hypothesis 3.1.8. Then thee0 ) lie in X • (ZG ∩ Ze0 )[2], and π admits animages of µv and νv under X • (T ) → X • (ZG ∩ Z
L-algebraic lift if and only if these images are independent of v|∞.
The basic example in which an L-algebraic representation does not lift to an L-algebraic representation is given by a mixed-parity Hilbert modular form π on GL2 (AF ) (F a totally real field notequal to Q). That is, π is the cuspidal automorphic representation associated to a classical Hilbert
modular cusp form whose weights (in the classical sense) at two different infinite places of F havedifferent parities: such a π restricts to an L-algebraic representation of SL2 (AF ), but no twist of
π itself is L-algebraic. In contrast Proposition 1.1.5 tells us that, when F is totally real, there isno obstruction to finding L-algebraic lifts for G a simple split group of type A2n , E6 , E8 , F4 or G2 .
For other groups the obstructions are in fact realizable: Corollary 3.1.15 applies limit multiplicity
formulas (as in [Clo86]) to produce many discrete-series examples generalizing the basic case of
mixed-parity Hilbert modular forms. These examples imply that Arthur’s conjectural construction([Art02]) of a morphism LF → GF requires modification (see Remark 3.1.16).
Propositions 1.1.4 and 1.1.5 give the basic answer to our refined lifting question for automorphic representations. We next turn to the corresponding question for geometric Galois representations; Conjecture 1.1.1 leads us to expect conclusions analogous to those of these two automorphic
results. First, we recall the definition (due to Fontaine-Mazur) of a geometric Galois representation:
Definition 1.1.6. A semi-simple representation ρ : ΓF → GLn (Q` ) is geometric if it is unramified at all but a finite set of primes and is de Rham at all places v|`. (See §2.2.1 for further
discussion of the de Rham condition.) More generally, if H is a linear algebraic group over Q` ,then a representation ρ : ΓF → H(Q` ) is geometric if it is almost everywhere unramified, de Rhamat all places above `, and if the Zariski closure of ρ(ΓF ) in H is reductive.
The Galois question parallel to our refined automorphic lifting results has been raised by Brian
Conrad in [Con11]. In that paper, Conrad, building on previous work of Wintenberger ([Win95])
addresses lifting problems of the forme `) ,
H(Q
=

ρ̃

ΓF

ρ


/ H(Q )`e  H is a surjection of linear algebraic groups with central kernel. He discusses existence
where H(a local-global principle), ramification control, and `-adic Hodge theory properties, and the results
are comprehensive, except for one question (see Remark 1.6 and Example 6.8 of [Con11]):e  H is a torus. If ρ is geometric, when does
Question 1.1.7. Suppose that the kernel of H
there exist a geometric lift ρ̃?
3Note that we use somewhat different notation in §3.1; see the introduction to that section.
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Version 4 - Submitted to MEMOe=Ge∨ , is the natural Galois analogue of the question addressed
This, in the case H = G∨ , H
by Propositions 1.1.4 and 1.1.5; indeed, it provided much of the motivation to understand those
automorphic questions.
Conrad’s Example 6.8 shows the answer is at least ‘not always;’ he produces a character×ψ̂ : ΓL → Q` over certain CM fields L (with F the totally real subfield) such that IndFL (ψ̂) reduces
to a geometric projective representation that has no geometric lift. We give a general solution
to Question 1.1.7, assuming the representations in question satisfy a certain ‘generalized Hodgesymmetry’ property, which is automatic in automorphic or motivic contexts. We formalize thisgeneralized Hodge-Tate weight symmetry in Hypothesis 3.2.4 and then prove (Theorem 3.2.10
and remarks following):
Theorem 1.1.8. Let F be totally imaginary, and let ρ : ΓF → H(Q` ) be a geometric representae ` ).
tion satisfying Hypothesis 3.2.4. Then ρ admits a geometric lift ρ̃ : ΓF → H(QThere is an analogue in the totally real case (Corollary 3.2.8 and Remark 3.2.11), which isparallel to Proposition 1.1.5. Again, the reader may want to keep in mind the key example ofmixed parity Hilbert modular representations: to these one can associate geometric PGL2 -valued
representations that do not lift to geometric GL2 -valued representations. In the preliminary Chapter 2, we work this particular example (and, more generally, similar examples for automorphic
representations of symplectic groups) through in detail, since it is instructive to see precisely how
the automorphic and Galois theories match one another. Then in Chapter 3 we proceed to thegeneral arguments, which of course cannot rely on any such automorphic-Galois correspondence,as this remains entirely conjectural in general. Finally, we remark that it is somewhat surprisingthat the Galois theory is not more complicated than the automorphic theory: roughly speaking,the automorphic input of temperedness is missing, and the obstruction to finding a geometric lift,
say for the simple yet decisive case GLn → PGLn , seems to be a question of 1n -integrality rather
than 12 -integrality; this first appearance is, however, a red herring–it is precisely Hodge symmetry that provides the Galois-theoretic substitute for temperedness–and W-algebraicity remains thecondition of basic importance on the Galois side as well.
We also include a local version of Theorem 1.1.8. Namely, for K/Q` finite, we can ask the same
sorts of lifting questions with ΓK in place of ΓF . A theorem of Wintenberger ([Win95]), in the casee  H a central isogeny, generalized by Conrad to He  H with kernel of multiplicative type,of H
asserts that for any `-adic Hodge theory property P (i.e. Hodge-Tate, crystalline, semi-stable, or deRham) a ρ of type P admits a type P lift if and only if ρ restricted to the inertia group IK admits a
Hodge-Tate lift. This Hodge-Tate lift need not exist in the isogeny case, but we can complete thisstory by showing it exists unconditionally for central torus quotients: see Corollary 3.2.12. The
argument is a simpler version of that of Theorem 3.2.10, requiring no additional assumption on‘Hodge-Tate symmetry.’ This question too was suggested by Conrad.
Having addressed the refined (“algebraic” and “geometric,” respectively) lifting problems in
the automorphic and Galois-theoretic settings–and, reassuringly, having found the answers matchup under the correspondence of Conjecture 1.1.1 (or, more precisely, Conjecture 1.2.1)–it remains
to consider the analogous motivic lifting problem. Before formulating this precisely, however, let
us explain how quite concrete Galois-theoretic considerations already point to a fundamental gap
in the Galois-theoretic story, whose implications will demand that we embrace the formalism of
the motivic Galois group. If ` , `0 are distinct primes, recall that, in spite of the incompatibilityof the `- and `0 -adic topologies, there is sometimes a meaningful way to compare a pair of Galois
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Version 4 - Submitted to MEMOrepresentations ρ` : ΓF → GLn (Q` ) and ρ`0 : ΓF → GLn (Q`0 ). Namely, having fixed embeddings
Q ,→ Q` and Q ,→ Q`0 , ρ` and ρ`0 are said to be weakly compatible if for all but a finite set ofprimes S of F, both representations are unramified outside S , and for v < S the characteristic
polynomials of ρ` (frv ) and ρ`0 (frv ) have coefficients in Q (with respect to the fixed embeddings)
and are in fact equal. We then extend this to a notion of weak compatibility for any collection of`-adic representations as the prime ` varies. Deligne’s work on the last of the Weil conjectures([Del74]) implies that this compatibility holds for the Galois representations arising from `-adic
cohomology (for all `) of smooth projective varieties over F.
Similarly, we can formulate a notion of compatibility for a pair of representations ρ` : ΓF →H(Q` ) and ρ`0 : ΓF → H(Q`0 ), where now H is any linear algebraic group over Q. In fact, there area few possible notions, which may not be equivalent for groups other than GLn ; so as not to burden
the discussion, we put off precise formulations until §1.2.4 and §3.3.1. In light of Theorem 1.1.8,
it is then natural to ask the following question:e → H be a surjection of linear algebraic groups over Q with kernel
Question 1.1.9. Let H
equal to a central torus. When does a weakly compatible system of geometric representationse ` )?
ρ` : ΓF → H(Q` ) lift to a weakly compatible system of geometric representations ρ̃ : ΓF → H(QeEven when, as for instance ensured by Theorem 1.1.8, all ρ` lift to geometric H-valued
representations, compatible lifts may not exist! See Example 2.1.10 for the prototypical counterexample; this phenomenon is quite familiar from the automorphic theory, where it is closely related
to questions of automorphic multiplicities and endoscopy. How, then, can we address Question
1.1.9? The only way I know involves both assuming and proving much more: granting that the ρ`are `-adic avatars of a (pro-algebraic) representation of the motivic Galois group GF,E of motiveseover F with coefficients in a number field E (implicit here are specified E-forms of H and H),
e possibly after enlarging E. We haveand showing that this motivic Galois representation lifts to H,
therefore come full-circle to the motivic lifting-problem raised at the beginning of this introduction. This is far more difficult than the corresponding automorphic and Galois questions, but wecan unconditionally treat certain examples.
In order to pose the question precisely, we need a category of ‘motives’ in which the Galois
formalism of GF is unconditional. There are two common constructions, one based on Deligne’s
theory ([DMOS82]) of absolute Hodge cycles, the other based on André’s theory ([And96b]),much inspired by Deligne’s work, of motivated cycles. We work with motivated cycles, since theinclusions
algebraic cycles ⊆ motivated cycles ⊆ absolute Hodge cycles ⊆ Hodge cycles
more or less imply that results about motivated cycles (eg, ‘Hodge cycles are motivated’) follow
a fortiori for absolute Hodge cycles. Note that the Hodge conjecture asserts that each ⊆ is anequality; Deligne’s ‘espoir’ ([Del79, 0.10]) asserts that the last ⊆ is an equality. We let MF,E denotethe category of motives for motivated cycles over F with coefficients in E (see §4.1.3 and 4.1.4).
This is a semi-simple E-linear Tannakian category, and, choosing an embedding F ,→ C, we canassociate (via the E-valued Betti fiber functor) a (pro-reductive) motivic Galois group GF,E . Notethat Serre has asked ([Ser94, 8.3]) whether homomorphisms GF → PGL2 (note the algebraicallyclosed ground field) lift to GL2 . This is closely related to the question, also raised by Serre, of
whether the derived group of GF is simply-connected, but Serre notes that questions of this sort
do not have obvious conjectural answers, even if we assume we are in le paradis motivique. Our
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geometric Galois and algebraic automorphic lifting results (in combination with Conjecture 1.1.1
or Conjecture 1.2.1) suggest the following sharpening and generalization of Serre’s question, that
this motivic lifting phenomenon is utterly general:e  H be a surjection, with
Conjecture 1.1.10. Let F and E be number fields, and let H
central torus kernel, of linear algebraic groups over E. Suppose we are given a homomorphismρ : GF,E → H. Then if F is imaginary, there is a finite extension E 0 /E and a homomorphismeE 0 lifting ρ ⊗E E 0 . If F is totally real, then such a lift exists if and only if the Hodgeρ̃ : GF,E 0 → H
number parity obstruction of Corollary 3.2.8 vanishes.
But what does this conjecture mean? There is in fact one classical construction in complex
algebraic geometry, of a very special Hodge-theoretic nature, that fits into this framework. Namely,
Kuga and Satake ([KS67]) showed in the late 1960’s how, given a complex K3 surface X, one could
construct a complex abelian variety KS (X) related to X by an inclusion of rational Hodge structures
H 2 (X, Q) ⊂ H 1 (KS (X), Q)⊗2 .
Implicit in Deligne’s work ([Del72]) on the Weil conjectures for K3 surfaces, in which he used the
Kuga-Satake construction to reduce the case of K3 surfaces to the (known) case of abelian varieties,
is the perspective that the Kuga-Satake construction arises from lifting an SO-valued representation(of the Deligne torus of Hodge theory, for instance) through the surjection GSpin → SO. André([And96a]), building on work of Deligne, showed that the above inclusion of Hodge structures ismotivated, i.e. derives from a morphism in MC , making the motivic Galois group appear more
explicitly in the discussion of the Kuga-Satake construction. He also proved a weak arithmetic
descent result, showing that if the K3 X is defined over some subfield F ⊂ C, then KS (X) canbe descended to some finite extension of F. Our Conjecture 1.1.10 suggests a sharpening of thisarithmetic descent (down to the field F itself), and we will indeed take up this question in Chapter
4.
Before describing these results, however, let us pause to consider the scope of Conjecture
1.1.10. The classical Kuga-Satake construction depends on a lucky numerical coincidence: it isstraightforward to lift the Hodge structure associated to the K3 to a GSpin-valued Hodge structure,and it turns out we get exactly the right weights (in the natural representation of GSpin on the Clifford algebra) to have a Hodge structure that looks like it comes from H 1 of an abelian variety. And
then Riemann’s theorem tells us that all such Hodge structures do come from abelian varieties!
Riemann’s remarkable theorem, however, is a miraculous exception; in general, we understand almost nothing about the essential image of the Hodge-Betti realization functor from MC to rational
Hodge structures. For example, the reader could write down almost any smooth projective variety
X/F, say of even dimension d, and the analogous motivic lifting problem (or even its base-change
to C, let alone the arithmetic refinement)
GSpin(H d (X)( d2 ))
8

ρ̃

GF

ρ


/ SO(H d (X)( d ))2

would be totally out of reach. (Here we write H d (X)( d2 ) for the object of MF corresponding tod-fold Tate twist of the degree d cohomology of X; and we have possibly made a quadratic base2change on F to obtain an SO-valued rather than O-valued ρ.) In essence, each new case of the7

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“generalized Kuga-Satake” Conjecture 1.1.10 requires us, in light of our Galois-theoretic lifting
results (Theorem 1.1.8) to prove a new case of the Fontaine-Mazur conjecture.
That said, let us turn to some positive results toward Conjecture 1.1.10. The crucial limitation
shared by all results to be discussed here is that they take place entirely in the Tannakian categoryof motives generated by abelian varieties (and Artin motives), the reliance on Riemann’s theorem
being the essential crutch. We will return at the end of the introduction to mention some later workthat begins to push the boundaries of the conjecture. In Chapter 4 we prove a lifting result of theform
GSpin(VE )
9

ρ̃

GF,E


/ SO(VE )ρfor certain ρ arising from degree 2 primitive cohomology VQ of a hyperkähler variety (see Definition 4.2.1) over F (or, rather, for ρ having analogous formal properties); the extension of scalars
VE = VQ ⊗Q E is essential for the lifting result to hold. The starting-point for this refined motivic
lifting result is, again, the work of André ([And96a]) on the ‘motivated’ theory of hyperkählers; asmentioned above, his results imply a version of this lifting statement with F replaced by a large but
quantifiable (with considerable work: see Theorem 8.4.3 of [And96a]) finite extension F 0 /F. Ourcontribution is the arithmetic descent from F 0 to F, and for this we generalize a technique introduced by Ribet ([Rib92]) to study so-called ‘Q-curves,’ elliptic curves over Q that are isogenous
to all of their ΓQ -conjugates. Passing from a softer geometric statement (compare: ‘a variety overF has a point over some finite extension’) to a precise arithmetic version (compare: ‘when does avariety over F have a point over F?’) typically requires some deep input; in this case, Faltings’s
isogeny theorem ([Fal83]) does the hard work. The method requires a case-by-case analysis (depending on the motivic group of the transcendental lattice), and I have decided not to pursue all
the cases here, but here is a partial result (see Theorem 4.2.13, Theorem 4.2.31, and Proposition
4.2.33 and following for more cases and more precise versions):
Theorem 1.1.11. Let (X, η) be a polarized variety over F satisfying André’s conditions Ak , Bk , B+k(see §4.2.1). For instance, with k = 1, X can be a hyperkähler variety with second Betti number
greater than 3. Suppose that the transcendental lattice T Q ⊂ VQ = Prim2k (XC , Q)(k) satisfies either• EndQ-HS (T Q ) = Q; or• T Q is odd-dimensional.
For simplicity, moreover assume that det V` = det T ` = 1 as ΓF -representations.4 Then after somefinite scalar extension to VE = VQ ⊗Q E there is a lifting of the motivic Galois representation
ρV : GF → SO(VQ ):GSpin(VE )
9

ρ̃

GF,E


/ SO(VE ).

ρV

4This can be achieved in each case after a quadratic, independent-of-` extension on F. It is only so we canwork with SO(V` ) rather than O(V` ), but since our abstract Galois-lifting results apply to non-connected groups (seeTheorem 1.1.8, for instance), this hypothesis is not essential.
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Moreover, this lifting gives rise on λ-adic realizations (for all finite places λ of E) to lifts ρ̃λ : ΓF →GSpin(Vλ ) whose compositions with the spin (or sum of half-spin) representations form a weakly
compatible system.
For more cases of Theorem 1.1.11 in which T Q is even-dimensional, see §4.2.7; the omitted
cases should yield to similar methods. We have not pursued in depth whether Question 1.1.9 canbe answered in this setting, i.e. whether the ρ̃λ can actually be taken to be weakly compatible
as GSpin-valued representations (in the sense of Definition 1.2.3 below). For a partial result,
see Corollary 4.2.11. We stress that this compatibility of λ-adic realizations is not automatic forAndré’s motives (nor for absolute Hodge motives), so we need to exploit an explicit descriptionof the lift. Moreover, the excluded case b2 (X) = 3 of Theorem 1.1.11 is related to a more general
result for potentially abelian motives: we prove such a lifting result across an arbitrary surjectione → H with central torus kernel: see Proposition 4.1.30 and Lemma 4.2.5.
H
Our next example seems to be a novel result even in complex algebraic geometry, where many
authors have studied variants of the Kuga-Satake construction for Hodge structures of ‘K3-type’(see, for example, [Mor85], [Voi05], [Gal00]). Generalizing the well-known case ([Mor85]) of H 2of an abelian surface–which has a Hodge structure of K3 type, so falls within the ken of classicalKuga-Satake theory5–we prove an analogous GSpin → SO motivic lifting result for H 2 of any
abelian variety (see Corollary 4.3.3). Moreover, we can describe the corresponding motive (in thespin representation) as a Grothendieck motive, without assuming the Standard Conjectures.
In all, the Galois-theoretic, automorphic, and limited motivic evidence (the cases of potentiallyabelian motives, hyperkähler motives, and abelian varieties) should encourage optimism about
Conjecture 1.1.10. The crucial next step would be to verify interesting “non-classical” examplesof Conjecture 1.1.10. After the initial writing of this book, I found two ways to produce such
examples, both relying on the theory of rigid local systems: the interested reader can consult [Pat]and [Pat14a] for these developments. There is clearly vast terrain waiting to be discovered here,some of which is not entirely hostile to exploration.
I have also made progress, in [Pat16], on a weaker version of Question 1.1.9. In the present
book, we do not emphasize questions of ramification control. The geometric representationsρ : ΓF → H(Q` ) of Theorem 1.1.8 are unramified outside some finite set S of primes of F, and thee ` ) produced by that theorem are unramified outside some, possibly larger, finite
lifts ρ̃ : ΓF → H(Qset Se ⊇ S . Achieving sharp (quantitative) control of the ramification set Se is a deep problem, but asofter version still provides a more demanding Galois-theoretic test of the generalized Kuga-Satakeconjecture. Namely, if {ρι : ΓF → H(Q` )}ι : Q,→Q` is the collection of `-adic realizations of a motivic
Galois representation ρ : GF,Q → H valued in a Q-group H, then there is a finite set S such that eachρι is unramified outside the union of S and the places S ` above ` (where ι : Q ,→ Q` ). Certainly
a necessary condition for Conjecture 1.1.10 to hold is that the representations ρι should all admit
geometric lifts ρ̃ι unramified outside Se ∪ S ` for a finite, independent-of -`, set Se. A strengtheningof this necessary condition, including the fact that the lifts ρ̃ι can be taken to be crystalline outsideof the bad set Se, is proven in [Pat16, Corollary 1.1]; the basic difficulty of such a result is that one
has to kill cohomological obstructions lying in infinitely many different cohomology groups, one
5

The paper [Gal00] treats the case of a particular family of abelian four-folds, where a constraint on the MumfordTate group allows one to extract a Hodge structure of K3-type; as we show, though, the lifting phenomenon is completely general.
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Version 4 - Submitted to MEMOfor each ι, but in an independent-of-ι manner. This result provides the natural generalization of thetheorem of Wintenberger (Theorem 2.1.8 below) to the case of central torus quotients; and indeed
it implies (with a different proof) Theorem 2.1.8: see [Pat16, Corollary 3.20, Corollary 3.22]. We
also remark that the method of proof yields a new demonstration ([Pat16, Corollary 3.12]) of thebasic theorem of Tate, whose classical proof is recalled in Theorem 2.1.1 below.
In the remainder of this introduction, we describe a few other general themes that recur throughout these notes, in automorphic, Galois-theoretic, and motivic variants. The first is the systematic
exploitation of ‘coefficients,’ and the general principle that our arithmetic objects will naturally
have coefficients in CM fields. In §2.4, we use this principle to formulate (and prove, for regular
representations; see Proposition 2.4.8) a conjectural generalization of Weil’s result that a type A
Hecke character of a number field F descends, up to finite order twist, to the maximal CM subfield
Fcm ; this generalization asserts that the infinity-type of an L-algebraic cuspidal automorphic representation of GLn (AF ), over a totally imaginary field F, necessarily descends to Fcm . In §4.1.5, we
use this principle–descent of infinity-types to the CM subfield, which in the motivic context follows from the Hodge index theorem–to show that André’s motives satisfy the Hodge-Tate weightsymmetries needed for the Galois lifting Theorem 1.1.8. Finally, Corollary 3.4.14 gives an example of how having CM coefficients can be exploited even in the study of abstract compatible
systems of `-adic representations. Since the writing of this paper, the same principles have been
applied in [PT15] to establish new results on the potential automorphy of regular motives, and theirreducibility of automorphic Galois representations.
Corollary 3.4.14 depends on the following abstract independence-of-` result, which may be of
independent interest (see Proposition 3.4.9 for a more general statement):
Proposition 1.1.12. Let ρλ : ΓF → GLn (Eλ ) be a compatible system of irreducible, Lie-multiplicityfree representations of ΓF with coefficients in a number field E, so that ρλ  IndFLλ (rλ ) for someLie-irreducible representation rλ of ΓLλ , where the number field Lλ a priori depends on λ. Thenthe set of places of F having a split factor in Lλ is independent of λ. If we further assume that theLλ /F are Galois, then Lλ is independent of λ.
By Lie-multiplicity-free, we mean multiplicity-free after all finite restrictions. One application is a converse to a theorem of Rajan (Theorem 4 of [Raj98]), also generalizing a result of Serre(Corollaire 2 to Proposition 15 of [Ser81]). See §3.4 for further discussion, as well as the aformentioned application (Corollary 3.4.14), which is a weak Galois-theoretic shadow of the automorphic
Proposition 2.4.8. In §3.4, we also record a variant for number fields of a result of Katz (for affine
curves over a finite field; see [Kat87]), which in particular clarifies the place in the general theory
occupied by Lie-multiplicity-free (or more specifically, Lie-irreducible) representations:
Proposition 1.1.13. Let ρ : ΓF → GLn (Q` ) be an irreducible representation. Then either ρ isinduced, or there exists d dividing n, a Lie irreducible representation τ of dimension dn , and an
Artin representation ω of dimension d such that ρ  τ ⊗ ω. This decomposition is unique up to
twisting by a finite-order character. Consequently, any (irreducible) ρ can be written in the formρ  IndFL (τ ⊗ ω)
for some finite L/F and irreducible representations τ and ω of ΓL , with τ Lie-irreducible and ω
Artin.
This is a very handy result, which despite its basic nature does not seem to be widely-known.
Some of our descent arguments in §4.2 rely on it.
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Finally, time and again our arguments are buttressed by some explicit knowledge of the constraints on infinity-types of automorphic representations, and on Hodge-Tate weights of Galois
representations. More generally, we find that asking finer structural questions about the interaction of functoriality and algebraicity naturally leads to existence (and non-existence) problems
that are often dissociated from functoriality; as we will see in §2.1, this theme is present even in
the proof of Tate’s original vanishing theorem H 2 (ΓF , Q/Z) = 0. We certainly have many morequestions than answers (scattered through §’s 2.3-2.7; see eg Corollary 2.7.8 for a curious positive result), although fortunately our main results depend largely on detailed study of GL1 (§2.3),
which provides both the technical ingredients for later arguments and the motivation for higherrank results (namely, after a close reading of Weil’s [Wei56], the definition of W-algebraicity andthe key descent principle of Proposition 2.4.8). In this latter respect, I do not believe that GL1 has
yielded all of its fruit–see the discussion surrounding Corollary 2.3.9–but the extreme difficultyof establishing the (non-)existence of automorphic representations with given infinity-types, evenin qualitative form, necessarily tempers further conjecture. We also use the GL1 -theory, in combination with Conjecture 1.2.1, to show how the conjectural automorphic-Galois correspondence
for the group GLn implies an analogous correspondence for SLn : see §3.3.2. For an unconditional
construction, building on the arguments of §3.3.2, of geometric projective Galois representations
associated to certain ‘mixed-parity’ automorphic representations on higher-rank Spin-groups, see[Pat14b].
For a reader interested only in certain aspects of these notes, I hope that the table of contents
is a clear reference to the points of interest. Roughly, Chapter 2 contains preliminary material andmany examples arising from the groups GL1 and GL2 . Chapter 3 contains the general automorphic
and Galois-theoretic lifting theorems, as well as ‘abstract’ results about monodromy groups ofGalois representations. Chapter 4 contains the discussion of motivic lifting problems, culminatingin the statement of the generalized Kuga-Satake conjecture, Conjecture 4.3.1.
1.2. What is assumed of the reader: background references
Because this book studies all three vertices–and, to a far lesser extent, the edges–of the mysterious Galois-automorphic-motivic triangle, the reader will need some limited familiarity with all
three subjects. In this subsection, we indicate what background will be assumed, give some usefulreferences, and also explain that while the background required may be broad, it is not terribly
deep; indeed, the arguments are largely elementary, once some basic definitions are assimilated.
Let us deal individually with each of these three topics.
1.2.1. Automorphic representations. Preliminary to the study of automorphic representations, one must have some familiarity with the theory of reductive algebraic groups. For simplicity, we always restrict to split groups, so it is sufficient to know the theory over algebraicallyclosed fields. Many arguments in this paper rest on the manipulation of root data; Springer’s survey [Spr79] and his book [Spr09] (especially chapters 7-10) are very clear, and provide morethan enough background. We also assume familiarity with the algebraic representation theory of(connected) reductive groups (i.e., elementary highest-weight theory).
We use a little of the representation theory of reductive groups over local fields. Familiarity
with unramified representations of split p-adic groups and the Satake correspondence ([Gro98]
is a beautiful guide; [Car79, III] is a thorough treatment of unramified representations; [Cas] isthe best general introduction to the theory of admissible smooth representations), and with some11

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Version 4 - Submitted to MEMOaspects of the formalism of the archimedean local Langlands correspondence will suffice. As forthe global theory of automorphic representations themselves, and the theory of the L-group, thestandard and more than adequate reference is [Bor79]; [Bor79, §9-11] would be particularly usefulbackground reading, sketching the archimedean theory and, crucially for our purposes, explaining
the desired, in some cases proven, connection between central characters and L-parameters.
1.2.2. `-adic Galois representations. Apart from some very elementary notions, we only
require some familiarity with the formal, non-geometric, aspects of `-adic Hodge theory, whichis to say the study of `-adic representations of ΓK for a finite extension K/Q` (when studied in apurely local context, ` is traditionally replaced by p, as in ’p-adic Hodge theory’). An excellent
overview, with references to detailed proofs, is [Ber04, §I-II]. A ‘text-book-style’ reference (veryuseful but not quite in final form) for everything we do is [BC].6 Finally, we should remark to areader new to p-adic Hodge theory that the subject has been much in flux in the last 10 years, andthat one should take heed of recent conceptual advances (eg [Bei12], [Bha12], [FF12], [Sch13])
before reading too deeply into the ‘classical’ theory.
1.2.3. Motives. Here we recommend some familiarity with the notion of motive, as conceived
by Grothendieck, and with Grothendieck’s Standard Conjectures on algebraic cycles. Kleiman’s
articles [Kle68] and [Kle94] provide a clear, careful, and concise introduction. Certainly familiarity with the various cohomological realizations (Betti, de Rham, `-adic), and the relations between
them, of a smooth projective variety is necessary; the first section, ‘Review of Cohomology,’ ofDeligne’s article [DMOS82, §I.1] will provide quick and easy orientation for someone new to thesubject. We will also require (some of) the theory of Tannakian categories; we will give a brief
introduction, which should be enough for a reader to follow all of our arguments, but a thoroughtreatment is [DM11] (originally [DMOS82]). Finally, although not necessary for the present work,
the excellent book [And04] provides a broad survey of the theory of motives, from its inception tomore recent developments.
1.2.4. Connecting the dots. These notes rely on the systematic transfer of intuition back andforth between these three areas. We now make precise the towering conjectures that dominate thisconceptual landscape. The following collection of conjectures (crudely stated before as Conjecture
1.1.1), and the consequences of combining the various parts, summarize the principal problems in
the field (in contrast to Conjecture 1.1.1, here we give an asymmetric formulation in order todistinguish the original Fontaine-Mazur conjecture from stronger versions).
Conjecture 1.2.1 (Fontaine-Mazur, Langlands, Tate, Grothendieck-Serre). Let ρ : ΓF → GLn (Q` )be an irreducible geometric Galois representation. Recall from the introduction that this means ρ
is almost everywhere unramified, and is de Rham at all places above `. Then:(1) (Fontaine-Mazur) There exists a smooth projective variety X/F and an integer r such that
ρ is isomorphic to some sub-quotient of H j (XF , Q` )(r).
(2) (Grothendieck-Serre) For any smooth projective variety X/F, each H j (XF , Q` ) is a semisimple ΓF -representation.
(3) (Above plus Tate) Such a ρ is motivic in the sense that it is cut out by Q` -linear combinations of (homological) algebraic cycles. More precisely, for any embedding ι` : Q ,→ Q` ,6

It is also highly recommended to visit Laurent Berger’s website to check on the status of the course-notes fromhis course at IHP in the Galois Trimestre of 2010.
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0
there is an idempotent project e in the algebra Chom
(X, X)Q (see [Kle68, §1.3.8], where
this is denoted A(X)) of self-correspondences of X with Q-coefficients, such thatρ  e(H ∗ (X)(r)Q ) ⊗Q,ι` Q`
as ΓF -representations.
(4) (Above plus Langlands) Fix an embedding ι∞ : Q ,→ C. There exists a cuspidal automorphic representation Π of GLn (AF ) such that for almost all v unramified for ρ, theeigenvalues of ρ(frv ) agree with the Satake parameters of Πv , viewed in Q` via the compo7
sition ι` ◦ ι−1∞ (recv (Πv )). Moreover for ι : F ,→ C, the archimedean Langlands parameterrecv (Πv )|Fv × : WFv → GLn (C),landing in a maximal torus T n , is of the form z 7→ zµι z̄νι for µι , νι ∈ X• (T n ), and for anyτ : F ⊂ Fv ,→ Q` , the Hodge-Tate co-character µτ of ρ|ΓFv is (conjugate to) µι∗∞,` (τ) .8
Conversely, given a cuspidal automorphic representation Π of GLn (AF ) with integral
archimedean Langlands parameters, there exists an irreducible geometric Galois representation ρΠ : ΓF → GLn (Q` ) (depending on ι∞ , ι` ) satisfying the above compatibilities.
Remark 1.2.2.
(1) Note that the Standard Conjectures are a special case of part 3 of Conjecture 1.2.1. We will rarely work directly with motives for homological equivalence inthese notes, substituting instead André’s category of motivated motives. Keeping one’s
faith in the Standard Conjectures, but wanting to assume less at the outset, one could
0
substitute the algebra of motivated correspondences Cmot
(X, X) (Definition 4.1.13 and fol0
lowing) for Chom (X, X) in Conjecture 1.2.1.
(2) This is not a historically faithful presentation of these conjectures; for our purposes inthese notes, however, this formulation is convenient.
(3) Part 4 implies a similar correspondence between semi-simple (not necessarily irreducible)
geometric Galois representations and (suitably algebraic) isobaric automorphic representations of GLn (AF ). Alternatively, one can begin from this more general conjecture anddeduce that cuspidal automorphic representations must correspond to irreducible Galois
representations using standard properties of Rankin-Selberg L-functions.
The Fontaine-Mazur-Tate conjecture challenges us, given a geometric `-adic representation ρ` ,to produce a motive with ρ` as `-adic realization, and in particular to produce a family of `0 -adic
realizations (for all `0 ) that are ‘compatible’ with ρ` . It is often convenient to abstract this notionof compatible system of Galois representations, as an intermediary between the isolated ρ` and therobust motive.
Definition 1.2.3. Let F and E be number fields, and let N be a positive integer. A rank N
weakly compatible system of λ-adic (or, informally, `-adic) representations of ΓF with coefficientsin E is a collection
R = {ρλ }λ , S , {Qv (X)}v<S ,Implicit in the conjecture is that the Satake parameters of Π are algebraic; of course this is not the case for generalautomorphic representations. A stronger version moreover asks that for all finite places v,
7

WD(ρ|ΓFv ) f r−ss  ι` ◦ ι−1∞ (recv (Πv ))ι∞,` (τ) is the pullback of τ to an archimedean embedding via ι` , ι∞ ; see §1.4.

8∗

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consisting of:(1) for each finite place λ of E, a continuous semi-simple geometric representation
ρλ : ΓF → GLN (Eλ );(2) a finite set of places S of F, containing the infinite places, such that for all v < S and forall λ of different residue characteristic from v, ρλ |ΓFv is unramified;(3) for all v < S , a polynomial Qv (X) ∈ E[X] such that for all λ of different residue characteristic from v, Qv (X) is the characteristic polynomial of ρλ (frv ).
We will sometimes use a similar notion where GLN is replaced by a, for simplicity, split connected
reductive group H over the number field of coefficients E. In this case, repeat the above definition
verbatim, except replace the characteristic polynomial of ρλ (frv ) with the analogous point of thespace of Weyl-invariant functions on a maximal torus in H, i.e. the image of ρλ (frv ) ∈ H(Eλ ) under
the Chevalley map induced by the map on coordinate rings∼

E[H] ⊃ E[H]H −−→ E[T ]W ,res

where T is an E-split maximal torus, and W is the Weyl group of (H, T ). That is, we ask that forall λ, the associated E λ -points of E[T ]W in fact arise from a common E-point.
Motivated again by [Del74], we will sometimes impose a purity hypothesis on R:
Definition 1.2.4. Fix an integer w. For any finite place v of F, let qv denote the order of theresidue field of F at v. We say that a weakly compatible system R is pure of weight w ∈ Z if for adensity one set of places v of F, each root α of Qv (X) in E satisfies |ι(α)|2 = qwv for all embeddingsι : E ,→ C.
Another standard variant of the definition of weakly compatible system would also specify
that the `-adic Hodge numbers are suitably ‘independent of `,’ but we will not require this; seefor instance [BLGGT14, §5.1]. Finally, an elementary argument (see, eg, [CHT08, Lemma
2.1.5]) shows that any continuous Eλ -representation of ΓF (a compact group) takes values in some
GLN (E 0 ) for some finite extension E 0 of Eλ .
1.3. Acknowledgments
Some of this work was carried out with the support of an NDSEG fellowship, and revisions
were made with the support of NSF grant DMS-1303928. I thank the Institut Henri Poincaré,
which I visited for the 2010 Trimestre Galois, and the Oxford University mathematics department,
which I visited in 2011 − 2012, for their hospitality. I of course am grateful to the Princeton mathdepartment, both for its official support, and for its singular camaraderie.
The anonymous referees made many suggestions which have greatly improved the readabilityof this book; I heartily thank them, and also thank Kevin Buzzard for helpful comments and corrections. I am especially grateful to the referee who twice took the time to report on substantiallydifferent versions of this book.
Above all, it gives me great pleasure to thank the following people. Brian Conrad’s paper[Con11] catalyzed much of the present work: indeed, I was stumbling towards the questions studied in [Con11] when Brian posted his paper, and the clarity and generality of the questions heraised put my ideas into the sharpest focus. Having posed such fruitful questions, Brian moreover
generously stepped back to allow me the time to pursue my ideas. He further provided very helpful
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comments on an earlier draft. I am grateful to Peter Sarnak for his encouragement and for several
enjoyable conversations about this material, and to Christopher Skinner for reading an earlier draft.
Most of all, I thank Andrew Wiles, my advisor, for his ongoing support and patient encouragement
during my graduate career.
1.4. Notation
We end this introductory chapter by collecting some of the most frequently-used notation inthis text. The reader may also refer to the notation index to see where a given piece of notation is
introduced. For a number field F, we always choose an algebraic closure F/F and let ΓF denote
Gal(F/F). We write C F = A×F /F × for the idele class group of F.
If L/F is a finite extension inside F and W a representation of ΓL (= Gal(F/L)), we abbreviateIndΓΓFL (W) to IndFL (W). For g ∈ ΓF , we write (gW) for the conjugate representation of gΓL g−1 .
We fix separable closures Q and Q` of Q and Q` for all `. We fix throughout embeddingsι` : Q ,→ Q` and ι∞ : Q ,→ C. An archimedean embedding ι : F ,→ C thus induces an `-adic
embedding τ∗`,∞ (ι) = ι` ◦ ι−1∞ ◦ ι : F ,→ Q` ; and similarly an `-adic embedding τ : F ,→ Q` induces∗ι∞,` (τ) : F ,→ C. If there is no risk of confusion, we just write τ∗ (ι) and ι∗ (τ). These embeddings
will be invoked (often implicitly) whenever we associate automorphic forms and Galois representations.
For a connected reductive F-group G, we construct dual and L-groups G∨ and LG over Q(having chosen a maximal torus, Borel, and splitting, although we will only ever make the maximal
torus explicit), and then use ι` and ι∞ to regard the dual group over Q` or C as needed.
If v is a place of F, we denote by recv the local reciprocity map from irreducible admissible
smooth representations (v finite) or irreducible admissible Harish-Chandra modules (v infinite) to(frobenius semi-simple) representations of the Weil(-Deligne) group of Fv . We use this in theunramified and archimedean cases, and for GLn . For the group GL1 , this is local class field theory,
normalized so that uniformizers correspond to geometric frobenii, which we denote frv .
It will be convenient to have a short-hand for the assertion that an automorphic representation
and Galois representation ‘correspond’ under the conjectural global Langlands correspondence.
We will elaborate on a number of related notions in §3.3.1, but for now we give two that will comeup frequently. Recall that we have fixed embeddings ι` and ι∞ of Q into Q` and C, respectively.
If π is an automorphic representation of G(AF ), and ρ : ΓF → LG(Q` ) is a continuous, almost
everywhere unramified, representation, then we write ρ ∼w π if for almost every unramified placev of F (at which we may assume ρ is unramified), recv (πv ) : WFv → LG(C) can be realized (up toι∞G∨ -conjugation) over Q −→ C, and that then
fr−ss(recι` ◦ ι−1
(π
))
∼
ρ|
,
v v
ΓFv
∞
where ∼ here denotes G∨ (Q` )-conjugacy, and ‘fr-ss’ means we replace ρ(frv ) with its semi-simple
part. More concretely, we restrict to places v at which both π and ρ are unramified, so that recv (πv )and ρ|ΓFv are both just given by their evaluations at frv ; after using ι` ◦ ι−1∞ to regard them as defined∨over the same field, we ask that these two elements be G (Q` )-conjugate.
As in the Fontaine-Mazur-Langlands Conjecture 1.2.1, we also often want to compare thearchimedean component of π with the Hodge-Tate weights of ρ; when π is L-algebraic (see Definition 1.1.2), and ρ is de Rham, we may write, for all ι : F ,→ C, the archimedean Langlands
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parameter recv (πv )|F ×v in the formz 7→ ι(z)µι ῑ(z)νιfor µι , νι ∈ X• (T ∨ ). Then write ρ ∼w,∞ π if ρ ∼w π, and moreover for all such ι : F ,→ C, and forall τ : F ⊂ Fv ,→ Q` , the τ-labeled Hodge-Tate co-character (see §2.2.2) of ρ|ΓFv is (conjugate to)µι∗∞,` (τ) .
By a CM field we mean as usual a quadratic totally imaginary extension of a totally real field;
these, and their real subfields, are the number fields on which complex conjugation is well-defined,
independent of the choice of complex embedding. For any number field, we write Fcm for the
maximal subfield of F on which complex conjugation is well-defined; thus it is the maximal CM
or totally real subfield, depending on whether F contains a CM subfield. We also write Qcm for theunion of all CM extensions of Q inside Q.
For any topological group A, we denote by AD the group Homcts (A, S1 ) (S1 is the unit circle);when A is abelian and locally compact, we topologize this as the Pontryagin dual.
For any ground field k (always characteristic zero for us) with a fixed separable closure k s , ande for the normal (Galois) closure of Ka (separable) algebraic extension K/k inside k s , we write Ksover k inside k .
We always denote base-changes of schemes by sub-scripts: thus, Xk0 is the base-change X×Spec k
Spec k0 for a k-scheme X and a base-extension Spec k0 → Spec k. Notation such as M ⊗k k0 will bereserved (see §4.1.4) to describe extension of scalars from an object M of a k-linear abelian category to one of a k0 -linear abelian category; when there is no risk of confusion we will sometimes
denote this also by Mk0 .
Finally, we will signal either a significant change in running hypotheses, or an essential yeteasily-overlooked point, with the ‘dangerous bend’ symbol 

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Version 4 - Submitted to MEMOCHAPTER 2Foundations & examples
This chapter discusses the motivating examples and technical ingredients that underly the general arguments of Chapter 3. After (§2.1) recalling foundational lifting results of Tate, Wintenberger, and Conrad, and (§2.2) setting up the tools we will need from `-adic Hodge theory, we
undertake a detailed discussion (§2.3) of Hecke characters and `-adic Galois characters, especiallywith reference to their possible infinity-types and Hodge-Tate weights. In §2.4.1 we first explain
an abstract principle that is important for ‘doing Hodge theory with coefficients;’ in §2.4 we apply
this to generalize Weil’s result on descent of type A Hecke characters. Drawing further inspirationfrom [Wei56], we then discuss (§2.5) what seems to be the most useful generalization of the type
A condition to higher-rank groups, which we call W-algebraic representations. The subsequent
sections (§2.6 and §2.7) discuss the simplest non-abelian example, that of W-algebraic Hilbert
modular forms, their associated Galois representations, and the first interesting cases of Conrad’s
geometric lifting question (Question 1.1.7); although certain results in these sections are superseded by the general theorems of Chapter 3, we can also prove much more refined statements in
the Hilbert modular case, as well as some complementary results about the GL2 × GL2 →− GL4functorial lift.1 Another case where more precise results are accessible with the current technology
is for certain automorphic representations on symplectic groups, and their associated (orthogonal
and spin) Galois representations (§2.8).
2.1. Review of lifting results
In this section, we review the basic Galois lifting results due to Tate, Wintenberger, and Conrad. We then highlight some of the basic problems that remain unaddressed by these foundational
results. The two main elements of Tate’s proof will come up repeatedly throughout these notes,
so they are worth making explicit: he requires information coming both from the ‘automorphicGalois correspondence’ and from the bare automorphic theory, which in our work amounts to somequestion of what infinity-types automorphic representation can have. In Tate’s proof, the former
is global class field theory–in the form of the local-global structure of the Brauer group–and thelatter takes the form of precise results about the structure of the connected component of the idele
class group C F . We will give a slightly different proof that emphasizes the continuity with some of
our other arguments, especially Lemma 2.3.6.
Theorem 2.1.1 (Tate; see [Ser77, Theorem 4]). Let F be a number field. Then H 2 (ΓF , Q/Z) = 0.
Proof. It suffices to prove H 2 (ΓF , Q p /Z p ) = 0 for all primes p, and then an easy inflationrestriction argument shows we may assume that F contains µ p . Since H 2 (ΓF , Q p /Z p ) is p-power
1

Some of the results of §2.7, among other things, have been independently obtained by Tong Liu and Jiu-Kang
Yu in their preprint [LY].
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torsion, to show it is zero we may instead show that multiplication by p is injective, or equivalently
that the boundary mapδ

H 1 (ΓF , Q p /Z p ) →− H 2 (ΓF , Z/p)  Br(F)[p]is surjective (the identification with the Brauer group is possible since µ p ⊂ F). First we notethat the corresponding claim holds for the completions Fv : since Br(Fv )[p]  Z/p (for v finite;
for v archimedean the argument is even simpler), we only need the corresponding (at v) boundarymap δv : H 1 (ΓFv , Q p /Z p ) → H 2 (ΓFv , Z/p) to be non-zero, and thus only that multiplication by p on
H 1 (ΓFv , Q p /Z p ) should not be surjective. But a character Fv× → Q p /Z p is a pth -power if and only if
it is trivial on µ p (Fv ) ⊂ Fv× , and there are obviously characters not satisfying this condition. Notealso that for φv ∈ H 1 (ΓFv , Q p /Z p ), the image δv (φv ) depends only on the restriction φv |µ p (Fv ) .
For global F, let α be a p-torsion element of Br(F) ⊂ ⊕v Br(Fv ), with local components αv .
By the local theory, for all v we have characters φv : Fv× → Q p /Z p such that δv (φv ) = αv , and thecollection of αv is equivalent to the collection of restrictions φv |µ p (Fv ) , i.e. to the corresponding
character φ : µ p (AF ) → Q p /Z p . The fact that the αv arise from a global Brauer class α implies
that φ factors through µ p (F)\µ p (AF ), and we need only produce a finite-order extension to a Hecke
character φ̃ : C F → Q/Z. The archimedean classes αv are trivial for v complex, so Lemma 2.3.6
implies the existence of such an extension (note that it suffices to produce, as in the Lemma, an
extension to a character φ̃ : C F → Q/Z, since we can then project down to Q p /Z p and still have an
extension of φ).

Example 2.1.2. As mentioned in the introduction, Tate’s result certainly breaks down with finite
coefficients. For example, consider the projectivization of the `-adic Tate module of an elliptic
curve over a totally real field F. The obstruction to lifting to SL2 (Q` ) lives in H 2 (ΓF , Z/2), and it
vanishes if and only if the `-adic cyclotomic character admits a square-root, which cannot happen
for F totally real.
The papers [Con11] and [Win95] both study the problem of lifting Galois representation ΓF →e → H with central kernel. The problem
H(Q` ) through a surjection of linear algebraic groups H
naturally breaks into two cases: that of finite kernel (an isogeny), and that of connected (torus)
kernel. It is helpful to contrast these cases using an algebraic toy model:e  H be a surjection of tori over an algebraically closed field of characExample 2.1.3. Let H
teristic zero, with kernel D, and suppose we are given an algebraic homomorphism ρ : Gm → H.
Via the anti-equivalence of categories between diagonalizable algebraic groups and abelian groups(taking character groups), we see that the obstruction to lifting ρ as an algebraic morphism lives in
Ext1 (X • (D), Z); in particular, when D is a torus, all such ρ lift, but they do not all lift when D is
finite. For instance, we can fill in the dotted arrow in the diagram
>

G2m


(w,z)7→wr z s

Gm z7→zn / Gm
precisely when gcd(r, s)|n. The kernel of G2m → Gm is connected precisely when gcd(r, s) = 1, so
there is no obstruction in that case, and in general there is an explicit congruence obstruction.
Since it is the basis of all that follows, I recall the application of Theorem 2.1.1 to lifting
through central torus quotients (see Proposition 5.3 of [Con11]).
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Version 4 - Submitted to MEMOe  H be a surjection of linear algebraic groups over
Proposition 2.1.4 (5.3 of [Con11]). Let H∨ 2
Q` with kernel a central torus S . Then any continuous representation ρ : ΓF → H(Q` ) lifts toe ` ), i.e. we can fill in the diagram
H(Q
e `)
H(Q
<

ρ̃

ΓF

ρ


/ H(Q ).
`

e1 in He
Proof. By induction, we may assume S ∨ = Gm . There is an isogeny complement H∨∨
e1  H, but now H
e1 ∩ S is finite, and thus equal toto S : that is, we still have a surjection H
e1 to Hen := He1 · S ∨ [n],µn0 ⊂ Gm for some n0 . For any integer n divisible by n0 , we can enlarge H
which now surjects onto H with kernel S ∨ [n]  µn . These isogenies yield obstruction classescn ∈ H 2 (ΓF , Z/n) (as ΓF -module, the µn ⊂ S ∨ [n] is of course trivial) that are compatible under thenatural maps Z/n → Z/n0 for n|n0 . Tate’s theorem (Theorem 2.1.1) tells us thatlim H 2 (ΓF , Z/n) = H 2 (ΓF , Q/Z) = 0,−−→nen (Q` ).
so for sufficiently large n, there exists a lift ρ̃ : ΓF → H

Remark 2.1.5.
• Note that crucial to lifting here is the ability to enlarge the coefficients:e
if H  H is a morphism of groups over Q` and ρ lands in H(Q` ), then we only obtain ae ` (µn )) for sufficiently large n.
lift to H(Q• For example (and similarly in general; compare Lemma 3.2.2), in the case of GLn →PGLn , this proof produces lifts with finite-order determinant. If ρ is Hodge-Tate, it ispossible that no such lift is also Hodge-Tate. These non-Hodge-Tate Galois lifts haveno parallel in the ‘toy model’ of Example 2.1.3, but we will use them as a steppingstone toward finding geometric lifts, just as in the automorphic context we can choose
an initial, possibly non-algebraic, lift, and then modify it to something algebraic (seeProposition 3.1.12). By contrast, in the ‘motivic’ version of this problem (see §4.2), themotivic Galois group does not have the extra flexibility to admit such ‘non-algebraic’ lifts.
• If ρ is almost everywhere unramified, then it is easy to see that ρ̃ is almost everywhereunramified; see Lemma 5.2 of [Con11].
e1 → H (or any isogeny H
e → H) is taken up
The problem of lifting through the isogeny H
in [Win95]. Before describing this work, we state results of Wintenberger and Conrad related tolifting p-adic Hodge theory properties through central quotients (Wintenberger treated isogenies;
the general case is due to Conrad). Since all of our Galois representations will be `-adic (rather
than p-adic), from now on we will refer to “`-adic Hodge theory,” rather than p-adic Hodge theory.
We will use the phrase “basic `-adic Hodge theory property” to refer to any of the usual properties Hodge-Tate, de Rham, semistable, or crystalline. In the current subsection, we will only
formally apply the following result; in the next subsection (§2.2), we will recall the more specificbackground from `-adic Hodge theory needed later on in this book.
2

I use this awful notation to remain consistent with the ‘dual picture’ that we will later use to think about thesequestions.
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Version 4 - Submitted to MEMOe  H be a surjection of linear algebraic
Theorem 2.1.6 (Corollary 6.7 of [Con11]). Let H
groups over Q` with central kernel of multiplicative type. For K/Q` finite, let ρ : ΓK → H(Q` ) bea representation satisfying a basic `-adic Hodge theory property P. Provided that ρ admits a lifte ` ), it has a lift satisfying P if and only if ρ|IK has a Hodge-Tate lift IK → H(Qe ` ).
ρ̃ : ΓK → H(Qe → H is a central torus, Proposition 2.1.4 shows that ρ
Remark 2.1.7. When the kernel of H
always has some lift. We will see later (Corollary 3.2.12) that it even always has a Hodge-Tate lift,e ` ) satisfying P. Even simple cases of thisso there is no obstruction to finding a lift ρ̃ : ΓK → H(Qe = GLn1 ×GLn2 → H ⊂ GLn1 n2 , fortheorem yield very interesting results: for instance, applied to H
H the image of the exterior tensor product, it shows that if a tensor product of ΓK -representations
satisfies P, then up to twist they themselves satisfy P. We will use this example in Proposition
3.3.8.
We now sketch the proof of Wintenberger’s main global result. This provides an occasion tosimplify and generalize the arguments using subsequent progress in p-adic Hodge theory; it also
serves to make clear what these methods do and do not prove, and to set up a contrast with thequite different techniques that we will use.
πe→
Theorem 2.1.8 (2.1.4 of [Win95]). Let H− H be a central isogeny of linear algebraic groups
over Q. Let S be a finite set of non-archimedean places of F. Then there exist two extensions of
number fields F 00 ⊃ F 0 ⊃ F, and a finite set of finite places S 0 of F 0 such that for any prime number` and any representation ρ` : ΓF → H(Q` ) satisfying• ρ` has “good reduction,” i.e. is unramified for places not dividing ` and crystalline forplaces above `, outside S ;• For all v|`, the one-parameter subgroups µv : Gm → HCFv giving the Hodge-Tate structure
eCF ;3of ρ` |ΓFv lift to Hve ` ); if ρ` isthen the restriction ρ` |ΓF0 admits a geometric lift ρ̃` , unramified outside S 0 , to H(Qcrystalline (resp. semistable) at places above `, then the lift may be chosen crystalline (resp.
semistable). Moreover, the restriction ρ̃` |ΓF00 is unique, i.e. independent of the initial choice of liftρ̃` .

Proof. We follow Wintenberger’s arguments in detail, simplifying where the technology ale ` ) → H(Q` ) needlows. First we explain some generalities: since the map on Q` -points π` : H(Q11
not be surjective, we have an initial obstruction O (ρ` ) in Hom(ΓF , H (ΓQ` , ker π)) given by thecompositionρ`ΓF −→ H(Q` )/ im(π` ) ,→ H 1 (ΓQ` , ker π(Q` )).
We have the choice of killing this obstruction by enlarging Q` to some finite extension, or by
restricting F; following Wintenberger, we will do the former. Having dealt with this obstruction,
we will face the more serious lifting obstruction O2 (ρ` ) in H 2 (ΓF , ker π(Q` )).
With these preliminaries aside, we construct the field F 0 , first dealing with all but the (finite)
set of ` lying below a place in S . Let a(π) be the annihilator of ker π. The field F 0 (and set of places
S 0 ) will be defined, independent of `, in the following three steps:3e ` ) exists, this condition is equivalent to the existence of a local Hodge-Tate lift.
When some local lift IFv → H(QWhen ρ` |IFv is moreover crystalline, or semistable, it is equivalent to the existence of a crystalline, or semistable, lift,by Theorem 2.1.6.
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Version 4 - Submitted to MEMO• Let F1 be the maximal abelian extension of exponent a(π), unramified outside S . For
all ` and ρ` , the class O1 (ρ` ) dies after restriction to ΓF1 : outside of S ` = S ∪ {v|`}, itis of course unramified, and if v|` but v < S (i.e. ρ` |ΓFv is crystalline), then Théorèmee ` ), of ρ` |IF .
1.1.3 of [Win95]) shows the existence of a crystalline lift, valued in H(Qv(Since weakly-admissible is now known to be equivalent to admissible, Wintenberger’s
argument applies without restriction on the Hodge-Tate weights, or the ramification of thelocal field Fv /Q` .)• Let F2 /F1 be a totally imaginary extension containing Q(ζa(π) ) and such that for all placesv0 above a place v in S , a(π) divides the local degree [F2,v0 : F1,v (ζa(π) )]. If we write S 2 forthe set of places of F2 above S , `, or a(π` ) (the annihilator of ker π(Q` )), then this implies
local triviality of the obstruction classes, as long as we assume S contains no places above`. That is,


M


O2 (ρ` ) ∈ ker H 2 (ΓF2 ,S 2 , ker π(Q` )) →H 2 (ΓFv , ker π(Q` )) .
v∈S 2 ∪S ∞At v0 above S , the fact that F2 contains ζa(π) makes the local obstruction class a Brauer
obstruction, which is killed over F2 by the assumption a(π)|[F2,v0 : F1,v (ζa(π) )]. At places
not in S but above a(π` ) there is no obstruction, since unramified reprsentations always
lift (granted that O1 (ρ` ) is trivial). Finally, at places above `, which we have assumed for
now do not lie in S , Wintenberger’s local result implies the existence of a crystalline lift:on the full decomposition group ΓFv , this is [Win95, Proposition 1.2]–note that for theextension from IFv to ΓFv , one must use the fact that a crystalline lift on inertia is unique.
• Let F3 be the Hilbert class field of F2 . By global duality, the above kernel is Pontryagin
dual to


M


ker H 1 (ΓF2 ,S 2 , ker π(Q` )∗ ) →H 1 (ΓF2 ,S 2 , ker π(Q` )∗ )
v∈S 2 ∪S ∞The Galois module ker π(Q` )∗ is trivial since F2 contains ζa(π) , so this is just the space ofhomomorphisms ΓF2 → ker π(Q` ) that are unramified everywhere and trivial at the places
S 2 . Restriction of the class O2 (ρ` ) to F3 corresponds to (via global duality for F3 now)
the transfer on the Pontryagin dual of the H 1 ’s, so by the Hauptidealsatz (trivialization ofideal classes upon restriction to the Hilbert class field), O2 (ρ` ) dies upon restriction to F3 .
We conclude that, independent of ` not below places of S , and of ρ` satisfying the hypothesesof the theorem, there exist lifts over the number field F3 . To deal with ` for which the local
representations are not crystalline, we apply Proposition 0.3 of [Win95] to find a finite extension
F(`)/F, unramified outside S ` , and such that all ρ` |ΓF(`) will lift.4 Enlarging F3 by the composite of
the F(`) (for ` below places of S ), we obtain a number field F4 and a set of places S 4 equal to allplaces above S or a(π), such that for all `, any ρ` as in the theorem has a lift with good reduction
outside S 4 after restriction to ΓF4 .
This is a ‘soft’ result. It is easy to see that for fixed `, all ρ` unramified outside S lift after restriction to some F(`).
The point of Wintenberger’s theorem is that, with the `-adic Hodge theory hypotheses, F(`) can be taken independent
of `.
4

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Version 4 - Submitted to MEMOThe final step of the argument makes one final extension to show that these lifts can be taken
crystalline (or semistable) whenever ρ` is, and even unramified outside all primes above S ` (including those above a(π), although this is not necessary for the theorem’s conclusion); we omit thedetails, but see [Win95, Lemme 2.3.5ff.]. This will be the number field F 0 , and the set of places
S 0 can then be taken to be those above S . Finally, all such lifts ρ̃` (with good reduction outside S 0 )are the same over the maximal abelian extension of exponent a(π), unramified outside S 0 , of F 0 ;this is our F 00 .
Remark 2.1.9. We can apply the theorem to a system of representations ρ` all having good reduction outside S , and all having liftable Hodge-Tate cocharacters (which in the ‘motivic’ case we
may assume all arise from a common underlying geometric Hodge structure), to obtain a common
number field F 0 , with finite set of places S 0 , and a further extension F 00 such that the whole system
lifts to F 0 (with good reduction outside S 0 ), and such that all the lifts are unique over F 00 . The crucial limitation of this result is that, even if the ρ` are assumed to be the system of `-adic realizationsof some motive (for absolute Hodge cycles, as in [Win95], or for motivated cycles), the proof of
the theorem produce lifts ρ̃` that are neither (proven to be) motivic nor even weakly compatible(almost everywhere locally conjugate). So, even with the strong uniqueness properties ensured byWintenberger’s theorem, it is still not at all clear how to lift a compatible (or even motivic) systemof `-adic representations to another compatible (or motivic) system.
Here is the simplest cautionary example of how weakly compatible systems need not lift to
weakly compatible systems; it arises from an example due to Serre (see [Lar94]) involving projective representations that are everywhere locally, but not globally, conjugate:
Example 2.1.10. For any positive integer n, let Hn denote the Heisenberg group of uppertriangular unipotent elements in GL3 (Z/nZ). This is a non-split extension
1 → Z/n → Hn → Z/n × Z/n → 1,
and writing A and B for lifts of the generators of the quotient copies of Z/n, and Z for a generatorof the center, we have the commutation relation [A, B] = Z. Now let ζ be a primitive nth root ofunity; for any α ∈ Z/n, we can then define a representation ρα : Hn → GLn (C) by, in the standard
basis ei , i = 1, . . . , n, and abusively letting e0 denote en as well,A(ei ) = ei−1
B(ei ) = ζ (i−1)α ei
Z(ei ) = ζ α ei
for i = 1, . . . , n. One can check that for all α ∈ (Z/nZ)× , the associated projective representations
ρα : Hn → PGLn (C) are everywhere locally conjugate. Nevertheless, for distinct α, they are notglobally conjugate: if they were, then the corresponding GLn (C)-representations would be twistequivalent. The implications for our problem are the following: take ρα and ρβ for α , β, andform a weakly-compatible system ρ` : ΓF  Hn → PGLn (Q` ) (Hn certainly arises as Galois groupof a finite extension of number fields.) in which some ρ` are formed from ρα and others from ρβ .
There is then no system of weakly-compatible lifts ρ̃` : ΓF → GLn (Q` ), since any character of ΓF
is trivial on Z (viewed as an element of Gal(F ab F 0 /F), where F 0 /F is the given Hn -extension; note
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that Gal(F ab F 0 /F ab ) is generated by Z). Also note that we can even produce such examples where
the lifts to GLn (C) have the same determinant:
A 7→ (−1)n−1 ;det(ρα ) : 
B 7→ 1 if n is odd, or if α and n are even; −1 otherwise;Z 7→ 1.
×

Thus, even weakly compatible systems ΓF → PGLn (Q` ) × Q` need not have compatible lifts
through the isogeny GLn → PGLn × Gm .
To connect this example more explicitly with endoscopy, note that CentPGLn (C) (ρα ) is the (cyclic
order n) subgroup generated by ρα (A), whereas ρα itself is irreducible. These centralizers partiallygovern the multiplicities of discrete automorphic representations in Arthur’s conjectures. Explicitly, ρα (ABA−1 ) = ζ α B.

2.2. `-adic Hodge theory preliminaries
2.2.1. Basics. In this section, we recall some background and prove some simple lemmas in `adic Hodge theory. Throughout, let K/Q` be a finite extension. Choose an algebraic closure K/K,
and let CK denote the completion of K. Since ΓK acts by isometries on K, CK inherits a continuousΓK -action. We will mainly use only the simplest of Fontaine’s period rings, BHT and BdR , and eventhese only formally. Again, we refer the reader to the references in §1.2.2 for background, but here
we quickly recall the little needed to follow future arguments. BHT is the graded CK -algebraM
CK (i),BHT =i∈Z

with the obvious continuous CK -semilinear action of ΓK , CK (i) denoting a twist of CK by the ith
power of the cyclotomic character. BdR is the fraction field of a complete DVR and K-algebra B+dR
whose residue field is CK . We do not recall the construction, but note that BdR is a filtered Kalgebra (the filtration associated to the maximal ideal of B+dR ) with a continuous K-linear action ofΓK . Two of the fundamental results of the theory assert that gr• (BdR )  BHT (as graded CK -algebras
Kwith semi-linear ΓK -action) and BΓdRK = BΓHT
= K. From this second point it follows that if V is a
finite-dimensional representation of ΓK over Q` , then
DHT (V) = (BHT ⊗Q` V)ΓK
is a K-vector space of dimension at most dimQ` (V), and similarly
DdR (V) = (BdR ⊗Q` V)ΓK
is a K-vector space of dimension at most dimQ` (V).
Definition 2.2.1. Let V be a finite-dimensional representation of ΓK over Q` . V is said to be
Hodge-Tate if dimK DHT (V) = dimQ` (V), and de Rham if dimK DdR (V) = dimQ` (V).
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If V is de Rham, then it is Hodge-Tate (see, eg, [BC, Proposition 6.3.2]), but not conversely.5
However, the converse holds for characters ([Ser98, III-A6]), a fact to which we will frequently
appeal.
We will usually consider ΓK -representations V with coefficients in some extension E/Q` , typically either some unspecified finite extension, or E = Q` . Then DdR (V) (mutatis mutandis for
DHT ) is a filtered K ⊗Q` E-module; if V, viewed as a Q` representation by forgetting the E-linear
structure, is de Rham, then DdR (V) is a free K ⊗Q` E-module. To see this, we may assume E is(finite-dimensional over Q` and) large enough to contain all embeddings of K into Q` . Then for
all Q` -embeddings τ : K ,→ E, we have orthogonal idempotents eτ ∈ K ⊗Q` E, giving rise to an
isomorphismM(eτ )K ⊗Q` E −−→E∼τ : K,→Ex ⊗ α 7→ (τ(x)α)τ .
As K ⊗Q` E-module,
DdR (V) Meτ DdR (V) τ

M(BdR ⊗K,τ V)ΓK

τ(compare Lemma 2.2.7), and each space on the right-hand side has E-dimension at most dimE (V).
The sum of their E-dimensions is then at most dimQ` (K) dimE (V), so the Q` -dimension of the righthand side is at most dimQ` (K) dimQ` (V), which is, since V is de Rham, the Q` -dimension of theleft-hand side. Therefore equality holds everywhere, and it easily follows that DdR (V) is free overK ⊗Q` E  ⊕τ E.
Example 2.2.2. Note that even when V is de Rham, other steps of the filtration need not be freeover K ⊗Q` E. We sketch a basic example, omitting the details because they require introducing
more about BdR than we will subsequently need. Take any extension to a character ψ of ΓK of thecharacter×
recKτ
IK −−−→ O×K →− Q` ,
where τ : K ,→ Q` is any fixed Q` -embedding. If K = Q` , so that there is only one such embedding,then ψ|IK is just the restriction to inertia of the cyclotomic character, and DdR (ψ) has filtration
concentrated in degree -1. If K , Q` , then for all embeddings τ0 , τ, the filtered one-dimensionalQ` -vector space eτ0 DdR (ψ) has its filtration concentrated in degree zero, whereas the τ piece isconcentrated in degree -1.
It will be crucial for us to have a refined notion of Hodge-Tate weight that takes into accountthe different embeddings τ : K ,→ Q` :Definition 2.2.3. Let V be a de Rham representation of ΓK on a Q` -vector space of dimensiond, so that DdR (V) is a free K ⊗Q` Q` -module of rank d. Then for each τ : K ,→ Q` , we define
The standard example is a non-split extension of ΓK - representations

5

0 → Q` → V → Q` (1) → 0.
Standard results on Galois cohomology of local fields show such a V exists. It’s not so hard to show V is Hodge-Tate,
but to show that it cannot be de Rham seems to require quite deep input; see for instance [BC, Example 6.3.5].
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the d-element multi-set of τ-labeled Hodge-Tate weights, HTτ (D), to be the multi-set of integers hsuch thatgrh (eτ (DdR (V))) , 0,
where h has multiplicity dimQ` grh (eτ DdR (V)).
We could also make this definition for V having coefficients in a subfield E ⊂ Q` , possibly
then having to enlarge E in order to contain the embeddings of K (and thus define the labeled
Hodge-Tate weights). For example, for any embedding τ : K ,→ Q` , the cyclotomic characterκ : ΓK → Q×` has a single Hodge-Tate weight of −1. In §2.4.1, we will discuss the elementary but
significant implications of E not containing all embeddings of K (which in the cyclotomic example
just described, is that HTτ (κ) is independent of τ).
2.2.2. Labeled Hodge-Tate-Sen weights. Later we will work with ΓK -representations that are
not even Hodge-Tate and will need a generalized notion of (labeled) Hodge-Tate weight. Moreover,
we will need an analogue of the sets HTτ for representations of ΓK valued in linear algebraic groups
other than GLN . Both of these flexibilities will be required to address Question 1.1.7. First, again
let V be a de Rham representation of ΓK on a Q` -vector space. The identification of gr• (BdR )with BHT implies that the multiplicity of q in HTτ (V) also equals dimQ` (V ⊗τ,K CK (−q))ΓK . We use
this formulation to extend the definition of τ-labeled weights to the Hodge-Tate case. Note that
if K 0 /K is any finite extension, and τ0 : K 0 ,→ Q` is any embedding extending τ : K ,→ Q` , thenHTτ0 (V|ΓK0 ) = HTτ (V).
Now we drop the assumption that V is Hodge-Tate. To analyze such V, we use a constructionof Sen ([Sen81]): for Sen’s theory, [Ber04, II.1.2] is a very brief, and very illuminating, overview;
see the references there or [BC, Part IV] for a detailed introduction. Sen’s theory gives much more
precise results than what we use here, and we will not need any details of the construction. We will
therefore only summarize the properties of the construction that we will formally apply:
Theorem 2.2.4 (Sen). For any CK -semilinear representation V of ΓK , there is a CK -linear endomorphism ΘV , the ‘Sen operator’ of V, satisfying the following functorial properties for a pair
V1 , V2 of CK -semilinear representations:T• If V1 −→ V2 is a ΓK -equivariant map of CK -semilinear representations, then Lie(T )◦ΘV1 =ΘV2 ◦ Lie(T ).
• ΘV1 ⊕V2 = ΘV1 ⊕ ΘV2 .
• ΘV1 ⊗V2 = ΘV1 ⊗ idV2 + idV1 ⊗ΘV2 .
The representation V is is Hodge-Tate6 if and only if ΘV is semi-simple with integer eigenvalues.
In general, we define the ‘Hodge-Tate-Sen’ weights of V to be the eigenvalues of ΘV .
Proof. See [Sen81] or, more specifically, [BC, Corollary 15.1.6, Theorem 15.1.7, Exercise
15.5.4].

We will now be more precise about how to work with ΓK -representations with coefficients inQ` . We will also describe Sen operators for representations valued in a general linear algebraic
group. For some finite extension E/Q` inside Q` , our (arbitrary) ΓK -representation V descends
to an E-linear representation VE . For any embedding ι : Q` ,→ CK , we have the Sen operator
6

Strictly speaking, in §2.2.1 we only defined Hodge-Tate representations on finite-dimensional Q` -vector spaces.
This is a special case of a more general notion for CK -semi-linear ΓK -representations: see [BC, Definition 2.3.4].
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Version 4 - Submitted to MEMOΘρ,ι ∈ EndCK (V ⊗Q` ,ι CK ). More precisely, let K 0 = ι(E)K, so that VE ⊗E,ι CK is a CK -semilinear
representation of ΓK 0 , to which we can in the usual way associate the Sen operator Θρ,ι . Notethat the Sen operator of a CK semi-linear representation is insensitive to finite restriction, so thisconstruction is independent of the choice of (sufficiently large) K 0 . Moreover, choosing a model
VE 0 of V with coefficients in some finite extension E 0 /E inside Q` also yields the same Θρ,ι , since
we don’t change the CK -semilinear representation of ΓK 0 (this may be a bigger K 0 , having enlarged
E). This allows the following Tannakian observation about Sen operators (§6 of [Con11]).
Lemma 2.2.5. Let G be a linear algebraic group over Q` with Lie algebra g and ρ : ΓK → G(Q` )a (continuous) representation. Then for each embedding ι : Q` ,→ CK , there exists a unique Sen
operator Θρ,ι ∈ g ⊗Q` ,ι CK such that for all linear representations r : G −−→ GLV , Lie(r)(Θρ,ι ) is/Q`

equal to the previously-defined Θr◦ρ,ι .

Proof. This follows from Tannaka duality for Lie algebras (a precise reference is [HC50]).
Namely, for each Q` -representation G → GLV we get an element ΘV,ι ∈ glV ⊗ι CK , and these satisfy
the functorial properties
T• If V1 −→ V2 is a G-morphism, then Lie(T ) ◦ ΘV1 ,ι = ΘV2 ,ι ◦ Lie(T );• ΘV1 ⊕V2 ,ι = ΘV1 ,ι ⊕ ΘV2 ,ι ;• ΘV1 ⊗V2 ,ι = ΘV1 ,ι ⊗ idV2 + idV1 ⊗ ΘV2 ,ι .
(If different fields E and K 0 as above are needed to define the ΘVi ,ι , we can pass to a common
extension and apply the remarks preceding the lemma.) Tannaka duality then implies that all ΘV,ιarise from a unique element Θρ,ι ∈ g ⊗ι CK .
Remark 2.2.6. The necessary functorial properties of the Sen operators are not automatic. All
the relevant statements are in §15 of [BC].
As mentioned previously, V ⊗Q` ,ι CK is Hodge-Tate if and only if Θρ,ι is semi-simple with integer
eigenvalues. We will need the following comparison:
Lemma 2.2.7. Suppose ρ : ΓK → AutQ` (V) is Hodge-Tate. Any embedding ι : Q` ,→ CK inducesτι : K ,→ Q` , and we then haveHTτι (ρ) = {eigenvalues of Θρ,ι }.
Proof. Let E and K be as above. The ΓK -representation VE is Hodge-Tate, and there is anatural isomorphism of graded E ⊗Q` K 0 -modules
0∼(VE ⊗Q` BHT )ΓK ⊗K K 0 →− (VE ⊗Q` BHT )ΓK0 .
Restricting to the qth graded pieces, we get an E ⊗Q` K 0 -isomorphism∼(VE ⊗Q` CK (−q))ΓK ⊗K K 0 →− (VE ⊗Q` CK (−q))ΓK0 ,which in turn is an E ⊗Q` K 0 -isomorphism
M
M∼(VE ⊗τ,K CK (−q))ΓK ⊗K K 0 →−(VE ⊗E,ι CK (−q))ΓK0 .
τ : K,→E

ι : E,→CK

Projecting to the τ-component, we obtain an E ⊗τ,K K 0 -isomorphism
M∼(VE ⊗E,ι CK (−q))ΓK0 .
(VE ⊗τ,K CK (−q))ΓK ⊗K K 0 →−ι : E,→CK :τι =τ

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Writing mq,τ for the multiplicity of q in HTτ (V), the left-hand side is a free E ⊗τ,K K 0 -module ofrank mq,τ . Meanwhile, the K 0 -dimension of the ι-factor of the right-hand side is by definition themultiplicity of q as a Hodge-Tate weight of VE ⊗E,ι CK , hence
Q is q’s multiplicity as an eigenvalueof Θρ,ι . We deduce (applying the ι-projection in E ⊗τ,K K 0  ι:τι =τ K 0 ) that for all ι the eigenvaluesof Θρ,ι match the multi-set HTτι (V).
Remark 2.2.8. We will only apply this when V is in fact de Rham, with the exception ofCorollary 3.2.12.
2.2.3. Induction and `-adic Hodge theory. We will later need a result on compatibility of
Fontaine’s functors with induction. The following lemma is certainly well-known, but I don’tknow of a proof in the literature, so I record some details. For a finite extension K/Q` , we let (asis standard notation in the subject) K0 ⊂ K denote the maximal unramified sub-extension of K.
Lemma 2.2.9. Let L/K/Q` be finite, and let W be a de Rham representation of ΓL . Then V =K
IndL W is also de Rham, and DdR (V) is the image under the forgetful functor FilL → FilK of DdR (W).
Moreover, IndKL W is crystalline if and only if W is crystalline and L/K is unramified.
Proof. This follows almost immediately from Frobenius reciprocity if one uses contravariant
Fontaine functors:
D∗dR (V) := HomΓK (V, BdR )  HomΓL (W, BdR |ΓL ) = D∗dR (W).
Since D∗dR (V)  DdR (V ∗ ), and IndKL (W ∗ )  IndKL (W)∗ , this shows that DdR (V) is simply the filteredK-vector space underlying DdR (W). Comparing dimensions, it is clear that V is de Rham if andonly if W is. In the crystalline case, the same observation yields D∗cris (V)  D∗cris (W), so
dimK0 Dcris (V) = dimK0 Dcris (W) = dimL0 Dcris (W)[L0 : K0 ],and comparing dimensions we see that V is crystalline if and only if W is crystalline and [L0 :K0 ] = [L : K], i.e. L/K is unramified.

2.3. GL1
A detailed analysis of the GL1 theory provides both motivation and technical tools for addressing the more general Galois and automorphic lifting problems.
2.3.1. The automorphic side. We begin by reviewing some basic facts about Hecke characters ψ : A×F /F × → C× . Any such ψ is the twist by | · |rAF , for some r ∈ R, of a unitary Hecke
character. At each place v of F, ψv = ψ|Fv× can be decomposed via projection to the maximal
compact subgroup in Fv× ; at v|∞, this lets us write any unitary ψv in terms of a given embedding
ιv : Fv ,→ C as
ψv (xv ) = (ιv (xv )/|ιv (xv )|)mv |ιv (xv )|itCv ,with mv = mιv an integer, and tv a real number. To avoid cumbersome notation, we will sometimes
omit reference to the embedding ιv . Hecke characters encode finite Galois-theoretic information
as well as rather subtle archimedean information; the latter will be particularly relevant in what
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Version 4 - Submitted to MEMOLemma 2.3.1. There is a (unitary, say) Hecke character ψ of F with archimedean parameters{mv , tv }v|∞ , as above, if and only if for some positive integer M, all global units α ∈ O×F satisfyMY ιv (α) !mv

itv 

 = 1.
|ι
(α)|
v


|ι
(α)|
v
v|∞
Equivalently, the inside product is trivial for all α in some finite-index subgroup of O×F .
It follows that if F is a CM field, and {mv }v|∞ is a set of integers indexed by the archimedean
places of F , then there is a unitary Hecke character ψ of F with archimedean components givenby ψv (xv ) = (ιv (xv )/|ιv (xv )|)mv , for all v|∞.
Using the notation we have just established, we recall Weil’s notions of type A0 and type A
Hecke characters:
Definition 2.3.2. A Hecke character ψ is said to be of type A if r ∈ Q and tv = 0 for all v|∞. ψ
is said to be type A0 if tv = 0 and m2v + r is an integer for all v|∞. Equivalently, ψ is type A0 if for
all v|∞, there exist integers pv and qv such that ψv (z) = ιv (z) pv ῑv (z)qv .
In particular, note that Lemma 2.3.1 implies the existence of all the type A Hecke charactersover CM fields that one could hope for. The situation is more interesting over general number
fields, as we will see in Lemma 2.3.4. First we note another result of Weil ([Wei56]), which
reveals the central role in the algebraic theory of automorphic forms (for GL1 , here) played by thenotion of ‘type A.’
Lemma 2.3.3 (Weil). Let χ be a type A Hecke character locally trivialized by the modulus m(i.e. unramified outside m, and, if pvfv is the exact power of a prime ideal pv dividing m, trivial on
1 + pvfv ). The character χ then induces a character χm of I m , the group of non-zero, prime to m,
fractional ideals, whose values are all algebraic numbers.
Later we will be interested in a higher-rank version of the type A condition; in current discussions of algebraicity of automorphic representations, people tend to focus on analogues of type A0 .
We recall a basic result of Weil-Artin ([Wei56]), which will later inspire much of our discussion
in higher-rank as well. Recall that for Hecke characters, base-change (in the sense of Langlands
functoriality) from a field F to a field L containing F is simply precomposition with the norm NL/F ;we denote the base-change of a Hecke character ψ of F to one of L by BCL/F (ψ).
Lemma 2.3.4 (Weil-Artin). Let ψ be a type-A Hecke character of a number field F. Write Fcm
for the maximal CM subfield of F. Then there exists a finite-order Hecke character χ0 of F and a
type-A Hecke character ψ0 of Fcm such that ψ = χ0 · BCF/Fcm (ψ0 ). If Fcm is totally real, then ψ is a
finite-order twist of some power | · |r , r ∈ Q, of the absolute value.
Remark 2.3.5. We emphasize that the true content of this result, which should fully generalize
to higher-rank (see Proposition 2.4.8), is that the infinity-type of a type-A Hecke character descends
to the maximal CM subfield.
First Proof: Weil leaves it as an exercise, so we give a proof. We then indicate a second proof,
which we will apply more generally in Proposition 2.4.8. By twisting by a rational power of theabsolute value, and then if necessary squaring, we may assume that ψ is type A0 . First assume F/Q
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is Galois, and set G = Gal(F/Q). We may regard F as a subfield of C (i.e., choose an embedding),
so the above relation becomesYg(α)mg = 1
g∈G

for all α in some finite-index subgroup of O×F and some integers mg . Now take any embeddingι : F ,→ C, using it to define a complex conjugation cι on F, and thus on G. In particular, when F
is totally imaginary we can rewrite the above asYg(α)mg (cι g)(α)mcι g = 1.
G/cι

Applying log |ι(·)|, we get (for all α)
X(mg + mcι g ) log |ι(g(α))|C = 0,G/cι

which by the unit theorem is only possible when mg + mcι g equals a constant wι independent of g;
the analogous argument in the totally real case shows that the mg themselves are independent of g,
concluding that case of the proof. But wι is independent of ι as well,7 hence mg +mcι g is independentboth of g and the choice of conjugation. This implies that for any two ι, ι0 , mg = mcι cι0 g , and since
Fcm is precisely the fixed field (in F) of the subgroup generated by all products cι cι0 , we are done.
e denote
Now let F be arbitrary, and let ψ be a type A Hecke character of F. As usual letting F
e cm .
the Galois closure, we know from the Galois case that ψ|Fe has infinity-type descending to (F)ee F)e cm ), the values mιv (for embeddings ιv : F
e ,→ C) are
Setting H1 = Gal(F/F)
and H2 = Gal(F/(constant on H1 -orbits (by construction) and H2 -orbits (by the Galois case), hence on H1 H2 -orbits.
e cm = Fcm , so H1 H2 = Gal(F/F
e cm ), and the infinity-type descends all the way down to
But F ∩ (F)Fcm , proving the lemma in general.
Second Proof: Later, we will generalize the following argument (see Proposition 2.4.8 formore details) in the totally imaginary case: for all σ ∈ Aut(C), there is a Hecke character σ ψwhose finite part is σ ψ f (x) = σ(ψ f (x)) and whose infinity-type is given by the integers mσ−1 ιv (thekey but easy check is F × -invariance). For some integer r, the twist |·|r/2 ψ is unitary, so c ψ = |·|−r ψ−1 ,and therefore the fixed field Q(ψ f ) ⊂ Q of all σ ∈ Aut(C) such that σ ψ f  ψ f is contained in Qcm .
The result follows from some Galois theory.
As pointed out to me by Brian Conrad, a third, more algebraic, approach is possible, where
infinity-types of algebraic Hecke characters are related to (algebraic) characters of the connected
Serre group; compare the discussion of potentially abelian motives in §4.1.30. Of course, in allarguments, the unit theorem is the essential ingredient.
We will have to analyze Hecke characters in the following setting: An automorphic representation π of an F-group G has a central character ωπ : ZG (F)\ZG (A) → C× , and we will want tounderstand the possible infinity-types of extensions
ωπ

ZG (F)\ZG (A)


/ C×
9

ω̃

e
e
Z(F)\
Z(A)
For instance, whatever ι, 12 |G|wι =7Pg∈G mg .

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Version 4 - Submitted to MEMOe is an F-torus containing ZG . The basic case to which this is reduced, for split groups atwhere Ze = Gm , where we have to extend a character from µn (F)\µn (A) to theleast, is where ZG = µn and Z
××
full idele class group F \A .
Lemma 2.3.6. Let F be any number field. Given a continuous character ω : µn (F)\µn (A) → S1 ,with archimidean component ω∞ , fix embeddings ιv : Fv ,→ C and writeYω∞ : (xv )v|∞ 7→
ιv (xv )mvv

for some set of residue classes mv ∈ Z/nZ. Then:• There exist type A (unitary) Hecke character extensions ω̃ : F × \A× → S1 if and only if
the images mv ∈ Z/nZ depend only on the restriction v|Fcm . There exists a finite-order
extension ω̃ if and only if for all complex v|∞, the classes mv ∈ Z/nZ are all zero.
• The extension ω̃ is unique up to the nth power of another type A Hecke character, except
in the Grunwald-Wang special case (see the proof), where one first has two choices ofextension to C F [n], and then the nth power ambiguity.
• In particular, if F is totally real or CM, then (unitary) type A extensions of ω always exist.
If F is totally real, they are finite-order, and if F is CM, a finite-order extension can bechosen if and only if ω|F∞× is trivial.
Proof. For the time being, let F be arbitrary. The cokernel of µn (F)\µn (A) → C F [n] is either
trivial or order 2, the latter case being the Grunwald-Wang special case (see [Con11, AppendixA]), so we first choose an extension ω0 to C F [n] (we will use this flexibility in Proposition 3.1.4).
Then Pontryagin duality gives an isomorphism∼− (C F [n])D ,C FD /nC FD →×
so there exists some ω̃ ∈ C FD extending ω0 , and it is unique up to nth powers. Restricted to F∞, wecan write (implicitly invoking ιv )Yω̃∞ : (xv )v|∞ →(xv /|xv |)mv |xv |itCvv|∞

for integers mv and real numbers tv . Then a type A lift of ω0 exists if and only if there is a Hecke
character ψ with infinity-typeYψ∞ : (xv ) 7→
(xv /|xv |) fv |xv |itCv /nv|∞

for some integers fv (the desired type A character is then ω̃ψ−n ). Following Weil (Lemma 2.3.1),
the existence of a Hecke character ω̃ corresponding to the infinity-data {mv , tv }v is equivalent to theexistence of some integer M such that for all α ∈ O×F ,Y(α/|α|)mv M |α|itv v M = 1,v|∞

and, similarly, such a ψ can exist if and only if there exists M 0 ∈ Z such thatY00(α/|α|) fv MM n |α|itv v MM = 1v|∞

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0for all α ∈ O×F . Substituting, the left-hand side is v|∞ (α/|α|)( fv n−mv )MM , which can equal 1 for allα ∈ O×F if and only if there exists a type A Hecke character with infinity-type
Y(xv ) 7→(xv /|xv |) fv n−mv .
v|∞

By Weil’s classification of type A Hecke characters, this is possible if and only if fv n − mv depends
only on v|Fcm . This in turn is possible (for some choice of fv ) if and only if mv ∈ Z/nZ depends
only on v|Fcm . In particular, over CM and totally real fields F, there is no obstruction, so a type A
lift of ω always exists.
The claim about existence of finite-order extensions is a simple variant: if all mv (v complex)
are zero in Z/nZ, then there is a type A lift ω̃ with infinity-typeYω̃∞ : (xv )v|∞ →(xv /|xv |)mv
v|∞

for some integers mv , all divisible by n and only depending on ιv |Fcm . But then there is also a type
A Hecke character ψ of F with infinity-typeYψ∞ : (xv )v|∞ →(xv /|xv |)mv /n ,v|∞

so ω̃ψ is a finite-order extension of ω.
As for uniqueness, once a type A lift of ω0 is chosen, any other is (by the discussion at thestart of the proof) a twist by the nth power of another Hecke character, which must itself clearly beof type A. This is then the ambiguity in extending ω itself, except in the Grunwald-Wang specialcase, in which there is the additional Z/2Z ambiguity noted above.
−nRemark 2.3.7.
• Of course any quasi-character of µn (A) is unitary, and to understand all
extensions it suffices (twisting by powers of | · |A ) to understand unitary extensions–hence
the restriction to S1 instead of C× .
• This raises a tantalizing question: certainly over a non-CM field F we can produce characters ω that have no type A extensions (although they certainly all have some extension toa Hecke character, by Pontryagin duality). What if ω actually arises as the central character of a suitably algebraic cuspidal automorphic representation (on SLn (A), for instance)?
We return to this in §2.4.
A few consequences, either of the result or the method of proof, will follow; first, though, let
us make a definition:
Definition 2.3.8. Let F be any number field, and let ψ ∈ C FD be a unitary Hecke character. We
say that ψ is of Maass type if for all v|∞, the restriction ψv : Fv× → S1 has the form x 7→ |x|itv forreal numbers tv .
Corollary 2.3.9. Let ψ : C F → C× be a Hecke character of a number field F.
• If F is totally real or CM, then ψ can be decomposed asψ = ψalg ψMaass ψfin | · |w ,where w is the unique real number twisting ψ to a unitary character, ψalg is unitary type
A, ψMaass is of Maass type, and ψfin is finite order. The last three characters are all unique
up to finite-order characters.
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Version 4 - Submitted to MEMO• If ψ is of Maass type (after twisting to a unitary character), with F arbitrary, then ψ is‘nearly divisible’: for any n ∈ Z there exist Hecke characters χ of Maass type and χ0 offinite order such that χn χ0 = ψ. In particular, after a finite base-change ψ is n-divisible.
• Write AF (1) for the space (topological group) of all Hecke characters of F. Suppose F isCM, of degree 2s over Q. Then there is an exact sequence
1 → ΓDF → AF (1) → R × Z s × Q s−1 → 1.
(As always, ΓDF denotes Homcts (ΓF , S1 ); it is the space of Dirichlet characters.) For the totally realsubfield F+ of F, we have a similar sequence
1 → ΓDF+ → AF+ (1) → R × Q s−1 → 1.
It is also possible (via the unit theorem) to compute all possible infinity-types of Hecke characters of any number field, but we will make no use of this calculation. Roughly speaking, the
new transcendentals that arise in the ‘mixed’ case where algebraic and Maass parts cannot be separated are the arguments (angles) of fundamental units. It is very tempting to ask whether for CM(or totally real fields) algebraic and spherical (‘Maass’) infinity-types on higher-rank groups cantwist together in a non-trivial way. If this question is too naı̈ve, is there any other higher-rank
generalization of part 1 of Corollary 2.3.9?
We include a couple other related useful results. For the first, note that in contrast with thesituation for `-adic Galois characters (see Lemma 2.3.15 below), a general Hecke character cannot
be written as an nth -power up to a finite-order twist.8
Lemma 2.3.10. Let ψ be a Hecke character of F, and suppose that L/F is a finite Galois
extension over which BCL/F (ψ) = ψ ◦ NL/F is an nth -power. Then up to a finite-order twist, ψ itself
is an nth -power.
Proof. Let ω be a Hecke character of L such that ψ ◦ NL/F = ωn . Then for any σ ∈ Gal(L/F),
ω−1 · (ω ◦ σ) has finite order. After passing to a further finite extension e
L that kills all thesencharacters, we see that BCeL/F ψ = BCeL/L (ω) , where BCeL/L (ω) is now Gal(eL/F)-invariant. The
next lemma shows that ω descends to F up to a finite-order twist.
The next lemma completes the previous one, and also enables a slight refinement of a result ofRajan (see Remark 2.3.12):
Lemma 2.3.11. Let L/F be a Galois extension of number fields, and let ψ be a Gal(L/F)invariant Hecke character of L. Then there exists a Hecke character ψF of F and a finite-order
Hecke character ψ0 of L such that ψ = (ψF ◦ NL/F ) · ψ0 .
Proof. We may assume ψ is unitary, and we may choose, for all infinite places w of L, embeddings ιw : Lw → C such that all ιw for w above a fixed place v of F restrict to the same embedding
ιv : Fv → C. Gal(L/F) acts transitively on the places w|v, so when we write!mιw (xw ) w
ψw (xw ) =· |ιw (xw )|itw ,|ιw (xw )|Gal(L/F)-invariance implies that the mw and tw depend only on the place v below w (from now onin the proof, the embeddings ιw will be implicit). We therefore denote these by mv and tv . Lemma
8

Consider, for example, a type A Hecke character whose integral parameters at infinity are not divisible by n.
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2.3.1 implies there is a Hecke character of F with infinity-type given by the data {mv , tv }v . Namely,
there is an integer M such that for all α ∈ O×L ,MY α !mw
1 = 
|α|itCw  .
|α|w|∞
Restricting to α ∈ O×F , this becomesM  M#{w|v}Y α !mv
Y Y α !mv
1 = 
|α|itCv  = |α|itCv 
,
|α|
|α|
v|∞ w|v
v|∞(#{w|v} is independent of v) which is simply the criterion for there to be a Hecke character of F
with infinity-type given by the collection {mv , tv }v . Any such character has base-change differingfrom ψ by a finite-order character, so we are done.
Remark 2.3.12. A theorem of Rajan describes the image of solvable base-change: precisely,
for a solvable extension L/F of number fields and a Gal(L/F)-invariant cuspidal automorphic
representation π of GLn (AL ), Theorem 1 of [Raj02] asserts that there is a cuspidal representation
πF of GLn (AF ) and a Gal(L/F)-invariant Hecke character ψ of L such that
BCL/F (π0 ) ⊗ ψ = π.
Lemma 2.3.11 shows that this ψ may be chosen to have finite-order; in particular, for some cyclic
extension L0 /L (so L0 /F is still solvable), BCL0 /L (π) descends to F.
×2.3.2. Galois GL1 . We now discuss (continuous) Galois characters ψ̂ : ΓF → Q` . For the time
being, F is any number field. These are necessarily almost everywhere unramified, and we focus
on `-adic Hodge theory aspects. The following is well-known, and is proven in [Ser98, Chapter
III]:
Theorem 2.3.13. Suppose that ψ̂ is de Rham. Then for all places v not dividing `, ψ̂|ΓFv assumesι`→ Q` , and there exists a type A0 Hecke character ψ : AF /F × → C× correalgebraic values in Q −sponding to ψ̂ (via ι∞ : Q ,→ C). Moreover, ψ̂ is motivic: the Fontaine-Mazur conjecture holds forGL1 /F.
Proof. We indicate how this follows from the arguments of [Ser98, III]; see Remark 2.3.14
for the explanation of a simple case. Since ψ̂ is Hodge-Tate at all v|`, it is (by a theorem of Tate:see [Ser98, III-A6]) locally algebraic in the sense of [Ser98, III-1.1] (and therefore de Rham in
the language that post-dates [Ser98]). Then the argument of [Ser98, III.2.3 Theorem 2] implies
that, for a suitable modulus m, ψ̂ is the `-adic Galois representation associated to an algebraic
homomorphism Sm,Q` → Gm,Q` , where Sm is the Q-torus of [Ser98, II-2.2]. Up to isomorphism,
this algebraic representation can be realized over some finite extension E of Q inside Q` . Taking
the ‘archimedean realization’ as in [Ser98, II-2.7], we obtain the desired type A0 Hecke character.
It then follows by Lemma 2.3.3 that ψ̂|ΓFv assumes algebraic values for v - `. The Fontaine-Mazur
conjecture for ψ̂ follows from a theorem of Deligne ([DMOS82, Proposition IV.D.1]), which shows
the somewhat stronger statement that ψ̂ is the `-adic realization of a motive for absolute Hodge
cycles.

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Remark 2.3.14. Consider the simple case in which F = Q. Then ψ̂|ΓQ` has a single labeled
Hodge-Tate weight; it therefore has the same (labeled) Hodge-Tate weights as an integer power ωr`of the `-adic cyclotomic character. It follows then from the above-cited theorem of Tate ([Ser98,
III-A6]) and global class field theory that ψ̂ω−r` is finite-order, hence is the `-adic realization (via a∼fixed isomorphism C →− Q` ) of some finite-order Hecke character χ. The Hecke character corresponding to ψ̂ is then χ| · |rAQ . It is also easy to show that ψ̂ is motivic: ωr` is the `-adic realization of
the Tate motive Q(r), and if ψ̂ω−r` factors through a finite quotient Gal(F/Q), then it is a suitable di0
rect factor of H (XQ , Q` ), for the zero-dimensional variety X = ResF/Q (Spec F) whose Q-points arenaturally indexed by embeddings τ : F ,→ Q, with the ΓQ -action on H 0 arising from permutationof the set of embeddings (for more details, see the discussion of Artin motives in §4.1.2).
More generally, the idea behind establishing the Fontaine-Mazur conjecture in the abelian caseis to find a sub-quotient of the cohomology of a CM abelian variety having ‘labeled Hodge numbers’ matching those of ψ̂; that this is possible essentially follows from the constraints on the ∞type of the type A0 Hecke character underlying ψ̂ (as in Lemma 2.3.4). If the CM abelian variety
were actually defined over F, this would realize ψ̂ inside some finite-order twist of its cohomology.
The subtle part of the theorem is the need to have some control over the field of definition of the
CM abelian variety.
We make repeated use of the following well-known observation (see for instance Lemma 3.1
of [Con11]).
×Lemma 2.3.15. Let χ : ΓF → Q` be an `-adic Galois character. For any non-zero integer m,×
there are characters χ1 , χ0 : ΓF → Q` , with χ0 finite-order, such that χ = (χ1 )m χ0 .
We will need to consider Galois characters somewhat outside the algebraic range. As is customary, we write HTτ (ρ) to indicate the τ : F ,→ Q` -labeled Hodge-Tate weights of an `-adic Galois
representation ρ; see §2.2 for details, as well as for what we mean when we write non-integral
τ-labeled Hodge-Tate-Sen weights.
Corollary 2.3.16. Let F be a totally imaginary field, and n a non-zero integer. For all×τ : F ,→ Q` , fix an integer kτ . Then there exists a Galois character ψ̂ : ΓF → Q` with HTτ (ψ̂) = knτ
(respectively, with HTτ (ψ̂) ≡ knτ mod Z) if and only if(1) kτ depends only on τ0 := τ|Fcm (respectively, only depends modulo n on τ0 );(2) and there exists an integer w such that kτ0 +kτ0 ◦c = w (respectively, kτ0 +kτ0 ◦c ≡ w mod n)
for all τ, and c the complex conjugation on Fcm .9
Proof. Suppose such a ψ̂ exists, with weights knτ . Then ψ̂n is geometric (the de Rham and
Hodge-Tate conditions are equivalent for characters, as noted in Theorem 2.3.13), and so (via ι`∼and ι∞ ) there exists a type A0 Hecke character ψ of F corresponding to ψ̂n . For v|∞ and ιv : Fv →− C,
ψ|Fv× is given by
ψv (xv ) = ιv (xv )kτ∗ (ιv ) ῑv (xv )kτ∗ (ῑv ) ,where τ∗ (ιv ) = τ∗`,∞ (ιv ) denotes the embedding F ,→ Q` induced by ιv , ι` , and ι∞ . By purity for
Hecke characters–immediate from the description of characters of R× and C× –there is an integer w
If F is Galois, we can rephrase this as kτ + kτ◦c = w for all choices of complex conjugation c on F.

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such that kτ∗ (ιv ) + kτ∗ (ῑv ) = w; moreover, by Weil’s descent result, kτ∗ (ιv ) depends only on ιv |Fcm . The
constraint on the weights follows.
Conversely, given a set of weights satisfying the purity constraint, we form a putative infinitytype pιv = kτ∗ (ιv ) for a Hecke character of F; that this is in fact an achievable infinity-type follows
from Weil. We form the associated geometric Galois character and then use the fact that, up to a
finite-order twist, we can always extract nth roots of `-adic characters.
The mod n statement is a simple modification. For instance, to construct a character withgiven weights satisfying the purity constraint modulo n, proceed as follows. Choose a maximal set
kτ
modulo c of embeddings τ0 : Fcm ,→ C. Choose lifts to Q of the congruence classes n0 mod Z,
and declare kτ = kτ0 for all τ lying above these τ0 . Then choose w ∈ Z lifting kτ0 + kτ0 ◦c (mod n),which is possible by hypothesis, and set kτ0 ◦c = w − kτ0 .

The same technique yields an easy example of the constraints on Galois characters over totallyreal fields:
×

Lemma 2.3.17. Suppose F is totally real, and ψ̂ : ΓF → Q` is a character with all Hodge-TateSen weights in Q. Then all of these weights are equal. Conversely, for x, d ∈ Z, there are global
characters with all HTS weights equal to dx .
Proof. For some integer d, all the HTS weights lie in d1 Z, so ψ̂d is geometric, and therefore
corresponds to a type A0 Hecke character. F is totally real, so ψ̂d must be, up to a finite-order twist,
an integer power of the cyclotomic character.
It is worth remarking that the most general answer to the question ‘what Galois characters×ψ̂ : ΓF → Q` exist’ is essentially Leopoldt’s conjecture.
Finally, we will need a lemma refining the construction of certain Galois characters over CM
fields:Lemma 2.3.18. Let F be a totally real field, and ` , 2 a prime unramified in F. Let L = KF be
its composite with an imaginary quadratic field K in which ` is inert, and let ψ be any unitary type
A Hecke character of L. Then:×(1) There exists a Galois character ψ̂ : ΓL → Q` such that ψ̂2 corresponds to ψ2 (which is
type A0 ).
(2) Moreover, the frobenius eigenvalues ψ̂(frv ) at all unramified v lie in Qcm .
Proof. First, let mιv as before denote the integers giving the infinity-type of ψ. Using the
known algebraicity of ψ (Lemma 2.3.3) and the fixed embeddings ι∞ , ι` , we can define the `-adic
representation associated to ψ2 :×× ×
(ψb2 )` ◦ rec−1
L : AL /(L L∞ ) → Q`

(xw ) 7→Yw-`∞

Y 
ψw (xw )2
ψw (xw )2w|`Yτ : Lw ,→Q`mι∗ (τ) ∞,`τ(xw )
 .

By the dual Grunwald-Wang theorem ([Con11, Remark 1.1]), to show that (ψb2 )` is the square of×some character ΓL → Q` , it suffices to check that for all places w of L, the above character is locally35

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on Lw× a square. This in turn immediately reduces to seeing whether, for each w|`, the character×χw : Lw× → Q` given byYm∗τ(xw ) ι∞,` (τ)χw (xw ) =τ : Lw ,→Q`

is a square. Writing
Lw× = h`i × µ∞ (Lw× ) × (1 + `Ow ),it suffices to check on each component of this factorization. On the h`i factor, this is clear (choosea square root of χw (`)). On the 1 + `Ow factor, the `-adic logarithm, using our hypotheses that
` , 2 is unramified, lets us define a single-valued square-root function, and thus extract a squareroot of χw |1+`Ow . Now note that the product over τ : Lw ,→ Q` is a product over pairs of complexconjugate embeddings extending a given Fv ,→ Q` , and mι∗ (τ̄) = mι∗ (τ) = −mι∗ (τ) (ψ is unitary).
µ∞ (Lw× ) is isomorphic to Z/(q − 1)Z, where q is the order of the residue field at w. Let ζ 7→ 1 givethe isomorphism, for ζ a primitive (q − 1) st root of unity. Complex conjugation must identify tomultiplication by an element r ∈ Z/(q − 1) satisfying r2 ≡ 1 mod q − 1; in particular, r is an odd
residue class. Then x 7→ τ(x)mι∗ (τ) · (τ ◦ c)(x)−mι∗ (τ) takes ζ to an even power of τ(ζ), hence χw |µ∞ (Lw× )is a square.10
The second part of the lemma follows from the construction of ψ̂ and the corresponding statement (Lemma 2.3.3) for the type A Hecke character ψ.

2.4. Coefficients: generalizing Weil’s CM descent of type A Hecke characters
We now begin to pursue higher-rank analogues of two aspects of Weil’s paper [Wei56]. Thissection extends to higher rank the important observation that type A Hecke characters of a number
field F descend, up to a finite-order twist, to the maximal CM subfield Fcm (see Lemma 2.3.4).
Of interest in its own right, this generalization also provides some of the intuition necessary for ageneral solution to Conrad’s lifting question (Question 1.1.7).
2.4.1. Coefficients in Hodge theory. The guiding principle that allows us to reinterpret, andcorrespondingly generalize, Weil’s result is that careful attention to the ‘field of coefficients’ of anarithmetic object can yield non-trivial information about its ‘field of definition,’ or that of certainof its invariants. We begin by recording in an abstract setting a lemma whose motivation is ‘doing
Hodge theory with coefficients.’ Let k be a field of characteristic zero, and let F and E be extensionsof k with F/k finite. Let D be a filtered F ⊗k E-module that is free of rank d. Let E 0 be a Galois
extension of E large enough to split F over k. For all (k-embeddings) τ : F ,→ E 0 , we define
as in Definition 2.2.3 the τ-labeled Hodge-Tate weights HTτ (D) as follows: for such τ we haveorthogonal idempotents eτ ∈ F ⊗k E 0 giving rise to projections(eτ ) Y 0
F ⊗k E 0 −−→E∼τx ⊗ α 7→ (τ(x)α)τ .
0

The projection eτ (D ⊗E E ) is then a filtered E 0 -vector space of dimension d, and we define HTτ (D)to be the collection of integers h (with multiplicity) such thatgrh (eτ (D ⊗E E 0 )) , 0.
Note also that if ` is split in L/Q, χw cannot be a square when mι∗∞,` (τ) is odd.

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Version 4 - Submitted to MEMOIn §2.2.1 we applied this formalism to the filtered K ⊗Q` Q` -module DdR (V), when V was a representation of ΓK (K/Q` finite) on a Q` -vector space. In this section, however, we emphasize thegeneral formalism: one should really keep in mind not `-adic Hodge theory, but rather the de Rhamrealization of a motive over F with coefficients in E.
Lemma 2.4.1. Let D as above be a filtered F ⊗k E-module that is free of some rank d. Then theset of integers HTτ (D) depends only on the Gal(E 0 /E)-orbit of τ : F ,→ E 0 . If E/k is Galois, thenHTτ (D) depends only on the restriction τ|τ−1 (E) .
Q
Proof. We decompose F ⊗k E into a product of fields Ei , writing qi : F ⊗k E  Ei for thequotient map. This yields filtered Ei -vector spaces Di for all i. Any E-algebra homomorphism
τ : F ⊗k E → E 0 factors through qi(τ) for a unique index i(τ), and then HTτ (D) is simply the multiset of weights of Di(τ) . This implies the first claim, since the Gal(E 0 /E)-orbit of τ is simply allembeddings τ0 for which i(τ0 ) = i(τ).
Now we assume E/k Galois and address the second claim. Having fixed a ‘reference’ embedding τ0 : F ,→ E 0 , it makes sense to speak of F ∩ E := F ∩ τ−1
0 (E) inside E. Obviously E splits
F ∩ E over k, so we can writeYF ⊗k E  F ⊗F∩E (F ∩ E ⊗k E) 
F ⊗F∩E,σ E,
σwhere the σ range over all embeddings F ∩ E ,→ E. These factors F ⊗F∩E,σ E are themselves fields,
since E/k is Galois, and so this decomposition realizes explicitly the decomposition of F ⊗k E into
a product of fields. In particular, by the first part of the Lemma, HTτ (D) depends only on i(τ), i.e.
only on the σ : F ∩ E ,→ E to which τ restricts.
Definition 2.4.2. We call a filtered F ⊗k E-module D regular if the multi-sets HTτ (D) are
multiplicity-free.
A very simple application that we will use later is:
Corollary 2.4.3. Let D be a filtered L ⊗k E-module (globally free) for some finite extension
e Then the restriction of scalars (image under the forgetful
L/F, and suppose L does not embed in E.
functor) ResL/F (D) is not regular.
2.4.2. CM descent. We will see that Weil’s result (Lemma 2.3.4) is the conjunction of Lemma
2.4.1 with the fact that algebraic Hecke characters have CM ‘fields of coefficients.’ This observation will lead us naturally to the desired higher-rank generalization.
We recall some notation from §1.4. Let G be a connected reductive F-group. For each v|∞∼− C. For π an automorphic representation of G(AF ), we can write (infix an isomorphism ιv : F v →Langlands’ normalization) the restriction to WF v of its L-parameter asrecv (πv ) : z 7→ ιv (z)µιv ῑv (z)νιv ∈ T ∨ (C).
with µιv , νιv ∈ X • (T )C and µιv − νιv ∈ X • (T ). For v imaginary, µῑv = νιv . Unless there is risk ofconfusion, we will omit reference to the embedding ιv , writing µv = µιv , etc. We also recall thefollowing terminology, introduced in [BG11]:
Definition 2.4.4. In the above notation, the automorphic representation π is L-algebraic if for
all v|∞, µv and νv lie in X • (T ); it is C-algebraic if µv and νv lie in ρ + X • (T ), where ρ denotes thehalf-sum of the positive roots (with respect to our fixed Borel containing T used to define a basedroot datum of G).
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Version 4 - Submitted to MEMOOur starting point is a result (and, more generally, conjecture) of Clozel. We take G = GLn /Fand collect the data of π’s archimedean L-parameters as M = {µι }ι , which we will loosely refer toas the ‘infinity-type’ of π. For σ ∈ Aut(C), define the action σ M = {µσ−1 ι }ι . Recall that π is said tobe regular if all roots of GLn are non-vanishing on all of the co-characters µι .
Theorem 2.4.5 (Théorème 3.13 of [Clo90]). Let F be any number field, and suppose π isa cuspidal, C-algebraic automorphic representation of GLn (AF ) that is moreover regular. Thenπ f has a model over the fixed field Q(π f ) ⊂ Q ⊂ C of all automorphisms σ ∈ Aut(C) suchthat σ π f  π f , and Q(π f ) is in fact a number field. For each σ ∈ Aut(C) there is a cuspidal
representation σ π with finite part σ π f = π f ⊗C,σ C and with infinity-type σ M.
More generally, Clozel conjectures this for any C-algebraic isobaric automorphic representation of GLn (AF ). Note that for GLn , the notions of C- and L-algebraic are equivalent when n isodd, but they differ by a twist when n is even. I’m grateful to Kevin Buzzard for pointing out tome that this twist suffices to make Theorem 2.4.5 fail for L-algebraic representations; our interest,
however, is a consequence (Proposition 2.4.8) with which this twist does not interfere.
Hypothesis 2.4.6. Throughout the rest of this section, we will assume that π is a cuspidal
C- or L-algebraic automorphic representation of GLn (AF ) whose C-algebraic twists satisfy theconclusion of Clozel’s theorem (if one such twist has this property, then all do).
We first note an elementary but crucial refinement of Hypothesis 2.4.6:
Corollary 2.4.7. Assume π satisfies Hypothesis 2.4.6. Continue to denote by Q(π f ) ⊂ Q ⊂ C
the fixed field of all σ ∈ Aut(C) such that σ π f = π f ⊗C,σ C  π f (this may no longer be a number
field). Then the field Q(π f ) is contained in Qcm , the union of all CM extensions of Q inside Q.
Proof. The conclusion of the corollary remains unchanged if we replace π by its twist by | · |r/2√
for some integer r, since for all primes p, Q( p) is a CM field. Therefore we may assume that
π is C-algebraic, and that either π or π| · |1/2 is unitary. Using the L2 inner product we deducethat in the former case, c π  π∨ , and in the latter case c π| · |  π∨ (note that for any automorphic
representation Π, c Π makes sense as an automorphic representation–in contrast to twists by moregeneral σ ∈ Aut(C), which seem only to exist for C-algebraic Π). Either way, we see that for allσ ∈ Aut(C), cσ π  σc π, and therefore that the fixed field Q(π f ) is contained in Qcm .
We can now formulate (and prove under Hypothesis 2.4.6) the appropriate higher-rank generalization of Weil’s result that type A Hecke characters of F, up to a finite-order twist, descend toFcm .
Proposition 2.4.8. Assume π satisfies Hypothesis 2.4.6. Then the infinity-type of π descends toFcm .
Proof. This statement too is invariant under twisting by powers of the absolute value, so we
may assume π is C-algebraic. Let E denote the Galois closure (in C) of Q(π f ); E is also a CM field.
e ,→ C of the Galois closure F.
e The image ι̃(F)e ⊂C
Extend each ι : F ,→ C to an embedding ι̃ : F
e is linearly disjoint from E over ι̃((F)
e cm ),
does not depend on the extension. By the corollary, ι̃(F)e F)e cm );and therefore we can find σ ∈ Aut(C/E) restricting (via ι̃) to any element we like of Gal(F/(the collection of such σ acts transitively on the set of embeddings F ,→ C lying above a fixed38

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e cm = Fcm ,→ C.11 For any two such ι, ι0 : F ,→ C (related by σ ∈ Aut(C/E)), we deduce,F ∩ (F)
since σ π  π and hence σ M = M, that µι = µσ−1 ι = µι0 .
Again, we remark that Hypothesis 2.4.6 is expected to hold for all cuspidal C- or L-algebraic
automorphic representations of GLn (AF ). Any study of algebraic automorphic forms over non-CM
fields will need to take this result into account.
Remark 2.4.9.
• If π is regular, this yields an unconditional descent result for its infinitytype.
• Let F be a CM field and π be a regular L or C-algebraic cuspidal representation ofGLn (AF ). Suppose that π = IndFL (π0 ) for some extension L/F. Then L is CM. This modest
consequence of the proposition suggests that in the study of regular automorphic representations/motives/Galois representations over CM fields, we will never have to grapple
with the (less well-understood) situation over non-CM fields. In §3.4.2 we discuss anabstract Galois-theoretic analogue.
• If we know more about Q(π f ) than that it is CM (an extreme case: Q(π f ) = Q), the proofof the proposition yields a correspondingly stronger result. For the related formalism, see§2.4.1.
Remark 2.4.10. Let us also note that Proposition 2.4.8 is a ‘seed’ result (under Hypothesis
2.4.6); if we moreover assume functoriality (in a form that requires functorial transfers to be strongtransfers–i.e. of archimedean L-packets–at infinity), then it implies:• Let G be a connected reductive group over a totally imaginary field F, and let π be a
cuspidal tempered L-algebraic automorphic representation of G(AF ). Then the infinitytype of π descends to Fcm .
e ⊃ G be a connected reductive F-group obtained by enlarg• Let G and π be as above. Let Ge (see §3.1). Then the central character ωπ : ZG (F)\ZG (AF ) →ing the center of G to a torus Z×e F ). (Compare Proposition 13.3.1.)
C extends to an L-algebraic Hecke character of Z(A2.5. W-algebraic representations
The current section generalizes a second aspect of [Wei56], discussing a higher-rank analogueof the type A, but not necessarily A0 , condition, and begins to motivate its arithmetic significance.
As always, let G be a connected reductive group over a number field F. We continue with theinfinity-type notation of §2.4.
For many interesting questions about algebraicity of automorphic representations, and especially the interaction of algebraicity and functoriality, the framework of C and L-algebraic representations does not suffice. Motivated initially by Weil’s study of type-A Hecke characters, we
make the following definition:
Definition 2.5.1. Let π be an automorphic representation of G(AF ). We say that π is Walgebraic if for all v|∞, µιv and νιv in fact lie in 12 X • (T ).
Example 2.5.2. For GL1 , a unitary W-algebraic representation is precisely a unitary Hecke
character of type A in the sense of Weil ([Wei56]). Weil’s type A characters also include arbitrarye cm ) is generated by H = Gal(F/F)
eeF
ecm ), with H 0 normal. The set HomFcm (F, C) is
G = Gal(F/Fand H 0 = Gal(F/permuted transitively by G, with H acting trivially, so for any such embedding x, H 0 x = H 0 Hx = Gx = HomFcm (F, C).
11

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Version 4 - Submitted to MEMOtwists | · |r for r ∈ Q, since these also yield L-series with algebraic coefficients. The L- and Calgebraic representations are the Hecke characters of type A0 .
Example 2.5.3. Consider a Hilbert modular form f on GL2 /F with (classical) weights {kτ }τ : F,→R ,where the positive integers kτ are not all congruent modulo 2; these are called ‘mixed-parity’. fthen gives rise to an automorphic representation (in the unitary normalization, say) that is Walgebraic but neither L- nor C-algebraic. The previous example yields a special case: choose aquadratic CM (totally imaginary) extension L/F, and let ψ be a unitary Hecke character of L thatis type A but not type A0 . Then the automorphic induction IndFL ψ yields an example on GL2 /F.
We will elaborate on the case of mixed-parity Hilbert modular forms in §2.6.
We are led to ask (compare §2.4 and Conjectures 3.1.5 and 3.1.6 of [BG11]):
Question 2.5.4.
• Let π be a W-algebraic automorphic representation on G. Does theG(AF, f )-module π f have a model over Q (or Qcm )?. Note that we do not seek a model overa fixed number field. Alternatively, is πv defined over Q for almost all finite places v? Byanalogy with the terminology of [BG11], let us call the latter condition W-arithmetic, andask whether some condition (which will not quite be W-algebraicity!) on infinity-types
characterizes W-arithmeticity.
• Similarly, we can ask the CM descent question of §2.4 for W-algebraic representations.
Example 2.5.6 below will be a cautionary tale: if non-induced cuspidal Π with infinitytypes as in that example exist, then there is no evidence that they would satisfy the analogue of Clozel’s algebraicity conjecture. If they do not exist, or if they miraculously
still satisfy the conclusion of Clozel’s theorem, then we could confidently extend the CM
descent conjectures to the W-algebraic case.
For tori, the most optimistic conjecture holds:
Lemma 2.5.5. For arbitrary F-tori, W-algebraic implies W-arithmetic.
Proof. After squaring, this follows from the corresponding result for L-algebraic/L-arithmetic(§4 of [BG11]).
Continuing with Example 2.5.2, let us emphasize that the W-algebraic condition does not
capture all automorphic representations with algebraic Satake parameters; we nevertheless wantto make a case for isolating this condition, rather than allowing arbitrary rational parametersµv , νv ∈ X • (T )Q , which would be the naı̈ve analogue of type A. First, note that if a unitary π
is tempered at infinity with real parameters µv , νv ∈ X • (T )R , then these parameters in fact lie in
1 •
X (T ). In particular, the Ramanujan conjecture implies that all cuspidal unitary π on GLn /F with2
real infinity-type are in fact W-algebraic. It is easy to construct non-cuspidal automorphic representations with rational Satake parameters that do not twist to W-algebraic representations, but the
point is that all such examples will be degenerate, so W-algebraicity is the condition of basic importance. A more interesting question, concerning the difference between W, L, and C-algebraicity, isthe following:
Example 2.5.6. Let G = GL4 /Q (for example; there are obvious analogues for any totally real
field). Let F/Q be real quadratic, and let π as in Example 2.5.3 be a mixed parity (unitary) Hilbert
modular representation. It cannot be isomorphic to its Gal(F/Q)-conjugate, so Π = IndQF (π) is
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cuspidal automorphic on GL4 /Q, and at infinity its Langlands parameter (restricted to C× ) looks
like


k −1

(z/z̄) 12
0
0
0


1−k1

 0
2
(z/z̄)00
z 7→ 
k2 −1
 0
0 
0(z/z̄) 2

1−k2 000(z/z̄) 2 ,where k1 and k2 are the classical weights of π at the two infinite places of F. This exhibits the‘parity-mixing’ within a single infinite place, which implies the following:r

Lemma 2.5.7. For any functorial transfer L GL4 →− L GLN arising from an irreducible representation r, not equal to any power of the determinant, of GL4 (C), the transfer Liftr (Π) cannot be
L-algebraic.
Proof. We have stated the lemma globally, and therefore conjecturally, but of course the analogous archimedean statement (which is well-defined since the local transfer of Π∞ via r is known
to exist) is all we are really interested in. The proof is elementary highest-weight theory.

We will see (in §2.6) that mixed-parity Hilbert modular representations are W-arithmetic, using
the fact that they have L-algebraic functorial transfers; by contrast, if a representation Π with theinfinity-type of the above example is W-arithmetic, it may be harder to establish. Of course, in theabove example, Π is automorphically induced, so its W-arithmeticity is an immediate consequenceof that of π. We are led to ask whether there exist non-automorphically induced examples of such
Π, or more generally of cuspidal, non-induced Π on GLn /F that exhibit the ‘parity-mixing’ within
a single infinite place. The trace formula does not easily yield them, since such a Π is not thetransfer from a classical group of a form that is discrete series at infinity (in these cases there is aparity constraint on regular elliptic parameters).
One further motivation for considering W-algebraic representations comes from studying thefibers of functorial lifts, and their algebraicity properties. For instance, given two mixed-parity
Hilbert modular forms π1 and π2 , with the (classical) weight of π1,v and π2,v having the same parity
for each v|∞, the tensor product Π = π1  π2 is L-algebraic. The πi are not themselves twists of Lalgebraic automorphic representations, so Π cannot be expressed as a tensor product of L-algebraic
representations (this is proven in Corollary 2.6.3). This is, however, the ‘farthest’ from L-algebraicthat the πi ’s can be:
Proposition 2.5.8. Let Π be an L-algebraic cuspidal automorphic representation of GLnn0 (AF )for some number field F. Assume that Π = ππ0 is in the image of GLn ×GLn0 →− GLnn0 for cuspidal
automorphic representations π and π0 of GLn (AF ) and GLn0 (AF ).12 If F is totally real or CM, thenthere are quasi-tempered (i.e., tempered up to a twist) W-algebraic automorphic representations π
of GLn (AF ) and π0 of GLn0 (AF ) such that Π = π  π0 .
Proof. The first part of the argument is similar to, and will make use of, Lemme 4.9 of [Clo90](Clozel’s ‘archimedean purity lemma’), which shows that Π∞ itself is quasi-tempered. First assume
F is totally imaginary. Let wΠ (the ‘motivic weight’) be the integer such that Π| · |−wΠ /2 is unitary,
To be precise, we want Π to be a weak lift that is also a strong lift at archimedean places.

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and write zµv z̄νv and zµv z̄νv for the L-parameters at v|∞ of π and π0 . Temperedness of Π implies thatfor all i, j = 1, . . . , n,00Re(µv,i + µ0v, j + νv,i + νv,0 j ) = wΠ .
Fixing j and varying i, we find that Re(µv,i + νv,i ) is independent of i–call it wπv . Similarly, for wπ0v =0) = wπ0v for all i. For each of π and π0 there is a unique real numberwΠ − wπv , we have Re(µ0v,i + νv,i0
r, r0 such that each of π| · |−r/2 , π0 | · |−r /2 is unitary. Thus, zµv −r/2 z̄νv −r/2 (likewise for π0v ) is a generic(by cuspidality) unitary parameter.13 This implies (see the proof of Lemme 4.9 of [Clo90]) thatthese parameters are sums of the parameters of unitary characters and 2 × 2 complementary series
blocks (what Clozel denotes J(χ, 1) and J(χ, α, 1), where α ∈ (0, 1/2) is the complementary series
parameter). Since Re(µv,i +νv,i ) is independent of i, we can immediately rule out any complementary
series factors, and we deduce that each zµv −r/2 z̄νv −r/2 is a direct sum of (parameters of) unitarycharacters, and that wπv = r is independent of v (and wπ0v = r0 ), and r + r0 = wΠ ∈ Z. We may
r−r0
r0 −r
replace π by π|·| 2 and π0 by π0 |·| 2 , so that still ππ0 = Π but now each has half-integral ‘motivic
weight’ (namely, 12 wΠ ∈ 12 Z). In particular, Re(µv ), Re(νv ), Re(µ0v ), and Re(νv0 ) all now consist ofhalf-integers (it is easily seen that all of these half-integers are moreover congruent modulo Z).
Now, Im(µv,i + µ0v, j ) = 0 for all i, j, so there exists a tv ∈ R such that0
Im(µv,i ) = Im(νv,i ) = −Im(µ0v,i ) = −Im(νv,i) = tv

for all i. To conclude the proof, note that if there exists a Hecke character of F with infinitycomponenets |z|itCv for all v|∞, then we can twist π and π0 to arrange that both be W-algebraic (infact, either both L-algebraic or both C-algebraic). Looking at central characters, we are handed a
Hecke character with infinity-typez 7→ zdet(Re(µv ))+intv z̄det(Re(µv ))+intv ,and by Corollary
P 2.3.9, the desired Hecke character exists if and only if the set of half-integers
det(Re(µv )) := i Re(µv,i ) is the infinity-type of a type A Hecke character of F, i.e. descends to
Fcm . Of course, when F itself is CM, this is no obstruction, and the proof is complete.
Now assume F is totally real. By a global base-change argument (as in Clozel’s Lemme 4.9),
we may deduce temperedness of π and π0 from the totally imaginary (or even CM) case, andto determine whether there are W-algebraic descents π and π0 , as usual it suffices to look at thearchimedean L-parameters restricted to WF v . The purity constraint (see Corollary 2.3.9) on thehalf-integers det(Re(µv )) forces these to be independent of v, and we can argue as in the CM caseto finish the proof.
In the non-CM case, we can describe the infinity-types of possible Hecke characters; this gives
rise to an explicit relation between the det(Re(µv )) and tv that needs to be satisfied if π and π0
are to have W-algebraic twists. Conversely, note that the tensor product transfer (only known
to exist locally, of course) GLn × GLm →− GLnm clearly preserves L- or W-algebraicity (but not
C-algebraicity).
e → LG,In §3.1, we take up in detail the (algebraicity of the) fibers of the functorial transfer LG
e with torus quotient.
where G ⊂ G
13

On GLn , L-packets are singletons, and we can identify which parameters correspond to generic representations.
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2.6. Further examples: the Hilbert modular case and GL2 × GL2 →− GL4In this section we will elaborate on the example of mixed-parity Hilbert modular forms: we
discuss W-arithmeticity in this context and make some initial forays on the Galois side. First,
however, we recall known results on the automorphic tensor product GL2 × GL2 →− GL4 andprovide a description of its fibers.


2.6.1. General results on the GL2 ×GL2 →− GL4 functorial transfer. Ramakrishnan ([Ram00])
has proven the existence of the automorphic tensor product transfer in the case GL2 × GL2 →− GL4 .
Moreover, he establishes a cuspidality criterion:
Theorem 2.6.1 (Ramakrishnan, [Ram00, Theorem M]). Let π1 and π2 be cuspidal automorphic
representations on GL2 /F. Then π1  π2 is automorphic. The cuspidality criterion divides into twocases:• If neither πi is an automorphic induction, π1  π2 is cuspidal if and only if π1 is not
equivalent to π2 ⊗ χ for some Hecke character χ of F.
• If π1 = IndFL (ψ) for a Hecke character ψ of a quadratic extension L/F, then π1  π2 is
cuspidal if and only if BCL (π2 ) is not isomorphic to its own twist by ψθ ψ−1 , where θ is the
non-trivial automorphism of L/F.
Here is the Galois-theoretic heuristic for the first part of the cuspidality criterion. Assume V1
and V2 are two-dimensional, irreducible, non-dihedral representations. Non-dihedral implies thateach Sym2 Vi is irreducible. Suppose that V1 ⊗ V2  W1 ⊕ W2 . Taking exterior squares, we get(Sym2 V1 ⊗ det V2 ) ⊕ (det V1 ⊗ Sym2 V2 )  ∧2 W1 ⊕ (W1 ⊗ W2 ) ⊕ ∧2 W2 .
We may therefore assume dim W1 = 3, dim W2 = 1 (so rename W2 as a character ψ), and thus
V1 ⊗ V2 (ψ−1 )  W1 (ψ−1 ) ⊕ 1,
which implies that V1 and V2 are twist-equivalent.
We now describe the fibers of this tensor product lift, beginning with a Galois heuristic. Suppose we have four irreducible 2-dimensional representations (over Q` , say) satisfying
V1 ⊗ V2  W1 ⊗ W2 .
Taking exterior squares as above, we get(Sym2 V1 ⊗ det V2 ) ⊕ (det V1 ⊗ Sym2 V2 )  (Sym2 W1 ⊗ det W2 ) ⊕ (det W1 ⊗ Sym2 W2 )Assume V1 is non-dihedral, so Sym2 V1 is irreducible. Then we may assume
Sym2 V1 ⊗ det V2  Sym2 W1 ⊗ det W2
anddet V1 ⊗ Sym2 V2  det W1 ⊗ Sym2 W2 .
Comparing determinants in this and the initial isomorphism, we find det V1 det V2 = det W1 det W2 ,and so (for i = 1, 2)
Ad0 (Vi )  Ad0 (Wi ).
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Version 4 - Submitted to MEMOVi and W1 therefore give rise to isomorphic two-dimensional projective representations, and so, bythe (not very) long exact sequence in continuous cohomology associated to×

1 → Q` → GL2 (Q` ) → PGL2 (Q` ) → 1, 14
Vi and Wi are twist-equivalent: we find a pair of characters ψi such that V1  W1 (ψ1 ) and V2 
W2 (ψ2 ). If V2 is also non-induced, then ψ2 = ψ−1
1 , else V1 ⊗ V2 , and hence one of V1 or V2 , is aninduction.
Sym2

∧2

Because the functorial transfers GL2 −−−→ GL3 ([GJ78]) and GL4 −−→ GL6 ([Kim03]) are
known, this argument works automorphically until the last step. To conclude a purely automorphic
argument, we have to know the fibers of Ad0 . These were determined by Ramakrishnan as aconsequence of his construction of the tensor product lift (and his cuspidality criterion):
Theorem 2.6.2 (Ramakrishnan, [Ram00] Theorem 4.1.2). Let π and π0 be unitary cuspidal
automorphic representations on GL2 /F such that Ad0 π  Ad0 π0 . Then there exists a Hecke
character χ of F such that π  π0 ⊗ χ.
It would be remiss not to mention the special nature of this fiber description: by [LL79], itis essentially multiplicity one for SL2 . In any case, from this and the preceding calculation, we
conclude:Corollary 2.6.3. Let π1 , π2 , π01 , π02 be unitary cuspidal automorphic representations on GL2 /Fsatisfying π1  π2  π01  π02 (it suffices to assume this almost everywhere locally). Assume thatπ1  π2 is cuspidal (equivalently, satisfies Ramakrishnan’s criterion). Up to reordering, we have
Ad0 πi  Ad0 π0i for i = 1, 2, and there are Hecke characters χi of F such that πi  π0i ⊗ χi (i = 1, 2).
Moreover, χ1 = χ−1
2 , so the fibers of the lift are just twists by Hecke characters.
Proof. If at least one, say π1 , of the πi is non-dihedral, then it only remains to check χ1 = χ−1
2 .
Since π1  π2  (π1  π2 ) ⊗ (χ1 χ2 ), we deduce from a theorem of Lapid-Rogawski–which weelaborate on in Lemma 2.6.4 below–that either χ1 χ2 = 1, or π1  π2 is an automorphic induction
from the (quadratic or quartic) cyclic extension of F cut out by χ1 χ2 . In the latter case, there is a
cyclic base change L/F such that BCL (π1  π2 ) = BCL (π1 )  BCL (π2 ) is non-cuspidal. If both π1and π2 are non-dihedral, or if π2 = IndFK (ψ2 ) with K a quadratic extension of F not contained in L,then the cuspidality criterion implies there exists a Hecke character ψ of L such that BCL (π1 ) ⊗ ψ 
BCL (π2 ). Since π1 is non-dihedral, ψ is invariant under Gal(L/F); for cyclic extensions, there isno obstruction to descending invariant Hecke characters, so ψ descends to a character of F that we
also write as ψ, i.e. BCL (π1 ⊗ ψ)  BCL (π2 ). By cyclic descent (also Theorem B of [LR98]), π1
and π2 are twist-equivalent, contradicting cuspidality of π1  π2 . We conclude that χ1 χ2 = 1.
We resume the above argument when π2 is dihedral with K ⊂ L. If L = K, then χ1 χ2 is the0−1
quadratic character χ of Gal(K/F), so π2 = π02 ⊗ (χ−11 χ), and thus π2 = π2 ⊗ χ1 since π2 ⊗ χ = π2 .
If L/F is quartic, then the result of Lapid-Rogawski still implies π1  π2 = IndFL (ψ), where ψ is
now a Hecke character of L. Base-changing to L and comparing isobaric constituents, we again
contradict the assumption that π1 is non-dihedral.
Finally, we treat the case where both π1 and π2 are dihedral. Base-change implies that both π01
and π02 are dihedral as well, and moreover that we may assume that πi and π0i are both induced from
14

Note that the surjection admits a topological section.
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the same field Li , say by characters ψi and ψ0i . Write σi for the non-trivial element of Gal(Li /F).
Then




IndFL1 ψ1 ⊗ BCL1 (IndFL2 (ψ2 )) = IndFL1 ψ01 ⊗ BCL1 (IndFL2 (ψ02 )) ,and up to replacing ψ1 by ψσ1 1 , we may assume that on GL2 /L1 we have
ψ1
⊗ BCL1 (IndFL2 (ψ2 )) = BCL1 (IndFL2 (ψ02 )).
ψ01
If L1 , L2 , then we find
IndLL11 L2 (ψ1
· ψ2 ) = IndLL11 L2 (ψ02 ),ψ01and possibly replacing ψ2 by its conjugate ψσ2 2 , we may assume that as Hecke characters of L1 L2 ,
ψ1 /ψ01 = ψ02 /ψ2 . Let α be the Hecke character of L1 given by ψ1 /ψ01 . This equality shows α isσ1 -invariant, and therefore descends to a Hecke character of F. Consequently, π1 = π01 ⊗ α, and
π2 = π02 ⊗ α−1 as automorphic representations on GL2 /F. The case L1 = L2 is similar, but easier,and we omit the details.

The above corollary made use of a characterization of the image of cyclic automorphic induction (in the quadratic and quartic case). This characterization in the case of non-prime-degree does
not follow from the results of [AC89], but it is now known thanks to work of Lapid-Rogawski.
The following is stated without proof in [LR98], as an easy application of their Statement B15. We
fill in the details:
Proposition 2.6.4. [Lapid-Rogawski] Let Π be a cuspidal automorphic representation on GLn /F,and let L/F be a cyclic extension. Then Π is automorphically induced from L if (and only if)Π  Π ⊗ ω for a Hecke character ω of F that cuts out the extension L/F.
Proof. Lapid-Rogawski establish the following, which implies the Proposition:16
Theorem 2.6.5 (Statement B, part (a) of [LR98]). Let E/F be a cyclic extension of number
fields, with σ a generator of Gal(E/F), and let ω be a Hecke character of E. Denote by ωF its
restriction to A×F ⊂ A×E . Suppose Π is a cuspidal automorphic representation on GLn /E satisfying
Πσ  Π ⊗ ω. Let K/F be the extension (necessarily of order dividing n) of F cut out by ωF , and let
L = KE. Then E ∩ K = F, and [K : F] divides n.
Suppose ω cuts out L/F, cyclic of order m, and suppose m is divisible by a prime ` (and m , `).
We see that Π  Π ⊗ ωm/l as well, and letting E/F be the (cyclic degree `) extension cut out by
ωm/l , the case of prime degree implies that Π = IndFE (π) for some cuspidal representation π on
GLn /E. Moreover,
IndFE (π)  IndFE (π ⊗ (ω ◦ NE/F )),so the description of the fibers in the prime case implies π and π ⊗ (ω ◦ NE/F ) are Galois-conjugate:
writing τ for a generator of GalL/F (or for its restriction to E), there exists an integer i such thatπτ  π ⊗ (ω ◦ NE/F ).
i

15

The proof of which was conditional on versions of the fundamental lemma that are now known.
Which is in turn used to prove the more refined parts (b) and (c) of Statement B of [LR98].

16

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Version 4 - Submitted to MEMOPart (a) of Theorem 2.6.5 implies that the restriction of ω ◦ NE/F to A×F /F × , which is just ω` , cuts
out an extension K/F that is linearly disjoint from E/F,17 so in particular (ω◦NE/F )` is a non-trivial
Hecke character of E. But now, since τ` is trivial on E, we can iterate the previous conjugation
relation to deduceπ  π ⊗ (ω ◦ NE/F )` .
It now follows from induction on the degree [L : F] that π  IndEL (π0 ) for some cuspidal π0 on
n /L, so we’re done.
GL [L:F]
2.6.2. The Hilbert modular case. Now we take up the case of Hilbert modular forms, starting with an observation of Blasius-Rogawski ([BR93]): while a ‘mixed parity’ Hilbert modular
representation π (Definition 2.6.6) does not itself twist to an L-algebraic representation, its basechanges to every CM field will.18 Since we work up to twist, we may assume the parameters of π×at places v|∞ have the form (restricted to F v ⊂ WFv and implicitly invoking ι as above)k −1(z/z̄) v2
0 
z 7→ 
1−kv 
.
0(z/z̄) 2
Definition 2.6.6. In the above notation, π is mixed parity if as v varies over all v|∞, kv takes
both even and odd values.
For a quadratic CM extension L/F, let ψ : A×L /L× → S1 be a Hecke character whose infinity
type at v is given byz 7→ (z/|z|)kv −1
for all v. Then BCL (π) ⊗ ψ is L-algebraic, with infinity type!
(z/z̄)kv −1 0z 7→01Given two such π1 and π2 , both W- but not L-algebraic, and with Π = π1  π2 L-algebraic, BlasiusRogawski (see Theorem 2.6.2 and Corollary 2.6.3 of [BR93]) can then associate an `-adic Galois
representation ρΠ,` to Π via the identity
BCL (Π)  (BCL (π1 )(ψ1 ))  (BCL (π2 )(ψ2 )) ⊗ (ψ1 ψ2 )−1 ,since all three tensor factors on the right-hand side are L-algebraic with associated Galois representations. (To get a Galois representation for Π over F itself, they vary L and use the now-famouspatching argument.) We pursue this a little farther, emphasizing the ‘W-algebraic’ and non-de
Rham/geometric aspects of this construction.
Proposition 2.6.7. Let Π be the tensor product π1  π2 for πi as above corresponding to mixed
parity Hilbert modular forms (but π1 and π2 having common weight-parity at each infinite place).
Assume that Π is cuspidal, and for simplicity assume that the πi are non-dihedral.
(i) The `-adic Galois representation ρΠ,` is Lie irreducible.
The point is that Π is cuspidal. For instance, in the simple case `2 - m, there’s no need to appeal to the
Lapid-Rogawski result: ` divides m` i, so `|i, and then π  π ⊗ (ω ◦ NE/F ).
18
This observation of Blasius-Rogawski is the only time I have seen substantive use made of type A but not A0
Hecke characters.
17

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(ii) ρΠ,` is isomorphic to a tensor product ρ1,` ⊗ ρ2,` of two-dimensional continuous, almost
everywhere unramified, representations of ΓF . No twist of either ρi,` is de Rham (or even
Hodge-Tate), and ρΠ,` is not a tensor product of geometric Galois representations (although it is after every CM base change, by construction).19(iii) Ad0 (ρi,` ) is geometric, corresponding to the L-algebraic Ad0 (πi ) (suitably ordering therepresentations). In particular, a mixed-parity Hilbert modular representation π gives
rise to a geometric projective Galois representations ρ̄π,` : ΓF → PGL2 (Q` ); these are
the representations predicted by Conjecture 3.2.2 of [BG11] for irreducible SL2 (AF )constituents of π|SL2 (AF ) .
(iv) With the πi normalized to be W-algebraic, they are also W-arithmetic (see Question 2.5.4).
Proof. To show ρΠ,` is a tensor product, observe that it is essentially self-dual, since−1−1Π∨  π∨1  π∨2  (π1 ⊗ ω−1π1 )  (π2 ⊗ ωπ2 )  Π ⊗ (ωπ1 ωπ2 ) .

On the Galois side, we deduce that ρ∨Π,`  ρΠ,` ⊗ µ−1
Π , where µΠ is the (geometric) Galois charactercorresponding to the (L-algebraic) Hecke character ωπ1 ωπ2 . Since after one of these quadraticbase changes L/F ρΠ,` |ΓL is irreducible and orthogonal (rather than symplectic), we see that it (asΓF -representation) is orthogonal, i.e.
ρΠ,` : ΓF → GO4 (Q` ).
Writing µ for the multiplier character on GO4 , and observing that det2 = µ4 on this group, we findan exact sequence of algebraic groups

det ·µ−2

1 → Gm → GL2 × GL2 →− GO4 −−−−−→ {±1} → 1.
The image of ρΠ,` is contained in the kernel GSO4 of det ·µ−2 : for this it suffices to know that ρΠ,` isisomorphic to a tensor product after two disjoint quadratic base changes, for then (det ·µ−2 ) ◦ ρΠ,` isa character of ΓF that is trivial on a set of primes of density strictly greater than 1/2, and therefore it
is trivial. We can then apply Proposition 2.1.4 to deduce that ρΠ,` lifts across GL2 (Q` )×GL2 (Q` ) →GSO4 (Q` ), i.e. it is isomorphic to a tensor product ρ1,` ⊗ ρ2,` . The same exact sequence implies
that any expression of ρΠ,` as a tensor product has the form (ρ1,` ⊗ χ) ⊗ (ρ2,` ⊗ χ−1 ) for some×
continuous character χ : ΓF → Q` . Lie-irreducibility also follows since the Zariski closure of
the image of each ρi,` contains SL2 (otherwise ρΠ,` , and hence Π, is automorphically induced), andthus the Zariski closure of the image of ρΠ,` contains SO4 , which acts Lie-irreducibly in its standard
4-dimensional representation.
Now we show all twists ρi,` ⊗ χ are almost everywhere unramified, but none are geometric(they fail to be de Rham). This follows from the Galois-theoretic arguments of the next section,
but here we give a more automorphic argument. Note that both the de Rham and almost everywhereunramified conditions can be checked after a finite extension. By construction, for any L = FK
for K/Q imaginary quadratic, there exist (L-algebraic) cuspidal automorphic representations τi(i = 1, 2) on GL2 /L with associated geometric Galois representations σi : ΓL → GL2 (Q` ) such thatρ1,` |ΓL ⊗ ρ2,` |ΓL  ρΠ,` |ΓL  σ1 ⊗ σ2 .
19Compare Proposition 3.3.8.
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×This implies there is a Galois character χ : ΓL → Q` such that
σ1 (χ)  ρ1,` |ΓL ,σ2 (χ−1 )  ρ2,` |ΓL ,and every abelian `-adic representation is unramified almost everywhere (easy), so each ρi,` isalmost everywhere unramified. We then have the equivalences
ρi,` is geometric ⇐⇒ ρi,` |ΓL is geometric ⇐⇒ χ is geometric.20
But if χ is geometric, then (Theorem 2.3.13) there exists a type A0 Hecke character χA of L suchthat
τ1 ⊗ χA ∼w,∞ ρ1,`
τ2 ⊗ χ−1
A ∼w,∞ ρ2,` ,where ∼w,∞ is the notation of §1.4. Gal(L/F)-invariance and cyclic base change then imply there
are cuspidal automorphic representations τ̃i on GL2 /F lifting τ1 ⊗ χA and τ2 ⊗ χ−1
A , and therefore
BCL (τ̃1  τ̃2 )  BCL (Π). This implies τ̃1  τ̃2 and Π are twist-equivalent, and Corollary 2.6.3
ensures that (up to relabeling) πi and τ̃i are twist-equivalent. But τ̃i is L-algebraic since τi and χA
are, yet it is easy to see (because π is mixed-parity and F is totally real) that no twist of πi can be
L-algebraic.
That, suitably ordered, Ad0 (ρi,` ) is a geometric Galois representation whose local factors match
those of the L-algebraic cuspidal representation Ad0 (πi ) follows easily from the earlier ∧2 argument(see §2.6.1)and the fact that our forms (representations) are all non-dihedral.
Part (iv): Normalize πi to be W-algebraic. Then πi  πi = Sym2 (πi )  ωπi is L-algebraic, and by[BR93] it has Satake parameters in Q at unramified primes. Writing {αi,v , βi,v } for the parametersof πi,v (when unramified), we conclude that α2i,v , β2i,v ∈ Q, hence αi,v , βi,v ∈ Q. This implies πi,v hasa model over Q for all such v,21 answering the weaker version of Question 2.5.4. In this case thestronger version (that πi, f is defined over Q) follows as well, by a check (omitted) at the ramified
primes.
Now we prove a more refined result in the case π1 = π2 :Proposition 2.6.8. Let π be a unitary, cuspidal, non-induced, mixed-parity Hilbert modular
representation, so that Π = π  π has an associated Galois representation, the sum of Galois
representations associated to the L-algebraic representations Sym2 (π) and ωπ . As before, thereare `-adic representations ρi,` such that ρΠ,` = ρ1,` ⊗ ρ2,` .
• We can normalize the ρi,` so that ρi,` (frv ) has characteristic polynomial in Q[X] for allunramified v. Moreover, its eigenvalues in fact lie in Qcm , and are pure of some weight,
which we may take to be zero.
• For quadratic CM extensions L = KF with K/Q a quadratic imaginary field in which` is inert, there is a ρ` : ΓF → GL2 (Q` ) for which Sym2 (ρ` ) ∼w,∞ Sym2 (π). Later on(Corollary 2.7.8), we will see this is not possible over a totally real field.
Moreover, ρi,` is Hodge-Tate ⇐⇒ ρi,` |ΓL is Hodge-Tate ⇐⇒ χ is Hodge-Tate ⇐⇒ χ is geometric!
Note that the fields of definition of πv and its Satake parameters may be different; but if one is algebraic, thenboth are.
20

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Version 4 - Submitted to MEMO• Nevertheless, we can choose ρ` and a finite-order character χ such that Sym2 (ρ` ) : ΓF →
GO3 (Q` ) is, viewing GO3 (Q` ) as the dual group (Gm × SL2 )∨ (Q` ), associated to an automorphic representation (ωπ χ, π0 ) of (Gm × SL2 )(AF ), as in 3.2.2 of [BG11], with π0 any
irreducible constituent of π|SL2 (AF ) .
Proof. If we take π1 = π2 = π in the unitary normalization (this will ensure π  π has ‘motivic
weight zero,’ with an implicit–and provable–application of Ramanujan at infinity) and decompose
the Galois representation ρΠ,` associated to Π = π  π as ρ1 ⊗ ρ2 , then the now-familiar ∧2 argumentshows that Ad0 (ρ1 )  Ad0 (ρ2 ). We can therefore write ρΠ,`  ρ ⊗ (ρ ⊗ χ) for some Galois character
χ. This χ need not be a square, but we can normalize ρ so that χ is finite order (by Lemma 2.3.15).
Now, the Ramanujan conjecture is known for Π22, and of course for χ, so ρ⊗ρ is pure of weight zero(at unramified primes, say). Let αv and βv be frobenius eigenvalues of ρ(frv ) at an unramified primev, so that α2v and β2v (being eigenvalues of Sym2 (ρ)) have absolute value one under any embeddingQ` ,→ C. The same then holds for αv and βv , so ρ itself is pure of weight zero. Therefore thefrobenius eigenvalues of ρ lie in Qcm .
We conclude by showing that, after certain CM base-changes, we can take the finite-ordercharacter χ to be trivial. Restricting to L = KF as above, we can find a type A Hecke characterψ of L such that π ⊗ ψ is L-algebraic, and then letting (ψb2 )` be the geometric `-adic representation
associated to ψ2 , we claim that, possibly replacing ψ by ψ−1 , there exists a Galois character λ of Lsuch that
ρ|ΓL ⊗ λ  ρ|ΓL ⊗ (χ(ψb2 )` λ−1 ).
Over L, π · ψ−1 has associated Galois representation r, and then r · (ψb2 )` corresponds to π · ψ. Then
there exists a λ such that either r = ρ · λ and r · (ψb2 )` = ρ(χλ−1 ), or r = ρ · (χλ−1 ) and r · (ψb2 )` = ρ · λ.
Either way, plugging one equation into the other we get the claim.
b2
ρ is not an induction, so χ|ΓL = λ2 (ψb2 )−1` . In particular, χ|ΓL is a square if and only if (ψ )` admitsa Galois-theoretic square-root. This was discussed in Lemma 2.3.18 above: it need not always be
the case (for instance, if ` is split in L/Q), but it is if ` is inert in K. Choosing such an L = KF, we
can now find χL : ΓL → Q` such thatρΠ,` |ΓL  (ρ|ΓL ⊗ χL )⊗2 .
Here χL = λψ̂ for ψ̂ a square-root of (ψb2 )` , and we’re done.
For the final point, note that the inclusion Gm × SO3 → GO3 is an isomorphism, with inverse(g), det(g)−1 g). In particular, Sym2 : GL2 → GO3 becomes g 7→ (det(g), Ad0 (g) inmap g 7→ ( detµµthese coordinates. The SL2 (AF )-constituents of π have Langlands parameters corresponding to
the projectivization of those of π, and since Ad0 : GL2 → SO3  PGL2 is just the quotient map
GL2  PGL2 , the claim follows.

Remark 2.6.9. In particular, the Proposition tells us that there are pure `-adic Galois representations no twist of which are geometric. They do not, however, live in compatible systems.
These propositions would have no content if the only examples of mixed parity Hilbert modular
forms were the inductions described in Example 2.5.3; we end this section by showing
Lemma 2.6.10. Over any totally real field, non-CM mixed parity Hilbert modular forms exist.
Of course, Π is not cuspidal, so literally this is for Sym2 (π) and ωπ .

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Proof. Considering the (semi-simple, connected) Q-group G = ResF/Q (SL2 /F), we can apply
Theorem Q
1B from §4 of Clozel’s paper [Clo86]. We fix a discrete series representation π∞ ofG(R)  τ : F,→R SL2 (R) corresponding to the desired mixed-parity infinity-type. Then fix an
auxiliary supercuspidal level K p0 ⊂ G(Q p0 ) for some prime p0 . At some other finite prime p, let π p
be a fixed (twist of) Steinberg representation. Finally, let S be a set of finite primes (disjoint from
p0 , p) at which we will let the level go to infinity in the manner of Clozel’s paper (written KS → 1).
Letting K be any fixed compact open subgroup away from S , p, p0 , ∞, Clozel proves thathi2lim inf vol(KS ) · mult π∞ ⊗ π p , Lcusp(G(Q)/G(AF ))K p0 ×KS ×K > 0.
KS →1

(He also determines an explicit but non-optimal constant.) This suffices for the Lemma: it implies
the existence of (infinitely many) cuspidal automorphic representations on G that are Steinberg at
p, and therefore not automorphically induced, and have the desired mixed parity at infinity.
Remark 2.6.11. In [Shi12], Shin derives, as a corollary of very general existence results forautomorphic representations, an exact limit multiplicity formula for cohomological Hilbert modular forms. Unfortunately, this excludes precisely the mixed-parity forms we are interested in. It
seems, however, that his methods should extend to cover all discrete series infinity types.
2.6.3. A couple of questions. In Proposition 2.6.7, the key point was an understanding of
which type A Hecke characters–or their Galois analogues–could exist. I would like to raise herea couple questions in higher rank about the existence of automorphic forms with certain infinitytypes.
Question 2.6.12. Are there automorphic representations π on GL2 /F (F totally real) such thatat two infinite places v1 and v2 ,• πv1 is discrete series and πv2 is limit of discrete series, but excluding the easily-constructeddihedral cases, as in Example 2.5.3? Clozel’s result does not apply to this case; but after
the initial√ writing of this paper, Kevin Buzzard informed me that one such example, for
F = Q( 5), has been produced by computer calculation in [MS14];• πv1 is discrete series (or limit of discrete series), and πv2 is principal series (looks like a
Maass form)?
• as in the previous item, but with algebraic infinity-type (so a ‘Plancherel density’-type
result will not suffice)?
2.7. Galois lifting: Hilbert modular case
2.7.1. Outline. We continue to elaborate on the examples of the previous section, now turning
to some preliminary cases of a problem raised by Conrad in [Con11]. Recall from the introductionthat he addresses lifting problemse `) ,
H(Q
=

ρ̃

ΓF

ρ


/ H(Q )`e  H is a surjection of linear algebraic groups with central kernel, and the key remaining
where H
question is:50

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Version 4 - Submitted to MEMOe  H is a torus. If ρ is geometric, when does
Question 2.7.1. Suppose that the kernel of H
there exist a geometric lift ρ̃?
We will address this question in stages. Here is an outline of the coming sections:• First we address, in the regular case, with F totally real, Conrad’s question for lifting
across GL2 (Q` ) → PGL2 (Q` ). Modulo the difference between potential automorphy andautomorphy, we find that all examples of ρ not having geometric lifts are accounted forby mixed-parity Hilbert modular forms (a converse to Propositions 2.6.7 and 2.6.8).
• To give a higher-rank example in which potential automorphy theorems still allow us tolink the automorphic and Galois sides, we then carry out in §2.8 an analogous discussion
for GSpin2n+1 (Q` ) → SO2n+1 (Q` ) (§2.2 introduces the necessary background about Sen
operators and ‘labeled Hodge-Tate-Sen weights’ in `-adic Hodge theory; in the current
section, we use more elementary terminology, although some manipulations are justified
by the general theory).
• Before proceeding to the general solution (§3.2) of Conrad’s question, we (§3.1) discuss
the automorphic analogue. This is in some sense more intuitive, and it motivates the
Galois-theoretic solution. Note that in general we have neither constructions of automorphic Galois representations nor (potential) automorphy theorems, so we cannot unconditionally bridge the automorphic-Galois chasm. Proving a ‘modular lifting theorem’ frome in this context is an important remaining question.
H to H
2.7.2. GL2 (Q` ) → PGL2 (Q` ). In this subsection we prove:
Theorem 2.7.2. Let F be totally real. Suppose ρ : ΓF → PGL2 (Q` ) is a geometric representation with no geometric lift to GL2 (Q` ). Then
• There exists a lift ρ̃ : ΓF → GL2 (Q` ) such that for all CM L/F, ρ̃|ΓL is the twist of ageometric Galois representation (in particular, ρ|ΓL has a geometric lift, which ought tobe automorphic).
Now assume moreover that Ad(ρ) : ΓF → SO3 (Q` ) ⊂ GL3 (Q` ) satisfies the hypotheses of (thepotential automorphy theorem) Corollary 4.5.2 of [BLGGT14]23 Then:• After a totally real base-change F 0 /F, Ad(ρ) is automorphic, and more precisely• we may normalize ρ̃ such that Sym2 (ρ̃)|ΓF0 is a geometric Galois representation corresponding to Sym2 (π) ⊗ χ for some mixed parity Hilbert modular representation π on
GL2 /F and non-trivial finite order character χ. Furthermore, ρ̃ is totally odd.
Remark 2.7.3. If Ad0 (ρ) is not regular, there are two possibilities: either it fails to be regular
everywhere, in which case Fontaine-Mazur conjecture that ρ (up to twist) has finite image. Such
ρ always have finite-image (hence geometric) lifts by Tate’s theorem (Proposition 2.1.4). If ρ istotally odd, and Ad0 (ρ) exhibits a mixture of regular and irregular behavior, it should be related tothe existence of a mixed-parity Hilbert modular form that is limit of discrete series at some infinite
places, and genuine discrete series at others. I know of no such examples not arising from Hecke
characters. The question of whether there exist non-dihedral ρ that are even at some and odd atother infinite places is related to Question 2.6.12.
Essentially: there is a lift ρ̃ of ρ such that (ρ̃ mod `)|ΓF(ζ` ) is irreducible and ρ̃ is regular (if true for one lift, thisis true for all), and ` is sufficiently large.
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Version 4 - Submitted to MEMOWe now make our first use of non-Hodge-Tate Galois representations; recall that in §2.2 wehave described the theory of (non-integral) Hodge-Tate-Sen weights. In fact, a reader unfamiliar
with this general theory will have no difficulty following the arguments of the current section.
Lemma 2.7.4. For any number field F and any geometric ρ : ΓF → PGL2 (Q` ), there is a liftρ̃ : ΓF → GL2 (Q` ) of ρ whose Hodge-Tate-Sen (HTS) weights at all v|` belong to 21 Z. In this case,Sym2 (ρ̃) and det(ρ̃) are geometric.
Proof. Proposition 2.1.4 ensures that we can find some lift ρ̃. By assumption, Ad0 (ρ̃) is geometric, soρ̃ ⊗ ρ̃ ⊗ det(ρ̃)−1  Ad0 (ρ̃) ⊕ 1
is geometric. We ask whether det(ρ̃) is a square. There at least exists a Galois character ψ : ΓF →×
Q` such that det(ρ̃) = ψ2 χ with χ finite order (Lemma 2.3.15). Thus[ρ̃ ⊗ ψ−1 ]⊗2  (Ad0 (ρ̃) ⊗ χ) ⊕ χ
is also geometric, and in particular Hodge-Tate. Replacing ρ̃ with the new lift ρ̃ ⊗ ψ−1 we conclude that Sym2 (ρ̃) and det ρ̃ are Hodge-Tate. The theorem of Wintenberger and Conrad (Theorem
2.1.6) then proves they must be de Rham, since the original projective ρ was. Any lift ρ̃ is almost
everywhere unramified by Lemma 5.3 of [Con11], so the proof is complete.

We now proceed in the same spirit as in the construction of W-algebraic forms via type A
Hecke characters. The following lemma is an ad hoc version of Theorem 3.2.7; it may help innavigating that rather formal proof:
Lemma 2.7.5. Consider ρ̃ : ΓF → GL2 (Q` ) as produced by Lemma 2.7.4. Let L/Q be a quadratic CM extension. Then for all such L, the restriction of ρ̃ to ΓL is the twist of a geometricGalois representation.
ι∞ι`

Proof. Recall that our fixed embeddings Q ,−→ Q` yield a map→ C and Q ,−[ι∗∞,` :
HomQ` (Fv , Q` ) → HomQ (F, C).
v|`

Sym2 (ρ̃) is de Rham, so for each place v|` of F and each τ : Fv ,→ Q` , it has τ-labeled Hodge-Tateweights24Aτ + BτHTτ Sym2 (ρ̃|ΓFv ) = {Aτ ,, Bτ },
2
where the Aτ and Bτ are integers, necessarily having the same parity. We can distinguish the subset{Aτ , Bτ } ⊂ HTτ Sym2 (ρ̃|ΓFv )(trivially in the non-regular case, easily otherwise). It follows that ρ̃|ΓFv is Hodge-Tate (hence
de Rham) if and only if all Aτ and Bτ are even. Consider the map ι∗∞,` for L as well as F; thetwo versions are compatible, so conjugate archimedean embeddings ι, ῑ : L ,→ C pull back toembeddings L ,→ Q` that lie above a common τ : F ⊂ Fv ,→ Q` . To each τ we then unambiguously
24

By duality, or by the Hodge-Tate-Sen theory.
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associate the parity τ ∈ {0, 1} of Aτ (and Bτ ), and at the unique archimedean place v∞,τ of L aboveι∗∞,` (τ) we define a character
ψv∞,τ : Lv×∞,τ → C×kτkτz 7→ ι(z) 2 ῑ(z)− 2 ,where kτ is any integer with parity τ .
L is a CM field, so Weil ([Wei56]) tells us that there exists a type A Hecke character ψ of
L with components at archimedean places given by these ψv∞,τ . We wish to twist ρ̃|ΓL by a Galois
realization of ψ, shifting all of its HTS weights to be integers (recall this can be checked by looking
at Sym2 (ρ̃|ΓL )), but care is needed: there is no Galois representation canonically associated to a typeA (not A0 ) Hecke character. Instead, we associate (canonically, having specified ι∞ and ι` ) an `adic representation (ψb2 )` to the type A0 Hecke character ψ2 , and then we non-canonically extract,
up to a finite-order character, a (Galois-theoretic, non-geometric) square root ψ̂. Then ρ̃|ΓL ⊗ ψ̂ is
Hodge-Tate, since the analogues of the Aτ and Bτ , but now for Sym2 (ρ̃|ΓL ⊗ ψ̂) = Sym2 (ρ̃|ΓL ) ⊗ (ψb2 )` ,are all even. Thus ρ̃|ΓL ⊗ ψ̂ is geometric (again by Theorem 2.1.6).
This completes the proof of the first part of Theorem 2.7.2. We need deeper tools to makefurther progress. Let ρ̃ be the lift as above, with both Sym2 (ρ̃) and det(ρ̃) geometric. Sym2 (ρ̃)
factors through GO3 (Q` ) with totally even multiplier character det(ρ̃)2 . We now make further
assumptions on ρ̃ (or ρ) in order to apply a potential automorphy theorem:
Hypothesis 2.7.6.
(1) Assume ` > 7.
(2) Assume Sym2 (ρ̃) is ‘potentially diagonalizable’ and regular at all v|`. For instance, we
require that for each v|` it be crystalline with τ-labeled Hodge-Tate weights distinct andfalling in the Fontaine-Laffaille range for all τ : Fv ,→ Q` .
(3) Assume that the reduction mod ` of Sym2 (ρ̃)|ΓF(ζ` ) is irreducible. (We have taken ` , 2, so
this is equivalent to the analogous assumption for ρ̃.)
Then we can can apply Theorem C of [BLGGT14] to deduce that after base-change to sometotally real field F 0 , Sym2 (ρ̃)|ΓF0 is automorphic, corresponding to a RAESDC automorphic representation Π on GL3 /F 0 . Let us abusively denote by det(ρ̃) the Hecke character (of type A0 )corresponding to det(ρ̃). Then Π ⊗ det(ρ̃)−1 is self-dual with trivial central character, and (by, forinstance, [Art13]–details, in greater generality, appear in Lemma 2.8.6–although in this case theresult is older) there exists a cuspidal π on GL2 /F 0 such that Ad0 (π)  Π ⊗ det(ρ̃)−1 . We may
assume that ωπ has finite order (applying Lemma 2.3.6 and Proposition 3.1.4, since F 0 is totallyreal), because the descent is originally to SL2 /F 0 , and we then have control over the choice ofextension to GL2 /F 0 .25
Lemma 2.7.7. Such a π on GL2 /F 0 satisfying Ad0 (π)  Π ⊗ det(ρ̃)−1 is necessarily a mixed
parity Hilbert modular representation.
Proof. It is clear from the archimedean L-parameters that π is W-algebraic. If it were Lalgebraic, there would be a corresponding Galois representation ρπ 26 with Ad0 (ρπ )  Ad0 (ρ̃), andthus ρπ and ρ̃|ΓF0 would be twist-equivalent. Lemma 2.3.17 then implies ρ̃ has all integral or all
half-integral Hodge-Tate weights, which contradicts the assumption that ρ has no geometric lift25

We only sketch this here, because these matters will be discussed in greater detail and generality in §3.1.
By [BR93].

26

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(note that both F 0 and F are totally real). Similarly, if π were C-algebraic, then there would be ageometric representation corresponding to π|·|1/2 , and we can use the same argument. We conclude
that π must be mixed parity.

Corollary 2.7.8. For any ρ : ΓF → PGL2 (Q` ) as in Theorem 2.7.2, and satisfying the auxiliary
potential automorphy Hypothesis 2.7.6, there exists a lift ρ̃ : ΓF → GL2 (Q` ) and, after a totally realbase change F 0 /F, a mixed-parity Hilbert modular representation π on GL2 /F 0 , such that
Sym2 (ρ̃)|ΓF0 ∼w,∞ Sym2 (π) ⊗ χ
for some finite order Hecke character χ of F 0 . Moreover, any lift ρ̃ is totally odd, and the character χ is necessarily non-trivial. In contrast, restricting to L0 = F 0 K where K/Q is a quadratic
imaginary extension in which ` is inert, we can find a lift ρ̃ such that Sym2 (ρ̃) corresponds to
BCL0 /F 0 (Sym2 (π)).
Conversely, starting with a mixed-parity π, we can produce such a χ and ρ̃.
Proof. In the notation of the previous Lemma, we haveSym2 (π) ⊗det(ρ̃) Π ∼w,∞ Sym2 (ρ̃)|ΓF0 .
ωπ

The character det(ρ̃)/ωπ is L-algebraic, hence equals χ| · |m for a finite-order character χ and an
integer m. Replacing π by π ⊗ | · |m/2 , which is still mixed-parity, we are done except for the lastpart.
Assume instead that we can take χ = 1. In particular, ωπ ∼w,∞ det(ρ̃), and thus det(ρ̃)(cv ) =ωπv (−1). By the description of discrete series representations of GL2 (R) and purity of ωπ , we knowthat the Langlands parameter φv : WR → GL2 (C) takes the formk −1(z/z̄) v2
0  −w/2
z 7→ 
1−kv 
 |z|C0(z/z̄) 2!
0 (−1)kv −1
j 7→
.
1
0
Hence ωπv (−1) = det(φv )( j) = (−1)kv . Since Ad0 (ρ̃)|ΓF0 ∼w,∞ Ad0 (π) is regular algebraic self-dual
cuspidal, Proposition A of [Tay12] shows that Ad0 (ρ̃) is odd, and thus for all v|∞, the eigenvaluesof Ad0 (ρ̃)(cv ) are {−1, 1, −1}. We conclude that det(ρ̃)(cv ) = −1 for all v|∞, proving the oddness
claim for ρ̃, and contradicting, when we assume χ = 1, the fact that the kv are both odd and even.
The remaining claims, constructing from π such a ρ̃ and χ, were established in Proposition
2.6.8.
This concludes the proof of Theorem 2.7.2.
2.8. Spin examples
To address Conrad’s question in general, we will have to re-cast the arguments of §2.7 interms of root data. In the meantime we give another pleasantly concrete example, but one that
relies on root-theoretic manipulation, in the hopes of easing the transition to the general case;moreover, specializing to discrete series/regular examples, we can also still exploit known results
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Version 4 - Submitted to MEMOabout automorphic Galois representations. The previous example (lifting across the surjection
GL2 (Q` ) → PGL2 (Q` )) is really just the first case of a family of spin examples,
GSpin2n+1 (Q` )
9

ρ̃

ΓF

ρ


/ SO2n+1 (Q ),`where ρ is geometric but has no geometric lift ρ̃. F will be a totally real field, and we will again
see that over CM fields, geometric lifts do in fact exist. The case n = 1 will amount to the contentsof §2.7.
We first recall the basic setup for (odd) Spin groups. We will also make heavy use of thisnotation, and its obvious analogues for even Spin groups, later on in some of our examples of‘motivic lifting’ (see §4.2 and §4.3).
1

1

1


/ {±1}

/ Spin

1


/ Gm


/ GSpin

2n+1

z7→z2



2n+1

/1

/ SO2n+1

/1

ν



Gm

/ SO2n+1

Gm





1

1

where ν is the Clifford norm. We can then identify
GSpin2n+1 o∼

Gm ×Spin2n+1{1,(−1,c)}ν(x,g)7→ x2





Gm

Gm

where c is the non-trivial central element of Spin2n+1 . This yields an identification of Lie algebras∼

gspin2n+1 ←− gl1 × so2n+1 ,where gl1 is identified with the center of gspin2n+1 . Alternatively, GSpin2n+1 is the dual group to
GSp2n , and it will be convenient to keep both normalizations of (based) root data for GSpin2n+1 –∨∨one from the spin description, one from the dual description–in
 0 mind.
 Let (X, ∆, X , ∆ ) be the1nbased root datum for GSp2n (say defined with respect to J2n = −1n 0 ), with its diagonal maximal
torus T and the Borel B ⊃ T for which ei − ei+1 and 2en − e0 form a set of positive simple roots,
withei : diag(t1 , . . . , tn , νt1−1 , . . . , νtn−1 ) 7→ ti
e0 : diag(t1 , . . . , tn , νt1−1 , . . . , νtn−1 ) 7→ ν.
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Then for GSpin2n+1 , we have the based root datum (with X ∨ the character group)
X =
∨

∆

∨

n
M

Ze∗i

i=0
∨
∗
= {α∨i = e∗i − e∗i+1 }n−1i=1 ∪ {αn = en }
n
M

X=

Zei
i=0

∆ = {αi = ei − ei+1 }n−1i=1 ∪ {αn = 2en − e0 },with X and X ∨ in the duality hei , e∗j i = δi j .
Alternatively, since Spin2n+1 is the connected, simple, simply-connected group with Lie algebra
so2n+1 , we can write its character group as the weight lattice of so2n+1 , i.e., P
as the submoduleχ1 +...+χnn
of ⊕ni=1 Qχi spanned
,
and
its
co-characterlattice
as
those
by
χ
,
.
.
.
,
χ
,
1
n
i=1 ci λi such that2P
ci ∈ Z and ci ∈ 2Z. The duality is hχi , λ j i = δi j . Letting χ0 and λ0 generate X • (GL1 ) andGL1 ×Spin2n+1
X• (GL1 ), respectively, the description of GSpin2n+1 as {1,(−1,c)}then leads to another descriptionof the root datum asnnMMχ1 + . . . + χn•Y =Zχi ⊕ Z(χ0 +
)⊂
Qχi
2
i=1
i=0
∆• = {χi − χi+1 }n−1i=1 ∪ {χn }
n
M
λ0
Y• =
Z(λi + ) ⊕ Zλ02
i=1∆• = {λi − λi+1 }n−1i=1 ∪ {2λn }.
We summarize by comparing the two descriptions:Lemma 2.8.1. There is an isomorphism of based root data
(X ∨ , ∆∨ , X, ∆)  (Y • , ∆• , Y• , ∆• )given bye∗i 7→ χi

for i = 1, . . . , n;
χ1 + . . . + χn;e∗0 7→ χ0 −2
λ0
ei 7→ λi +for i = 1, . . . , n − 1;2
e0 7→ λ0 .
For example, the center of GSpin2n+1 is P
generated by the co-character e0 ↔ λ0 . The Clifford∗
norm is given by the character 2χ0 ↔ 2e0 + n1 e∗i .
Returning to our representation ρ, recall that to each v|` and ι : Q` ,→ CFv we can associate theSen operatorΘρ,ι := Θρ|ΓFv ,ι ∈ so2n+1 ⊗Q` ,ι CFv .
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(The place v is implicit in ι.) Since ρ is Hodge-Tate, we can identify Θρ,ι up to conjugation with adiagonal element
 m1

 .

..




mn

 ,
0
−mn




...
−m1

where m j = m j (ι) are all integers.

27

Proposition 2.8.2. Let ρ : ΓF → SO2n+1 (Q` ) be a geometric representation as above, with F
totally real. Then ρ lifts to a geometric, GSpin2n+1 -valued representation if and only if the parity
Pof j m j (ι) is independent of v|` and ι : Q` ,→ CFv .
Remark 2.8.3. By itself, this result is formal. Later we will see how such ρ arise, at least when
ρ is regular; but note that this Proposition makes no such assumption.
Proof. By Proposition 2.1.4, some lift ρ̃, necessarily unramified almost everywhere, exists.
×
All possible lifts differ from this ρ̃ by e0 ◦ χ for some χ : ΓF → Q` . As before, we can write the×multiplier character ν(ρ̃) : ΓF → Q` as χ2 χ0 , where χ0 has finite order, and then replace ρ̃ by atwist having ν(ρ̃) of finite-order (Lemma 3.2.2 will explain this procedure in general). Then, sincefinite-order characters have Hodge-Tate weights zero, functoriality of the Sen operator implies thatfor all ι : Q` ,→ CFv , Θρ̃,ι maps to (0, Θρ,ι ) in (gl1 )ι ⊕ (so2n+1 )ι .28
rspin

Now consider the spin representation GSpin2n+1 −−−→ GL2n . In the above root datum notationn. The image of Θρ̃,ι is then a semi-simple
this corresponds to the highest weight −e∗0 ↔ −χ0 + χ1 +...+χ
2
m1 +...+mn
element with one eigenvalue, and all eigenvalues congruent to this (half-integer) modulo
2
Z. In particular, rspin ◦ ρ̃|ΓFv is Hodge-Tate, and thus de Rham, if and only if for all ι : Q` ,→ CFv ,m1 (ι) + . . . + mn (ι) is even. If all (for all v|` and all ι) of these sums are odd, then we twist
by a character χ with all Hodge-Tate-Sen weights 1/2 (see Lemma 2.3.17) to get a new ρ̃, now
geometric, lifting ρ. On the other
P hand, Lemma 2.3.17 shows that for F totally real we cannot

twist ρ̃ in a similar way if some j m j (ι) are even and others are odd.
Corollary 2.8.4. Let ρ : ΓF → SO2n+1 (Q` ) be as in Proposition 2.8.2. Let L/F be a CM
extension. Then ρ|ΓL has a geometric lift.
Proof. With the framework of the previous proof, this follows by the same argument as Lemma
2.7.5.

If we make additional assumptions so that potential automorphy theorems apply, we can ofcourse say more. The next two lemmas merely exploit some very deep recent results.
Lemma 2.8.5. Assume that ρ : ΓF → SO2n+1 (Q` ) ⊂ GL2n+1 (Q` ) as in the previous proposition
moreover satisfies:• For all v|`, ρ|ΓFv is regular and potentially diagonalizable;• ρ̄|ΓF(ζ` ) is irreducible.
27

We choose the ‘anti-diagonal’ orthogonal pairing, so that SO2n+1 has a maximal torus consisting of diagonal
matrices.
28
Writing gι as a short-hand for g ⊗Q` ,ι CFv .
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Then after some totally real base change F 0 /F, there exists a regular L-algebraic self-dual cuspidal
automorphic representation π of GL2n+1 (AF 0 ) such that π ∼w,∞ ρ.
Proof. This is immediate from Theorem C (=Corollary 4.5.2) of [BLGGT14], since ρ is automatically totally odd self-dual.29
Lemma 2.8.6. Let ρ ∼w,∞ π be as in Lemma 2.8.5. Then π descends to a cuspidal automorphic representation on Sp2n (AF 0 ), in a way compatible with unramified and archimedean local
L-parameters.
Proof. This follows from Arthur’s classification of automorphic representations of classicalgroups ([Art13]). Namely, ρ∨  ρ implies that π∨  π, and det(ρ) = 1 implies that ωπ = 1. πis cuspidal, so the associated formal A-parameter φ (see §1.4 of [Art13]) is simple generic and socomes from a unique simple twisted endoscopic datum Gφ as in Theorem 1.4.1 of [Art13]; since
2n + 1 is odd, the parameter φ therefore factors through either SO2n+1 (C) or O2n+1 (C), but the lattercase is ruled out since ωπ = 1. It follows that Gφ = Sp2n /F.30 The local statement follows from
[Art13, Theorem 1.4.2].
Proposition 2.8.7. Continuing with the assumptions (and conclusions) of the previous two lemmas, ρ|ΓF0 has a geometric lift to GSpin2n+1 (Q` ) if and only if π admits an L-algebraic extension to
GSp2n (AF ). In the other direction, we have, as in Lemma 2.6.10, an automorphic construction of
infinitely many such ρ, whose local behavior31 we can additionally specify at any finite number ofplaces, that do not admit a geometric lift.
Proof. (Some details of this argument are omitted, deferring to more general arguments in thenext section.) At each v|∞, we can write the L-parameter of πv as
φv : WFv → SO2n+1 (C) × WFv
 zm1 z̄l1

..

.

zmn z̄ln1
φv |WFv : z 7→ 
z−mn z̄−ln


...

z−m1 z̄−l1 .





 ,




with all mi , li ∈ Z. Since the transfer of π to GL2n+1 (AF ) is cuspidal, Clozel’s archimedean purity
theorem (see Lemme 4.9 of [Clo90]) implies that mi + li = 0 for all i (and all v). To extend π inan L or W-algebraic fashion, the key point is to construct an appropriate extension of the central
character. Any lift φ̃v to GSpin2n+1 (C) of the parameter φv must, on WF v , take the form z 7→ zµ̃v z̄ν̃v ,29

Note that those authors always work with C-algebraic automorphic representations, so the statements of their
theorems always have an extra, but easily unraveled, twist.
30
One would like to say that since ρ lands in SO2n+1 (Q` ), Sym2 (ρ) contains the trivial representation and thereforeL(s, π, Sym2 ) has a pole at s = 1; this would allow us to descend π to Sp2n (AF ) using [CKPSS04]. Unfortunately,
nothing is known a priori about L(s, V) where Sym2 (ρ) = 1 ⊕ V; in particular, we can’t say immediately it is nonvanishing at s = 1, so this argument doesn’t seem to work. The argument we have given allows us to deduce that
L(s, π, Sym2 ) has a simple pole at s = 1, and therefore L(s, V) is non-vanishing at s = 1.
31
i.e. inertial type
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whereµ̃v =nXmv,i (λi +i=1ν̃v = −nX

λ0
) + µv,0 λ02mv,i (λi +i=1

λ0) + νv,0 λ0 .
2

The central character of this extension is (by pairing with 2χ0 , the Clifford norm; this procedure
for computing central characters is explained in general in [Lan89])
ωv : z 7→ z i=1 mv,i +2µv,0 z̄− i=1 mv,i +2νv,0 .
Pn

Pn

If we can choose an extension of π to an automorphic representation π̃ of GSp2n (AF 0 ) with finiteorder central character, then this calculation shows thatPφ̃v (the local L-parameter for π̃v ) is always‘W-algebraic,’32 and it is ‘L-algebraic’ if P
and only if ni=1 mv,i is even. The result would follow
easily, twisting our given π̃ by | · |1/2 if all i mv,i are odd (just as in the Galois case). That we canfind such an extension π̃ with finite-order central character will be proven in much more generality
in Proposition 3.1.14.
The construction of geometric ρ : ΓF → SO2n+1 (Q` ) having no geometric lift to GSpin2n+1 (Q` )(and with specified local behavior) follows as in Lemma 2.6.10, applying Clozel’s limit multiplicity
formula to G = Sp2n over F, transferring these forms to GL2n+1 (via [Art13]), and then invoking
the Paris Book Project, or, more precisely, Remark 7.6 of [Shi11].

We have therefore generalized some of the results of §2.7 (the case n = 1).

32

Using this term abusively for the obvious local analogue.
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NumberTheory

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Version 4 - Submitted to MEMOCHAPTER 3

Galois and automorphic liftingThis chapter begins (§3.1) by addressing the natural automorphic analogue of Conrad’s lifting
question. It is much easier to see what should be true in this setting, and the proofs are simpler
as well. Equipped with the intuition coming from the automorphic case, we address the original
Galois-theoretic question in §3.2. In §3.3, we combine the results of §3.1 and 3.2 to compare,modulo the Fontaine-Mazur-Langlands conjecture, descent problems for certain automorphic representations and Galois representations. The closing section, §3.4, is of a rather different nature,assembling a few results about the images of compatible systems of `-adic Galois representations.
The results of that section continue the attempt to compare aspects of the automorphic and Galoistheoretic formalisms.
3.1. Lifting W-algebraic representations
In this section, we make the simplifying assumption that G is a connected semi-simple split
group over F. This semi-simplicity assumption is not essential, and in §3.2 we will work more
e be an F-split torus containing ZG , and set G
e = (Ze×G)/ZG , as before, with maximal
generally. Let Z
e × T )/ZG and center Z.
e In each case ZG is embedded anti-diagonally. We will study
torus Te = (Ze and also from G to intermediate
lifting problems for automorphic representations from G to G,e here the reader should think of G ⊂ G as the general(connected reductive) groups G ⊂ G ⊂ G;
e whose center Z
e is a torus, simply by embedding the center Z
case, but any such G embeds in a G
e We focus on the case G ⊂ G
e because the general case reduces to this one.1of G into a torus Z.
Note that the cases of greatest interest are when G is simply-connected (so G∨ is adjoint), such as
SLn or Sp2n , but we do not assume this.
Now let π be a (unitary) cuspidal automorphic representation of G(AF ) that is W-algebraic. Wee F );are interested in the problem of lifting π to a W-algebraic automorphic representation π̃ of G(Awhen π is moreover L-algebraic, we similarly ask whether an L-algebraic lift exists. By ‘lift,’ wesimply mean the most naive thing: the restriction π̃|G(AF ) contains π. Corollary 3.1.6 justifies thise → LG, the L-packet of π is a functorial transfer
convention, showing that under the L-morphism LGof the L-packet of π̃.
3.1.1. Notation and central character calculation. In order to do computations in terms ofroot data, we choose coordinates (using invariant factors) such that
X (Z) =
• e

r
M

Zwi ⊕

i=1

s
M

Zw0j ,

j=1

To distinguish the two cases, consider for instance G = Spin2n ⊂ G = GSpin2n , which dualizes to a surjection
GSO2n  PSO2n . The center of G is not connected.
1

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di Zwi ⊕s
M

Zw0j .

j=1

i=1

e G . We then write X • (ZG ) = ⊕r (Z/di Z)w̄i . To relate parameters
This is X (S ) for the torus S = Z/Z
1e to those for G, we use the Cartesian diagramfor G•

X • (Te)/ X • (Z)e
/ X • (ZG ),X • (T )representing elements of X • (Te) as pairs (χT , χZe) that are congruent in X • (ZG ). Extending scalars∼e Q . We will usually (out of sad− X • (T )Q ⊕ X • (Z)to Q (or any characteristic zero field), X • (Te)Q →necessity) take F to be either CM or totally real. In the former case (or whenever an archimedean
place is complex) we compute local central characters (see page 21 ff. of [Lan89]) as follows:∼
Suppose v is complex, so there is an isomorphism ιv : Fv →− C. Let
recv (πv ) : z 7→ ιv (z)µv ῑ(z)νv ∈ T ∨ (C)2P
be the associated Langlands parameter, with µv , νv ∈ X • (T )C with µv − νv ∈ X • (T ). Write ri=1 [µv −νv ]i w̄i for the projection of µv − νv to X • (ZG ). Choose any µv,iP, νv,i ∈ C with µv,i − νv,iPan integer
projecting to [µv − νv ]i ∈ Z/di Z. We can then regard µ̃v = (µv , r1 µv,i wi ) and ν̃v = (νv , r1 νv,i wi ) as
elements of X • (Te)C parametrizing an extension of the local L-parameter of πv to a parameter fore v ). Identifying, via our chosen basis, Z(Fe v ) with (C× )r+s , the central character ωπ̃v of this lift isG(Fsimply
r
Y
µ −ν
ν
0
0
ωπ̃v : (z1 , . . . , zr , z1 , . . . , z s ) 7→
zi v,i v,i |zi |Cv,i .
i=1

Restricting to ZG (Fv )  µd1 × · · · × µdr ⊂ (C× )r , we find that the central character ωπv is given byrYµ −νωπv : (ζ1 , . . . , ζr ) 7→
ζi v,i v,i ,
i=1

where each ζi is a dith root of unity. Clearly this is independent of the choice of lifts µ̃v , ν̃v .
Definition 3.1.1. If F is a CM field, and π is a (unitary) automorphic representation of G(AF )with archimedean parameters µv , νv as above at each v|∞, then we define ω̃ = ω̃(π) to be any choicee lifting ωπ and such thatof Hecke character of ZrY(zi /|zi |)µv,i −νv,i .
ω̃v (z1 , . . . , z0s ) =i=1

This is a unitary, type A Hecke character whose existence is assured by Lemma 2.3.6.
When v is real, the lifting process is more complicated (see [Lan89]), and the central charactercomputation depends very much on where the L-parameter of πv sends an element of WR − WC .
We can avoid this, however:From now on, we write z, z̄ in place of ιv (z), ῑv (z).

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Version 4 - Submitted to MEMODefinition 3.1.2. Let π be as in the previous definition, but now suppose F is totally real. Thene extending ωπ (existence again by
we define ω̃ to be any choice of finite-order Hecke character of Z
Lemma 2.3.6).
e F ). We now try to find an automorphic
3.1.2. Generalities on lifting from G(AF ) to G(A32ee F ), ω̃) lifting π, where ω̃ is any (unitary) lift of the central
representation π̃ ⊂ Lcusp (G(F)\G(Acharacter ωπ . We say that we are in the Grunwald-Wang special case if one of the pairs (F, di ) ise F ) (a normal subgroup of G(Ae F )),
in the usual Grunwald-Wang special case. Let HAF = G(AF )Z(Aeand let HF = HAF ∩ G(F).
Lemma 3.1.3. We are in the Grunwald-Wang special case if and only if HF strictly containse
G(F)Z(F).
e (since G is semi-simple). These induce an
Proof. Recall the characters di wi ∈ X • (S ) = X • (G)isomorphismrQ
di wi Y  ×e
F ∩ (A×F )di /(F × )di .
HF /G(F)Z(F)
−−−−→
i=1


e F ); the argument we use here is
There are various ways to show forms on G(AF ) extend to G(Ae G) = (GSp2n , Sp2n ) (see Proposition 2.4.3 of [Fli06]).
modeled on one of Flicker for the case (G,Proposition 3.1.4. Let π be a cuspidal automorphic representation of G(AF ), with central character ωπ . If we are not in the Grunwald-Wang special case, then for any extension ω̃ of ωπ to ae there exists a cuspidal automorphic representation π̃ of G(Ae F ) lifting π, andHecke character of Z,
having central character ω̃. If we are in the Grunwald-Wang special case, then for at least one
extension of ωπ torr
YY
C F [di ] ⊃µdi (F)\µdi (AF ),
i=1

i=1

e there exists a cuspidal π̃ liftingand for any extension of this character to a Hecke character ω̃ of Z,
π with central character ω̃.
e any cuspidal automorphic repreFor any intermediate connected reductive group G ⊂ G ⊂ G,
sentation of G extends to a cuspidal automorphic representation of G.
Remark 3.1.5. In particular, in all cases, for all Hecke characters ω̃ extending ωπ , there exists ae F ) lifting
Hecke character ω̃0 having the same infinity-type, and a cuspidal representation π̃ of G(A0π, with central character ω̃ .
ee F ) → S1 of ωπ . By extending functions along Z(Ae F)
Proof. Choose an extension ω̃ : Z(F)\Z(Aevia ω̃, we embed the space Vπ of π into the space of cusp-forms on G(F)Z(F)\HAF , and thereby
obtain an extension of π to a representation πω̃ of HAF . We construct an intertwining map
U

2
F)
e
e F ), ω̃),
IndG(A→ Lcusp(G(F)\G(AHA (πω̃ ) −eFee F ) with (unitary) central character ω̃, and squareThat is, the space of measurable functions on G(F)\G(A0eintegrable modulo Z(F∞ ) –or, equivalently, modulo AGe(R)0 , where AGe is the maximal Q-split central torus ine
ResF/Q (G).
3

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e F ) → Vπ with the usual left-HAF -equivariance,
where the induction consists of functions F : G(Aand the requirement of compact support modulo HAF . Write δ1 : Vπ → C for evaluation at 1 of thecusp forms in Vπ , and set
X
L(F) =
δ1 (Fu ),ee
G(F)Z(F)\G(F)3uewhere we write Fu for the value at u of F ∈ Ind(πω̃ ). This sum is in fact finite: G(F)is discreteeeeein G(AF ), the fibers of G(F)Z(F)\G(F) → HAF \G(AF ) are finite, and we have assumed F hasecompact support modulo HAF . It is also well-defined because for γ ∈ G(F)Z(F),δ1 (Fγu ) = δ1 (γ · Fu ) = Fu (γ) = Fu (1) = δ1 (Fu ).
The map U is then given by
U(F) : g 7→ L(I(g)F),
e F )-action on the induction. U is clearly G(Ae F )-equivariant and has
where I(·) denotes the G(Aeoutput U(F) which is left-G(F)-equivariant(by its construction as an average) and has central
character ω̃; U(F) is a cusp form by a simple calculation using the fact that the unipotent radical
e is in fact contained in G.
of any parabolic of G
To see that U , 0, take a non-zero form f ∈ Vπ ; we may assume (by translating) f (1) , 0.
Then define F ∈ Ind(πω̃ ) by
Fh : h0 7→ f (h0 h)
if h ∈ HAF , and Fg = 0 for g < HAF . ThenXX f (u) if u ∈ HAF ;U(F)(1) =δ1 (Fu ) =0otherwise.
uuSo, if we are not in the special case, we just get U(F)(1) = f (1) , 0. If we are in the specialcase, then for each pair (F, di ) in the special case there is an element αi ∈ F × which is everywhere
locally a dith -power–say αi = βdi i , but not globally a dith -power. By abuse of notation, we also writee
αi for an element of G(F)such that (di wi )(αi ) is this element of F × and the other characters d j w je are trivial on αi .4 Regarding βi as an element of the ith componenet of Z(Ae F ),( j , i) and w0j of G
we can therefore write αi = βi · γi , where each γi lies in G(AF ). If only one pair (F, di ) is in thespecial case, then the above expression for U(F)(1) becomes f (1) + f (αi ) = f (1) + ω̃(βi ) f (γi ).
This time we normalize f so that f (γi ) , 0 (rather than f (1) , 0), and so necessarily either
f (1) + ω̃(βi ) f (γi ) , 0, or this expression is non-zero for any ω̃ extending the other extension of ωπ
to a character of C F [di ]. Thus we can choose at least one initial extension to C F [di ], and thereafter
the argument proceeds as in the non-exceptional case. If multiple (F, di ) are in the special case, thesame argument applies: arrange some f (γi ) , 0, and if U(F)(1) = 0, change the extension of ωπ
to C F [di ] just in this ith component.
e F)
Finally, given that U , 0, we take π̃ to be the image of any irreducible constituent of IndG(AHAF (πω̃ )that survives under U, and we claim this is the desired extension of πω̃ . The isomorphism classes
of HAF -representations appearing in the restriction to HAF of the full induction are precisely thee F ), and G(Ae F )-stability of π̃ implies that all of these in fact appear in
conjugates πgω̃ , for g ∈ G(Aπ̃|HAF . In particular, this latter restriction contains πω̃ .
4We use the assumption that G is split over F.
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For the final claim, we extend π from G to G,
restriction can be taken as the desired extension.
We need to supplement this by understanding to what extent π is a ‘functorial transfer’ of π̃,ηe  LG. Somewhat more generally, let G →e be awith respect to the L-homomorphism LG− Gmorphism of connected reductive groups over Fv , with abelian kernel and cokernel; here we either
e are unramified over Fv . By assumption,
take v to be archimedean, or finite such that G and G
LL
Le
there is a dual L-homomorphism η : G → G. In both the archimedean and unramified cases,
the L-parameters φv and φ̃v , as well as the corresponding L-packets Πφv and Πφ̃v , of πv and π̃v havebeen defined (as will be explained in Corollary 3.1.6).
Corollary 3.1.6. Continue in the setting of the proposition. Then π is a weak transfer of π̃,which is also a strong transfer at archimedean places. More precisely, for all places v of F that areeither archimedean or such that G is unramified at v, the restriction π̃v |G(Fv ) is a finite direct sum ofelements of Πφv , and φ̃v reduces to φv (i.e. L η ◦ φ̃v = φv up to G∨ (C)-conjugacy).
Proof. For infinite places v, the assertion follows from desideratum (iv) on page 30 of [Lan89].
Since the construction of the correspondence is inductive, the verification necessarily stretches out
through §3 of that paper; for the first (and most important) case of discrete series, see page 43.
Now we treat the unramified case. Suppose π̃v is unramified, i.e. there exists a hyperspecial
ev of G(Fe v ) such that π̃vKev , 0. Then for some unramified character
maximal compact subgroup Ke v)
G(Fe v )/ B(Fe v ) is anχ̃v of Te(Fv ), π̃v is a sub-quotient of IB(Fχ̃ . The natural map G(Fv )/B(Fv ) → G(Fe v) v
isomorphism, so restriction of functions gives an isomorphism of G(Fv ) representations
∼ G(Fv )
e v)
G(F
IB(F
χ̃ |
→
− IB(F(χ̃v |B(Fv ) ).
e ) v G(Fv )v)v

Write χv for χ̃v |B(Fv ) . It is an unramified character of T (Fv ), and under the identification of unramified characters of T (Fv ) with Hom(X• (T )ΓFv , C× ) (and the analogue for Te), and thus with the spaceof unramified L-parameters, χv corresponds to the parameter φv = L η ◦ φ˜v . By definition of unramified L-packets (see [Bor79, §10.4]), to see that any constituent πv of π̃v |G(Fv ) lies in the packet Πφv ,we need only check that πv has invariants under some hyperspecial maximal compact subgroup ofei(with the πi
G(Fv ). To see this, let u be a non-zero vector in π̃vKv . Decomposing π̃v |G(Fv ) = ⊕1M π⊕midistinct isomorphism classes; in fact, the multiplicities mi are all equal), we see that u lies in one ofev - inthe isotypic componenets πi⊕mi , since the induction I(χv ) can only have one Kv := G(Fv ) ∩ K
Pmi
thvariant line. Fixing a decomposition u = j=1 u j with u j in the j copy of πi , we find that the u j arethemselves also Kv -invariant, hence πi contains a Kv -invariant vector (this in turn implies the mule v ) acts transitively on the isomorphism classes πi , each πi ,tiplicities mi must equal 1). Since G(Fand in particular our given πv , contains some vector of the form π̃v (g)u, which is gKv g−1 -invariant;
gKv g−1 is also a hyperspecial maximal compact subgroup, so we’re done.
Remark 3.1.7. According to expected properties of local L-packets, π should be a strong transfer of π̃ (see §10.3 of [Bor79]), but of course this statement is meaningless until the local Langlandsecorrespondence is known for G and G.
Eventually, we will also want to understand the ambiguity in the choice of π̃. Strictly speaking, we only need this in one direction of Proposition 3.1.14 below, but as a general problem, its
importance is basic. We will need to assume something from local representation theory, which
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Version 4 - Submitted to MEMOe by Adler andwas originally conjectured always to hold, more generally for quasi-split G and G,
Prasad ([AP06]); in fact, in certain quasi-split, but not split, cases, Adler and Prasad have foundcounterexamples to their conjecture (these and related questions will be taken up in [APP16]).
e be as above. Then for any irreducible
Hypothesis 3.1.8. Let v be any place F, and let G and Ge v ), the restriction π̃v |G(Fv ) decomposes with multiplicity one.
smooth representation π̃v of G(FThe motivation for this hypothesis is the uniqueness of Whittaker models for quasi-split groups,
and indeed it always holds for generic π̃v . The analogous statement for v archimedean is straightforward in many cases: for instance, for a simple G, it is easy because R× /(R× )n is cyclic for all n.
Moreover, the structure of archimedean L-packets may be well-enough understood to resolve thequestion in general for archimedean v, but we do not pursue this here. For v non-archimedean, we
have verified it in the unramified case (see the proof of Lemma 3.1.6), and in general the hypothe G): (GLn , SLn ) (from the theory of, possibly degenerate,esis is known for the following pairs (G,Whittaker models; see [LL79] for the n = 2 case), and any pair (GU(V), U(V)) where (V, h, i) is avector space over Fv equipped with a non-degenerate symmetric or skew-symmetric form h, i, andGU(V), respectively U(V), denote the similitude and isometry groups of the pairing (see Theorem
1.4 of [AP06]). So, whatever the status of the general hypothesis, the discussion below applies tosome interesting cases.
Lemma 3.1.9. Assume Hypothesis 3.1.8. Suppose π̃ and π̃0 are two lifts of π (i.e. their restrictions contain π) with the same central character ω̃.
• There exist continuous idele characters αi : A×F → C× , for i = 1, . . . , r, such that, in thenotation of §3.1.1,
 r

Y

0 
π̃  π̃ · 
αi ◦ di wi  ,i=1
diand each αi factors as a genuine Hecke character αdi i : A×F /F × → C× .

• Since Hecke characters satisfy purity, so do the characters αi .

Remark 3.1.10. The point of the Lemma is that the idele characters αi need not be Hecke
characters, i.e. need not factor through C F . We are particularly interested in the constraint on theinfinity type of αi , which by the lemma must be just as rigid as the constraints for Hecke characters.
This will be applied in Proposition 3.1.14.
e F ), and write Hv = G(Fv )Z(Fe v ) for its local analogue.
Proof. As before, write HAF = G(AF )Z(A0
By hypothesis, both restrictions π̃|HAF and π̃ |HAF contain π  ω̃, so this holds everywhere locally as
well. Lemma 2.4 of [GK82] (applying Hypothesis 3.1.8) implies that for all finite v, there exists ae v )/Hv → C× such that π̃v  π̃0v · αv . By duality (for finite abelian groups),smooth character αv : G(Fwe can extend this to a character ᾱv :Q/ G(F/ r F × /(F × )die v )/Hv
1
i=1 v
v
αv



ᾱv

C×

v

We express αv in coordinates as
αv = αi,v ◦ (di wi ) ri=1
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 r

Y




π̃  π̃0 ·  (⊗0v αi,v ) ◦ di wi  .
i=1

Here the αi := ⊗0v αi,v are continuous5 characters A×F → C× , but they need not be Hecke characters.
Taking central characters, however, we see that each αdi i is in fact a Hecke character.
Remark 3.1.11. Under Hypothesis 3.1.8, one should be able to refine the arguments of thissection to produce a multiplicity formula for cuspidal automorphic representations of G(AF ) ine F ), as in Lemma 6.2 of [LL79]). One can also axiomatize the passage
terms of those of G(Ae this would include verifying compatibility
between local Langlands conjectures for G and for G;of the conjectural formulae for the sizes of L-packets (a template, in the case of (GSp, Sp), is givenin the paper of [GT10]; their arguments will clearly apply much more generally).
3.1.3. Algebraicity of lifts: the ideal case. Now we return to questions of algebraicity, handling the CM and totally real cases in turn. In each case, we carefully choose an extension of
the central character of π (as described in Definitions 3.1.1 and 3.1.2), and then find a π̃ as in
Proposition 3.1.4 with that central character, and whose restriction to G(AF ) contains π. This will
yield enough information about the archimedean L-parameters of π̃ to deduce the desired algebraicity statements. To give a clean argument with broad conceptual scope, we will assume that π
is tempered at infinity6. As remarked before, this is not a serious assumption for algebraic representations: it is satisfied for forms having cuspidal transfer to some GLN . For non-tempered forms,
analogous results for the discrete spectrum can be deduced from Arthur’s conjectures.
Proposition 3.1.12. Let F be CM, with π, π̃ as above, and with ω̃ the (type A) extension of ωπdescribed in Definition 3.1.1. Assume π∞ is tempered.
(1) If π is L-algebraic, then π̃ is L-algebraic. In particular, there exists an L-algebraic lift ofπ.
(2) If π is W-algebraic, then π̃ is W-algebraic.
Similarly, any L-algebraic (respectively, W-algebraic) π on G extends to an L-algebraic (respectively, W-algebraic) automorphic representation of G(AF ).
Proof. We use the notation of §3.1.1. We may of course take all µv,i and νv,i to be integers. By
the choice of central character ω̃, the archimedean L-parameter for π̃v corresponds toX µv,i − νv,ie C,wi ) ∈ X • (T )C ⊕ X • (Z)µ̃v = (µv ,2andX νv,i − µv,ie C.
ν̃v = (νv ,wi ) ∈ X • (T )C ⊕ X • (Z)2
We write this parameter as an obviously integral term plus a defect:
X
X µv,i + νv,i
µ̃v = (µv ,µv,i wi ) + (0, −wi ),2
Almost all αi,v are unramified, since the same holds for π̃v and π̃0v .
No assumption at finite places.

5

6

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µ +νand likewise for ν̃v . Note that this lies in X(Te) if and only if for all i = 1, . . . , r, we have v,i 2 v,i ∈ di Z.
The discussion is so far general; if we now assume πv is tempered, then for all λ ∈ X• (T ), thecharacterz 7→ zhµv ,λi z̄hνv ,λi
is unitary, i.e.
Re(hµv + νv , X• (T )i) = 0.
W-algebraic representations of course have real infinitesimal character, so µv = −νv , and therefore
in the initial choice of lift we may assume µv,i = −νv,i for all i. Then obviously µ̃v is L-algebraic.
P
If π is only W-algebraic, then we have to check that 2µ̃v = (2µv , (µv,i − νv,i )wi ) lies in X • (Te).
e represents an element of X • (Te) if and only if 2µv maps to Pi [µv −This element of X • (T ) ⊕ X • (Z)νv ]i w̄i in X • (ZG ). The latter is also the image of µv − νv , so it is equivalent to ask that µv + νv ∈ X • (T )map to zero in X • (ZG ). As above, temperedness of πv guarantees this, so we are done.
e and then restrict to G;
For the final claim, we again apply the result to the inclusion G ⊂ G,
restriction of course preserves L-algebraicity and W-algebraicity.

Remark 3.1.13. This result would immediately extend to totally imaginary fields if we knew
e F ). As noted in Remark 2.4.10, this wouldthat ωπ extended to a type A0 Hecke character of Z(Afollow from the (‘CM descent’) conjectures of §2.4.
We now turn to the case of totally real F. I am grateful to Brian Conrad for urging a coordinatefree formulation of the obstruction in part 2 of the proposition.
Proposition 3.1.14. Now suppose F is totally real, with π, ω̃, π̃ as before (ω̃ finite-order as
in Definition 3.1.2). Alternatively, let F be arbitrary, but assume that ωπ admits a finite-order
extension ω̃. Continue to assume π∞ is tempered.
(1) If π is L-algebraic, then it admits a W-algebraic lift π̃.
(2) Assume Hypothesis 3.1.8 for the ‘only if’ direction of this statement. Assume F is totallyreal, and consider this W-algebraic lift π̃. Then the images of µv and νv under X • (T ) →X • (ZG ) lie in X • (ZG )[2], and π admits an L-algebraic lift if and only if these images are
independent of v|∞.
Similarly, any L-algebraic π on G admits a W-algebraic lift to G, and it admits an L-algebraic
lift if (and, under Hypothesis 3.1.8, only if) the images of the µv in coker (X • (Z)tor → X • (ZG )) =X • (ZG ∩ Z0 ) are independent of v|∞.
Proof. We continue with the parameter notation of the previous proof. Let π be L-algebraic.
First we check that since ω̃ can be chosen finite-order (automatic in the totally real case, but anadditional assumption at complex places) µv − νv maps to zero in X • (ZG ). If v is imaginary, then ω̃cannot be finite order unless ωπv is trivial, hence µv and νv themselves are trivial in X • (ZG ). If v isreal, then in T ∨ ⊂ G∨ we have the relation
φ( j)zµv z̄νv φ( j)−1 = zνv z̄µv ,writing φ for the L-parameter and j ∈ WFv − WF v . Now, φ( j) ∈ NG∨ (T ∨ ) represents an element w of
the Weyl group of G∨ , and this yields the relations wµv = νv and wνv = µv . Thus µv − νv = µv − wµv ,which lies in the root lattice Q of G. [This holds for wχ − χ for any χ ∈ X • (T ): reduce to the case of
simple reflections, where it is clear from the defining formula.] Restricting characters to ZG factors
through a perfect duality ZG × X • (T )/Q → Q/Z, so µv − wµv ∈ Q has trivial image in X • (ZG ).
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Version 4 - Submitted to MEMOAny lift with finite-order central character has parameters µ̃v = (µv , 0) and ν̃v = (νv , 0); thispair yields a well-defined representation of WF v , i.e. µ̃v − ν̃v ∈ X • (Te), and its projections to T ande are what they have to be, so they are the only possible parameters; that this extends (possiblyZ
non-uniquely) to an L-parameter on the whole of WFv follows from general theory (Langlands’
Lemma), but we do not need the details of this extension. Now we use the assumption that πv istempered: as in the previous proposition, we see that µv = −νv . Hence 2µv maps to zero in X • (ZG ),and so 2(µv , 0) ∈ X • (Te), i.e. (µv , 0) is W-algebraic.
For F totally real, we now show the second part of the proposition. By Lemma 3.1.9, if asecond lift π̃0 is L-algebraic, then
 r

Y



π̃0  π̃ · αi ◦ di wi  ,i=1

where each αdi i is a Hecke character of our totally real field, and therefore (since π̃ and π̃0 both haverational infinitesimal character) takes the form χi · | · |di yi for a finite-order character χi and a rationalP
number yi . The infinitesimal character of π̃0v (for v|∞) then corresponds to (µv , yi di wi ) ∈ X • (Te)Q ,which is integral if and only if yi ∈ d1i Z and yi di is the image of µv in Z/di Z for each i. The claim
follows easily.
It follows immediately that an L-algebraic π on G extends to a W-algebraic automorphic representation on G. For the last claim, note that coker (X • (Z)tor → X • (ZG )) is canonically isomorphic
to X • (ZG ∩ Z0 ). Our previous arguments apply to the inclusion (enlarging a subgroup of ZG to a∼
torus) G ⊂ (G × Z0 )/(ZG ∩ Z0 ) →− G (this map is an isomorphism since G is connected reductivewith derived group G), showing the existence of an L-algebraic extension of π precisely when theimage of µv in X • (ZG ∩ Z0 ) is independent of v|∞.

Corollary 3.1.15. Let F be totally real and π be L-algebraic on the F-group G.
(1) Suppose π∞ is tempered. If all di are odd, then π̃ is L-algebraic. In particular, if the simple
factors of G are all type A2n (i.e. SL2n+1 ), or of type E6 , E8 , F4 , G2 ,7 then any L-algebraic
π has an L-algebraic lift.
(2) Assume F , Q (still totally real). For F-groups G that are simple simply-connected (andsplit) of type Bn , Cn , D2n , and E7 , there exist, assuming Hypothesis 3.1.8 for the paire L-algebraic π on G, tempered at infinity, that admit no L-algebraic lift to G.
e(G, G),
Proof. The first part is immediate from the proof of the proposition: for di odd, if the image of
2µv in Z/di Z is trivial, then so of course is the image of µv . For the second part, we use existence
results for automorphic forms that are discrete series at infinity as in Lemma 2.6.10 and Proposition
2.8.7. For a semi-simple, simply-connected group, ρ lies in the weight lattice, so discrete series
representations will be L-algebraic. Choose a quotient G  G0  Gad , with T 0 the induced maxi∼
mal torus of G0 , such that X • (T 0 )/Q  X • (ZG )[2] under the isomorphism X • (T )/Q →− X • (ZG ).Then
we can find a discrete series representation of G(F∞ ) such that at two infinite places v1 and v2 , theparameters have the form (on C× )z 7→ zρ+µvi z̄−ρ−µvi ,7

A2n and E6 are the interesting examples here, since in the other cases the adjoint group is simply-connected. The
index [X • (T ) : Q] for the simply-connected simple group of type E6 is Z/3Z.
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where the µvi are distinct in X • (T 0 )/Q.8 This ensures that the ρ + µvi have distinct images inX • (ZG )[2]. It remains therefore to check that the split real forms of these groups, except for A2n−1 ,actually admit discrete series. Type Cn was treated previously. For type Bn , the (real) split group isSpin(n, n + 1), which always admits discrete series (real orthogonal groups SO(p, q) have discrete
series whenever pq is even). Likewise, for type Dn , the split form Spin(n, n) has discrete series if
and only if n is even. The split real form of E7 also admits discrete series: its maximal compact
subgroup is SU(8), which contains a compact Cartan isomorphic to (S1 )7 (see the table in Appendix
4 of [Kna02]).

Remark 3.1.16.
• Arthur constructs in [Art02] a conjectural candidate for the automorphic Langlands group LF , also giving an ‘automorphic’ construction of a candidate forthe motivic Galois group GF . (In §4.1 we will discuss motivic Galois groups in detail,explaining both conjectural and unconditional constructions.) The ability to extend Le is essential to Arthur’salgebraic representations of G to L-algebraic representations of G
conjectural construction (see [Art02, §6]) of a morphism LF → GF . Roughly, the factorof the motivic Galois group GF associated to π (or rather its almost-everywhere system of
Hecke eigenvalues; this construction is restricted to suitably ‘primitive’ π) is defined as ae∨ ×TF , with TF the Taniyama group (see §4.1.6 for background on TF );sub-group Gπ of G
then LF and GF are constructed as suitable fiber products over varying pairs (G, π).9 The
natural motivic Galois representation corresponding to ‘the motive of π’ is the projectione∨ . Corollary 3.1.15
to G∨ , and implicit in the construction is the fact that this lifts to G
then gives concrete examples of automorphic representations (including ones that ought
to be ‘primitive,’ arranging the local behavior suitably) for which the Tannakian formalism cannot behave in this way; that is, while Arthur’s construction is consistent with our
discussion in the CM case (Proposition 3.1.12), it must be modified for totally real fields
F.
• If our lifting results were generalized to quasi-split groups, we could presumably include
Dn for n odd in the second part of the Proposition, since the non-split quasi-split form has
signature (n + 1, n − 1), for which the associated orthogonal group admits discrete series.
Similarly, we could include type A2n−1 by using the quasi-split unitary group SU(n, n)
instead of SL2n .
3.2. Galois lifting: the general caseNow we discuss a general framework for Conrad’s lifting problem. We consider lifting problems of the form
e `)
H(Q
<

ρ̃

ΓF

ρ


/ H(Q ),`We have shifted what was previously denoted µv by ρ in this case because of the description of discrete series
L-parameters as arising from elements of ρ + X • (T ). Note that ρ mapst to X • (ZG )[2] since 2ρ is in the root lattice Q.
9e∨ o TF , with TF acting via its projection to the global
For non-split G, one works with the semi-direct product G
Weil group WF .
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Version 4 - Submitted to MEMOe  H, with central torus kernel, of linear algebraic groups over Q` . In thisfor any surjection H
picture, we want to understand when a geometric ρ does or does not admit a geometric lift ρ̃. We
e and H are connected reductive, and so towill see that the general case reduces to the case where H
e=Ge∨ ,reap the psychological benefits of the work of §3.1, we will begin by considering the case H
e is an inclusion of connected reductive10 split F-groups, constructed by
H = G∨ , where G ⊂ Ge that isextending the center ZG of G to a central torus Z;
e = (G × Z)/Ze G.
Ge G is a torus S , and we get an exact sequence
The quotient Z/Ze∨ → G∨ → 1,
1 → S∨ → G
and a canonical isomorphism of Lie algebras g̃∨  g∨ ⊕ s∨ . We again caution the reader that, evene and H are assumed connected reductive, this is not the most general situation: namely, theif He need not be connected in order to have Ge∨  G∨ a surjection with central torus kernel.
center Z
Nevertheless, at least for F totally imaginary, we will reduce the general analysis to this case. To
start, we elaborate on some of the associated group theory, slightly recasting the notation of theprevious section to take into account the fact that G may have positive-dimensional center. The
exact sequencee→ S → 1
1 → ZG → Z
gives rise, by applying X • (·) and then Hom(·, Z), to an exact sequencee → X• (S ) → Ext1 (X • (ZG ), Z) → 1.
1 → Hom(X • (ZG ), Z) → X• (Z)e such that the inclusion X • (S ) ⊂ X • (Z)eUsing invariant factors, it is convenient to fix a basis of X • (Z)is in coordinates/ X • (Z)eX • (S )Lr
i=1

Zdi wi

/

Lr+s
i=1

/ X • (ZG )

/

Zwi

Lr
i=1

Z/di Zw̄i ⊕Ls
j=1

Zw̄r+ je∨ ), and let ν∗ denote the basis of X • (S ∨ ) dual to di wi ; inLet w∗i denote the dual basis for X • (Zie∨ ) → X • (S ∨ ).
particular, w∗i maps to di νi∗ under X • (Ze G of G,
e and deduce from its definition an exact
We work with the maximal torus Te = (T × Z)/Z
sequencee → X • (ZG ) → 1,
1 → X • (Te) → X • (T ) ⊕ X • (Z)and thus a (crucial) exact sequencee∨ ) → X • (Te∨ ) → Ext1 (X • (ZG ), Z) → 1.
1 → Hom(X • (ZG ), Z) → X • (T ∨ ) ⊕ X • (Z(1)
10

Unlike in §3.1, whose results were largely intended to motivate the results of this section, we no longer require
G to be semi-simple.
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Version 4 - Submitted to MEMO∼e∨ ) →e∨ ), which follows from exactness
Note also that there is a canonical isomorphism X • (Z− X • (Gof the row in the diagram:
1


S∨

/G
e∨

/ G∨

1

sc

/Z
e∨

/1



G∨


1
where G∨sc denotes the simply-connected cover of the derived group of G∨ . Recall that we index
Sen operators Θρ,ι by embeddings ι : Q` ,→ CFv , where v is a place above `. Given a geometricρ : ΓF → G∨ (Q` ), Conrad’s result (see Theorem 2.1.6, and Remark 2.1.5) implies that a lift ρ̃ isgeometric if and only if it is Hodge-Tate at all places above `; we will use this from now on withoute∨ )ι is conjugate
comment. That is, we need to arrange ρ̃ such that each Sen operator Θρ̃,ι ∈ Lie(Gto an element of Lie(Te∨ )ι that pairs integrally with all of X • (Te∨ )11 under the natural map
LieX • (Te∨ ) −−→ Hom(Lie(Te∨ ), Q` ).
Here is our starting-point:
Lemma 3.2.1. There exists some lift ρ̃ of ρ. Any other lift is of the formrrXYρ̃( νi ◦ χi ) : g 7→ ρ̃(g) ·(νi ◦ χi )(g),
i=1

i=1
×

where the νi = di wi range over the above basis of X• (S ∨ ), and each χi : ΓF → Q` is a continuous
character.
Proof. A lift exists by Proposition 2.1.4. Although continuous cohomology does not in general
have good δ-functorial properties, short exact sequences do give (not very) long exact sequenceson H 0 and H 1 , so any lift has the form ρ̃(χ) for some χ : ΓF → S ∨ (Q` ). We compose with the dual
characters νi∗ in X • (S ∨ ) to put χ in the promised form.
The following lemma is the general substitute for choosing lifts with finite-order Clifford normin the spin examples; this result is also implicit in the proof of 2.1.4, but a little warm-up with ournotation is perhaps helpful.
This condition is independent of the way Θρ̃,ι is conjugated into Lie(Te∨ )ι , since: (1) we already know that Θρ̃,ι
pairs integrally with the roots, which all lie in X • (T ∨ ); (2) the ambiguity in conjugating into Lie(Te∨ )ι is an element of
the Weyl group; (3) for any weight λ ∈ X • (Te∨ ) and w in the Weyl group, wλ − λ lies in the root lattice. Compare theproof of Proposition 3.1.14.
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Version 4 - Submitted to MEMOLemma 3.2.2. Let ρ : ΓF → G∨ (Q` ) be a geometric representation. Then there exists a lifte∨ (Q` ) such that, for all v|` and all ι : Q` ,→ CFv , the Sen operator Θρ̃,ι pairs integrallyρ̃ : ΓF → Ge∨ )  X • (Ge∨ ).
with all of X • (Ze∨ ) is Hodge-Tate.
Proof. It suffices to find a lift whose composition with all elements of X • (ZWe use the bases of the various character groups specified above. In particular, composing ane∨ ), i = 1, . . . , r, we can write
initial lift ρ̃ with the various w∗i ∈ X • (G×w∗i ◦ ρ̃ = χdi i χi,0 : ΓF → Q` ,where the χi and χi,0 are Galois characters with χi,0 finite-order. Then we consider the new liftrX0ρ̃ = ρ̃( (di wi ) ◦ χ−1
i ),
i=1
i
is finite-order for all i = 1, . . . , r. Moreover,which has the advantage that w∗i ◦ ρ̃0 = (w∗i ◦ ρ̃) · χ−di∗e∨ ), thefor the characters wr+ j , j = 1, . . . , s, namely, the sub-module Hom(X • (ZG ), Z) ⊂ X • (Ze∨ ),compositions w∗r+ j ◦ ρ̃0 are all geometric, since ρ is. Therefore α◦ ρ̃0 is geometric for all α ∈ X • (Zas desired.
Returning to equation (1) on page 71, we see that the obstruction to geometric lifts then comesfrom Ext1 (X • (ZG )tor , Z). For any weight λ ∈ X • (Te∨ ), there is a positive integer d ∈ Z such thate∨ ), so with a ρ̃ as produced by Lemma 3.2.2, Θρ̃,ι pairs integrally with dλ fordλ ∈ X • (T ∨ ) ⊕ X • (Zall ι. We obtain a well-defined class, independent of the choice of ρ̃ as constructed in the proof of
Lemma 3.2.2 (namely, with the compositions w∗i ◦ ρ̃ finite-order for i = 1, . . . , r),hλ, Θρ̃,ι i ∈ Q/Z,
which can also be interpreted as the common value modulo Z of the eigenvalues of Θrλ ◦ρ̃,ι =e∨Lie(rλ ) ◦ Θρ̃,ι ; here, and in what follows, we denote by rλ the irreducible representation of G
associated to the (dominant) weight λ ∈ X • (Te∨ ). This pairing factors through a map
Ext1 (X • (ZG ), Z) → Q/Z,
and since for any finite abelian group A, the long exact sequence associated to 0 → Z → Q →Q/Z → 0 yields an isomorphism∼

Hom(A, Q/Z) →− Ext1 (A, Z),
we can make the following definition:
Definition 3.2.3. Let θρ,ι be the element of X • (ZG )tor canonically corresponding to the above
map Ext1 (X • (ZG ), Z) → Q/Z.
To make further progress, we need to assume that ρ satisfies certain Hodge-Tate weight symmetries.
Hypothesis 3.2.4. Let H be a linear algebraic group and ρ : ΓF → H(Q` ) a geometric Galois representation with connected reductive algebraic monodromy group Hρ = (ρ(ΓF ))Zar . We
formulate the following Hodge-Tate symmetry hypothesis for such a ρ:
• Let r be any irreducible algebraic representation r : Hρ → GL(Vr ). Then:(1) For τ : F ,→ Q` , the set HTτ (r ◦ ρ) depends only on τ0 = τ|Fcm .
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(2) Writing HTτ0 for this set common to all τ above τ0 , there exists an integer w suchthat12HTτ0 ◦c (r ◦ ρ) = w − HTτ0 (r ◦ ρ),for c the unique complex conjugation on Fcm .
For ρ whose algebraic monodromy group is reductive but not necessarily connected, the corresponding hypothesis is simply that some finite restriction (with connected monodromy group) of ρ
satisfies the above.
In particular, we note for later use that the lowest weight in HTτ0 ◦c (r ◦ ρ) is w minus the highest
weight in HTτ0 (r ◦ ρ). We will see that to establish lifting results for geometric representationsρ : ΓF → G∨ (Q` ), we will only need to know Hypothesis 3.2.4 for some easily identifiable finite
collection of compositions rλ ◦ ρ, λ ∈ X • (T ∨ ), but we do not make that explicit here. Most important, Hypothesis 3.2.4 should in fact be no additional restriction on ρ, because of the following conjecture, which would follow from various versions of the conjectural Fontaine-Mazur-Langlands
correspondence and Tate conjecture (Conjecture 1.2.1):
Conjecture 3.2.5. Let F be a number field, and let ρ : ΓF → GLN (Q` ) be an irreducible geometric Galois representation. Then:(1) For τ : F ,→ Q` , the set HTτ (ρ) depends only on τ0 = τ|Fcm . (This will still hold if ρ isgeometric but reducible.)(2) Writing HTτ0 for this set common to all τ above τ0 , there exists an integer w such thatHTτ0 ◦c (ρ) = w − HTτ0 (ρ),
for c the unique complex conjugation on Fcm .
Unfortunately, for an abstract Galois representation, this conjecture will be extremely difficult to establish. The next lemma explains it in the automorphic case; for a motivic variant, seeCorollary 4.1.26.13
Lemma 3.2.6. Suppose ρ : ΓF → GLN (Q` ) is an irreducible geometric representation. If ρ isautomorphic in the sense of Conjecture 1.2.1, corresponding to a cuspidal automorphic representation π of GLN (AF ), and if we assume that Proposition 2.4.8 is unconditional for π (i.e., admit
Hypothesis 2.4.6), then Conjecture 3.2.5 holds for ρ.
Proof. This is immediate from the passage between infinity-types and Hodge-Tate weights,
Conjecture 2.4.8, and Clozel’s archimedean purity lemma (which was proven as part of Proposition
2.5.8).
We can now understand when geometric lifts ought to exist; the proof proceeds by reduction
to the following key case:
12Interpreted in the obvious way.
But note for now that in the most basic motivic cases, where the Galois representation is given by H j (XF , Q` )for some smooth projective variety X/F, the Hodge-Tate symmetries are immediate (even when this representation is
reducible) from the `-adic comparison isomorphism of [Fal89]; if j is even, the symmetries similarly hold for primitive
cohomology. What is not obvious is that if this Galois representation decomposes, that the irreducible factors all satisfy
the conjecture.
13

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Proposition 3.2.7. Let F be a totally imaginary field, and let ρ : ΓF → G∨ (Q` ) be a geometricrepresentation with algebraic monodromy group equal to the whole of G∨ . Assume ρ satisfiese∨ (Q` ).
Hypothesis 3.2.4. Then ρ admits a geometric lift ρ̃ : ΓF → G
Proof. Choose a lift ρ̃ as supplied by Lemma 3.2.2. Recall that our weight-bookkeeping isdone within the diagram0Oe∨ )X • (Z/ X • (S ∨ )O/ Ext1 (X • (ZG ), Z)

/0

e∨ )
X • (TO∨
X • (T)O

0
We have the elements νi∗ ∈ X • (S ∨ ) dual to di wi ∈ X • (S ). Their images in Ext1 (X • (ZG ), Z) form abasis. Let λi be a (dominant weight) lift to X • (Te∨ ) that also satisfies hλi , d j w j i = δi j . Note that thevalue hΘρ̃,ι , λi i ∈ Q/Z does not depend on the choice of lift of νi∗ , and it clearly lies in d1i Z/Z, so wekwrite it in the form dι,ii + Z for an integer kι,i . By considering the geometric representation rdi λi ◦ ρ̃,
we find that Θrdi λi ◦ρ̃,ι has eigenvalues that depend only on τι (by Lemma 2.2.7; see that lemma forthe notation τι as well), and thus we can write kτ,i in place of kι,i . (Equivalently, we can work withk
the elements θρ,τ ∈ X • (ZG )tor .) These classes dτ,ii mod Z serve as both highest and lowest τ-labeled
Hodge-Tate weights (modulo Z) for rλi ◦ ρ̃; we deduce that, modulo di Z, the highest and lowest
τ-labeled weights of rdi λi ◦ ρ̃ are both congruent to kτ,i 14.
Now, the geometric Galois representations (rdi λi ) ◦ ρ̃ for i = 1, . . . , r are irreducible, because wehave assumed that G∨ is the monodromy group of ρ. Applying Hypothesis 3.2.4, we deduce thatkτ depends only on τ0 = τ|Fcm , along with the symmetry relationkτ0 ,i + kτ0 ◦c,i ≡ wi

mod difor some integer wi (for all τ0 : Fcm ,→ Q` ).
×This relation allows us, by Lemma 2.3.16, to find Galois characters ψ̂i : ΓF → Q` with HTτ (ψ̂i ) ∈Pkτ,i+ Z for all τ. We then form the twist ρ̃0 := ρ̃( i (di wi ) ◦ ψ̂−1i ); recall that hdi wi , λ j i = δi j . Thisdinew lift is then geometric:kτι ,i kτι ,i−≡ 0 mod Z.
didi(Recall it suffices to check ρ̃0 is Hodge-Tate, by Conrad’s result, quoted here as Theorem 2.1.6.)hΘρ̃0 ,ι , λi i = hΘρ̃,ι , λi i + hΘP(d j w j )◦ψ̂−1j ,ι , λi i ≡Note that all eigenvalues of Θrλi ◦ρ̃ are congruent modulo Z; this does not imply that all elements of HTτ (rdi λi ◦ ρ̃)are congruent modulo di Z, but this congruence does hold for the highest and lowest weights.
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Corollary 3.2.8. Let ρ : ΓF → G∨ (Q` ) be geometric with algebraic monodromy group G∨ ;maintain the notation of the previous proof, but now suppose F is totally real. Then ρ has ageometric lift if and only if for varying ι, the elements θρ,ι (see Definition 3.2.3) in X • (ZG )tor are
independent of ι.
Remark 3.2.9. More concretely, to determine whether ρ has a geometric lift, apply the following criterion:(1) Ignore any i for which di is odd; these do not obstruct geometric lifting;(2) Then for fixed i the integers kτ,i mod di are all w2i translated by a two-torsion class in
Z/di Z;(3) ρ has a geometric lift if and only if each of these classes kτ,i (or, equivalently, the associatedtwo-torsion class) is independent of τ.
Proof. By the previous proof and Lemma 2.3.17, ρ has a geometric lift if and only if the classes
kτ,i mod di (fixed i, varying τ) are independent of τ. The weight-symmetry relation becomes
2kτ,i ≡ wi mod di , and the claim follows easily.
Over totally imaginary fields, we can now reduce the general lifting problem to the special caseof full monodromy:e  H be any surjection of linear
Theorem 3.2.10. Let F be totally imaginary, and let π : H
algebraic groups with central torus kernel. Suppose ρ : ΓF → H(Q` ) is a geometric representation,e ` ).
with arbitrary image, satisfying Hypothesis 3.2.4. Then ρ admits a geometric lift ρ̃ : ΓF → H(Qe and H are reductive since π induces an isomorphism on unipotent
Proof. We may assume H
parts (we take this observation from Conrad, who has exploited this reduction in the arguments of[Con11]).
Zareρ be
We next show that the theorem holds if Hρ := ρ(ΓF ) ⊂ H is connected. In that case, let H
e of Hρ . Then Heρ → Hρ is a surjection of connected reductive groups with central
the preimage in Heρ = Ge∨ where G ⊂ G
e is an inclusion of connectedtorus kernel, and we may write Hρ = G∨ , H
e = (G × Z)/Ze G for some inclusion of ZG into a multiplicative groupreductive groups of the form Geee into aZ. If Z is not connected, Proposition 3.2.7 does not immediately apply, so we embed Z
e Dually, Ge ∨  G∨ is a quotient to which
e with corresponding inclusions G ⊂ G
e ⊂ G.
torus Z,e ∨ to Ge∨ , we obtain a geometric lift
we can apply Proposition 3.2.7, and then projecting from Ge ` ) of ρ.
eρ (Q` ) ⊂ H(Qρ̃ : ΓF → H
For arbitrary ρ (i.e. Hρ not necessarily connected), let F 0 /F be a finite extension such thatZareρ (Q` ) ⊂ρ(ΓF 0 ) = Hρ0 is connected. Over F 0 , we can therefore find a geometric lift ρ̃F 0 : ΓF 0 → He ` ), letting Heρ as before denote the preimage of Hρ in H.
e Thus, for any lift ρ̃0 : ΓF 0 → H(Qe `)H(Qof ρ|ΓF0 , there exists a character ψ̂ : ΓF 0 → S ∨ (Q` ) such that ρ̃0 · ψ̂ is geometric. In particular,e ` ) be any lift over F itself (with rational Hodge-Tate-Sen weights), there isletting ρ̃ : ΓF → H(Q∨a ψ̂F 0 : ΓF 0 → S (Q` ) such that ρ̃|ΓF0 · ψ̂F 0 is geometric. But by Corollary 2.3.16, there is a Galois
character ψ̂ : ΓF → S ∨ (Q` ) whose labeled Hodge-Tate-Sen weights descend those of ψ̂F 0 . It follows
e ` ) is a geometric lift of ρ.
immediately that ρ̃ · ψ̂ : ΓF → H(QRemark 3.2.11.
(1) The theorem also lets us make explicit precisely which sets of labeled
Hodge-Tate weights can be achieved in a geometric lift ρ̃. We will exploit this in §4.2.
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(2) We will not treat the general totally real case here, since a somewhat different approachseems more convenient in that case. For a succinct, coordinate-free treatment of the totallyreal case, and another perspective on the arguments of this section, see [Pat14b], wheree  H as in our lifting setup, we write
the following general result is obtained: for any H
0
∨
0
∨
e =G
e , where G
e = (G × Z)/Ze G ; here Z
e is not necessarily connected. We
H = G and (H)can define in this generality elements
e tor → X • (ZG )tor ,θρ,τ ∈ coker X • (Z)e if and only if the θρ,τ are independent of τ. The argumentand ρ admits a geometric lift to H
described in [Pat14b] has the disadvantage of not making explicit the parity obstruction(assuming ‘Hodge symmetry’) found in Proposition 3.2.8; for this reason we have retained
the two different expositions of the totally real case.
The method of proof of Proposition 3.2.7 and Theorem 3.2.10 also implies the following local
result, which emerged from a conversation with Brian Conrad. Here local algebraicity of HodgeTate representations replaces the appeal to the theory of algebraic Hecke characters. In fact, thisconnection holds more generally when K is the fraction field of a complete discrete valuation ringwith perfect residue field: see [CCO14, §3.9].
e  H be a central torus quotient, and let ρ : ΓFv → H(Q` ) be a
Corollary 3.2.12. Let H
Hodge-Tate representation of ΓFv , for Fv /Q` finite. Then there exists a Hodge-Tate lift ρ̃ : ΓFv →e ` ).
H(Q
Proof. We sketch a proof, replacing appeal to Hypothesis 3.2.4 and the existence of certain
Hecke characters by the simpler observation that local class field theory lets us find the necessary
×twisting characters ψ̂ : ΓFv → Q` by hand. That is, OFv sits in IFabv in finite index, and on the former
we can define the characterYτ(x)kτ
x 7→
τ : K,→Q`

for any integers kτ ; then up to a finite-order character we take a dth root for an integer d to build(having extended to all of ΓFv ) characters with any prescribed set of rational τ-labeled Hodge-Tateweights (τ running over all Fv ,→ Q` ). This observation suffices (invoking Lemma 2.2.7 in the fullgenerality of Hodge-Tate, rather than merely de Rham, representations) for the previous arguments
to carry through.
This Corollary combined with [Con11, Proposition 6.5] implies the following stronger result:e  H be a central torus quotient, and let ρ : ΓFv → H(Q` ) be a
Corollary 3.2.13. Let H
representation of ΓFv , for Fv /Q` finite, satisfying a basic p-adic Hodge theory property15 P. Thene ` ) also satisfying P.
there exists a lift ρ̃ : ΓFv → H(Q3.3. Applications: comparing the automorphic and Galois formalisms
In §3.2, we took the Fontaine-Mazur-Langlands conjecture (or rather its weakened form Hypothesis 3.2.4) relating geometric Galois representations ΓF → GLn (Q` ) to L-algebraic automorphic representations of GLn (AF ) as input to establish some of our lifting results. Now we want to15

Namely: crystalline, semi-stable, de Rham, or Hodge-Tate.
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apply these lifting results to give some evidence for the relationship between automorphic forms
and Galois representations on groups other than GLn . We will touch on the Buzzard-Gee conjecture, certain cases of the converse problem, and some general thoughts about comparing descentproblems on the (`-adic) Galois and automorphic sides. We first digress to discuss what ‘automorphy’ of an LG(Q` )-valued representation even means.
3.3.1. Notions of automorphy. As usual we have fixed ι∞ : Q ,→ C and ι` : Q ,→ Q` . It issometimes more convenient simply to fix an isomorphism ι`,∞ : C → Q` , and to regard Q as thesubfield of algebraic numbers in C. G is a connected reductive F-group, and we take a Q-formof the L-group LG. For an automorphic representation π of G(AF ) and ρ : ΓF → LG(Q` ), always
assumed continuous and composing with ΓF -projection to the identity, there are (at least) four
relations between such ρ and π that might be helpful, of which only the first two can be statedunconditionally. First we describe analogues that restrict to either the automorphic or Galois side,borrowing some ideas from [Lap99] (and, inevitably, [LL79]). For automorphic representations π
and π0 of G(AF ), three notions of similarity are:• π ∼w π0 if almost everywhere locally, the (unramified, say) L-parameters are G∨ (C)conjugate.
• π ∼w,∞ π0 if almost everywhere locally, and at infinity, the L-parameters are conjugate.
This also makes sense unconditionally, since archimedean local Langlands is known.
• π ∼ew π0 if everywhere locally, the L-parameters are G∨ (C)-conjugate; this only makes
sense if one knows the local Langlands conjecture for G.
• π ∼ s π0 is the most fanciful condition: if the conjectural automorphic Langlands group LF
exists, so π and π0 give rise to representations LF → LG(C), then this condition requires
these representations to be globally G∨ (C)-conjugate.
A fifth notion would compare L-parameters in a particular finite-dimensional representation of G∨ .
We can unconditionally make the same sort of comparisons between `-adic Galois representations, writing ρ ∼w ρ0 , ρ ∼w,∞ ρ0 , ρ ∼ew ρ0 , and ρ ∼ s ρ0 . By equivalence ‘at infinity’ here, we meanthat at real places the actions of complex conjugation are conjugate, and at places above `, theassociated Sen operators (i.e. labeled Hodge-Tate data) are conjugate. Since it is conjectured, but
totally out of reach, that frobenius elements act semi-simply in a geometric Galois representation,
we should only compare ‘frobenius semi-simplifications’ in these definitions of local equivalence.
For some nice examples, Lapid’s paper ([Lap99]) studies the difference between ∼w , ∼ew , and ∼ sfor certain Artin representations (and, when possible, the corresponding comparison on the automorphic side).
We then have corresponding ways to relate an `-adic ρ and an automorphic π:• ρ ∼w π• ρ ∼w,∞ π• ρ ∼ew π• ρ ∼s π∨First, write ρ ∼w π if for almost all unramified v (for ρ and π), ρ|ss
WFv is G (Q` )-conjugate torecv (πv ) : WFv → G∨ (C)oΓF ; implicit is the assumption that the local parameter lands in G∨ (Q)oΓF ,so that we can apply ι` ◦ ι−1∞ . Note that this definition does not distinguish between π and otherelements of its (conjectural) global L-packet L(π). The other relations are straightforward modifications (for compatibility with complex conjugation, we take the condition in Conjecture 3.2.1 of78

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Version 4 - Submitted to MEMO[BG11]), except for ρ ∼ s π. Writing GF,E (σ) for the motivic Galois group for motives16over F with
E-coefficients, using a Betti realization via σ : F ,→ C, one might hope that after fixing E ,→ C
as well, one would obtain a map of pro-reductive groups over C, LF → (GF,E )(σ) ⊗E C (compare
Remark 3.1.16). If π corresponds to a representation rec(π) of LF , and ρ arises from (completingat some finite place of E) a representation ρE of GF,E (σ), it makes sense to ask whether rec(π) factors through GF,E (σ)(C), and whether the resulting representation is globally G∨ (C)-conjugate to(the complexification via E ,→ C of) ρE . Of course, any discussion of the automorphic Langlands
group and its relation with the motivic Galois group is pure speculation; but these heuristics do
provide context for the basic problems raised in Question 1.1.9 and Conjecture 1.1.10, as well as
the work of §4.2.
Although we don’t actually require it, it is helpful to keep in mind a basic lemma of Steinberg,
which implies that ∼w can be checked by checking in all finite-dimensional representations:Lemma 3.3.1. Let x and y be two semi-simple elements of a connected reductive group G∨ overan algebraically closed field of characteristic zero. If x and y are conjugate in every (irreducible)
representation of G∨ , or even merely have the same trace, then they are in fact conjugate in G∨ .
Proof. For semi-simple groups, this is Corollary 3 (to Theorem 2) in Chapter 3 of [Ste74];
the proof extends to the reductive case (and even more generally, see Proposition 6.7 of [Bor79]).
The key point is that the characters of finite-dimensional representations of G∨ restrict to a basis of
the ring of Weyl-invariant regular functions on a maximal torus; these in turn separate conjugacy
classes in the torus.

3.3.2. Automorphy of projective representations.

Throughout this section, we assume the Fontaine-Mazur-Langlands conjecture on automorphyof geometric (GLN -valued) Galois representations. It suffices to take a version that matches unramified (almost everywhere) and Hodge-theoretic parameters; to be precise, assume Part 3 of Conjecture 1.2.1, and note that this includes the requirement that cuspidality is equivalent to irreducibility
under the automorphic-Galois correspondence. We will show how our lifting results–both automorphic and Galois-theoretic–give rise to a ‘Fontaine-Mazur-Langlands’-type correspondence
between algebraic automorphic representations of SLN (AF ) and PGLN (Q` )-valued geometric ΓF representations.17 The starting point is the following consequence of the results of §3.2:
Corollary 3.3.2. Let F be totally imaginary. Then any geometric ρ : ΓF → PGLn (Q` ) is weakly
automorphic, i.e. there exists an L-algebraic automorphic representation π of SLn (AF ) such thatρ ∼w,∞ π (or ρ ∼ew π, if we assume a form of Fontaine-Langlands-Mazur that matches local factors
everywhere).
Proof. We have seen that ρ lifts to a geometric ρ̃ : ΓF → GLn (Q` ), which by assumption is
automorphic, corresponding to some π̃ on GLn /F. The irreducible constituents of π̃|SLn (AF ) form a
global L-packet whose local unramified parameters correspond to those of ρ.

16

Either for absolutely Hodge cycles, for motivated cycles, or, assuming the standard conjectures, for homologicalcycles. Again, we will deal more precisely with motivic Galois groups in §4.1.
17
For some unconditional results in this direction, see [Pat14b, §4].
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Version 4 - Submitted to MEMOWe now give descent arguments that extend this automorphy result to F totally real. First weneed a couple of elementary lemmas.
Lemma 3.3.3. Let L/F be a cyclic, degree d, extension of number fields, with σ a generator of
Gal(L/F). Let χ be a Hecke character of L, and let δ be any Hecke character whose restriction
to C F ⊂ C L is δ = δL/F , a fixed order d character that cuts out the extension L/F. Assume thatd−1
χ1+σ+...+σ = 1. Then for a unique integer i = 0, . . . , d − 1, χδi is of the form ψσ−1 for a Hecke
character ψ of L.
Proof. We may assume χ is unitary. Write C D as usual for the Pontryagin dual of a locallycompact abelian group C. We have the following exact sequences:
1O
Gal(L/F)O
1/ C L σ−1 / C L

/ CF
O
NL/F

,

CL
dualizing to1Gal(L/F)D1o
/ CD ores

FC LD o

σ−1

C LD

NL/F

.

C LD

By assumption, NL/F ◦ res(χ) = 1, so res(χ) = δ−i ∈ Gal(L/F)D for some integer i, unique modulo
d. Then res(χδi ) = 1, and we are done by exactness of the horizontal diagram.

We need a special case of an `-adic analogue of the remark after Statement A of [LR98];18 that
remark is in turn the (much easier) analogue, for complex representations of the Weil group, of themain result of their paper. We first record the simple case that we need, and then out of independentinterest we prove a general `-adic analogue of the Lapid-Rogawski result.
Lemma 3.3.4. Let L/F be a quadratic CM extension of a totally real field F, with σ ∈ ΓF×generating Gal(L/F) Suppose ψ̂ : ΓL → Q` is a Galois character such that ψ̂1−σ is geometric.
Then there exists a unitary, type A Hecke character ψ of L such that ψ1−σ is the type A0 Hecke
character associated to ψ̂1−σ .
18In the proof of Corollary 3.3.6 below, we could replace appeal to this lemma by simply citing Statement B of
[LR98].
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Proof. Write (ψ̂1−σ )A for the Hecke character associated to ψ̂1−σ . By the previous lemma, itsuffices to check that (ψ̂1−σ )A is trivial on C F ⊂ C L . If a finite place v of L is split over a placevF of F, and unramified for ψ̂, then for a uniformizer $v of FvF (embedded into the Lv and Lσv
components of A×L ),(ψ̂1−σ )A ($v , $v ) = ψ̂1−σ (frv )ψ̂1−σ (frσv ) =ψ̂(frv ) ψ̂(frσv )·= 1.
ψ̂(frσv ) ψ̂(frv )ψ̂(frv )Similarly for v inert, (ψ̂1−σ )A ($v ) = ψ̂(σfr= 1. The Hecke character (ψ̂1−σ )A |CF is therefore−1vσ )trivial. To see that we may choose ψ to be unitary and type A, we invoke Corollary 2.3.9: decomposing ψ as in that result, both the | · |w and ‘Maass’ components descend to the totally real subfieldF, so dividing out by them yields a new ψ, now unitary type A, and with ψ1−σ unchanged.
Lemma 3.3.5. Let L/F be cyclic of degree d, with σ ∈ ΓF restricting to a generator of Gal(L/F).
Suppose ρ : ΓL → GLn (Q` ) is an irreducible continuous representation satisfying ρσ  ρ · χ for×
this holds if ρ issome character χ : ΓL → Q` . Suppose further that χ is geometric (in particular,Qgeometric), necessarily of weight zero, so we may regard it as a character χA = w∈|L| χw : C L →C× . Then the (finite-order) restriction of χA to C F ⊂ C L cannot factor through a non-trivial
character of Gal(L/F).
Proof. Iterating the relation ρσ  ρ · χ, we obtain ρ  ρ · χ1+σ+...+σ , so that χ1+σ+...+σi
is finite-order. Each of the characters χσ has some common weight w, so d · w = 0, and thus
w = 0. Moreover, writing as usual pιw for the P
algebraic parameter of χw (with respect to a choiceιw : L ,→ C representing the place w), we have pιw = 0 as ιw ranges over a Gal(L/F)-orbit of such×
embeddings. If L has a real embedding, then χ has finite-order, and the passage from χ : Γab
L → Q`
190
to χA : C L → C× is simply via the reciprocity map C L  Γab
L . In particular, χ(x) = χA (x ) for any
0
abrepresentative x in C L of the image of x in ΓL . If on the other hand L is totally imaginary, thencontinuing to write x0 = (xw0 )w∈|L| , we haveYY Ypι∗ (τ) 

0
0
0

∞,`
τ(xw )
χ(x) =
χw (xw ) .
χw (xw )
d−1

w-`∞

d−1

τ : Lw ,→Q`w|`

0

If we further assume that the representative x can be chosen in C F ⊂ C L , with elements xv0 ∈ Fv×giving rise to all xw0 for w|v, then we can rewriteY Yp∗τ(xw0 ) ι∞,` (τ)
w|` τ : Lw ,→Q`

as

Y Y

Y

τ(xv0 ) =

Y Y

P

p∗

τ(xv0 ) w|v,τ̃|τ ι∞,` (τ̃) = 1.

v|` τ : Fv ,→Q`v|` τ : Fv ,→Q` w|v,τ̃|τThe last equality follows since χ1+σ+...+σ is finite-order, and we are summing pι∗∞,` (τ̃) over a fullGal(L/F)-orbit. A similar argument shows thatYχw (xw0 ) = 1.
d−1

w|∞

Throughout this argument we implicitly use our fixed embeddings ι` , ι∞ , but omit any reference to them fornotational simplicity.
19

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Version 4 - Submitted to MEMOWe conclude that in this case (x0 ∈ C F ), χ(x) can be computed simply as χA (x0 ).
Let V be the space on which ρ acts. Then the isomorphism ρσ  ρ · χ yields an operator
A ∈ Aut(V) satisfying Aρσ = ρ · χA. Fix g ∈ ΓL , and for any x ∈ ΓL we computetr(ρ(g)A) = tr(ρσ (x)ρ(g)Aρσ (x)−1 )= tr(ρσ (x)ρ(g)ρ(x−1 )χ(x−1 )A)= χ(x−1 ) tr(ρ(σ xgx−1 )A).
(Here σ x = σxσ−1 .) So, if we can find an x ∈ ΓL such that σ xgx−1 = g and χ(x−1 ) , 1, then we
will have tr(ρ(g)A) = 0. Doing this for all g ∈ ΓL , we see that by Schur’s Lemma ρ cannot beirreducible, else A = 0. Now, y = g−1 σ ∈ ΓF satisfies σ ygy−1 = g, so x = yd ∈ ΓL does as well. It
suffices to show that if χ|CF cuts out the extension L/F, then χ(x) , 1. In fact, the image of x in ΓabLablies in the image of the transfer Ver : Γab→Γ
.
Explicitly,
F
L
Ver(y) =d−1
Yσi (g−1 σ)φ(σi g−1 σ)−1 ,i=0

where φ : ΓF → {σi }i=0,...,d−1 records the representative of the ΓL -coset of an element of ΓF . It isthen easily seen20 that
Ver(y) = (g−1 σ)d = x,0so by class field theory x ∈ Γab
L is represented by an element x of C F ⊂ C L under the reciprocity
map recL . This element is a generator of C F /NL/F C L , since y lifts a generator of Gal(L/F), andthus χ(x) = χA (x0 ) , 1 if χA |CF factors through a non-trivial character of Gal(L/F).

Finally we can (conditionally) prove automorphy of geometric projective representations overtotally real fields.
Corollary 3.3.6. Let F be totally real. Continue to assume Part 3 of Conjecture 1.2.1 (FontaineMazur-Langlands). Then for any geometric ρ : ΓF → PGLn (Q` ), there exists an L-algebraic π on
SLn /F such that ρ ∼w π.
Proof. We will first treat the case of ρ having irreducible lifts to GLn (Q` ). Choose a lift ρ̃ withfinite-order determinant, a CM quadratic extension L/F, and, by Theorem 3.2.7, a Galois character×ψ̂ : ΓL → Q` such that ρ̃|ΓL · ψ̂−1 is geometric. Let π̃ be the cuspidal automorphic representation ofGLn (AL ) corresponding to this geometric twist. Write σ for the nontrivial element of Gal(L/F),
so that π̃σ  π̃ · χ where χ is the Hecke character corresponding to the geometric Galois characterψ̂1−σ . Appealing either to Lemma 3.3.4 or Lemma 3.3.5, we can write χ = ψ1−σ for a unitary
type A Hecke character ψ. Then π̃ · ψ is σ-invariant, and by cyclic (prime degree) descent, there
is a cuspidal representation π of GLn (AF ) whose base-change is π̃ · ψ. We want to compare theprojectivization of the unramified parameters of π with the unramified restrictions ρ|ΓFv .
To do so, we repeat the argument but instead with infinitely many (disjoint) quadratic CM
extensions Li /F, showing that in all cases the descent π to GLn (AF ) gives an L-packet for SLn (AF )that is independent of the field Li . ρ̃ is still a fixed lift with finite-order determinant, and we canwrite, for each τ : F ,→ Q` , HTτ (ρ̃) ∈ knτ + Z for some integer kτ , which we fix (rather than justits congruence class mod n). For some integer w, we have the purity relation 2kτ ≡ w mod n,
Note that φ(σd−1 g−1 σ) = 1, while otherwise φ(σi g−1 σ) = σi+1 .

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Version 4 - Submitted to MEMOas follows, for instance, from (geometric) liftability after a CM base-change;21 as with kτ , we
fix an actual integer w, not just the congruence class. Now, for each such τ, let ι : F ,→ C be
the archimedean embedding associated via ι∞ , ι` (elsewhere denoted ι∗∞,` (τ)). For each Li , fix anembedding τ(i) : Li ,→ Q` extending τ, so that the other extension is τ(i)◦c. Likewise write ι(i) andι(i) ◦ c = ι(i) for the corresponding complex embeddings. We can then construct Galois characters×ψ̂i : ΓLi → Q` such thatkτHTτ(i) (ψ̂i ) =nw − kτ
HTτ(i)◦c (ψ̂i ) =,nsuch that ρ̃|ΓLi · ψ̂−1is geometric, corresponding to an L-algebraic cuspidal π̃i on GLn /Li . Asiibefore, we find a Hecke character ψi of Li such that ψ1−σ
is the type A0 Hecke character associatedi
1−σi
to ψ̂i ; here we write σi for the non-trivial element of Gal(Li /F), but of course all the σi are just
induced by complex conjugation. Again, for all i we find cuspidal automorphic representations πiof GLn (AF ) such that BCLi /F (πi ) = π̃i · ψi . Restricting to composites Li L j , we have the comparison!
ψ̂i−1|Li L j = ρ̃|Li L j ψ̂−1
ρ̃|Li L j ψ̂i |Li L j ·j |Li L j ,ψ̂ j
and thus!
!
ψjψ̂i· BCLi L j= BCLi L j /L j (π̃ j ψ j ), 22
BCLi L j /Li (π̃i ψi ) · BCLi L jψiψ̂ jso finally!
ψ j ψ̂i
BCLi L j (πi ) · BCLi L j·= BCLi L j (π j ).
ψi ψ̂ jψ

If the character ψij · ψ̂ψ̂ij is finite-order–in the next paragraph, we check that we may assume
this–it cuts out a cyclic extension L0 /Li L j , and we have BCL0 (πi ) = BCL0 (π j ). L0 /F is solvable,
however, so the characterization of the fibers of solvable base-change in [Raj02] implies that πi
and π j are twist-equivalent, hence that πi |SLn (AF ) and π j |SLn (AF ) define the same L-packet of SLn (AF ).
Let us denote by π0 any representative of this global L-packet. Now consider places v of F thatare split in a given Li /F. The semi-simple part ρ(frv )ss is equal (in PGLn (Q` )) to (ρ̃ψ̂−1 (frw ))ss
for any w|v, and this is conjugate in GLn (Q` ) to ι`,∞ (recw (π̃w )(frw )), whose projectivization lies
in the same PGLn (Q` )-conjugacy class as ι`,∞ recv (π0,v )(frv ) . This verifies that for all such v,
ρ(frv )ss is PGLn (Q` )-conjugate to ι`,∞ recv (π0,v )(frv ) . Varying Li /F, and remembering that π0 is
independent of this variation, we get the same result for all v split in any single quadratic CM
extension Li /F (we have to throw out a finite number of such Li to ensure our representations
remain cuspidal/irreducible), we conclude that ρ ∼w π.
ψTo finish the proof, we must check that ψij · ψ̂ψ̂ij may indeed be assumed finite-order. First,
recall that each ψi may be taken unitary and type A; in this case, the infinity-type is determined
After such a base-change L/F, the character ψ̂ twisting ρ̃ to a geometric representation will have Hodge-TateSen weights congruent to knτ ∈ Q/Z at both embeddings L ,→ Q` above τ; the integer w is then the weight of the Hecke
character associated to ψ̂n .
22Here ψ̂ψ̂i restricted to ΓLi L j is geometric, so we abusively write this for the associated Hecke character as well.
21

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iby the relation ψ1−σ= ψ̂i1−σi . Explicitly (using the above notation for the various embeddings),inψ̂i corresponds to a Hecke character of Li with infinity-type (where we abusively denote ι(i)(z) bysimply z)nrecι(i) (ψ̂i ) : z 7→ zkτ z̄w−kτ ,so2kτ −ww−2kτirecι(i) (ψ1−σ) : z 7→ z n z̄ n .
i(Recall that 2kτ ≡ w mod n.) We then have, under our assumptions,! 2kτ −wz n.
recι(i) (ψi ) : z 7→|z|τOf course, recι(i)◦c is the same but with w−2k
in the exponent. To make the parameter comparisonnafter restriction to a composite Li L j , we use the following notation for embeddings of Li L j into Q`
and C, lying above the given τ and ι:

τ1 extends τ(i) and τ( j),τ2 extends τ(i) and τ( j) ◦ c,
ι1 extends ι(i) and ι( j),ι2 extends ι(i) and ι( j) ◦ c.
We then have the conjugate embeddings τ1 ◦ c, etc. Computing the τk -labeled weights of ψ̂ψ̂ij ,and translating them to the infinity-type at the place corresponding to ιk , with ιk as the chosen
embedding Li L j ,→ C, we then find!
ψ̂i
: z 7→ 1
recι1
ψ̂ j
!
2kτ −w
ψ̂irecι2: z 7→ (z/z̄) n ,ψ̂ jwhereas
recι1
recι2

!
ψj
: z 7→ 1
ψi
!
ψj
w−2kτ
: z 7→ (z/z̄) n .
ψiψ

We conclude that ψij · ψ̂ψ̂ij is, with our normalization of the ψi , in fact finite-order, and the proposition
follows.
Finally, we quickly treat the case of general ρ, having reducible lifts. If ρ̃ as above decomposesρ̃ = ⊕mi=1 ρ̃i , say with ρ̃i of dimension ni , geometricity of ρ implies that over any CM L/F the same
Galois character ψ̂ twists ρ̃i , for all i, to a geometric representation. We can therefore use, for alli, the same Hecke character ψ such that ψ1−σ = ψ̂1−σ . As above, we invoke automorphy of ρ̃i ψ̂−1 ,and, twisting by ψ, descend to a cuspidal automorphic representation Πi of GLni /F. The same
local check (for v split in L/F) as above, but now crucially relying on the fact
 that ψ and ψ̂ werem
independent of i, shows ρ(frv ) is PGLn (Q` )-conjugate to ι recv (i=1 Πi,v )(frv ) .
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Version 4 - Submitted to MEMOBy a similar argument, we can ‘construct’ the Galois representations (assuming of course the
GLN correspondence) associated to (tempered) L-algebraic π on SLn /F for F CM (or imaginary,assuming Conjecture 2.4.8) or totally real. By Proposition 3.1.12, we are reduced to the case of F
totally real.
Proposition 3.3.7. Continue to assume Fontaine-Mazur-Langlands. Let F be a totally real
field, and let π be an L-algebraic cuspidal automorphic representation of SLn (AF ). Assume that
π∞ is tempered. Then there exists a (not necessarily unique) projective representation ρ : ΓF →PGLn (Q` ) satisfying ρ ∼w π.
Proof. By Proposition 3.1.14, there exists a W-algebraic cuspidal (and tempered at ∞) π̃ on
GLn /F lifting π. For all but finitely many quadratic CM L/F, we can find a type A Hecke characterψ such that BCL/F (π̃) · ψ is L-algebraic and cuspidal on GLn /L, hence corresponds to an irreducible[σ−1 ), the twist
geometric representation ρ̃L : ΓL → GLn (Q` ). Conjugating, we find ρ̃σL ≡ ρ̃L · (ψσ−1being by the geometric character associated to the type A0 Hecke character ψ . We wish to×[σ−1 ) in the form ψ̂σ−1 for some Galois character ψ̂ : Γ → Q (reversing the process
write χ̂ = (ψL`in Corollary 3.3.6). Once this is managed, we have (ρ̃L · ψ̂−1 ) is σ-invariant, hence descends
to ρ : ΓF → PGLn (Q` ). Arguing as in Corollary 3.3.6, we find that this projective descent is
independent of L/F, and, again as in that proof, by varying L/F we obtain the compatibilityρ ∼w π.
d2 ) up
To construct ψ̂, we take as first approximation a Galois character ψ̂ such that ψ̂2 equals (ψ2d2 ).
to a finite-order character χ0 ; recall that ψ is type A0 , so we can attach the Galois character (ψThend2 )χ )σ−1 = χ̂2 χσ−1 ,(ψ̂σ−1 )2 = (ψ̂2 )σ−1 = ((ψ00σ−1and consequently χ̂ agrees with ψ̂
up to a finite-order character. Twisting ρ̃L , we may therefore
assume that χ̂ has finite-order. It still satisfies χ̂1+σ = 1, so invoking Lemma 3.3.3 and (a simple
case of) Lemma 3.3.5 we find a finite-order Hecke character,23 which may therefore be directly
regarded as a Galois character, that casts χ̂ in the desired form.
Here is another example comparing the Tannakian formalisms:
Proposition 3.3.8. Continue to assume Fontaine-Langlands-Mazur and, in the totally imaginary but non-CM case, Conjecture 2.4.8. Let Π be a cuspidal L-algebraic representation ofGLn (AF ), and suppose that ρΠ  ρ1 ⊗ ρ2 , where ρi : ΓF → GLni (Q` ). Then there exist cuspidal
automorphic representations πi of GLni (AF ) such that Π = π1  π2 .
Remark 3.3.9. As the examples in §2.6 show, sometimes the πi cannot be taken L-algebraic.
Nevertheless, by Proposition 2.5.8, they can always be taken W-algebraic.
Proof. First suppose F is totally imaginary. For all v|`, the fact that ρ1 |ΓFv ⊗ ρ2 |ΓFv is de
Rham implies that locally these ΓFv -representations are twists of de Rham representations. To
see this, we apply Corollary 3.2.12 (to find a Hodge-Tate lift) and Theorem 2.1.6 (to find a deRham lift, given that a Hodge-Tate lift exists) to the lifting problem (with central torus kernel)
GLn1 × GLn2 →− Gn1 ,n2 ⊂ GLn1 n2 , where Gn1 ,n2 denotes the image of the tensor product map. In
Writing χ̂ = ψσ−1 where the infinity-components of ψ have the form z p z̄q , we see z p−q z̄q−p is the corresponding
component of χ̂, hence that p = q at each infinite place. Twisting ψ by the base-change of a character of the totallyreal field F, we can then assume it is finite-order.
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particular, the projectivizations of the ρi |ΓFv are de Rham, so the global projective representations
ρi : ΓF → PGLni (Q` ) are geometric. We know that these geometric projective representations havegeometric lifts, and we may therefore assume our original ρi were in fact geometric. They thencorrespond to L-algebraic πi , and we have Π = π1  π2 .
For F totally real, we perform a descent similar to previous arguments. Restricting to CM L/F,
we find a Galois character ψ̂ such that ρ1 · ψ̂−1 and ρ2 · ψ̂ are geometric, corresponding to L-algebraic
cuspidal πi on GLni /L. Writing ψ̂1−σ = ψ1−σ for a Hecke character ψ, we find π1 ·ψ and π2 ·ψ−1 are σinvariant, so descend to cuspidal representations π̄i of GLni /F. Since BCL/F (Π) = BCL/F (π̄1  π̄2 ),we deduce that Π and π̄1  π̄2 are twist-equivalent, from which the result follows. (The same
argument applies to F that are neither totally real nor totally imaginary: just replace the restriction
to CM extensions L/F with restrictions to totally imaginary quadratic extensions L/F.)
These examples (and, for instance, Corollary 2.7.8) motivate a comparison of the images ofr : L H → LG on the automorphic and Galois sides, when r is an L-morphism with central kernel.
The most optimistic expectation (for H and G quasi-split) is that if ker(r) is a central torus, then thetwo descent problems for (L-algebraic) Π and (geometric) ρΠ are equivalent; whereas if ker(r) is
disconnected, there is an obstruction to the comparison, that nevertheless can be killed after a finite
base-change. If G is not GLn /F, then one will have to decide whether weak equivalence (π ∼w ρ)
suffices to connect the descent problems, or whether some stronger link (the mysterious π ∼ s ρ)
must be postulated.
3.4. Monodromy of abstract Galois representations
In this section we discuss some general results about monodromy of `-adic Galois representations. Much of the richness of this subject comes from its blending of two kinds of representation theories, that of finite groups, and that of connected reductive algebraic groups. We will see(Proposition 3.4.1) that the basic lifting result (Proposition 2.1.4) allows us to some extent to understand how these two representation theories interact. In §3.4.2 we develop more refined resultsin the ‘Lie-multiplicity-free’ case (see Definition 3.4.5); this situation encapsulates the essential
difficulties of independence-of-` questions, such questions being trivial for Artin representations.
3.4.1. A general decomposition. The following result is a simple variant of a result of Katz([Kat87, Proposition 1]), which he proves for lisse sheaves on affine curves over finite fields. We
can replace Katz’s appeal to the Lefschetz affine theorem by Proposition 2.1.4. Recall that a Galois
representation is Lie irreducible if it is irreducible after restriction to every finite-index subgroup(i.e., the connected component or Lie algebra of its algebraic monodromy group acts irreducibly).
Proposition 3.4.1. Let F be any number field, and let ρ : ΓF → GLQ` (V) be an irreducible
representation of dimension n. Then either ρ is induced, or there exists d|n, a Lie irreducible
representation τ of dimension n/d, and an Artin representation ω of dimension d such that ρ τ ⊗ ω. Consequently, any (irreducible) ρ can be written in the formρ  IndFL (τ ⊗ ω)
for some finite L/F and irreducible representations τ and ω of ΓL , with τ Lie-irreducible and ω
Artin.
Proof. Let G denote the algebraic monodromy group of ρ, with G0 the connected componentof the identity. Abusively writing ρ for the representation G ,→ GL(V), we may assume ρ|G0 is
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isotypic (else ρ is induced, and we are done). If ρ|G0 is irreducible, then ρ itself is Lie-irreducible,
so again we are done. Therefore, we may assume that ρ|G0  τ⊕d
0 for some d ≥ 2, with τ0 an irreducible representation of G0 , and consequently a Lie-irreducible representation of ΓL for any L/F
sufficiently large that ρ(ΓL ) ⊂ G0 (Q` ). Since the irreducible ΓL -representation τ0 is ΓF -invariant,
it extends to a projective representation of ΓF . By the basic lifting result (Proposition 2.1.4), this×
projective representation lifts to an honest ΓF -representation τ1 , so for some character χ : ΓL → Q` ,
τ⊕d1 |ΓL  ρ|ΓL ⊗ χ.
−n
The character α := det(ρ)/ det(τ⊕d1 ) of ΓF has ΓL -restriction equal to χ , and over F itself we×
can find characters α1 , α0 : ΓF → Q` , with α0 finite-order, such that α = αn1 α0 . Then (χα1 |ΓL )n =(α−1 αn1 )|ΓL = α−1
0 |ΓL , and replacing τ1 by τ1 ⊗ α1 , and L by a finite extension trivializing α0 , we finda Lie-irreducible representation τ of ΓF and a finite extension L of F such that τ⊕d |ΓL  ρ|ΓL . The
ΓF -representation
ω := HomΓL (τ, ρ).
is therefore a d-dimensional Artin representation,24 and the natural map τ ⊗ ω → ρ (i.e. v ⊗ φ 7→
φ(v)) is an isomorphism of ΓF -representations.


Corollary 3.4.2. Let ρ : ΓF → GLQ` (V) be a semi-simple representation (not necessarily irreducible), and suppose that ρ is Lie-isotypic, i.e. for all F 0 /F sufficiently large, ρ|ΓF0 is isotypic. Then
there exists a Lie-irreducible representation τ and an Artin representation (possibly reducible) ω,both of ΓF , such that ρ  τ ⊗ ω.
Proof. Decompose ρ into irreducible ΓF -representations as ⊕r1 ρi . Each ρi is Lie-isotypic: thereiexists L/F and integers mi such that ρi |ΓL  τ⊕m
0 for all i, where τ0 is a Lie-irreducible representation
independent of i. By the argument of the previous proposition, after possibly enlarging L wefind a ΓF -representation τ whose restriction to L is isomorphic to τ0 , and then there are Artin
representations ωi of ΓF such that τ ⊗ ωi  ρi . Consequently,
r
M
ωi ).
ρτ⊗(
1


Remark 3.4.3.
• In general, the field L in Proposition 3.4.1 is not unique, even up toΓF -conjugacy. Examples of such non-uniqueness should not arise in the Lie-multiplicity
free case (see §3.4.2), but in the Artin case there are easy examples arising from therepresentation theory of finite groups. Consider, for instance, the quaternion group Q8 ={±1, ±i, ± j, ±k}. The (unique) irreducible two-dimensional representation of Q8 can be
Q8written in the form Indhxi(ε x ), where hxi denotes one of the subgroups generated by i, j, ork, and ε x is a generator of the character group of hxi. None of these subgroups is conjugate
to any of the others.25 It would be interesting to achieve a more systematic understandingof these ambiguities.
⊕dAs ΓL -representation, ω  HomΓL (τ|ΓL , τ|⊕d
Γ L )  Q` .
25
Although these groups are not conjugate, they are related by (outer) automorphisms of Q8 , but applying outerautomorphisms in this fashion will not in general preserve an irreducible induced character: consider the principal
series of GL2 (F p ) and the outer automorphism of A1 .
24

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Version 4 - Submitted to MEMO• These structure theorems for Galois representations should have an analogue on the automorphic side. In fact, Tate has shown (2.2.3 of [Tat79]) the analogue of Proposition 3.4.1
for representations of the Weil group WF , where it takes the particularly simple form that
any irreducible, non-induced ρ : WF → GLn (C) is isomorphic to ω⊗χ, where ω is an Artin
representation and χ : WF → C× is a character. A basic question is whether we should
expect Proposition 3.4.1 to hold for ‘representations of LF .’ If we had the formalism of
LF , then to carry out the argument of the proposition with complex representations of LF
in place of `-adic representations of ΓF requires two ingredients:– That a homomorphism LF → PGLn (C) lifts to a homomorphism LF → GLn (C); but
this is ‘implied’ by Proposition 3.1.4. I should mention in this respect the theoremof Labesse ([Lab85]), which establishes the analogue for lifting homomorphisms
e → LG with central torus kernel.
WF → LG across surjections LG– That the analogue of the character α in the proof of Proposition 3.4.1 can be written
as a finite-order twist α0 of the nth power of a character α1 . This is not automatic,
as it is for `-adic characters, but in this case we can exploit Lemma 2.3.10, whichapplies to the α of the Proposition, since there the restriction to L is (continuing with
the notation of the Proposition) χ−n .
This discussion motivates the following conjecture, whose formulation of course requires
assuming deep cases of functoriality:
Conjecture 3.4.4. Let π be a cuspidal automorphic representation of GLn (AF ); assume π is not automorphically induced from any non-trivial extension L/F. Then there exist cuspidal automorphic representations τ and ω of, respectively, GLd (AF ) and GLn/d (AF )such that π = τ  ω, with the following properties:– for all finite extensions L/F, the base-change BCL/F (τ) remains cuspidal;– for some finite extension L/F, BCL/F (ω) is isomorphic to the isobaric sum of n/dcopies of the trivial representation.
3.4.2. Lie-multiplicity-free representations. In this section, we focus on the cases antithetical to that of Artin representations, putting ourselves in the following situation. Let F be any
number field, and recall from definition 1.2.3 the definition of a (weakly) compatible system of
λ-adic representations of ΓF , with coefficients in a number field E. Letρλ : ΓF → GLn (Eλ )be such a (semi-simple, continuous) compatible system. Write Vλ for the space on which ρ` acts.
Definition 3.4.5. We say that Vλ is Lie-multiplicity free if after any finite restriction L/F, Vλ |ΓL
is multiplicity-free. Equivalently,
lim EndEλ [ΓL ] (Vλ )−−→L/Fis commutative. We will often abbreviate ‘Lie-multiplicity-free’ to ‘LMF.’
Cases to keep in mind are Hodge-Tate regular (see Definition 2.4.2 and the preceding discussion of §2.4.1) Vλ , or Vλ of the form H 1 (AF , Q` ) where A/F is an abelian variety with End0 (AF )a commutative Q-algebra (by Faltings’s proof of the Tate conjecture). Elementary representation
theory yields:
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Version 4 - Submitted to MEMOLemma 3.4.6.
written(1) Suppose Vλ is irreducible. Then Vλ is LMF if and only if it can beVλ  IndFL(λ) (Wλ ),where we write L(λ) to show the a priori dependence on λ if Vλ belongs to a compatible
system, and where Wλ is a Lie-irreducible Eλ -representation of ΓL(λ) , all of whose ΓF conjugates remain distinct after any finite restriction.
(2) Let Wλ be an irreducible representation of ΓL , and assume that Vλ = IndFL (Wλ ) is LMF.
Then Vλ is irreducible.
Proof. For (1), restrict to a finite-index subgroup of ΓF over which Vλ decomposes into a
direct sum of Lie-irreducible representations; take one such factor, and consider its stabilizer inΓF –Vλ is then induced from this subgroup. For (2), Mackey theory implies we need to check that
Wλ |gΓL g−1 ∩ΓL and (gWλ )|gΓL g−1 ∩ΓL are disjoint for all g ∈ ΓF − ΓL . These two representations occur as
distinct factors in the Vλ |gΓL g−1 ∩ΓL , so they are disjoint since Vλ is LMF.
For general (possibly reducible) LMF representations, there is a decomposition into a sum ofterms as in the lemma. If Vλ belongs to a compatible system, we expect that the number of suchfactors should be independent of λ; this is an extremely difficult problem (unlike the corresponding
question for Artin representations). Let us indicate the difficulties through an example.
Example 3.4.7. Suppose f is a holomorphic cuspidal Hecke eigenform on the upper half-planeof some weightlevel N and nebentypus . We normalize f so that its q-expansion at theP k ≥ 2 andncusp ∞, f = an ( f )q , has leading coefficient a1 ( f ) = 1. Then work of Eichler-Shimura-Deligne(see [Del71]) yields a compatible systemρ f,λ : ΓQ → GL2 (Eλ )of λ-adic representations (here E is the number field generated by the an ( f )) characterized by theproperty that for all p - N, the characteristic polynomial of ρ f,λ (fr p ) is equal to
X 2 − a p ( f )X + pk−1 (p).
Moreover, the eigenvalues of ρ f,λ (fr p ) are p-Weil numbers of weight k − 1, i.e. have absolute valuek−1
p 2 in all complex embeddings, and ρ f,λ is Hodge-Tate of weights 0, 1 − k. Since f is cuspidal,
we expect these Galois representations to be irreducible. This can be proven because the FontaineMazur-Langlands conjecture is known for the group GL1 . Suppose ρ f,λ  χ1 ⊕ χ2 . Then (up to
re-ordering χ1 , χ2 ) Theorem 2.3.13 implies26 that χ1 must be a finite-order character and χ2 must
be the product of a finite-order character and the (1 − k)th power of the cyclotomic character. This
contradicts the fact that the eigenvalues of ρ f,λ (fr p ) are p-Weil numbers of weight k − 1, so we have
proven the asserted irreducibility.
Therefore in this section we pursue a much more modest goal: restricting to the case of irreducible compatible systems, we will be able to say something about independence of λ of the fields
L(λ) of Lemma 3.4.6.
26This special case of Theorem 2.3.13 is notably easier than the general case. It suffices to show that a character×χ : ΓQ → Q` with all labeled Hodge-Tate weights equal to zero is finite-order; this follows from global class field×
theory and the corresponding statement for characters ΓQ` → Q` . This latter statement is a slight improvement of afundamental theorem of Tate: see [Ser98, §III.A-3].
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Version 4 - Submitted to MEMOOur basic strategy is that the places v for which tr(ρλ (frv ) equals zero should detect the field
L(λ). This is in marked contrast to the case of Artin representations: any irreducible (non-trivial)
representation of a finite group has elements acting with trace zero. Our main tool will be thefollowing (slight weakening of a) theorem of Rajan:
Theorem 3.4.8 (Theorem 3 of [Raj98]). Let E be a mixed characteristic non-archimedean
local field, and let H/E be an algebraic group. Let X be a subscheme of H (over E), stable under
the adjoint action of H. Suppose ρ : ΓF → H(E) is a Galois representation, unramified almost
everywhere, and let C = X(E) ∩ ρ(ΓF ). Denote by Hρ ⊂ H the algebraic monodromy group
Zar

ρ(ΓF ) , and let Φ = Hρ /Hρ0 denote its group of connected components. For φ ∈ Φ, we write H φfor the corresponding component, and we set Ψ = {φ ∈ Φ|H φ ⊂ X}. Then the density of the set ofplaces v of F with ρ(frv ) ∈ C is precisely |Ψ|/|Φ|.
Rajan applies this to prove27 (Theorem 4 of [Raj98]) that an irreducible, but Lie reducible,
representation necessarily has a positive density of frobenii acting with trace zero; note that thisalso follows immediately from Čebotarev and Proposition 3.4.1, which is a more robust versionof Rajan’s result (basically combining his argument with Proposition 2.1.4). Our next two results
establish a converse, also extending Corollaire 2 to Proposition 15 of [Ser81] to its natural level ofgenerality: that result handles the case of connected monodromy groups.
Proposition 3.4.9. Let ρλ : ΓF → GLn (Eλ ) be a continuous, semi-simple, LMF representation.
Decompose Vλ as above, so
rλ
M
Vλ 
IndFL(λ)i (Wλ,i )i=1

for Lie-irreducible representations Wλ,i of ΓL(λ)i . Then:(1) Up to a density zero set of places,{v ∈ |F| : tr(ρλ (frv )) = 0} = {v : frv <[ [i

σΓL(λ)i σ−1 },σ∈S λ,iwhere S λ,i is a set of representatives of ΓF /ΓL(λ)i .
(2) Further assume that ρλ belongs to a compatible system {ρλ } of λ-adic representationsof ΓF (although in contrast to definition 1.2.3, we need not assume here that the ρλ aregeometric). Then up to a set of density zero, the set of places of F which have a split factorin L(λ)i for some i is independent of λ. If we further assume that all Vλ are absolutely
irreducible (rλ = 1) and all L(λ)/F are Galois, then L(λ) is independent of λ.28
Proof. For the first part of the Proposition, we ignore the underlying ‘coefficient’ number field
E and just view ρλ as valued in GLn (E) for some sufficiently large finite extension E of Q` . The “⊇”direction follows from the usual formula for the trace of an induced representation.
To establish`
the reverse inclusion, let us consider, for each non-empty subset I ⊂ i S λ,i , the set XI of placesv such that tr(ρλ (frv )) = 0, and frv ∈ σΓL(λ)i(σ) σ−1 if and only if σ ∈ I [Notation: if σ ∈ I, thenIn addition to the main result of his paper, a beautiful ‘strong multiplicity one’ theorem for `-adic
representations.
28
In the non-Galois case, see Exercise 6 in [Cp86]!
27

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Version 4 - Submitted to MEMOσ ∈ S λ,i for a unique i =: i(σ)]. Also setΓI =\σΓL(λ)i(σ) σ−1 ,σ∈Iand let GI , resp. G, denote the algebraic monodromy group of ρλ |ΓI , resp. ρλ . To establish the “⊆”(up to density zero) direction, we must show that XI has density zero for every non-empty I. For all
I, GI contains G0 , the identity component of G. We apply Rajan’s Theorem (3.4.8 above) to ρλ |ΓI :if XI has positive density, then there is a full connected component T G0 ⊂ GI on which the trace
vanishes (here T is some coset representative for the component). Representing endomorphismsof Vλ in block-matrix form corresponding to the decomposition
M
Vλ |ρ−1=
σWλ,i ,0λ (G )i,σ

we have   ∗ 0 · · · 0   0 ∗ · · · 0  
tr T ·  .. .. . .
 = 0,  . .
. 0 
 

0 0 ··· ∗where the ∗’s represent arbitrary elements of the End(σWλ,i ). For this, we use the fact that these
constituents are (absolutely) irreducible and distinct: either apply Wedderburn theory to the semi0
simple Eλ -algebra Eλ [[ρλ (ρ−1λ (G ))]], or apply Schur’s lemma (using absolute irreducibility) and an
algebra version of Goursat’s lemma (using multiplicity-freeness). Then the above matrix equation
implies the automorphism T , written in block-matrix form, has all zeros along the (block)-diagonal.
But I is non-empty, so T ∈ GI preserves at least one subspace σWλ,i , and so its block-diagonal
entry corresponding to that sub-space must be non-zero (invertible). This contradiction forces
all components of GI to have non-zero trace (generically), for all non-empty I, and thus the “⊆”direction is established.
Part 2 now follows from independence of λ of tr(ρλ (frv )) and Čebotarev, noting that[ [{v : frv ∈Γσ(L(λ)i ) }iσ∈S λ,iis the set of places v of F that have at least one split factor in some L(λ)i .

We make a few remarks about the limitations of this method:
Remark 3.4.10.
(1) Without any information about how the various Vλ decompose into
irreducible sub-representations, this result yields frustratingly little, since it is easy to finddisjoint collections of number fields {Li }i∈I and {Li0 }i∈I 0 such that the union of primes witha split factor (or even split) in the various Li equals the union of those with a split factor√
in the various Li0 . For, the simplest example, take L1 = Q, L10 = Q(i), L20 = Q( 2),√
L30 = Q( −2).
(2) Even when the Vλ are irreducible, and even assuming that one field L(λ0 ) is Galois, caremust be taken when the other inducing fields L(λ) are not (known to be) Galois. We seethat L(λ0 ) is contained in L(λ) for all λ, but equality does not follow, as the following
example from finite group theory shows. We want an inclusion H ≤ K C G of groups
with K normal in G, H a proper subgroup of K, and ∪g∈G gHg−1 = K. Taking K C S 4 tobe the copy of the Klein four-group given by the (2, 2)-cycles, and H to be the subgroup91

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generated by one of these permutations, meets the requirements. Note that such examples
require K to have non-trivial outer automorphism group: if G-conjugation acts by Kinner automorphisms on K, then ∪G gHg−1 = ∪K kH k −1 = K implies H = K, since nofinite group is the union of conjugates of a proper subgroup.
(3) The group-theoretic counterexample of the previous item should not arise in practice: if
we assume that Wλ0 can be put in a compatible-system, say with λ-adic realization Wλ,0 ,then conjecturally Wλ,0 will be Lie irreducible as well, and then the isomorphism
IndFL(λ) (Wλ )  IndFL(λ0 ) (Wλ,0 )
implies, by Mackey theory, that there is a non-zero ΓL(λ0 ) -morphism
0)
IndL(λs(L(λ)) (sWλ )  Wλ,0for some s ∈ ΓF . By part (2) of Lemma 3.4.6, this induction is irreducible, so this map is
an isomorphism. But Wλ,0 is (conjecturally) Lie-irreducible, so L(λ0 ) = L(λ).
In any case, the following corollary is the promised converse to Rajan’s result:Corollary 3.4.11. An irreducible, LMF representations ρ` : ΓF → GLn (Q` ) is Lie irreducible
precisely when the set of v with tr(ρ` (frv )) = 0 has density-zero. In particular, in a compatible
system of irreducible, LMF representations, Lie-irreducibility is independent of λ.
Remark 3.4.12. If we know general automorphic base-change for GLn , we can formulate theconditions ‘Lie irreducible’ and ‘Lie multiplicity free’ on the automorphic side. This result thensuggests how to tell whether a ‘LMF’ automorphic representation is automorphically induced.
Finding an intrinsic characterization of the image of automorphic induction, even conjecturally,
is a mystery (in contrast to its close cousin base-change), so it may come as a surprise that there
should be such a simple condition at the level of Satake parameters, for this broad class of LMF
representations.
I originally developed Proposition 3.4.9 to prove that a regular compatible system of representations of ΓF for F a CM field, if induced, is necessarily induced from a CM field. See Remark 2.4.9
for an automorphic analogue. Here is a partial result; it is another application of the ’Hodge-theory
with coefficients’ in §2.4.1. For the most natural formulation of the result, recall the definition of
purity of a weakly compatible system (Definition 1.2.4).
Lemma 3.4.13. Let F be a CM field, and let R = {ρλ : ΓF → GLn (Eλ )} be a weakly compatible
system of λ-adic representations with coefficients in a number field E. Assume that the ρλ are(almost all) Hodge-Tate regular, and pure. Finally, assume that there is a single number field Lsuch that for λ above a set of rational primes ` of density one,ρλ  IndFL (rλ ),for some Eλ -representation rλ of ΓL . Then L is CM.
Proof. We may assume E/Q is Galois. In yet another variant of the theme of §2.4, purity
implies we may take the number field E to be CM: simply observe that for any choice c of complex
conjugation in Gal(E/Q), the characteristic polynomials Qv (X) of ρλ (frv ) satisfyc

Qv (X) = X n Qv (qwv /X)/Qv (0),92

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and thus for any two complex conjugations c, c0 , cc R  R. It follows that for a density one set ofv, Qv (X) has coefficients in Ecm . The Čebotarev density theorem then yields a well-defined weakly
compatible system (in the sense of Definition 1.2.3) of λ-adic representations with coefficientsin Ecm . (This argument is taken from [PT15, Lemma 1.1, 1.2].29). Thus we take E to be CM.
By regularity, we may also assume that there is a CM extension E 0 /E such that for all finite-index
subgroups H of ΓF , and all primes λ of E 0 , all sub-representations r ⊂ ρλ |H are actually defined overEλ0 : this is an elementary argument (see [BLGGT14, Lemma 5.3.1(3)], with the CM refinement of[PT15, Lemma 1.4]), the key idea being that regularity gives us an abundant supply of elementsin the image of ρλ with distinct eigenvalues, and that ρλ can then be defined over the extension of
E generated by these eigenvalues. Thus, after enlarging E, we may assume all ρλ and rλ act on
Eλ -vector spaces.
For simplicity enlarge E to contain Lcm , the maximal CM subfield of L, and take its Galois
closure–the result remains CM. If L , Lcm , then we can find a (positive-density) set of ` (unramified
in L and for the system ρλ ) which are split in E (with, say, λ|`) but not in (the Galois closure of L,
hence) L. Consider a non-split prime w|` of L, above a place v of F, and the restrictionrλ |ΓLw : ΓLw → GLn (Eλ ) = GLn (Q` );By Lemma 2.2.9, DdR (ρλ |ΓFv ) is the image under the forgetful functor (from filtered Lw -vector
spaces to filtered Fv -vector spaces) of DdR (rλ |ΓLw ). Since Lw does not embed in Eλ = Q` , we can
invoke Corollary 2.4.3 to show ρλ is not regular, a contradiction. Therefore L = Lcm .
Combining this with Proposition 3.4.9, since regular clearly implies LMF,30 we deduce:
Corollary 3.4.14. Let F be a CM field, and let R = {ρλ }λ be an absolutely irreducible, pure,
Hodge-Tate regular weakly compatible system of representations of ΓF . Suppose that when wewriteρλ  IndFL(λ) (Wλ ),where Wλ is Lie irreducible, the extensions L(λ)/F are Galois. Then the field L = L(λ) is independent of λ, and L is itself CM.
Remark 3.4.15. One way of interpreting this result is that to study regular motives, compatible
systems, or algebraic automorphic representations over CM fields, we will never have to leave the
comfort of CM fields; essentially all progress in the study of automorphic Galois representations
is currently restricted to this context. This is in particular the case for Galois representations
occurring as irreducible sub-quotients of the cohomology of a Shimura variety.

29

Our definition of compatible system is somewhat weaker than that given in [PT15]; for our purposes, we canjust ignore the part of the proof of [PT15, Lemma 1.1] that computes Hodge numbers.
30This is the one case in which the LMF condition is provably independent of `.
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Version 4 - Submitted to MEMOCHAPTER 4

Motivic lifting
As noted in the introduction (see Question 1.1.9), the results of Part 3 raise more questions
than they resolve. In this chapter, we discuss some cases of the motivic analogue of Conrad’s
lifting question; this is also the natural framework for the problem of finding compatible lifts of acompatible system of Galois representations. Ideally, we would be able to work in Grothendieck’s
category of pure motives, defined using the relation of homological equivalence on algebraic cycles(see §4.1.2 for a brief review). This category can only be proven to have the desirable categoricalproperties–namely, equivalence to the category of representations of some pro-reductive (‘motivic
Galois’) group–if we assume Grothendieck’s Standard Conjectures on algebraic cycles, which are
far out of reach. (For the basic formalism (as relevant for the theory of motives) of algebraic cycles
and precise statement of the Standard Conjectures, see [Kle68].) We therefore need an unconditional variant of the motivic Galois formalism, and we adopt André’s approach, using his theory of
motivated cycles ([And96b]). In §4.1 we review André’s theory, prove some supplementary results
needed for the application to motivic lifting, and then treat the motivic lifting problem in the potentially abelian case. In §4.2 we prove an arithmetic refinement of André’s work ([And96a]) on,
roughly speaking, the motivated theory of hyperkähler varieties. This provides a motivic analogueof Theorem 3.2.10 in many non-abelian examples. Finally, in §4.3, we speculate on a generalized
Kuga-Satake construction, of which the results of §4.2 are the ‘classical’ case; we then prove thisfor H 2 of an abelian variety, generalizing known results for abelian surfaces.

4.1. Motivated cycles: generalities
4.1.1. Lifting Hodge structures. In trying to produce a motivic analogue either of Wintenberger’s or of my lifting theorem, one is naturally led to try to lift all cohomological, rather thanmerely the `-adic, realizations of a ‘motive.’ The easiest such lifting problem is for real Hodgestructures, which are parametrized by representations of the Deligne torus S = ResC/R (Gm ). Recall
that X • (S) = Zα1 ⊕ Zα2 , where α1 and α2 are the first and second projections in the isomorphism∼

S(C) = (C ⊗R C)× →− C× × C×z1 ⊗ z2 7→ (z1 z2 , z1 z2 ).
Here S(R) ⊂ S(C) is C× = (C ⊗R R)× , and the Gal(C/R)-action, induced by z1 ⊗ z2 7→ z1 ⊗ z2 , isgiven by c : (w, z) 7→ (z̄, w̄). In particular,(c · α1 )(w, z) = c(α1 (z̄, w̄)) = z,i.e. (c · α1 ) = α2 , and similarly (c · α2 ) = α1 . The group of characters over R, denoted XR• (S), isthen Z(α1 + α2 ), where α1 + α2 = N is the norm, satisfying N(z) = zz̄ on R-points.
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Version 4 - Submitted to MEMOe0 → H0 be a surjection of linear algebraic groups over R with kernel equal to a central
Let H
torus Z0 . We are interested in the lifting problem for algebraic representations over R:
?

e0
H

h̃

π



S

h

/ H0 .

Any such h lands in some (typically non-split) maximal torus T 0 , and any lift will land in thee0 . We are therefore reduced to studying thepreimage π−1 (T 0 ) =: T 00 , which is a maximal torus of H
dual diagram of free Z-modules with Gal(C/R) = ΓR -action:
0O
X • (Z)O 0
X • (T 00 )Oz

X • (S) o

X • (T)
O 0
0

As short-hand, we denote the vertically-aligned character groups, from bottom to top, by Y, Y 0 ,and L, so we in fact are studying the sequence
0 → HomΓR (L, X • (S)) → HomΓR (Y 0 , X • (S)) → HomΓR (Y, X • (S)) → Ext1ΓR (L, X • (S)) → . . .
Any real torus is isomorphic to a product of copies of Gm , S, and S1 = ker(N : S → Gm ). The
character group X • (S1 ) is X • (S)/Z(α1 + α2 ). A generator is the image of α1 − α2 , which on R-points
is simply the character z 7→ z/z̄ of the (analytic) unit circle S1 . Complex conjugation acts as −1 on
X • (S1 ). We can therefore completely address the lifting problem by understanding morphisms andextensions between these three basic Z[ΓR ]-modules. The case of immediate interest to us will bewhen Z0 is split, so L is just some number of copies of Z with trivial ΓR -action. Now,
Ext1ΓR (Z, X • (S)) = H 1 (ΓR , X • (S)) = 0,
since ker(1+c) = im(c−1) = Z(α1 −α2 ). Therefore any h : S → H0 lifts, and the ambiguity in lifting
is a collection of elements (of order equal to the rank of Z0 ) of HomΓR (Z, X • (S))  Z(α1 + α2 ).1
Note that if we dealt with representations of S1 we would find an obstruction in H 1 (ΓR , X • (S1 )) = Z/2Z.

1

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Version 4 - Submitted to MEMOExample 4.1.1. In §4.2.2 we will consider the following setup: VR will be an orthogonal spacewith signature (m − 2, 2)2. Write m = 2n or m = 2n + 1. The lifting problem will be
GSpin(VR )
:

h̃



S

π

/ SO(VR ),hwhere h lands in a maximal anisotropic torus T 0  (S1 )n . We can write
 n

P
M


χi


•
0


X (T 0 ) = Zχi  ⊕ Z(χ0 +
),
2
i=1

where conjugation acts by −1 on each χi , i = 1, . . . , n, and trivially on χ0 . Here the characters χiare conjugate in SO(VC ) to the characters denoted χi in §2.8 (see page 56). The torus T 0 is built
out of copies of SO(2) embedded in SO(m − 2, 2), and these are just the usual characters!
cos(θ) sin(θ)
7→ eiθ .
− sin(θ) cos(θ)
Moreover, ⊕ni=1 Zχi is the submodule X • (T 0 ). A morphism S → T 0 ⊂ SO(VR ) is given in coordinatesby χi 7→ mi (α1 − α2 ), for some integers mi . A lift to a morphism S → GSpin(VR ) then amounts toan extension
Pn
Pn
mi
0
i=1 χiχ0 +
7→ (α1 + α2 ) + i=1 (α1 − α2 ),222P
where 0 is any integer having the same parity as mi . The Clifford norm N is given by thecharacter 2χ0 , so the composition of such a lift with the Clifford norm is 0 (α1 + α2 ). In the K3 (orhyperkähler) examples to be considered in the next section, m1 = 1 and all other mi = 0, so we findthere is a unique lift
GSpin(VR )
:

h̃



S

h

π

/ SO(VR ),where N ◦ h̃ : S → Gm is any odd power of the usual norm S → Gm . The Kuga-Satake theory (see§4.2.2 and following) takes the norm itself, which then gives rise to weight 1 Hodge structures.
4.1.2. Motives for homological equivalence and the Tannakian formalism. Our aim in thissub-section is to review Grothendieck’s construction of the category Mhomof pure motives for hoF
mological equivalence over a field F. As a preliminary, we provide some background on the theoryof neutral Tannakian categories. We can then describe the output of the Standard Conjectures, taking for simplicity F to be an abstract (i.e., not embedded) field of characteristic zero, small enoughto be embedded in C: Mhomis (conjecturally) a graded, semi-simple, Q-linear neutral Tannakian
F
category. With the exception of §4.3.2, in the rest of this paper we will not work directly withMhom
F , so much of this discussion serves only to orient the reader. For more background, the reader
should consult [And04], especially Chapter 4, or [Sch94].
2This part of the discussion applies to any signature (p, q) with at least one of p or q even, so that SO(VR ) has acompact maximal torus.
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4.1.2.1. Neutral Tannakian categories. We begin with an overview of the theory of neutral
Tannakian categories; [DM11] is a very readable and thorough introduction to which we refer
the reader for details (the original source is [SR72]). Note that we will always work with neutral
Tannakian categories, which simplifies the theory considerably; for deeper aspects in the nonneutral case, see [Del90]. Let E be a field. The prototypical neutral Tannakian category is thecategory VecE of finite-dimensional vector spaces over E. VecE is an abelian category with anotion of tensor product (namely, the usual tensor product of E-vector spaces) satisfying certain
natural requirements: associativity, commutativity, and existence of a unit (E regarded as an Evector space) for the tensor product. Moreover, every object of VecE has a dual (more generally,
internal Hom’s exist), and the endomorphisms of the unit object are just the field E itself. The
main theorem of the theory in this context is the trivial observation that VecE is equivalent to thecategory of finite-dimensional representations (over E) of the trivial group.
The general definition merely formalizes the notions in the previous paragraph; we will quickly
make the necessary definitions, and then give a number of examples (Example 4.1.6 below).
Definition 4.1.2. Let C be a category, and let ⊗ : C × C → C be a functor satisfying the usualaxioms of a symmetric monoidal category, namely:(1) (Associativity constraint) There is a functorial isomorphism
AX,Y,Z : X ⊗ (Y ⊗ Z) → (X ⊗ Y) ⊗ Z
satisfying the pentagon axiom ([DM11, 1.0.1]).
(2) (Commutativity constraint) There is a functorial isomorphism
C X,Y : X ⊗ Y → Y ⊗ Xsuch that CY,X ◦ C X,Y = idX⊗Y , and that is compatible with the associativity constraint in
the sense of the hexagon axiom ([DM11, 1.0.2]).
(3) (Unit object) There is a pair (1, e) consisting of an object 1 and an isomorphism e : 1 →1 ⊗ 1 such that the functor C → C given by X 7→ 1 ⊗ X is an equivalence of categories.
We call such a (C, ⊗), equipped with its associativity and commutativity constraints (but omitted
from the notation), a tensor category, for short.
If (C, ⊗) and (C0 , ⊗0 ) are tensor categories (whose associativity and commutativity constraints
we will write as A, C and A0 , C 0 , respectively; unit elements will be (1, e) and (10 , e0 )), then a tensor
functor from (C, ⊗) to (C0 , ⊗0 ) is a pair (F, k) consisting of a functor F : C → C and a functorial
isomorphism kX,Y : F(X) ⊗ F(Y) → F(X ⊗ Y) satisfying the following three compatibilities:(1) For all objects X, Y, Z of C, the diagram
FX ⊗ (FY ⊗ FZ)


id ⊗k

/ FX ⊗ F(Y ⊗ Z)

k

k⊗id

/ F(X ⊗ Y) ⊗ FZ

kA0

(FX ⊗ FY) ⊗ FZ

/ F(X ⊗ (Y ⊗ Z))
F(A)/ F((X ⊗ Y) ⊗ Z)

is commutative.
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(2) For all objects X, Y of C, the diagram
FX ⊗ FYk/ F(X ⊗ Y)

C0

FY ⊗ FX

k



F(C)

/ F(Y ⊗ X)

is commutative.
(3) If (1, e) is a unit object of C, then (F(1), F(e)) is a unit object of C0 .
We have spelled out the precise conditions for (F, k) to be a tensor functor because of its relevance for the structure of Mhom
F : the Künneth Standard Conjecture essentially ‘corrects’ the factthat the Betti realization
HB : Mhom→ VecQF∼− HB (X × Y) does not intertwine thewith the natural Künneth isomorphism kX,Y HB (X) ⊗ HB (Y) →(obvious) commutativity constraints on Mhom
and
Vec
.
See
the discussion surrounding ConjecQF
ture 4.1.8 for details.
∼∼
For any unit object (1, e), we obtain isomorphisms lX : 1 ⊗ X →− X and rX : X ⊗ 1 →− X.
For any additive tensor category (C, ⊗) (for which we require ⊗ to be bi-additive), and any
unit object (1, e), EndC (1) = R is a commutative ring, unique up to unique isomorphism, and C isnaturally an R-linear category.
Definition 4.1.3. Let (C, ⊗) be a tensor category. It is rigid if for every object X there is a‘dual’ object X ∨ along with evaluation and co-evaluation morphismsevX

X ∨ ⊗ X −−→ 1coevX

1 −−−−→ X ⊗ X ∨such that the composites (we suppress the unit isomorphisms and the associators)
coevX ⊗idX

idX ⊗evX

X −−−−−−−→ X ⊗ X ∨ ⊗ X −−−−−−→ X
idX ∨ ⊗coevX

evX ⊗idX ∨X ∨ −−−−−−−−→ X ∨ ⊗ X ⊗ X ∨ −−−−−−→ X ∨are idX and idX∨ , respectively.
Note that in the absence of a commutativity constraint, X ∨ is what would be called a ‘rightdual.’
Finally, we come to the main definition:
Definition 4.1.4. Let E be a field. A neutral Tannakian category over E is a rigid abelian tensor
category (C, ⊗) such that End(1) = E, and for which there exists a faithful, exact, E-linear tensor
functor ω : C → VecE . Such an ω is called a fiber functor.
The main theorem of the theory of neutral Tannakian categories just says that every Tannakian
category is equivalent to the category of representations of some affine group scheme:
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Theorem 4.1.5 ([DM11, Theorem 2.11]). Let (C, ⊗) be a neutral Tannakian category over E,
equipped with a fiber functor ω : C → VecE . Then the functor of E-algebras given by tensorautomorphisms of ω (see [Del90, 1.9, 1.11]) is represented by an affine group scheme G, and thefunctor C → RepE (G) defined by ω is an equivalence of tensor categories.3
Informally, we might call G the ‘Galois group’ of (C, ⊗), whence the later terminology ‘motivic
Galois group’ in the (conjecturally Tannakian) case of MhomF .
At last, some examples:
Example 4.1.6.
(1) The following example is crucial for the theory of motives. The category VecZ/2of
Z/2Z-graded
vector spaces over E, with the usual graded tensor product,E∼
and with commutativity constraint given by the Koszul sign rule, i.e. CV,W : V ⊗ W →−W ⊗ V given on homogeneous tensors by v ⊗ w 7→ (−1)deg(v)·deg(w) w ⊗ v, is a rigid E-linear
abelian tensor category. Note, however, that the forgetful functor VecZ/2Z→ VecE (withE
the obvious isomorphisms kX,Y ) is not a tensor functor, because it fails to intertwine thecommutativity constraints. Indeed, VecZ/2Z
is not a Tannakian category. Any rigid tensor
Ecategory has an intrinsic notion of rank for every object X, given by the element of End(1)arising as the composition
coevX

C X,X ∨evX

1 −−−−→ X ⊗ X ∨ −−−−→ X ∨ ⊗ X −−→ 1.
For a Z/2Z-graded vector space V = V0 ⊕ V1 , it is easy to check that the rank in VecZ/2Z
isE
dim V0 −dim V1 . But this notion of rank is preserved by any tensor functor, so an obviously
necessary condition to admit a fiber functor is that all objects have non-negative rank. (Infact, a deep result of Deligne gives a converse: [Del90, 7.1 Théorème].)(2) The category Q-HSpol of pure polarizable Q-Hodge structures is a neutral Tannakian category over Q, with fiber functor just given by taking the underlying Q-vector space ofa Hodge structure (here we take the commutativity constraint on Hodge structures to be
the same as the usual constraint on vector spaces). This is a useful ‘toy model’ for thetheory of pure motives over C. In particular, Theorem 4.1.5 identifies Q-HSpol with thecategory of representations of some affine group scheme MT over Q; MT is the ‘universal Mumford-Tate group.’ We use this example to explain how properties of a Tannakian category reflect those of its Galois group (see [DM11, §2 ‘Properties of G andRep(G)’]). From its Hodge-theoretic description, MT is obviously connected. This isequivalent ([DM11, Corollary 2.22]) to the condition that for every non-trivial representation X of MT, the strictly full subcategory of Q-HSpol whose objects are isomorphic
to sub-quotients of some X ⊕n is not stable under ⊗. But if X is not the trivial Hodge
structure Q (i.e. if X corresponds to a non-trivial representation of MT), then either it
has non-zero weight, in which case the result is clear, or some H p,−p (X) is non-zero, forsome p , 0. Then H mp,−mp (X ⊗m ) , 0, which for m large enough cannot be the case forany sub-quotient of any X ⊕n . In a similar spirit, MT is pro-reductive, since the category
Q-HSpol is semi-simple (by polarizability; see [DM11, Proposition 2.23]).
3

That is, it is a tensor functor that is also an equivalence of categories; this ensures ([DM11, Proposition 1.11])
that there exists an inverse of F that is also a tensor functor.
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4.1.2.2. Homological motives. We begin by sketching the construction of Grothendieck’s category of (pure) homological motives over a field F. References for more details are [Kle68, §1]and [Sch94, §1]. Recall that for simplicity we take F to be a field that can be embedded in C (oreven a specified subfield of C)–this does not affect the construction, but does affect whether the
category of motives is (conjecturally) neutral Tannakian. Recall that for a fixed Weil cohomology
theory H ∗ (see [Kle68, §1.2]), homological equivalence defines an adequate equivalence relationon algebraic cycles on smooth projective varieties; for example, since F ⊂ C, we will always
have at our disposal Betti cohomology, HB∗ . For a smooth projective X/F, we let A∗hom (X) denote
the Q-algebra of (Q-linear combinations of) algebraic cycles for homological equivalence, gradedby codimension. ‘Adequate’ ensures that the intersection product is well-defined as a linear map
Ar (X) ⊗ A s (X) → Ar+s (X). For any two smooth projective F-varieties X and Y, with X connectedof (equi-)dimension d, we can then define the space of degree r correspondencesr
Chom(X, Y) = Ad+r
hom (X × Y).

We obtain a composition of correspondences
r
s
r+s
Chom
(X, Y) ⊗ Chom
(Y, Z) → Chom
(X, Z)

given byf ⊗ g = g ◦ f 7→ p13,∗ p∗12 f · p∗23 g ,where the pi j are the projections from X × Y × Z to the products of two factors. In particular,0
Chom(X, X) is a Q-algebra.
Definition 4.1.7. The category Mhomof motives over F for homological equivalence has asF
objects triples (X, p, m), where X is a smooth projective variety over F, m is an integer, and p is an0
idempotent correspondence in Chom
(X, X). Morphisms in Mhom
are defined byFn−m
HomMhom((X, p, m), (Y, q, n)) = qChom(X, Y)p.
F

Mhomis an additive, Q-linear, pseudo-abelian category ([Sch94, Theorem 1.6]). There is aFbi-additive functor ⊗ : Mhom× Mhom→ Mhomgiven on objects by
FFF(X, p, m) ⊗ (Y, q, n) = (X × Y, p ⊗ q, m + n)(the fiber product is over F; we do not here specify ⊗ on morphisms). This satisfies associativity
and commutativity constraints induced by the tautological isomorphisms (X × Y) × Z  X × (Y × Z)
and C : X × Y  Y × X. A unit object for ⊗ is given by (Spec F, id, 0), and we can define dual
objects by (again for simplicity take X of equi-dimension d)(X, p, m)∨ = (X, t p, d − m);
In sum, these structures make (Mhom(1) =F , ⊗) into a rigid Q-linear tensor category with EndMhom
Fhom
Q. Grothendieck’s Standard Conjectures anticipate that moreover MF should be a semi-simple
neutral Tannakian category. One must prove that
• Mhomis abelian;F• the abelian category Mhom
is moreover semi-simple;
F
hom• MF possesses a fiber functor to VecQ .
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Version 4 - Submitted to MEMOWe begin by discussing the question of a fiber functor. The answer is simple: (MhomF , ⊗), with thecommutativity constraint defined above, does not have a fiber functor! For simplicity, let us takeF ⊂ C, so that there is a Betti realization
HB : Mhom→ VecQ .
F
HB is not a tensor functor, however: the diagram/ HB (X × Y)k

HB (X) ⊗ HB (Y) ∼


C0



HB (C)/ HB (Y × X),k

HB (Y) ⊗ HB (X) ∼given by applying the naı̈ve commutativity constraints and the Künneth isomorphism, only commutes up to sign, since cup-product is anti-commutative. More precisely, HB gives a tensor functor
valued in VecZ/2Z, and since objects of Mhom
can then have negative rank, it cannot be neutral
F
Q∼

Tannakian with the commutativity constraint induced by X × Y →− Y × X. The Künneth Standard Conjecture would give Mhomagrading(corresponding
tocohomologicaldegree), allowingFhom
for a modified commutativity constraint, for which HB : MF → VecQ is a tensor functor. Moreprecisely, for any Weil cohomology theory H ∗ , the isomorphism (again taking d = dim X)
M
M
M
H 2d (X × X)(d) H 2d−i (X) ⊗ H i (X)(d) H i (X)∨ ⊗ H i (X) 
End(H i (X))
iiigives a cohomology class πiX corresponding to the composition H ∗ (X)  H i (X) ,→ H ∗ (X).
Conjecture 4.1.8. For all i = 0, 1, . . . , 2d, the cohomology class πiX is algebraic, i.e. lies in theimage of the cycle class map Adhom (X × X) → H 2d (X × X)(d).
The modified commutativity constraint on Mhom
would then be given as follows: if the originalF∼
constraint is C : M ⊗ N →− N ⊗ M, we can decompose C = ⊕r,sC r,s with∼

C r,s : πr M ⊗ π s N →− π s N ⊗ πr M,
and then the correct commutativity constraint is C 0 = ⊕r,s (−1)rsC r,s .
The Künneth Standard Conjecture is in fact implied by a stronger conjecture, the Lefschetz
Standard Conjecture, which would also show that the primitive decomposition of cohomology’
makes sense in the category Mhom
F . To state this, let X be a smooth projective variety over F, andfix an ample line bundle η on X, giving rise to the Lefschetz operator
L = Lη,H∗ : H i (X)(r) → H i+2 (X)(r + 1),
and the hard Lefschetz isomorphisms∼

Ld−i : H i (X)(r) →− H 2d−i (X)(d − i + r)
for all i ≤ d. As always with taking cup-product with the class of an algebraic cycle, these isomorphisms are given by algebraic correspondences.
Conjecture 4.1.9 (Lefschetz Standard Conjecture). For all i ≤ d, the inverse of Ld−i is given
by algebraic correspondences.
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Note that a priori both the Künneth and Lefschetz Standard Conjectures depend on the choiceof Weil cohomology. For the classical cohomologies (Betti, étale, de Rham) that are related bycomparison isomorphisms (respecting cycle class maps), these conjectures do not depend on the
choice of H ∗ .
Now we move to the question of whether Mhom
is abelian. Here there is a marvelous theorem
Fof Jannsen:
Theorem 4.1.10 ( [Jan92, Theorem 1] ). For an adequate equivalence relation ∼, the categoryof motives for ∼-equivalence is abelian semi-simple if and only if ∼ is numerical equivalence. In
particular, Mhom
is abelian if and only if numerical and homological equivalence coincide.
F
Indeed, long before Jannsen proved his theorem, Grothendieck conjectured:
Conjecture 4.1.11. Homological and numerical equivalence coincide.
In summary, under the Standard Conjectures, we can equip Mhom
with a commutativity conF
straint for which it is a semi-simple neutral Tannakian category, hence:
Conjecture 4.1.12. Let F be a subfield of C. There is a pro-reductive Q-group Ghom
F , themotivic Galois group for pure motives over F, and an equivalence (induced by the Betti fiber
functor HB∗ : Mhom
→ VecQ )F
Mhom RepQ (GhomFF ).
4.1.3. Motivated cycles. In the present Part 4 of these notes, most of our results can be statedsolely in terms of abelian varieties, but both the strongest assertions and the proofs will require invoking some version of the motivic Galois formalism; in the absence of the Standard Conjectures,
we use André’s theory of motives for motivated cycles ([And96b]). In this subsection, we provide
a brief review, elaborating on some points for later application. Although André’s theory is developed over any field F, to simplify we will always take F to be an abstract (i.e., not embedded) fieldof characteristic zero, small enough to be embedded in C, and which we will eventually specify to
be a number field.
In [And96b], André defines a Q-linear category of motives for ‘motivated cycles’ whose construction mirrors the classical construction of (Grothendieck) motives for homological equivalence,but circumvents the standard conjectures by formally enlarging the group of ‘cycles’ to include theLefschetz involutions. Here is the precise definition of the space of motivated cycles:
Definition 4.1.13. Fix a reference Weil cohomology H • . A motivated cycle on X with coefficients in E is an element of H • (X) of the form
prXYX,∗ (α ∪ ∗L β),
where• Y is a smooth projective F-scheme, with polarization ηY giving rise to a ‘product’ polarization ηX×Y = [X] ⊗ ηY + ηX ⊗ [Y], with corresponding Lefschetz involution ∗L , on
X × Y;• α and β are algebraic cycles mod H • -homological equivalence with E-coefficients on
X × Y.
We denote by A•mot (X)E the E-vector space of motivated cycles on X with E-coefficients. A•mot (X)will always mean the case E = Q.
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Version 4 - Submitted to MEMO•As in §4.1.2.2, we can then define spaces Cmot
(X, Y)E of motivated correspondences (see also
[And96b, Définition 2]); for X = Y, this construction yields a graded E-algebra containing theLefschetz involutions and the Künneth projectors πiX .
The first main result of this theory asserts that the spaces of motivated cycles in a precise sense
do not depend on the Weil cohomology H • used to define them:

Theorem 4.1.14 (Théorème 0.3 of [And96b]). For any smooth projective F-scheme X, letA•mot (X) be the graded Q-algebra of ‘motivated cycles,’ constructed with respect to a fixed Weilcohomology H • . A•mot (X) contains the classes of algebraic cycles modulo homological equivalence,and there is a Q-linear injection
clH : A•mot (X) → H 2• (X)extending the cycle class map for H. A•mot (X) has the following properties:• A•mot (X) depends bifunctorially (push-forward and pull-back) on X, satisfying the usual
projection formula.
• (See [And96b, §2.3]) Two Weil cohomologies related by comparison isomorphisms yield
canonically and functorially isomorphic algebras A•mot (X) of motivated cycles. In particular, all the classical cohomology theories yield the same A•mot (X).
From here, André defines a category MF of motives for motivated cycles exactly as in Definition 4.1.7, replacing algebraic correspondences with motivated correspondences.4 André shows theendomorphism algebras of objects of MF are semi-simple (finite-dimensional) Q-algebras, fromwhich it follows (see [Jan92]) that MF is an abelian semi-simple category. Since the Künneth
projectors πiX are motivated cycles, we can define the modified commutativity constraint describedin §4.1.2.2, thereby making MF into a neutral Tannakian category over Q:
Theorem 4.1.15 (Théorème 0.4 of [And96b]). MF is a neutral Tannakian category over Q. It
is graded, semi-simple, and polarized. Every classical cohomology factors through MF .
Remark 4.1.16. We will often use the short-hand H(X) for the object (X, id, 0), and H i (X) for(X, πiX , 0) (recall the notation from Definition 4.1.7). It will be clear from context when H(X) refers
to an object of MF , and when it refers to the output of any particular Weil cohomology theory H ∗ .
In particular, this sets in motion the formalism of motivic Galois groups, and the theory becomes a very useful circumvention of the standard conjectures. Perhaps its most unsatisfactoryfeature–present also in the theory of absolute Hodge cycles–is that its ‘motives’ are not known togive rise to compatible systems of `-adic representations.
Now we specify fiber functors and no longer regard F as an abstract field. For any σ : F ,→C, MF is Tannakian and neutralized by the σ-Betti fiber functor (denoted Hσ ), so we obtain its
Tannakian group, ‘the’ motivic Galois group, GF (σ). For the `-adic fiber functor X 7→ Hét (XF , Q` ),we denote by GF,` the corresponding motivic group (over Q` ). It is more convenient to choose anembedding σ : F ,→ C, since this allows us, via the comparison isomorphisms∼

Hét (XF , Q` ) −−→Hét (XF ⊗F,σ C, Q` )  Hσ (X, Q) ⊗Q Q` ,∗σ

4André defines more generally a category of motives ‘modeled on’ a full sub-category V of the category of smoothprojective F-schemes, with V assumed stable under products, disjoint union, and passage to connected components.
This amounts to restricting the auxiliary varieties Y permitted in Definition 4.1.13. We will always take V to be allsmooth projective F-schemes.
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Version 4 - Submitted to MEMOto deduce an isomorphism GF (σ) ⊗Q Q`  GF,` . Eventually, F will simply be regarded as a subfieldof C, with F its algebraic closure in C; in that case, we will omit the σ from the notation. For now,
however, we retain it.
As a variant on this formalism, to any object (or collection of objects) M of MF , we canassociate the smallest Tannakian subcategory hMi⊗ generated by M (fully faithfully embeddedin MF ), and we can then look at its Tannakian group, denoted GFM (σ). A particularly important
example comes from taking M to range over (the objects of MF associated to) all finite étale
F-schemes; this defines the sub-category of Artin motives. The fully faithful inclusion of thesubcategory Mart
F of Artin motives over F induces a surjection
GF (σ)  Gart
F (σ)  ΓF ,and when combined with the base-change functor MF → MF , we obtain an exact sequence5 ofpro-algebraic groups
1 → GF (σ) → GF (σ) → ΓF → 1.
By Proposition 6.23d of [DM11], for all ` there are continuous sections (homomorphisms) ΓF →GF (Q` ) (after unwinding everything, this is simply the statement that ΓF acts on `-adic cohomology).
We can similarly compare the ‘arithmetic’ and ‘geometric’ versions of the motivic Galois groupof any object M of MF : there is a commutative diagram, where the vertical morphisms are surjective:1/ G MF (σ)
F O/ G M (σ)
FO

1

/ G (σ)/ GF (σ)

F

/ 1.

/ ΓF

M

Lemma 4.1.17. Assume that GF F (σ) is connected. Then there exists a finite extension F 0 /F

such that GFM0F0 (σ) is connected.

Proof. Fix a prime `, a section s` : ΓF → GF (Q` ), and a finite extension F 0 /F such that the
Zariski closure of the `-adic representation ρ M` : ΓF 0 → GL(M` ) is connected. The image of
Zar
MF 0
0
G
 FM0 ⊗Q Q` in GL(M` ) is then equal to the product of the two connected groups ρ M` (ΓF ) andGF F ⊗Q Q` , hence is itself connected.
M

Remark 4.1.18. If in hMF i⊗ all Hodge cycles are motivated, GF F (σ) is connected.
The assertion that Hodge cycles are motivated is a (still fiercely difficult) weakened versionof the Hodge conjecture. We now describe what is essentially the only understood case of thisproblem. Many of the results of [And96a] rest on earlier work of André (Théorème 0.6.2 of[And96b]) showing that on a complex abelian variety, all Hodge cycles are motivated (a variant ofDeligne’s result that Hodge cycles on a complex abelian variety are absolutely Hodge). One usefulconsequence is:5

See Proposition 6.23a, c of [DM11]. This is a corrected, TeXed version of the original article in [DMOS82]; it isavailable at http://www.jmilne.org/math/xnotes/index.html. Note that part b is a modification of the (only conjectural)
statement in the original article.
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Corollary 4.1.19 (André). Let A/C be an abelian variety, and let M be the motive H 1 (A) (anobject of MC ). Then the motivic group GCM (for the Betti realization) is equal to the Mumford-Tategroup MT (A), and in particular is connected.
Proof. MT (A), recall, is the smallest Q-sub-group of GL(HB1 (AC , Q)) whose R-points containthe image of the S-representation corresponding to the R-Hodge structure HB1 (AC , R); by the general theory of Mumford-Tate groups, this is equal to• the Tannakian group for the Tannakian category of Q-Hodge structures generated by MB =HB1 (AC , Q); and• the subgroup of GL(MB ) fixing exactly the Hodge tensors in every tensor construction
T m,n (MB ) := (MB )⊗m ⊗ (MB∨ )⊗n .
Similarly, the motivic group GCM is the subgroup of GL(MB ) fixing exactly the motivated cycles inevery tensor construction T m,n (MB ). Of course (on any variety) all motivated cycles are Hodge cycles, so there is a quite general inclusion MT (A) ⊂ GCM . Applying André’s result to all powers of A,
we can deduce the reverse inclusion: if t ∈ T m,n (MB ) is a Hodge cycle, then (by weight considerations) m = n, and viewing this tensor space (via Künneth and polarization) inside H 2m (A2m , Q)(m),we see that t is motivated.

The following point is implicit in the arguments of [And96a], but we make it explicit in partto explain an important foundational point in the theory of motivated cycles.
Corollary 4.1.20. Let F be a subfield of C, with algebraic closure F in C, and let A/F be an
abelian variety. Let M be the object of MF corresponding to H 1 (AF ). Then GFM (for the F ⊂ C
Betti realization) is connected, equal to MT (AC ).
Proof. We deduce this from the previous result and the following general lemma, which is ofcourse implicit in [And96b]:
Lemma 4.1.21. Let L/K be an extension of algebraically closed fields, and let M be an object ofMK , with base-change M|L to L. Then via the canonical isomorphism Hét (M, Q` )  Hét (M|L , Q` ),M|LMthe `-adic motivic groups GK,`and GL,`agree.
Proof. This follows from the fact (2.5 Scolie of [And96b]) that the comparison isomorphism∼(for M = H ∗ (X)) identifies the spaces of motivated cycles A∗ (X) →− A∗ (XL ). This follows from standard spreading out techniques, but we provide some details, since they are omitted from [And96b].
Writing L = limKλ as the directed limit of its finite-type K-sub-algebras allows all the data required−−→
λto define a motivated cycle on XL (a smooth projective Y/L, algebraic cycles on XL × Y, the Lefschetz involution on cohomology of XL × Y) to be descended to some S λ = Spec(Kλ ). The generalmachinery allows us to assume (enlarging λ) that we have Yλ /S λ smooth projective (of course
Xλ = X ⊗K Kλ is smooth projective), a relatively ample invertible sheaf ηλ of ‘product-type’ on
Xλ ×S λ Yλ , and various closed S λ -subschemes
Z ,→ X := Xλ ×S λ Yλthat are smooth over S λ and whose linear combinations define spread-out versions of the algebraic
cycles on XL ×Y. The purity theorem in this context (Théorème 3.7 of Artin’s Exp. XVI of [sga73])
yields an isomorphism∼
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Version 4 - Submitted to MEMOhence a cycle class [Z] in H 2 j (X, Q` )( j). Regarding x̄ : Spec(L) → S λ as a geometric point
over some scheme-theoretic point x, and letting s̄ : Spec(K) → S λ be a geometric point over a∼scheme-theoretic closed point s lying in the closure of x, the cospecialization map H 2 j (X s̄ , Q` ) →−H 2 j (X x̄ , Q` ) is an isomorphism (X/S λ being smooth proper), and it carries the cycle class [Z s̄ ] tothe cycle class [Z x̄ ], since these are both restrictions of [Z], and the diagram
H 27 j (X s̄ , Q` )H 2 j (X, Q` )'

H 2 j (X x̄ , Q` )
commutes.


4.1.4. Motives with coefficients. We will need the flexibility of working with related categories of motives with coefficients. Let E be a field of characteristic zero. Given any E-linear
abelian (or in fact just additive, pseudo-abelian) category M, and any finite extension E 0 /E, we candefine the category ME 0 of objects with coefficients in E 0 as either of the following (this discussion
is taken from §2.1 of [Del79] and 3.11, 3.12 of [DM11]):(1) The category of ‘E 0 -modules in M,’ whose objects are pairs (M, α) of an object M of M
and an embedding α : E 0 → EndM (M), and whose morphisms are those commuting withthese E 0 -structures.
(2) The pseudo-abelian envelope of the category whose objects are formally obtained fromthose of M (writing ME 0 for the object in ME 0 arising from M in M), and whose morphisms are
HomME0 (ME 0 , NE 0 ) = HomM (M, N) ⊗E E 0 .
This construction is valid for infinite-dimensional E 0 /E.
To pass from the first to the second description, let (M, α) be as in (1), so that EndME0 (ME 0 ) =EndM (M) ⊗E E 0 contains, via α, E 0 ⊗E E 0 . This E 0 -algebra (via the left factor) is isomorphic to a
product of fields, and there is a unique projection eid : E 0 ⊗E E 0  E 0 in which x ⊗ 1 and 1 ⊗ x both
map to x. Then eid (ME 0 ) is the object of (2) corresponding to (M, α).
There is a functor M → ME 0 , which in the first language is M 7→ (M ⊗E E 0 , id M ⊗ idE 0 ).
(See 2.11 of [DM11] for a precise description of M ⊗E E 0 .) If M is semi-simple, then so is ME 0 .
If M is a neutral Tannakian category over E with fiber functor ω, then we can make ME 0 intoa neutral Tannakian category over E 0 . Define an E 0 -valued fiber functor ωE 0 : M → VecE 0 by
ωE 0 (M) = ω(M) ⊗E E 0 . There is a diagram commuting up to canonical isomorphism,/ ME 0

M
ωE 0

"



ω0E 0

VecE 0 ,107

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whereω0E 0 (M, α) := ωE 0 (M)⊗α⊗idE 0 ,E 0 ⊗E E 0

E0.

This tensor product (and similar tensor products) means that E 0 ⊗ E 0 acts on E 0 by the usual bimodule structure, and acts on ωE 0 (M) via α on the left-hand copy of E 0 and via E 0 -multiplicationon the right-hand copy (these two actions coincide on E). This ω0E 0 neutralizes ME 0 , and so we
can define the associated Tannakian group GE 0 = Aut⊗ (ω0E 0 ). We wish to compare GE 0 with G =Aut⊗ (ω). Note that ME 0 is equivalent to RepE 0 (G) and G ⊗E E 0 = Aut⊗ (ωE 0 ). Then the compositionof functors
ωF
ME 0 −→ RepE 0 (G) −→ RepE 0 (G ⊗E E 0 )∼∼is given on ME 0 by (F just sends an object V of RepE 0 (G), corresponding to an E-homomorphism
G → ResE 0 /E (GLE 0 (V)), to E 0 ⊗E 0 ⊗E E 0 V)
(M, α) 7→ (ω(M), ω(α)) 7→ E 0⊗

E 0 ⊗E 0 ,ω(α)⊗1(ω(M) ⊗ E 0 )This composite is naturally isomorphic to ω0E 0 , whence a tensor-equivalence∼

RepE 0 (GE 0 ) →− RepE 0 (G ⊗E E 0 ).
We apply these constructions to André’s category of motivated motives over F, to form thevariant MF,E , motives over F with coefficients in E, for E any finite extension of Q. We denote the corresponding motivic Galois group (for the σ-Betti realization) by GF,E (σ); its E-linear(pro-algebraic) representations correspond to objects of MF,E , and it is naturally isomorphic to
M
GF,Q (σ) ⊗Q E. For any object M of MF,E , we also have the corresponding motivic group GF,E(σ).
As a more concrete variant, we can start with the (Q-linear, semi-simple) isogeny category AVF00
of abelian varieties over F, and form the E-linear, semi-simple category AVF,E
of isogeny abelian0varieties over F with complex multiplication by E. There is a (contravariant) functor AVF,E→
MF,E . Faltings’ theorem implies the following E-linear variant: fix an embedding E ,→ Q` , and0
consider an object A of AVF,E; then the natural map
10 (A) ⊗ E Q` → EndEndAVF,EQ` [ΓF ] (H (AF , Q` ) ⊗E Q` )is an isomorphism: simply take the usual isomorphism with Q in place of E, restrict to thoseendomorphisms commuting with E → EndAVF0 (A), and then project to the E ,→ Q` component of
the resulting E ⊗Q Q` -module.
4.1.5. Hodge symmetry in MF,E . Our next goal is to show that objects of MF,E satisfy (unconditionally) the Hodge-Tate weight symmetries, and the more fundamental symmetries in therational de Rham realization, needed for our abstract Galois lifting results. I expect these results
are well-known to experts, but they do not seem to have been explained in their natural degree ofgenerality, and in a context in which they can be proven unconditionally6. So, let M be an objectof MF,E , of rank r. The de Rham realization MdR is a filtered F ⊗Q E-module, which is moreover6

Our discussion is an analogue of the standard discussion in the theory of motives of CM type–see §7 of [Ser94].
The Galois symmetries themselves have also been postulated in §5 of [BLGGT14], but since that paper is only
concerned with Galois representations arising from essentially conjugate self-dual automorphic representations, it has
no need to deal with this symmetry as a general principle.
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free of rank r. As in §2.4.1, we define the τ : F ,→ E-labeled weights of M as follows: HTτ (M) is
an r-tuple of integers h, with h appearing with multiplicity

dimE grh MdR ⊗F⊗Q E,τ⊗1 E .
In talking of τ-labeled weights, there is alway an ambient over-field, in this case E, but we will attimes want to change this to either C or Q` ; that will require fixing an embedding ι : E ,→ Q` orι : E ,→ C. In either case, we can then speak of HTιτ (M), with E embedded into C or Q` via ι, andthere is an obvious equality HTτ (M) = HTιτ (M).
The essential point is that, as in the automorphic analogue Corollary 2.4.7, “motives have CM
coefficients”; compare the assertion in [And96b, §4.6(iii)] that GF splits (i.e., for any algebraic
quotient GF  H over Q, the connected component H 0 has a maximal torus that splits) over Qcm .
For lack of reference for the proof, we give some details:
Lemma 4.1.22. Let N be an object of MF,E . Then there exists an object N0 of MF,Ecm such thatN  N0 ⊗Ecm E. Consequently, HTτ (N) depends only on the restriction of τ to Fcm .
Proof. By the formalism of Lemma 2.4.1, the second claim follows from the first. By Propo0
sition 3.3 (the analogue of the Hodge index theorem) of [And96b], Cmot
(X, X) is endowed, via a
choice of ample line bundle on X, with a positive-definite, Q-valued symmetric form, which wecall h·, ·i. For any sub-object M ⊂ H(X) in MF , there follows a decomposition H(X) = M ⊕ M ⊥ ,by positivity and the fact that M is a semi-simple abelian category. h·, ·i therefore restricts to a
positive form on M itself, and in particular every simple object of MF carries a positive definite
form. Again by semi-simplicity, any such simple object M has End(M) isomorphic to a division
algebra D, on which we now have an involution 0 (transpose with respect to h·, ·i) and a trace form
tr M : D → Q (given by φ 7→ tr(φ|HB (M) ) such that tr M (φφ0 ) > 0 for all non-zero φ ∈ D. As in the
proof of the Albert classification (see [Mum70, §21, Theorem 2]), this implies that the center of
D is either a totally real or a CM field. Thus, End(M) splits over a CM field; in particular, for
any number field E and any factor N of M ⊗Q E (in MF,E ), N can in fact be realized as (the scalar
extension of) an object of MF,Ecm .

Remark 4.1.23. When E = Q, Lemma 2.4.1 shows that HTτ (N) is independent of τ; this isessentially the assertion that for a smooth projective X/F, the Hodge numbers of X do not dependon the choice of embedding F ,→ C. The next few results (culminating in Corollary 4.1.26) all
have corresponding strengthenings when E = Q.
We first record the de Rham-Betti version of the desired symmetry:Lemma 4.1.24. Let N be an object of MF,E lying in the k-component of the grading. Then for
any choice of complex conjugation c in Gal(E/Q),
HTc◦τ (N) = {k − h : h ∈ HTτ (N)}.
Proof. Any such object is built by taking one of the form M = H k (X) ⊗ E, for X/F a smooth0
projective variety and applying an idempotent α ∈ Cmot
(X, X)E .7 Fix an embedding ι : E ,→ C, so
that ι ◦ τ : F ,→ C. There is a functorial (Betti-de Rham) comparison isomorphism
MdR ⊗F,ιτ C  MB,ιτ ⊗Q C,
7

And a Tate twist, but the statement of the lemma is obviously invariant under Tate twists.
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0which commutes with the action of Cmot
(X, X)E on MdR and MB,ιτ , in particular making this an
isomorphism of free E ⊗Q C-modules. It induces an isomorphism on the corresponding gradeds(with q = k − p):
gr p (MdR ⊗F,ιτ C)  Hιτp,q (M)  H p,q (Xιτ ) ⊗Q E,
again E ⊗Q C-linear. Just for orientation amidst the formalism, this says thatHTτ (MdR ) = HTιτ (MdR ) = p : H p,q (Xιτ ) , 0, counted with multiplicity dimC H p,q (Xιτ ) .

Now, the projection Xιτ ×C,c C → Xιτ induces a transfer of structure isomorphism on rational Betti
cohomology, and then an E ⊗Q C-linear isomorphismq,pF∞ : Hιτp,q (M) → Hcιτ(M).
0Let α ∈ Cmot
(X, X)E be an E-linear motivated idempotent correspondence defining an object N =αM of MF,E . There is a commutative diagram of E ⊗Q C-linear morphismsHιτp,q (M)αιτHιτp,q (M)F∞/ H q,p (M)
cιτ

F∞



αcιτ

/ H q,p (M).
cιτConsider the images of the two vertical maps. Since the horizontal maps are isomorphisms, andall the maps are E ⊗Q C-linear, we deduce an isomorphism∼q,pαιτ Hιτp,q (M) ⊗E⊗Q C,ι C →− αcιτ Hcιτ(M) ⊗E⊗Q C,ι C.

The left-hand side is isomorphic to
gr p (αM)dR ⊗F⊗Q E,ιτ⊗ι C ,and letting c0 be a complex conjugation in Gal(E/Q) such that ιc0 τ = cιτ, the right-hand side isisomorphic to
grk−p (αM)dR ⊗F⊗Q E,ιc0 τ⊗ι C .
It follows that
HTc0 τ (αM) = {k − h : h ∈ HTτ (αM)} .
But by the previous lemma, HTc0 τ (αM) = HTcτ (M), so we are done.
Next we observe (as in §2.4 of [And96b], but with a few more details) that ‘p-adic’ comparison
isomorphisms hold unconditionally in MF,E ; this is one of the main reasons for working withmotivated cycles rather than absolute Hodge cycles.
Lemma 4.1.25. Let N be an object of MF,E . Then for all v|`, the free E ⊗Q Q` -module M` isa de Rham representation of ΓFv , and there is a functorial (with respect to morphisms in MF,E )isomorphism of filtered Fv ⊗Q E  Fv ⊗Q` (Q` ⊗Q E)-modules
Fv ⊗F MdR  DdR (M` |ΓFv ).
Proof. It suffices to check for objects of the form M = H(X) ⊗Q E, for X/F smooth projective.
Faltings’ theorem (of which there are now several proofs, including the necessary corrections to
the original [Fal89]; a ‘simple’ recent proof is [Bei12]) provides the comparison


∗
HdR
(XFv ) ⊗Q E  DdR H ∗ (XFv , Q` ) ⊗Q E,
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0which is an isomorphism of filtered Fv ⊗Q E-modules. Let α ∈ Cmot
(X1 , X2 )E be a motivatedcorrespondence. Let d1 = dim(X1 ). The class α has realizations αdR and α` in cohomology groups
that are also compared by Faltings (note that α` is fixed by ΓFv ):



2d1
HdR
((X1 × X2 )Fv )(d1 )) ⊗Q E  DdR H 2d1 ((X1 × X2 )Fv , Q` )(d1 ) ⊗Q E,
and the claim is that αdR maps to α` . The comparison isomorphism is compatible with cycle
class maps (see part (b) of the proof of [Bei12, Theorem 3.6]), and our cycle α is spanned by
×X2 ×Yelements of the form prXX11 ×X(β ∪ ∗γ), with Y/F another smooth projective variety and β and γ
2 ,∗algebraic cycles on X1 × X2 × Y. Applying the comparison isomorphism also to X1 × X2 × Y,
and applying compatibility with cycle classes, cup-product, (hence) Lefschetz involution, and the
2 ×Y
projection prXX12×X–this last point because of compatibility with pull-back induced by morphisms,∗of varieties and, again, Poincaré duality–we can deduce that αdR maps to α` .
Corollary 4.1.26. Let M be an object of MF,E . Fix an embedding ι : E ,→ Q` , and use this to
identify E as a subfield of Q` . Then for all τ : F ,→ Q` , HTτ (M` ⊗E Q` ) depends only on τ0 = τ|Fcm .
If M is moreover pure of some weight k, thennoHTτ0 ◦c (M` ⊗E Q` ) = k − h : h ∈ HTτ0 (M` ⊗E Q` ) .
Proof. Write ι−1 τ for the embedding F ,→ E induced by τ and ι. By the previous lemma,HTτ (M` ⊗E Q` ) = HTι−1 τ (M). The latter depends only on τ|Fcm , and when M is pure satisfies therequired symmetry, by Lemmas 4.1.22 and 4.1.24.
Finally, we can show that the Galois lifting results of §3.2 apply to representations arising fromobjects of MF,E . The result that follows is clumsily proven and has been superseded (and greatly
strengthened) by [Pat16, Corollary 1.1]. The reader would do better to turn there, but here is a
simpler argument in the special case in which motivic Galois groups over F are connected, whichis expected always to be the case; the complication in the present proof arises from the fact that theproof of Theorem 3.2.7 requires an initial reduction to the case of connected monodromy group.
Fix an embedding E ,→ Q` , inducing a place λ of E. For short-hand, we denote by GF,Q` thebase-change to Q` of the Eλ -group GF,λ defined by the λ-adic étale fiber functor on the category
MF,E , and we make the analogous definition of G M for objects M of MF,E .
F,Q`

Corollary 4.1.27. Let F be a totally imaginary field. Let M be an object of MF,E and E ,→ Q`
a fixed embedding, to which we associate the `-adic representation M` ⊗E Q` , which is a representation both of ΓF and of the motivic group G M . Make the following assumption
F,Q`

• G

MF
F,Q`

is connected.

e → H is any central torus quotient of linear algebraic groups over Q` , and that
Suppose that H
there is a factorization
M
GF,Q,→ H ,→ GLQ` (M` ⊗E Q` ),`111

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where of course the composition of these inclusions is the natural representation. Then there exists
a geometric lift ρ̃
5 H(Q` )

e

ρ̃

ΓF

ρ

/ GM

F,Q`


/ H(Q ).

(Q` )`Similarly, if F is any number field, but E is Q, then without any assumption on the motivic groupM
GF F , such a ρ̃ exists.
M

Remark 4.1.28. The assumption that G F is connected is expected always to hold. In particF,Q`ular, this should not be necessary for the conclusion of the corollary to hold.
Proof. Corollary 4.1.26 tells us that composing ρ with any irreducible algebraic representationof G M yields an `-adic representation satisfying the Hodge symmetries of Conjecture 3.2.5. But
F,Q`
the arguments of Proposition 3.2.7 and Theorem 3.2.10 do not apply directly, because they assume
these symmetries after composition with irreducible representations of the connected component
Zar
M(ρ(ΓF ) )0 .8 The simplest (but imperfect) way around this is to assume G F is connected, and thenF,Q`

run through the arguments of Proposition 3.2.7 and Theorem 3.2.10 with G M0F0 , for F 0 sufficiently
F ,Q`

Zar

large (see Lemma 4.1.17), in place of ρ(ΓF 0 ) : reduce to the connected case (replacing F by F 0 )as in Theorem 3.2.10, and then note that the proof of Proposition 3.2.7 only requires identifying
a connected reductive group containing the image of ρ such that composition with irreducible
representations of this group yields Galois representations with the desired symmetry (whether or
not these Galois representations are themselves irreducible).
The second assertion (when E = Q) follows by the same argument: by Remark 4.1.23, the τlabeled Hodge-Tate co-character µτ of ρ is independent of τ. It can be interpreted as a co-character
Zarof (ρ(ΓF ) )0 , and composing with finite-dimensional representations of this group, we obtain thenecessary Hodge-Tate symmetries to apply the argument of Proposition 3.2.7.

Remark 4.1.29. Similarly, there is a motivic version of Corollary 3.2.8.
4.1.6. Motivic lifting: the potentially CM case. One case of the desired motivic lifting result
is immediately accessible, when the corresponding Galois representations are potentially CM. We
denote by CMF the Tannakian category of motives (for motivated cycles) over F generated byArtin motives and potentially CM abelian varieties. As before an embedding σ : F ,→ C yields,
through the corresponding Betti fiber functor, a Tannakian group TF (σ) = Aut⊗ (Hσ |CMF ). For anumber field E, we can also consider, as in §4.1.4, the category CMF,E of potentially CM motives
over F with coefficients in E, with its Tannakian group TF,E (σ).9 As with GF , there is a sequence
1 → TF,E → TF,E → ΓF → 1,
M

8

If this group were, contrary to conjecture, not equal to G F , then it is not obvious in general how to relate theF,Q`

two putative Hodge symmetries.
9
From now on, σ will be implicit.
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where again the projection TF,E → ΓF has a continuous section sλ on Eλ -points. Deligne showed in[DMOS82] (Chapter IV: ‘Motifs et groupes de Taniyama’) that TQ is isomorphic to the Taniyamagroup constructed by Langlands as an explicit extension of ΓQ by the connected Serre group.10
e → H be a surjection of linear algebraic
Proposition 4.1.30. Let F be totally imaginary. Let H
groups over a number field E with central torus kernel. For any homomorphism ρ : TF,E → H,
there is a finite extension E 0 /E such that ρ lifts to a homomorphism<

ρ̃

TF,E 0

ρ

eE 0
H


/ HE 0 .

Proof. Fix a finite place λ of E and consider the λ-adic representation ρλ = ρ◦sλ : ΓF → H(Eλ ).
Since TF is isomorphic to the connected Serre group, we can unconditionally apply Corollarye 0 ) of ρλ . Note that
4.1.27 to find, for some finite extension Eλ0 /Eλ a geometric lift ρ̃λ : ΓF → H(Eλeρ̃λ is potentially abelian, since ρλ is, and the kernel of H → H is central. Proposition IV.D.1
of [DMOS82] implies that ρ̃λ arises from a homomorphism (of groups over Eλ0 ) TF ⊗Q Eλ0 e ⊗E E 0 . This in turn must be definable over some finite extension E 0 /E (E 0 is thus
TF,E ⊗E Eλ0 → H
λe ⊗E E 0 whose extension to E 0 gives
embedded in Eλ0 ), i.e. there is a homomorphism ρ̃ : TF,E 0 → H
λrise to ρ̃λ . We explain this technical point in Lemma 4.1.31 below. To check that ρ̃ is actually a liftof ρ ⊗E E 0 , it suffices to observe:• The restriction to TF,E 0 is a lift: the restrictions of ρ̃ and ρ are simply the algebraic homomorphisms of the connected Serre group (tensored with E 0 ) corresponding to the labeled
Hodge-Tate weights of ρ̃λ and ρλ .
• The λ0 -adic lift, for the place λ0 of E 0 induced by E 0 ,→ Eλ0 , is a lift (by construction).
• It suffices to check that ρ̃ is a lift on Eλ0 0 -points. First, it suffices to check that ρ̃ ⊗E 0 Eλ0 0 liftsρ ⊗E Eλ0 0 . This in turn can be checked on Eλ0 0 -points: we are free to replace TF,E 0 ⊗E 0 Eλ0 0
by some finite-type quotient T in which the Eλ0 0 -points are Zariski-dense. Then the closed
subscheme T × H ,→ T (where the two maps to H × H are the diagonal H → H × H
H×Hand the product of the two maps T → H × H given by ρ and ρ̃ followed by the quotiente → H) contains T (E 0 0 ), hence equals T .
HλThen we are done, since TF,E 0 (Eλ0 0 ) = TF,E 0 (Eλ0 0 ) · sλ0 (ΓF ).
Here is the promised lemma showing that ρ̃ may be defined over a finite extension E 0 of E:
Lemma 4.1.31. Let K/k be an extension of algebraically closed fields of characteristic zero.
e → H be a surjection
Let T be a (not necessarily connected) reductive group over k, and let π : H(defined over k) of reductive k-groups. Suppose that ρ : T → H is a k-morphism, and that theeK . Then ρ lifts to a k-morphism T → H.
escalar extension ρK lifts to a K-morphism ρ̃ : T K → H
Proof. This is a simple spreading-out argument. Namely, writing K as the direct limit of its
finite-type k-sub-algebras, we can descend the relation πK ◦ ρ̃ = ρK to some finitely-generated
More precisely, equipped with the data of the projection to ΓQ , a section on AF, f -points, and the co-character
Gm,C → TC giving the Hodge filtration on objects of CMC , TQ is uniquely isomorphic to the Taniyama group,
equipped with its corresponding structures.
10

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Version 4 - Submitted to MEMOeR such that ρ̃R ⊗R K = ρ̃. We then choose
k-sub-algebra R of K, in particular obtaining ρ̃R : T R → H
a non-zero k-point α : R → k (a morphism of k-algebras), and specializing the relation πR ◦ ρ̃R = ρR
via α, we obtain a lift ρ̃R ⊗R,α k of ρ over k.
Remark 4.1.32.
• We have stated Proposition 4.1.30 only for imaginary fields for simplicity, but of course there is a variant for general number fields taking into account Corollary 3.2.8. When F is totally real, there will be such ρ that do not lift: simply take atype A Hecke character ψ of a quadratic CM extension L/F as in Example 2.5.3. Then
Ad0 (IndFL (ψ)) will have a potentially abelian Galois `-adic realizations ΓF → SO3 (Q` ),which arises from a representation of TF by Proposition IV.D.1 of [DMOS82]. As wehave seen, these representations do not lift geometrically to GSpin3 .
• The method of checking that ρ̃–once we know it exists–actually lifts ρ will recur in §4.2;
this division into a geometric argument (lifting ρ|TF,E and a Galois-theoretic argument(lifting ρ◦ sλ ) seems to be the natural way to make arguments about motivic Galois groups
over number fields (compare Lemma 4.1.17).
• On the automorphic side, representations of the global Weil group ought to parametrize‘potentially abelian’ automorphic representations. In that case, the analogous lifting result
is a theorem of Labesse ([Lab85]).
Before proceeding to more elaborate examples, let us clarify here that, even though we study
lifting through central quotients, the nature of these problems is in fact highly non-abelian. ThateE of ρ : GF,E → HE (we may assume He and H are reductive;
is, suppose we have a lift ρ̃ : GF,E → H
0
let us further suppose they are connected), and let r and r be irreducible faithful representationseE and HE . If the derived group of H is not simply-connected, but its simply-connected coverof He then typically r0 ◦ ρ̃ will not lie in the Tannakian subcategory of MF,E generated
Hsc injects into H,
by r ◦ ρ and all potentially CM motives, i.e. objects of CMF,E (of course, it does in the example of
Proposition 4.1.30). We make this precise at the Galois-theoretic level:e → H, r0 , and r be as above. Let ρ : ΓF → H(Q` ) be a geometric Galois
Lemma 4.1.33. Let He ` ). Assume that
representation having a geometric lift ρ̃ : ΓF → H(Q• the algebraic monodromy group of ρ is H itself;
Zar• the kernel of Hsc ∩ (ρ̃(ΓF )) → H is non-zero.
Then r0 ◦ρ̃ is not contained in the Tannakian sub-category of semi-simple geometric Q` -representationsof ΓF generated by r ◦ ρ and all potentially abelian geometric representations.
Proof. If r0 ◦ ρ̃ were contained in this category, then there would exist an irreducible potentiallyabelian representation τ of ΓF and an injection r0 ◦ ρ̃ ,→ τ ⊗ (r1 ◦ ρ) for some irreducible algebraic
representation r1 of H. By Proposition 3.4.1, τ has the form IndFL (ψ · ω) for some character ψand irreducible Artin representation ω of ΓL . By Frobenius reciprocity, there is a non-zero map(r0 ◦ ρ̃)|ΓL → ψ · (r1 ◦ ρ)|ΓL ⊗ ω; both sides are irreducible,11 so ω is one-dimensional, and absorbing∼ω into ψ we may assume r0 ◦ ρ̃|ΓL →− ψ · (r1 ◦ ρ)|ΓL . Comparing algebraic monodromy groups, weZarobtain a contradiction, since by assumption Hsc ∩ (ρ̃(ΓF )) cannot inject into Gm × H.

11For the right-hand side, see Proposition 3.4.1: the tensor product of a Lie-irreducible and an Artin representation
is irreducible; of course, for the purposes of this lemma, we could just further restrict L.
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Version 4 - Submitted to MEMOIn §4.2, we will see many examples of ρ with full SO monodromy group; their lifts to GSpin
will satisfy the conclusions of Lemma 4.1.33.
4.2. Motivic lifting: the hyperkähler case
4.2.1. Setup. The aim of this section is to produce a lifting not merely at the level of a single`-adic representation, but of actual motives, in a very special family of cases whose prototype is
the primitive second cohomology of a K3 surface over F. So that the reader has some examples tokeep in mind, we recall the following definitions:Definition 4.2.1. Let F be a subfield of C. A hyperkähler variety X over F is a geometricallyconnected and simply-connected smooth projective variety over F of even dimension 2r such that
Γ(X, Ω2X ) is one-dimensional, generated by a differential form ω for which ωr is non-vanishing at
every point of X (i.e., as a linear functional on the top wedge power of the tangent space). A K3
surface over F is a hyperkähler variety X/F of dimension 2.
Example 4.2.2. The simplest example of a K3 surface is a smooth quartic hypersurface in P3 .
Higher-dimensional examples of hyperkählers are notoriously difficult to produce. One standard
family is gotten by starting with any K3 surface X, and then for any integer r ≥ 1 considering
the Hilbert scheme X [r] of r points on X (more precisely, the moduli of closed sub-schemes oflength r); each X [r] is a hyperkähler variety, with of course X [1] = X. (This construction is due to
Beauville: see [Bea83, §6].)
More generally, we work in the axiomatized setup of [And96a]. By a polarized variety over a
subfield F of C we mean a pair (X, η) consisting of variety X/F and an F-rational ample line bundle
η on X. For a fixed k ≤ dim X, we will consider the motive of primitive cohomology (omitting theη-dependence from the notation),
Prim2k (X)(k) = ker ηdim X−2k+1 : H 2k (X)(k) → H 2 dim X−2k+2 (X)(dim X − k + 1) ,as an object of MF . Cup-product with η lets us endow the motive H 2k (X)(k) with the quadratic
form
hx, yiη = (−1)k x ∪ y ∪ ηdim X−2k ∈ H 2 dim X (X)(dim X)  Q,
and we can equivalently define Prim2k (X)(k) as the orthogonal complement of H 2k−2 (X)(k − 1) ∪ η.
The two Weil cohomologies we use are `-adic (with coefficients in Q` or some extension insideQ` ) and Betti cohomology; we will occasionally take integral Betti cohomology, where we will
always work modulo torsion, so that Prim2k (XC , Z)(k) is a sub-module of H 2k (XC , Z)(k)/(torsion).
In that case, the primitive lattice defines a polarized (by h·, ·iη ) integral Hodge structure of weight0; we write h p,q for the Hodge numbers. André proves his theorems, which include versions of the
Shafarevich and Tate conjectures, under the following axioms:
Ak : h1,−1 = 1, h0,0 > 0, and h p,q = 0 if |p − q| > 2.
Bk : There exist a smooth connected F-scheme S , a point s ∈ S (F), and a smooth projective
morphism f : X → S such that:– X  Xs;an– the Betti class ηB ∈ H 2 (XCan , Z)(1)/(torsion) extends to a section of R2 fC∗
Z(1)/(torsion);e– letting S denote the universal cover of S (C), and D denote the period domain of
Hodge structures on VZ := Prim2k (XC , Z)(k) polarized by h·, ·iη , we require that theimage of Se → D contain an open subset.
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Ak is essential to the method, which relies on studying the associated Kuga-Satake abelian variety,which exists for Hodge structures of this particular form. Bk is a statement about deforming X intoa ‘big’ family. B+k is of course a case of the Hodge conjecture, which is always known when k = 1(the theorem of Lefschetz). But we provide these axioms merely for orientation. Of interest is thefollowing collection of varieties for which they are known to hold:
Proposition 4.2.3 (See §2-3 of [And96a]). The axioms A1 , B1 , and B+1 are satisfied by: abeliansurfaces; surfaces of general type with ample canonical bundle KX , h2,0 (X) = 1, and KX · KX = 1;
and polarized hyperkähler varieties with b2 > 3 (in particular, K3 surfaces). Cubic fourfolds(polarized via OP5 (1)) satisfy A2 , B2 , and B+2 .
This relies on the work of many people; see §2 and §3 of [And96a]. Finally, André observes
that these axioms are independent of the choice of embedding F ,→ C; note that for Ak this is a
special case of Remark 4.1.23.
Remark 4.2.4. The proposition does not apply to hyperkähler varieties with Betti number 3,
since its argument (building on that of [Del72]) requires the variety X to have sufficiently robust(projective) deformation theory, whereas a hyperkähler with b2 (X) = 3 is rigid. In fact, it isbelieved that such hyperkählers do not exist. Regardless, in that case one can still say something
about the motivic lifting problem, since the `-adic representation H 2 (XF , Q` ) is potentially abelian.
In principle, this should force the underlying motivic Galois representation to factor through TF ,at which point we would apply Proposition 4.1.30; but this argument would require checking thattwo GF -representations with isomorphic `-adic realizations are themselves isomorphic, i.e. an
unknown case of the Tate conjecture. At least we have (as in Proposition 4.1.30):
Lemma 4.2.5. Let X/F be a smooth projective variety with b2 (X) = 3. If necessary, replace
F with a quadratic extension trivializing det H 2 (XF , Q` ). Then the `-adic realization ρ` : GF →SO(H 2 (XF , Q` )) has a lift ρ̃` to GSpin(H 2 (XF , Q` )) that arises as the `-adic realization of a representation of TF,E for suitable E.
Notation 4.2.6.
• From now on we view F as a subfield of C, with F its algebraic closure in C. These embeddings will be used implicitly to define Betti realizations, motivic
Galois groups (for Betti realizations), and étale-Betti comparisons, for varieties (or motives) over extensions of F inside F.
• We write VQ for Prim2k (XC , Q)(k), and for a prime `, we will write V` for the `-adic realization Prim2k (XF , Q` )(k). As before, Prim2k (X)(k) will be used to indicate the underlying
motive, i.e. object of MF .
• For fields E containing Q, we will sometimes denote the extension of scalars VQ ⊗Q E by
VE (or similarly for Q` and V` ). This is the Betti realization of an object Prim2k (X)(k)E of
MF,E .
• The same subscript conventions will hold for other motives we consider, especially the
direct factors of Prim2k (X)(k) given by the algebraic cycles (to be denoted Alg) and its orthogonal complement, the transcendental lattice (T ). These motives will be discussed in§4.2.4. In the meantime, we recall that the rational Hodge structure Prim2k (XC , Q)(k) has
a Q-subspace spanned by its Hodge classes; the orthogonal complement T Q , the transcendental subspace, of this space of Hodge cycles is again a polarized Q-Hodge structure.
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Version 4 - Submitted to MEMO• For an object M of MF , we will sometimes write ρ M for the associated motivic Galois
representation. For the `-adic realization of this motivic Galois representation (i.e., therepresentation of ΓF on H` (M)), we write ρ`M .
• First assume dim VQ = m is odd. The group GSpin(VQ ) may not have a rationally-defined
spin representation, but it does after some extension of scalars, and over any suitably
large field E, we denote by rspin : GSpin(VE ) → GL(WE ) this algebraic representation. If
m is even, we similarly denote by WE = W+,E ⊕ W−,E the direct sum of the two half-spin
representations. In contrast to VE = VQ ⊗Q E, this WE is not necessarily an extension ofscalars from an underlying Q-space.
Possibly after replacing F by a quadratic extension, we may assume that the ΓF -representationV` is special orthogonal, i.e. ρ` : ΓF → SO(V` ) ⊂ O(V` ). Since for almost all finite places v,
the frobenius frv acts on H 2k−2 (XF , Q` ) and H 2k (XF , Q` ) with eigenvalues that are independent of` ([Del74]), and trivially on the `-adic Chern class η` , the eigenvalues of frv on Prim2k (XF , Q` )(k)are independent of `. Consequently, if det ρ` is trivial for one `, then it is for all `. We will fromnow on assume, for technical simplicity, that this determinant condition is satisfied.
4.2.2. The Kuga-Satake construction. We review the classical Kuga-Satake construction
and outline André’s refinement, which implies a potential version (i.e., after replacing F by afinite extension) of our motivic lifting result. See §4 and §5 of [And96a]. His approach is inspiredby that of [Del72], in which Deligne used the Kuga-Satake construction (in families) to reduce
the Weil conjectures for K3 surfaces to the (previously known) case of abelian varieties. Let VZ
be the polarized quadratic lattice Prim2k (XC , Z)(k) of the previous subsection. Write m = 2n orm = 2n + 1 for its rank. Basic Hodge theory implies that the pairing on VR is negative-definite on
the sub-space (H 1,−1 ⊕ H −1,1 )R , and positive-definite on HR0,0 , hence has signature (2, m − 2). Recalling the discussion of §4.1.1, this real Hodge structure yields a homomorphism h : S → SO(VR )that lifts uniquely to a homomorphism h̃ : S → GSpin(VR ) whose composition Nspin ◦ h̃ with the
Clifford norm Nspin is the usual norm S → Gm,R .
Let C(VZ ) and C + (VZ ) denote the Clifford algebra and the even Clifford algebra associated tothe quadratic space VZ . Let LZ be a free left C + (VZ )-module of rank one. Denote by C + the ring
EndC + (VZ ) (LZ )op . Because of the ‘op,’ C + naturally acts on LZ on the right, and we then correspondingly have EndC + (LZ )  C + (VZ ). The choice of a generator x0 of LZ induces a ring isomorphism
φ x0

C + (VZ ) −−→ C +c 7→(bx0 7→ bcx0 ).
Via h̃ and the tautological representation GSpin(VR ) ,→ C + (VR )× , LR := LZ ⊗Z R acquires a Hodge
structure of type (1, 0), (0, 1); the Hodge type is easily read off from the fact that EndC + (LZ ) 
C + (VZ ) is also an isomorphism of Hodge-structures,12 or by recalling the remarks of §4.1.1 andlisting the weights of the spin representation. Concretely (as in the original construction of Satake[Sat66] and Kuga-Satake [KS67]), we can choose an orthogonal basis e1 , e2 of (H 1,−1 ⊕ H −1,1 )R ,normalized so that hei , ei i = −1 for i = 1, 2. Then the automorphism of C + (VR ) given by multiplication by e1 e2 is a complex structure, by the defining relations for the Clifford algebra. In fact, thisintegral Hodge structure is polarizable: this can be shown explicitly, or by a very soft argument
Giving C + (VZ ) the Hodge structure induced from that on the tensor powers of VZ .

12

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(apply Proposition 2.11.b.iii0 of [Del72]). We therefore obtain a complex abelian variety, the KugaSatake abelian variety KS (X) = KS (XC , η, k), associated to our original VZ . The right-action of C +on LZ commutes with the Hodge structure, so C + acts as endomorphisms of KS (X). Generically,
this will be the full endomorphism ring; we will have to be attentive later to how much bigger theendomorphism ring can be.
One of the main technical ingredients in [And96a] is the following descent result, which uses
rigidity properties of the Kuga-Satake construction:
Lemma 4.2.7 (Main Lemma 1.7.1 of [And96a]). Let (X, η) be a polarized variety over a subfield
F of C, satisfying properties Ak and Bk . Then there exists an abelian variety AF 0 over some finite
extension F 0 /F such that
• The base-change AC is the Kuga-Satake variety KS (X);• there is a subalgebra C + of End(AF 0 ) and an isomorphism of Z` [ΓF 0 ]-algebrasEndC + H 1 (AF , Z` )  C + (Prim2k (XF , Z` )(k)).
We subsequently write L` for the `-adic realization H 1 (AF 0 ⊗ F, Q` ). The main result of the‘motivated’ theory of hyperkählers, (the foundation of André’s results on the Tate and Shafarevich
conjectures) is:
Theorem 4.2.8 (see Theorem 6.5.2
of [And96a]).
For some finite extension F 0 /F, Prim2k (XF 0 )(k)
is a direct factor (in MF ) of End H 1 (AF 0 ) , and both Prim2k (XF 0 )(k) and H 1 (AF 0 ) have connected
motivic Galois group.
Write ρA : GF 0 → GL(LQ ) and ρV : GF → SO(VQ ) for the motivic Galois (for the Betti realization) representations associated to H 1 (AF 0 ) and Prim2k (X)(k). We will somewhat sloppily use thesame notation ρA and ρV for various restrictions of these representations (eg, to GF ). Note that by
Corollary 4.1.20, the image of ρA is the Mumford-Tate group of KS (X), which is easily seen to becontained in GSpin(VQ ) = {x ∈ C + (VQ )× : xVQ x−1 = VQ }. More precisely:
Corollary 4.2.9. For some finite extension F 0 /F, ρA factors through GSpin(VQ ) and lifts ρV |GF0 .
A fortiori, ρ`A lifts ρV` |ΓF0 .
Proof. Over C (or F), the image of ρA is MT (AC ), hence contained in GSpin(VQ ). We cantherefore compare the two maps
(ρV ,π◦ρA )GC −−−−−−→ SO(VQ ) × SO(VQ ) → GL C + (VQ ) × GL C + (VQ ) ,where π denotes the projection GSpin(VQ ) → SO(VQ ). Under the motivated isomorphism
C + (Prim2k (XC )(k))  End H 1 (A)the adjoint action of ρA onEnd(LQ ) agrees with the action of π ◦ ρA on C + (VQ ), so the two compositions GC → GL C + (VQ ) above coincide. Now, if m is odd, SO(VQ ) → GL(C + (VQ )) is injective,so ρV = π ◦ ρA ; that is, at least over C, ρA lifts ρV . If m is even, we deduce that the compositions(ρV ,π◦ρA )GC −−−−−−→ SO(VQ ) × SO(VQ ) → GL(∧2 VQ ) × GL(∧2 VQ )118

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agree.13 The kernel of ∧2 : SO(VQ ) → GL(∧2 VQ ) is {±1} (and central), so we see that ρV andπ ◦ ρA agree up to twisting by some character χ : GC → {±1} ⊂ SO(VQ ). But clearly this character
factors through GCM , where M = Prim2k (XC )(k) ⊕ H 1 (A), and by Theorem 4.2.8, GCM is connected.
Therefore χ is trivial, and ρA lifts ρV as GC -representations. By Lemma 4.1.21, the same holds
for the corresponding GF -representations, and then as in Lemma 4.1.17 the same holds over somefinite extension F 0 /F.
Corollary 4.2.9 does not always hold with F 0 = F, and it is our task in the coming sections
to achieve a motivic descent over F itself. The Kuga-Satake variety is highly redundant, and
it is technically convenient to work with a smaller ‘spin’ abelian variety, many copies of whichconstitute the Kuga-Satake variety. Since the rational Clifford algebra CQ+ may not be split, thisrequires a finite extension of scalars E/Q, after which we can work in the isogeny category AVF0 0 ,Eof abelian varieties over F 0 with E-coefficients. We take F 0 as in the Corollary, and now to ease
the notation, we write simply A for AF 0 .
Lemma 4.2.10. There exists a number field E and an abelian variety B/F 0 with endomorphisms
by E such that there is a decomposition in AVF0 0 ,E :n

if m = 2n + 1;

2n−1

if m = 2n.

A ⊗Q E  B2
A ⊗Q E  B

The `-adic realization H 1 (BF , Q` ) is isomorphic to the composite rspin ◦ρ`A as (E⊗Q` )[ΓF 0 ]-modules,
where as before ρ`A denotes the representation ΓF 0 → GSpin(V` ⊗ E) obtained from L` ⊗ E, and rspin
denotes either the spin (m odd) or sum of half-spin (m even) representations of GSpin(V` ⊗E). Whenm = 2n, B decomposes in AVF0 0 ,E as B+ × B− , corresponding to the two half-spin representations.
Proof. Choose a number field E splitting C + (VQ ). Then, letting WE denote either the spin (modd) representation or the direct sum W+,E ⊕ W−,E of the two irreducible half-spin representations
n
2n−1
2n−1
(m even), C + (VE ) is isomorphic as GSpin(VE )-representations to WE2 , or to WE,+
⊕ WE,−. As Ealgebra it is then isomorphic either to End(WE )  M2n (E) or End(W+,E ) ⊕ End(W−,E )  M2n−1 (E) ⊕M2n−1 (E).14 Using the orthogonal idempotents in C E+  C + (VE ), we decompose the object AF 0 ⊗Q E
of AVF0 0 ,E into 2n (when m is odd) or 2n−1 (when m is even) copies of an abelian variety B/F 0 withcomplex multiplication by E.
From this lemma and Corollary 4.2.9, we deduce a partial resolution of Question 1.1.9 in thissetting; this question clearly warrants further attention, but we will settle in the remainder of thisbook for a somewhat weaker result (see, eg, the statement of Theorem 4.2.13).
Corollary 4.2.11. Assume dim VQ is odd-dimensional. Then for some finite extension L/F 0 ,the `-adic realizations ρ`A : ΓL → GSpin(V` ) of ρA form a weakly-compatible system of GSpinvalued representations, in the sense of Definition 1.2.3.
Proof. Let dim VQ = 2n + 1. We must show that for ` , `0 , the semi-simple parts (under Jordan
decomposition) of ρ`A (frv ) and ρ`A0 (frv ) belong to the same GSpin2n+1 -conjugacy class (by [Del74],
The filtration on C + (VQ ) given by the image of V ⊗≤2i is motivated, since O(V)-stable, so ρV and π ◦ ρA coincideon the i = 1 graded piece, which is just ∧2 (VQ ).
14
Note that the field E can be made explicit if we know the structure of the quadratic lattice VZ . See Example
4.2.14 for a case where E = Q.
13

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these conjugacy classes are defined over Q, so we may compare them as classes in GSpin2n+1 (Q)under some fixed embeddings of Q into, respectively, Q` and Q`0 ). First we observe that a semisimple element x ∈ GSpin2n+1 is determined up to conjugacy by its conjugacy classes under the
spin representation, standard representation, and Clifford norm: a semi-simple conjugacy classin Spin2n+1 is determined by its conjugacy classes under the spin and standard representations(which together imply conjugacy in all the fundamental representations of the simply-connectedgroup Spin2n+1 ), and further knowing N(x) allows us (twisting by a choice of N(x)−1/2 ) to reduce to the Spin case. From the main theorem of [Del74], we know weak compatibility (as ΓF 0 representations) of the representations V` and H 1 (BF , Q` ). It remains to show that for some finite,
independent-of-` extension L/F 0 , {N ◦ ρ`A |ΓL }` is a weakly compatible system of characters.
To see this, note that by Lemma 4.1.25, each (Tate-twisted) character χ` = (N ◦ ρ`A )(1) is de
Rham with Hodge-Tate weights zero, hence has finite-order. It suffices to show that a common
finite extension L/F 0 trivializes χ` for all `, and for this it suffices to bound the conductors of thefinite-order characters χ` . Recall that the H 1 (BF , Q` ) are in fact known to form a strongly compatible system, in the sense that for each finite place v of F 0 , there is a Weil-Deligne representation(independent of `) of WFv0 corresponding via the `-adic monodromy theorem to (the Frobenius
semi-simplification of) H 1 (BFv0 , Q` ) (see [Fon94]). In particular, we obtain an independent-of-`bound on the conductor of χ` at places away from `, since as GSpin2n+1 -representation the Clifford⊗2⊗2
norm N is a direct summand of rspin(see page 135 for the full decomposition of rspin). Moreover,χ` is crystalline for almost all ` (wherever A has good reduction, for instance); it is therefore unramified (since the Hodge-Tate weights are zero) for almost all `, so we obtain an independent-of-`bound on the entire conductor of every χ` . Consequently, (after identification with complex characters) there are only finitely many possibilities for the characters χ` , and in particular they all
become trivial upon restriction to ΓL for some finite extension L/F 0 .

Remark 4.2.12.
• Under the Standard Conjectures, the motivic Galois representationAN ◦ ρ gives rise to a weakly compatible system of `-adic representations, so we expectthat we can take L = F 0 in the conclusion of Corollary 4.2.11.
• For n ≥ 3, it does not suffice to know the conjugacy classes of a semi-simple element x ∈GSpin2n+1 under the spin and standard representations–further knowledge of the Clifford
norm is essential. For example, let ζ be a primitive 8th root of unity, and, in the coordinatesof §2.8, let x = (λ1 − λ2 )(ζ 6 )(λ2 − λ3 )(ζ 2 )(2λ3 )(ζ). Then x and x · λ0 (ζ 2 ) have the same
eigenvalues in the spin (i.e., highest-weight −χ0 + χ1 +χ22 +χ3 ) and standard representations,but they do not have the same Clifford norm, and in particular are not GSpin7 -conjugate.
• More can be said about this question, but in what follows we will not descend this weakcompatibility result to a statement for ΓF -representations, even in situations where weachieve a motivic lift ρ̃ of ρV over F itself. In short, we will not know compatibility of the
Clifford norms N ◦ ρ̃` .
We state a simple case of the main result of this section; for the proof, see §4.2.5. More generalversions will be proven in stages, depending on the complexity of the transcendental lattice of VQ .
Theorem 4.2.13. Let (X, η) be a polarized variety over a number field F ⊂ C for whichPrim2k (XC , Z)(k) satisfies axioms Ak , Bk , and B+k . Possibly enlarging F by a quadratic extension,
we may assume as above that for all `, det V` = 1, so that the `-adic representation ρV` maps ΓF to
SO(V` ). We make the following hypothesis on the monodromy:
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Version 4 - Submitted to MEMO• The transcendental lattice T Q has EndQ-HS (T Q ) = Q.
• det(T ` ) = 1.15
Then there exists a lifting of motivic Galois representations
GSpin(VE )
9

ρ̃

GF,E

ρ


/ SO(VE ).

Moreover there exist:• a finite extension E/Q;
0
• an object B of AVF,E;• an Artin motive M in MF,E ;such that the composite rspin ◦ ρ̃ is isomorphic in MF,E to some (explicit) number of copies ofM ⊗E H 1 (B). In particular, for each prime λ of E, there are lifts ρ̃λGSpin(V` ⊗ Eλ )
ρ̃λ

ΓF

rspin

6

/ GLE (WE )λλ
/ SO(V` ) ⊂ SO(V` ⊗ E λ )such that the composites {rspin ◦ ρ̃λ }λ form a weakly (or even strongly: see the proof of Corollary
4.2.11) compatible system.
This theorem is an optimal (up to the O(V` ) ⊃ SO(V` ) distinction) arithmetic refinement of the(a priori highly transcendental) Kuga-Satake construction, showing the precise sense in which it
descends to the initial field of definition F.
4.2.3. A simple case. To achieve the refined descent of Theorem 4.2.13, the basic idea is
to apply our Galois-theoretic lifting results (which apply over F) to deduce ΓF -invariance of theabelian variety B that we know to exist over F 0 ; Faltings’s isogeny theorem ([Fal83]) implies
that this invariance is realized by actual isogenies; then we apply a generalization of a techniqueused by Ribet ([Rib92]; our generalization is Proposition 4.2.29) to study elliptic curves over Q
that are isogenous to all of their ΓQ -conjugates (so-called “Q-curves”). Ribet’s technique applies
to elliptic curves without complex multiplication, and we will have to keep track of monodromy
groups enough to reduce the descent problem to one for an absolutely simple abelian variety. In
some cases, a somewhat ‘softer’ argument than Ribet’s works–we give an example in Lemma
4.2.24–but in addition to being satisfyingly explicit, the Ribet method seems to be more robust.
But first we prove Theorem 4.2.13 in the simplest case, when the Hodge structure VQ is‘generic,’ dim VQ = 2n + 1 is odd, and the even Clifford algebra C + (VQ ) is split over Q. Our
working definition of ‘generic’ will be that VQ contains no trivial Q-Hodge sub-structures (i.e., it
is equal to its transcendental lattice), and that EndQ-HS (VQ ) = Q.
Example 4.2.14. If X/F is a sufficiently general K3 surface, then these hypotheses are satisfied.
The K3-lattice H 2 (XC , Z) is an even unimodular lattice of rank 22 whose signature over R is (by15

Unlike the first hypothesis, this determinant condition is a technicality; it can again be arranged after an (independent of `) quadratic extension. Of course, in the generic case in which VQ = T Q , it is no additional hypothesis.
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Hodge theory) (3, 19). The classification of even unimodular lattices implies it is isomorphic (overZ) to (−E8 )⊕2 ⊕ U ⊕3 , where E8 and U are the E8 -lattice and the hyperbolic plane lattice. Over Q,
the orthogonal complement of the ample class η is isomorphic to (−E8 )⊕2 ⊕ U ⊕2 ⊕ h−q(η)i, where q
is the quadratic form and hαi denotes the one-dimensional quadratic space with a generator whosesquare is α. Since Prim2 (XC , Q) is odd-dimensional, the basic structure theory of Clifford algebras(see Chapter 9, especially Theorem 2.10, of [Sch85]) implies that C + (Prim2 (XC , Q) is a central
simple algebra over Q, whose Brauer class is simply twice the Brauer class of E8 plus twice the(trivial) Brauer class of U. This is obviously the trivial class, so C + (VQ ) is in this case isomorphic
to M210 (Q). More generally, this argument applies to VQ = Prim2 (XC , Q)(1) if X is a hyperkählersatisfying:• b2 (X) > 3;
• the dimension of H 2 (XC , Z) is even;• the number of copies of the E8 -lattice is even.
Now, our ‘generic’ hypothesis implies that the Hodge group of VQ is the full SO(VQ ) (moregenerally, see Zarhin’s result, quoted as Proposition 4.2.21, below); it follows without difficultythat the Mumford-Tate group MT (AC ) is the full GSpin(VQ ), and End0 (AC ) = CQ+ . By our second simplifying hypothesis, this Clifford algebra is isomorphic to a matrix algebra M2n (Q), and,writing WQ for the spin representation of GSpin(VQ ), we have two isomorphisms of GSpin(VQ )representations:
C + (VQ )  WQ2 ;n

C + (VQ )ad  End(WQ ),where GSpin(VQ ) acts on the first C + (VQ ) by left-multiplication, and on C + (VQ )ad by conjugation(i.e., through the natural SO(VQ )-action).
We may certainly enlarge F 0 to a finite extension over which all endomorphisms of AC arendefined, and the complex multiplication by CQ+ then gives an isogeny decomposition A ∼ B2 , where
B/F 0 is an abelian variety with End0 (B) = Q. We can take the spin representation WQ to be equal to
H 1 (BC , Q) (and, extending scalars and invoking comparison isomorphisms, we get identifications
with other cohomological realizations of B–in particular, the `-adic realization Wλ = WQ ⊗Q Q` ).
Proof of Theorem 4.2.13 when VQ is generic. We sketch the argument, with some details postponed until later sections–the goal here is to review Ribet’s method and outline the argument ina simple case. We will now apply the technique of [Rib92] in combination with our abstract
Galois-theoretic lifting results. By Corollary 4.1.27, there exists a liftρ̃` : ΓF → GSpin(VQ` )of ρV` , and we can normalize this lift so that in the spin representation its labeled Hodge-Tateweights HTτ (rspin ◦ ρ̃` ) are of ‘abelian variety’-type, i.e. 2n−1 zeroes and 2n−1 ones, for each τ : F ,→Q` (for details, see Lemma 4.2.23). This normalization determines ρ̃` up to finite-order twist, and it
implies that ρ̃` |ΓF0 is a finite-order twist of ρ`A , since they both lift ρV` |ΓF0 with the same Hodge-Tatedata. We may therefore replace F 0 by a finite extension and assumeρ`A = ρ̃` |ΓF0 .
Since ρ̃` begins life over F, we see that ρ`A is ΓF -conjugation invariant. The composition of ρ`Awith the Clifford representation is 2n copies of the `-adic representation ρ`B associated to B, so ρ`B
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is also ΓF -invariant. By Faltings’s theorem, for each σ ∈ ΓF , there exists an isogeny µσ : σ B → B;
we can and do arrange that µσ is defined first for a (finite) system of representatives σi in ΓF for
Gal(F 0 /F), and then defined in general by µσi h = µσi for all h ∈ ΓF 0 . The collection of µσ yields an
obstruction class[cB ] ∈ H 2 (ΓF , End0 (B)× ) = H 2 (ΓF , Q× ), 16
given bycB (σ, τ) = µσ ◦ σ µτ ◦ µ−1στ .
That is, cB measures the failure of the diagram
στ
σµ
τ

σ

µστ

B

/B
=

µσ



B

to commute. Now, the class cB may be non-trivial, and the abelian variety B may not descend (upto isogeny) to F: in fact, that triviality of this class is equivalent to isogeny-descent is Theorem 8.2
×of [Rib92]. Nevertheless, Tate’s vanishing result H 2 (ΓF , Q ) = 0 tells us that there is a continous×
1-cochain α : ΓF → Q whose coboundary equals cB , i.e. cB (σ, τ) = α(στ)α(τ)−1 α(σ)−1 . By
continuity, α is locally constant with respect to ΓF 00 for some finite F 00 /F 0 , and it takes values
in some finite extension Q(α) of Q. We now consider the restriction of scalars abelian variety
C := ResF 00 /F (B) (here B is really the base-change to F 00 , but we omit this not to clutter thenotation). The endomorphism algebra R = End0 (C) of the abelian variety C is isomorphic as
Q-vector space to


 Y
 M
σ
HomF 00 B, B =Qµσ ,σ∈Gal(F 00 /F)

σwhere again we use the fact that End0 (BF ) = Q. Write λσ for the element of R corresponding toµσ under this isomorphism.
Lemma 4.2.15 (Lemma 6.4 of [Rib92]). The algebra structure of R is given byλσ λτ = cB (σ, τ)λστ ,so there is a Q-algebra homomorphism α : R → Q(α) given by the Q-linear extension of λσ 7→α(σ).
Since the isogeny category of abelian varieties over F is a semi-simple abelian category, we
can form the object
M = ResF 00 /F (B) ⊗R,α Q(α).
0We regard M as an object of AVF,Q(α), with Q(α)-rank (in the obvious sense) equal to the Q-rank of
B. Moreover, for any place λ|` of Q(α), the λ-adic realization

ρλM : ΓF → GLQ(α)λ (Mλ )The local constancy of the isogenies µσ implies this is a continuous cohomology class, with Q× equipped withthe discrete topology.
16

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has projectivization isomorphic to the canonical projective descent to ΓF of the ΓF -invariant, irreducible representation of ΓF 00 on H 1 (BF , Q(α)λ ).17 So, ρλM |ΓF00 and ρ`B |ΓF00  rspin ◦ ρ̃` |ΓF00 are
isomorphic up to twist, hence rspin ◦ ρ̃` and ρλM are twist-equivalent as ΓF -representations. The
representation rspin : GSpin(VQ ) → GL(WQ ) is the identity on the center, so after identifying ρλM
to a representation on Wλ ⊗ Q(α)λ , we see that it factors through GSpin(V` ⊗ Q(α)λ ) as a lift ofρV` ⊗ Q(α)λ . The required motivic lift is the representation of GF,Q(α) corresponding to H 1 (M) (formore details on how to check this carefully, see Corollary 4.2.20); and the various ρλM form acompatible system because they are formed from Tate modules of abelian varieties.

4.2.4. Arithmetic descent: preliminary reduction. Now we proceed to a more general argument, making first some preliminary reductions to the analogous lifting problem for the transcendental lattice. We must invoke André’s work on the Tate conjecture for X.
Theorem 4.2.16 (see Theorem 1.6.1 of [And96a]). Let (X, η) be a polarized variety over anumber field18 F satisfying Ak , Bk , and B+k . Then:• Prim2k (XF , Q` )(k) is a semi-simple ΓF -representation• the Galois invariants Prim2k (XF , Q` )(k)ΓF are all Q` -linear combinations of algebraic cycles;
The Tate conjecture for H 2k (X) then implies:
Lemma 4.2.17. There is an Artin motive Alg over F whose Betti realization is the subspaceof VQ spanned by algebraic cycle classes, and whose `-adic representation is the (Artin) ΓF representation on V`ΓF0 for any F 0 /F large enough (and Galois) that all of these cycle classes aredefined over F 0 . The transcendental lattice T likewise descends to an object of MF . In particular,
there is an orthogonal decompositionV` = Alg` ⊕ T `of ΓF -representations (not merely ΓF 0 -representations).
Proof. Giving an Artin motive over F is equivalent to giving a representation of ΓF on a (finitedimensional) Q-vector space. In our case, the space (Q-span) of cycles for homological equivalencek
Zhom(XF 0 ) ,→ H 2k (XF , Q` )(k),or rather its intersection with V` , does the trick. Since the object Prim2k (X)(k) of MF is polarized,
we can define T in MF as the orthogonal complement of Alg.

We will enlarge F 0 as in the Lemma, so that all algebraic classes in Prim2k (XC , Q)(k) are alreadydefined over F 0 , and so that the motivic group GTF 0 is connected (see Lemma 4.1.17). Note that T Q
is an orthogonal Hodge structure of type (1, −1), (0, 0), (−1, 1), with h1,−1 = 1, so the Kuga-Satake
construction applies to it as well (see Variant 4.1.5 of [And96a]). Since the Kuga-Satake variety
associated to VQ is simply an isogeny power of that associated to T Q , the latter, to be denoted A(T ),also descends to some finite extension F 0 /F.
Under the homomorphism R → EndΓF00 (⊕σ H 1 (B, Q` ), the λσ permute the factors in the direct sum; the projecαtion via R →− Q(α) collapses all the factors to a single copy with scalars extended to Q(α), i.e. to H 1 (B, Q` ) ⊗Q Q(α).
This implies the claim about projective descents.
18
Faltings’s theorem works for finitely-generated extensions of Q, so this does as well.
17

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Version 4 - Submitted to MEMOWe introduce a little more notation. After extending scalars to a sufficiently large field E (omitted from the notation unless we want to emphasize it) we denote by (rspin,V , WV ), (rspin,Alg , WAlg ),and (rspin,T , WT ) the spin representations of these three spin groups; as before, by ‘the’ spin representation in the Dn case we will mean the direct sum of the two half-spin representations.
Lemma 4.2.18. Suppose that ρV` factors through SO(Alg` ) × SO(T ` ) ,→ SO(V` ) (at worst, ensuring this requires making a quadratic extension of F). If we have found lifts ρ̃Algand ρ̃T` to`Alg
GSpin(Alg` ⊗ Q` ) and GSpin(T ` ⊗ Q` ) such that rspin,Alg ◦ ρ̃` and rspin,T ◦ ρ̃T` are motivic, thenρV` has a lift ρ̃` : ΓF → GSpin(V` ⊗ Q` ) such that rspin,V ◦ ρ̃` is also motivic. If the individual lifts
rspin,Alg ◦ ρ̃Alg
and rspin,T ◦ ρ̃T` belong to compatible systems of `-adic representations, then the same`
holds for rspin,V ◦ ρ̃` .19
ˆ
Proof. The isomorphism of graded algebras (for the graded tensor product ⊗)∼ˆC(AlgQ )⊗C(T− C(VQ )Q) →induces an inclusion C + (AlgQ ) ⊗ C + (T Q ) ,→ C + (VQ ), and then a map (not injective)
C + (AlgQ )× × C + (T Q )× → C + (VQ )× ,which in turn induces a commutative diagram
GSpin(AlgQ ) × GSpin(T Q )/ GSpin(VQ )/ SO(VQ ).

SO(AlgQ ) × SO(T Q )As long as ρV` : ΓF → SO(V` ) factors through SO(Alg` ) × SO(T ` ), this shows that we can lift Alg`and T ` in order to lift V` . We next want to understand the restriction to GSpin(AlgE )×GSpin(T E ) of
the spin representation of GSpin(VE ); Just for this argument, we will ignore the similitude factor,
i.e. work with weights of Spin rather than GSpin. We can write bases of the character lattices ofSO(AlgE ), SO(T E ), and SO(VE ) as, respectively, χ1 , . . . , χa , χa+1 , . . . , χa+t , andχ1 , . . . , χa+t if either dim(T ) or dim(Alg) is even;χ1 , . . . , χa+t+1 if both dim(T ) and dim(Alg) are odd.
The set of weights of the spin representation of so(AlgE ) (and similarly for the other cases) is thenall 2a weights of the form
a
X
±χi
.
2
i=1
In the case where at least one of dim(T ) and dim(Alg) is even, we see that the weights of WAlg  WT
are precisely those of WV |GSpin(AlgE )×GSpin(T E ) , and therefore WV  WAlg  WT as GSpin(AlgE ) ×
GSpin(T E )-representations. When both dim(T ) and dim(Alg) are odd, weight-counting gives WV 
(WAlg  WT )⊕2 .
Thus, if we have found lifts ρ̃Algand ρ̃T` (to GSpin(Alg` ⊗ Q` ) and GSpin(T ` ⊗ Q` )) such that`rspin,Alg ◦ ρ̃Alg
and rspin,T ◦ ρ̃T` are motivic (respectively, belong to compatible systems of `-adic`
representations), then the resulting lift to GSpin(V` ⊗ Q` ), in its spin representation, is a direct sum
19

Recall that objects of MF are not in general known to give rise to compatible systems.
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of tensor products of motivic Galois representations (respectively, Galois representations belonging
to compatible systems), hence is motivic.
Corollary 4.2.19. If det T ` = 1 as ΓF -representation, and if we can find a lift ρ̃T` of ρT` suchthat rspin,T ◦ ρ̃T` is motivic (respectively, belongs to a compatible system), then we can do the samefor ρV` .
Proof. Since Alg` is an Artin representation, Tate’s vanishing result allows us to lift to an Artin
representation ρ̃Alg` : ΓF → GSpin(Alg` ⊗ Q` ). The corollary follows.
Corollary 4.2.20. As in Corollary 4.2.19, assume that det T ` = 1, and that for some number
field E, and place λ|` of E, we can find a lift ρ̃Tλ : ΓF → GSpin(T ` ⊗Q` Eλ ) such that rspin,T ◦ ρ̃Tλ is
the λ-adic realization of an object M of MF,E whose base-change to some F 0 /F is one of the spin
direct factors of the Kuga-Satake motive (with scalars extended to E) associated to T Q (see Lemma
4.2.10). Then possibly enlarging E to a finite extension E 0 , there is a lifting of representations of
the motivic Galois group GF,E 0 :GSpin(VQ ⊗ E 0 )7

GF,E 0ρV/ SO(VQ ⊗ E 0 ).

Proof. The Artin representation ρAlg
` is definable over Q, and lifts to GSpin(AlgE1 ) after making
some finite extension E1 /Q. The conclusion of the corollary will hold with E 0 equal to the composite E1 E. We check it using the same principle as in the proof of Proposition 4.1.30: that is, we check‘geometrically’ (for the restriction to GF,E 0 ) and for the λ-adic realization. By Corollary 4.2.9, themotivic representation ρA(T ) of GF , and by extension of GF,E 0 , factors through GSpin(T E 0 ) and lifts
ρT : GF,E 0 → SO(T E 0 ). Since H 1 (A(T )) ⊗Q E 0 is just some number of copies20 of M, the same istrue of ρ M . By assumption, the λ-adic realization ρλM is just rspin ◦ ρ̃Tλ , so this also factors throughGSpin(T λ ), lifting ρTλ . Using the section sλ : ΓF → GF,E 0 (Eλ0 ), so that GF,E 0 (Eλ0 ) = sλ (ΓF ) · GF,E 0 (Eλ0 ),we conclude as in Proposition 4.1.30 that ρ M lifts ρT ⊗ E 0 .
Combining ρ M with the lift of ρAlg
` , as in Lemma 4.2.18, we similarly find a lift to GSpin(VQ ⊗Q
0VE ) of our given ρ : GF,E 0 → SO(VQ ⊗Q E 0 ).

4.2.5. Arithmetic descent: the generic case. To summarize, we have reduced the problemof finding motivic lifts of the motive (over F) Prim2k (X)(k) to the corresponding problem for thetranscendental lattice T . In this section we treat the ‘generic’ case in which the Hodge structure T Q
has trivial endomorphism algebra, using a variant of Ribet’s method (which will return in §4.2.6)
We isolate this case both to demonstrate a slightly different argument, and because the non-genericcases will require even deeper input, André’s proof of the Mumford-Tate conjecture in this context(see Theorem 4.2.27). The starting-point of the analysis of the motive T is Zarhin’s calculation in[Zar83] of the Mumford-Tate group:
Proposition 4.2.21 (Zarhin). Let T Q be a Q-Hodge structure with orthogonal polarization and
Hodge numbers h1,−1 = 1, h0.0 > 0, and h p,q = 0 if |p − q| > 2. Moreover assume that T Q contains
20 t

2 for dim T odd; 2t−1 for dim T even.
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no copies of the trivial Q-Hodge structure. Then ET = EndQ-HS (T Q ) is a totally real or CM field,
and T Q is a simple MT (T Q )-module. There is a non-degenerate ET -hermitian21 pairingh·, ·i : T Q × T Q → ETsuch that
MT (T Q ) = Aut(T Q , h·, ·iE ) ⊂ SO(T Q ).
For the rest of this section, we assume that EndQ-HS (T Q ) = Q; the Mumford-Tate group MT (T Q )is then the full SO(T Q ).
Lemma 4.2.22. Assume EndQ-HS (T Q ) = Q. Then the Mumford-Tate group, and therefore themotivic Galois group, of the Kuga-Satake variety A(T ) is equal to GSpin(T Q ). Consequently,
End0 (A(T )) = CQ+  C + (T Q ).
Proof. Easy.

We now choose a number field E/Q splitting C + (T Q ), and consider the decomposition in AVF0 0 ,Et

if dim(T Q ) = 2t + 1;

2t−1

∼ (B+ (T ) × B− (T ))2

A(T ) ∼ B(T )2
A(T ) ∼ B(T )t−1

if dim(T Q ) = 2t,

as in Lemma 4.2.10. We saw in Corollary 4.2.9 that ρA(T ) factors through GSpin(T Q ) and lifts
ρT ; viewing ρA(T ) in GSpin(T Q ), we then have the relation rspin ◦ (ρA(T ) ⊗ E) = ρB(T ) , taking theBetti realization of B(T ) to be our model for the spin representation. We let B0 equal B(T ) in
the odd case and B+ (T ) in the even case. Similarly, we let r0 denote rspin or one of the half-spin
representations (which we may assume corresponds to B+ (T )). We also for convenience fix anembedding E ,→ Q` .
Lemma 4.2.23. (Without any assumption on EndQ-HS (T Q )) There exist a liftρ̃T` : ΓF → GSpin(T ` ⊗ Q` )of ρT` and a finite extension F 00 /F 0 such thatrspin ◦ ρ̃T` |ΓF00  H 1 (B(T )F , Q` ) ⊗E Q` |ΓF00 .
Proof. As in the arguments of §3.2, we first choose a lift ΓF → GSpin(T ` ⊗ Q` ) with finiteorder Clifford norm. In the root datum notation of §2.8, the Hodge-Tate cocharacters µτ of ρT` , forall τ : F ,→ Q` , are (conjugate to) λ1 , and the finite-order Clifford norm lifts have Sen operators(up to conjugacy) corresponding to the elements µ̃τ = λ1 of the co-character lattice tensored with×
Q. We can modify this initial lift by twisting by λ0 ◦ ω0 , where ω0 : ΓF → Q` is a Galois character
whose square differs from the inverse ω−1 of the cyclotomic character by a finite-order twist (suchω0 exists for any F). This gives a new lift ρ̃T` whose composition with the Clifford norm (which,recall, is 2χ0 in our notation) is ω−1 , up to a finite-order twist, and whose τ-labeled Hodge-Tateweights in the spin representation are 1 and 0 with equal multiplicity (compare the argument of§4.1.1). Then rspin ◦ ρ̃T` |ΓF0 differs from H 1 (B(T )F , Q` ) ⊗E Q` by a finite-order twist, hence they are
isomorphic after some additional finite base-change F 00 /F 0 .

21In the case of totally real ET , this means symmetric.
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Lemma 4.2.24. There is a factor M of ResF 00 /F (B0 ) (viewed as an object of AVF,E), having00
endomorphisms by a finite extension E /E, and an embedding E ,→ Q` extending our fixed E ,→Q` such that the associated `-adic realization M` := H 1 (MF , Q` ) ⊗E 0 Q` is isomorphic as Q` [ΓF ]representation to r0 ◦ ρ̃T` .

Proof. By the (E-linear) Tate conjecture,
0
EndAVF,E
ResF 00 /F (B0 ) ⊗E Q`  EndQ` [ΓF ] IndFF 00 (H 1 (B0,F , Q` ) ⊗E Q` ) .
The Galois representation being induced is r0 ◦ ρ̃T` , and inside this endomorphism ring we canconsider
HomQ` [ΓF ] r0 ◦ ρ̃T` , IndFF 00 (r0 ◦ ρ̃T` ) ,which by Frobenius reciprocity is justEndQ` [ΓF00 ] r0 ◦ ρ̃T` = Q` ,

0
ResF 00 /F (B0 ) ⊗E Q`
since EndAV 0 00 (B0 ) = E. In other words, there is a unique Q` -line in EndAVF,E
F ,E
consisting of projectors onto the r0 ◦ ρ̃T` -isotypic piece of the `-adic representation. Decomposing0
the semi-simple E-algebra EndAVF,E
ResF 00 /F (B0 ) into simple factors, we see that this line lives ina unique simple component (tensored with Q` ), which itself must be just a finite field extension E 0of E (else the r0 ◦ ρ̃T` -isotypic piece would have multiplicity greater than 1); it then corresponds
to exactly one of the simple factors of E 0 ⊗E Q` , i.e. a particular embedding E 0 ,→ Q` . We
can therefore take M to be the abelian variety corresponding to this factor E 0 ; M has complex
multiplication by E 0 , and via this specified embedding E 0 ,→ Q` , the `-adic realization M` isisomorphic to r0 ◦ ρ̃T` .

Remark 4.2.25. This is at its core the same proof as given in §4.2.3; the latter proof is probably
more transparent, but this one is somewhat ‘softer.’ I don’t think it translates as well to the moregeneral context of §4.2.6, however.
Corollary 4.2.26. Theorem 4.2.13 holds for the motive Prim2k (X)(k) over F.
Proof. When dim(T Q ) is odd, we are done, by the previous lemma and Corollaries 4.2.19 and
0
4.2.20. When dim(T Q ) is even, we take the output M in AVF,E
0 of the previous lemma, view it∨
in MF,E 0 , and form the twisted dual M (−1). Here there are two cases: if dim T Q is not divisible
by four, this object corresponds to the composition of the other half-spin representation with ρ̃T` :P
the two half-spin representations of GSpin2n have highest weights −χ0 + 21 ( n−1i=1 χi + χn ) (for r0 )1 Pn−1and −χ0 + 2 ( i=1 χi − χn ) (for the other half-spin representation), so the lowest weight of r0∨ ⊗P(−2χ0 ) is −χ0 − 12 ( ni=1 χi ), which, when n is even, is visibly the lowest weight of the other halfspin representation. In that case, we can apply the earlier corollaries to M ⊕ M ∨ (−1). If dim T Q
is divisible by four (so the half-spin representations are self-dual), apply Lemma 4.2.24 to thecomposition of ρ̃T` with the other half-spin representation as well; so instead of a single motive
M, we now have two motives M+ and M− (enlarge the coefficient fields of M+ and M− , viewed
as subfields of Q` via the respective embeddings produced by Lemma 4.2.24, to some common
over-field) such that the `-adic realization (M+ ⊕ M− )` is isomorphic to rspin ◦ ρ̃T` . Then as before,
we may apply Corollary 4.2.20.

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4.2.6. Non-generic cases: dim(T Q ) odd. To study the non-generic case EndQ-HS (T Q ) , Q,
we have to understand the `-adic algebraic monodromy groups of the representations ρT` , i.e. the`-adic analogue of Zarhin’s result. André has proven ([And96a, Theorem 1.6.1]) the MumfordTate conjecture in this context; a gap in the argument of [And96a, §7.4] is completed in a preprint([Moo15]) of Ben Moonen, in the course of generalizing some of André’s work. Here is the precise
result:Theorem 4.2.27 (André). Let (X, η) be a polarized variety over a finitely-generated extension
Zar
F of Q satisfying Ak , Bk , and B+k . Then the inclusion ρ` (ΓF ) ,→ GVF ⊗Q Q` of the algebraic
monodromy group into the `-adic motivic Galois group of V = Prim2k (X)(k) is an isomorphism onconnected components of the identity.
Zar

Recall that over the field F 0 , we may assume the groups ρ` (ΓF 0 ) and GVF 0 are connected,
and therefore isomorphic. Combining Theorem 6.5.1 of [And96a] with André’s result that Hodge
cycles on abelian varieties are motivated, and with Zarhin’s description ([Zar83]) of the MumfordTate group of the transcendental lattice T 2k (XC , Q)(k), we obtain (see Corollary 1.5.2 of [And96a]):Corollary 4.2.28. The semisimple Q-algebra ET := EndGT 0 (T ) is a totally real or CM field,F
and there is a natural ET -hermitian pairing h·, ·iET : T × T → ET . The motivic group (which equals
the Mumford-Tate group, and equals, after ⊗Q Q` , the `-adic algebraic monodromy group) GTF 0 isthen isomorphic to the full orthogonal (ET totally real) or unitary (ET CM) group
Aut(T, h·, ·iET ) ,→ SO(T Q ).
Before continuing, we formulate a variant of Ribet’s method with coefficients:
Proposition 4.2.29. Let F 0 /F be an extension of number fields, and let E/Q be a finite extension. Suppose we are given an object B of AVF0 0 ,E such that for some embedding E ,→ Q` , theassociated `-adic realization B` = H 1 (BF , Q` ) ⊗E Q` satisfies the invariance condition σ B`  B`
for all σ ∈ ΓF . Further assume that EndAV 0 (BF ) = E (in particular, EndAV 0 0 (B) = E). Then
F ,E

F,E

there exist a finite extension E 0 /E, an extension E 0 ,→ Q` of the embedding E ,→ Q` , a number0
field F 00 /F 0 , and an object M of AVF,E
0 such that
H 1 (MF , Q` ) ⊗E 0 Q` |ΓF00  B` |ΓF00 .
That is, B, up to twist, has a motivic descent to F.
Proof. This is proven as on page 122, using the E-linear variant of Faltings’ theorem:∼− HomQ` [ΓF0 ] H 1 (σ BF , Q` ) ⊗E Q` , H 1 (BF , Q` ) ⊗E Q` .
HomAV 0 0 (σ B, B) ⊗E Q` →F ,EIn the notation of the earlier proof, M = ResF 00 /F (B) ⊗R,α Q(α) is the required motivic descent, toan isogeny abelian variety over F with Q(α)-multiplication (so in the statement of the proposition,
E 0 ,→ Q` is Q(α) ⊂ Q ,→ Q` , extending the initial E ,→ Q` ).

We now assume that dimQ T Q is odd, say of the form cd where d = 2d0 + 1 = [ET : Q]. In
particular, ET is totally real. Let ρ̃T` and B(T ) be as in Lemma 4.2.23, and replace F 0 by a large
enough extension to satisfy the conclusion of that lemma, and such that (ρT` (ΓF 0 ))Zar is connected.
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Recall that B(T ) is an object of AVF0 0 ,E for some finite extension E/Q large enough to split C + (T Q ).
Corollary 4.2.28 implies that
d
Y
Zar
T
SOc (Q` ) ⊂ SOcd (Q` ).
(ρ` ⊗ Q` )(ΓF 0 ) 
1QRestricting the spin representation Wcd of socd (Q` ) to d1 soc (Q` ), we obtain (via a weight calculation as in Lemma 4.2.18, and writing Wc for the spin representation of soc )d

Wcd |Q soc  (d1 Wc )2 0 .
The Lie algebra of the lift ρ̃T` is one copy of the additive group ga times the Lie algebra of ρT` ⊗ Q` ,and this ga acts by scalars in the spin representation, so rspin ◦ ρ̃T` |ΓF0 has an analogous decomposition
as 2d0 copies of some (absolutely, Lie) irreducible representation W 0 of ΓF 0 . For a suitable enlargement E 0 of E (and extension E 0 ,→ Q` ), and F 00 of F 0 , we can realize W 0 as H 1 (B(T )0 , Q` )⊗E 0 Q` foran object B(T )0 of AVF0 00 ,E 0 . More precisely, we need the decomposition of Wcd |Q soc to be definedover E 0 ; the first claim then follows from Faltings. The extension F 00 /F 0 is needed to decompose B(T )F ⊗E E 0 over some finite extension. Then EndAV 0 00 0 (B(T )0 ) = E 0 , and the isomorphism
F ,Erspin ◦ ρ̃T`  (W 0 )2 0 implies ΓF -conjugation invariance of W 0 . Thus we can apply Ribet’s method todeduce:dLemma 4.2.30. There exist a finite extension E 00 /E 0 , an embedding E 00 ,→ Q` extending the0
given E 0 ,→ Q` , and an object M of AVF,E
00 such that
H 1 (MF , Q` ) ⊗E 00 Q` |ΓF000  W 0 |ΓF000
for some still further finite extension F 000 /F 00 .
Let M` denote the associated `-adic realization (via E 00 ,→ Q` ). Since rspin ◦ ρ̃T` is Lie-isotypic,
and M` is a descent to ΓF of its unique (after finite restriction) Lie-irreducible constituent, Corollary
3.4.2 shows that there is an Artin representation ω of ΓF such thatrspin ◦ ρ̃T`  M` ⊗ ω.
Possibly enlarging the field of coefficients yet again, we deduce:
Theorem 4.2.31. Suppose dim T Q is odd, and that for some (hence for all) `, det T ` = 1 as
ΓF -representation. Then there exist
e and an embedding Ẽ ,→ Q` ;• a number field E
e of MF,Ee that is a tensor product of an Artin motive and (the image of) an• an object M
object of AVF,0 Ee;• and a lifting ρ̃T : GF,Ee → GSpin(T Ee) of ρT ;e On λ̃-adic realizations (for places λ̃ of E),
e ρ̃T gives risesuch that rspin ◦ ρ̃T is isomorphic to M.
eλ̃ such that {rspin ◦ ρ̃T }λ̃ is a weakly-compatible system. Moreover, the same
to lifts ρ̃Tλ̃ of ρT` ⊗Q` E
λ̃conclusions hold with the motive V = Prim2k (X)(k) in place of its transcendental lattice T (and,eagain, a possible enlargement of E).
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Version 4 - Submitted to MEMOe is the composite (inside the ambient Q` ) of the field E 00 and theProof. The number field E
field needed to define the Artin representation ω. To conclude the proof of the theorem, we makethree observations:e and ω are motivic);• MF,Ee is Tannakian (note that we already know that M ⊗E 00 E
• M and ω both give rise to compatible systems of `-adic representations;• Corollary 4.2.20 applies to lift the representations of motivic Galois groups.

4.2.7. Non-generic cases: dim T Q even. We do not fully treat the case of even-rank transcendental lattice, but here give a couple examples, describing the ‘shape’ of the Galois representations
in light of Proposition 3.4.1.
First, continue to assume ET is totally real. Let dim T Q = 2n = cd, with d = [ET : Q]. For any
N, denote by W2N,± (for each choice of ±) the two half-spin representations of so2N , and continue
to write W2N+1 for the spin representation of so2N+1 .
Lemma 4.2.32. When c is even, the restriction Wcd,+ |Q soc is given byM
Wcd,+ |Qd1 soc di=1 Wc,i ,d
=(Qi )∈{±}
i =1

where the indexing set ranges over all choices of signs with “−” occurring an even number oftimes. This is a direct sum of distinct Lie-irreducible representations.
When c is odd, so d = 2d0 is even,
Wcd,+ |Qd1 soc  2d0 −1 d1 Wc ,a single Lie-irreducible representation occurring with multiplicity 2d0 −1 .
Now, recall (Lemma 4.2.23) that after a sufficient base-change F 0 /F, we can find an abelianvariety B+ (T ) (with coefficients in a number field E, embedded in Q` ; we may by extending scalars
assume E is large enough that the above decomposition of spin representations is defined over E),
and a lift ρ̃T` such that r+ ◦ ρ̃T` |ΓF0 is isomorphic to H 1 (B+ (T ), Q` ) ⊗E Q` . We assume F 0 sufficiently
large that this Galois representation is a sum of Lie-irreducible representations.
Proposition 4.2.33. Suppose c is even. Then motivic lifting holds for ρT .
Proof. Since c is even, the previous lemma shows that r+ ◦ ρ̃T` is Lie-multiplicity-free, hence
is a direct sum of inductions of non-conjugate, Lie-irreducible Galois representations. If π0 (ρT` ) is
trivial, in which case ρ̃T` also has connected monodromy group,22 then no inductions occur in thisdecomposition, so each factor d1 Wc,i in Lemma 4.2.32 corresponds to a Lie-irreducible factor ofr+ ◦ ρ̃T` , as ΓF -representation. Each of these factors is, over F 0 , of the form
H 1 (B , Q` ) ⊗E Q` ,where B is an object of AVF0 0 ,E with endomorphism algebra just E itself (the usual application of
Faltings’ theorem, using the Lie-multiplicity-free property, and the fact that the spin representation
decomposition holds over E). By ΓF -invariance of each factor, we can apply Proposition 4.2.29 to1−kSince it contains the center of GSpin; the sort of example this avoids is ρ f,` ⊗ω 2 , where f is a classical modular
form of odd weight k.
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0
produce an object M of AVF,E, for some finite extension E /E inside Q` , with M,` |ΓF0 isomorphic
to a finite-order twist of H 1 (B , Q` )⊗E Q` . By Lie-irreducibility, some finite-order twist (a characterof Gal(F 0 /F), in fact, and, again, we may have to enlarge E ) M0 of M has `-adic realization
isomorphic to the corresponding factor of r+ ◦ ρ̃T` . Inside the ambient Q` , we take the compositum
E 0 of the various E , extending scalars on each M0 . Then the object ⊕ M0 of MF,E 0 satisfies the
hypotheses of Corollary 4.2.20, so we deduce the existence of the desired motivic lift.
We sketch the case of non-connected monodromy. Take the (motivic) Lie-irreducible factorsof r+ ◦ ρ̃T` |ΓF0 , and partition them into ΓF -orbits. Fix a representative Wi of each orbit, and consider
the stabilizer ΓFi in ΓF of Wi . Arguing as in Proposition 4.2.33, we can apply Ribet’s method todescend each Wi to an object Mi of MFi ,Ei for some extension Ei of E inside Q` . Moreover, twistingyields an Mi0 whose `-adic realization is isomorphic to a factor of r+ ◦ ρ̃T` : the isomorphisms
HomΓF0 (M1 , V) = HomΓFi (IndFFi0 (Mi ), V) = HomΓFi (Mi ⊗ Q` [Gal(F 0 /Fi )], V), with V = r+ ◦ ρ̃T` ,imply this claim, using the fact that Mi is Lie-irreducible and after finite restriction occurs withmultiplicity one in r+ ◦ ρ̃T` . Then we can induce (the representation of motivic Galois groups) fromFi to F to obtain our motives over F. This completes the case of even c (and ET totally real). 

Having demonstrated the available techniques in a couple of quite different situations (namely,
where the lifts range between the extremes of being Lie-multiplicity-free and Lie-isotypic), westop here, remarking only that when ET is CM, the analysis must begin not from Lemma 4.2.32 butfrom the restriction Wcd,+ |Q glc , the product ranging over pairs of complex-conjugate embeddings
ET ,→ Q` . This restriction is given by:
Lemma 4.2.34. Suppose d = 2d0 is even, with notation otherwise as above. We denote irreducible representations of glc by W(r), where W is an irreducible representation of slc , and (r)indicates that the restriction to the center ga ⊂ glc is multiplication by r. Then, letting Vi denotethe standard representation of the ith copy of slc
Mcc
Wcd,+ |Qd0 gl 
∧i1 V1∗ ( − i1 )  . . .  ∧id0 Vd∗0 ( − id0 ).
c122
i ,...,i :1Pd0

i j ∈2Z

Remark 4.2.35.
• Note that the representations occurring here do not necessarily extend
to representations of GLc , since c may be odd; they do extend on the (connected) doublecover of GLc .
• In particular, we see that when ET is CM, r+ ◦ ρ̃T` is Lie-multiplicity-free. This suggests
proceeding as in Proposition 4.2.33, although we will stop here.
4.3. Towards a generalized Kuga-Satake theory
4.3.1. A conjecture. It is fair to assume that one could establish a motivic lifting result for
the remaining hyperkähler cases. More important, these lifting results clamor for generalization.
Motivated by the Fontaine-Mazur conjecture, Theorem 3.2.10, Proposition 4.1.30, and Theorem
4.2.31, we are led to the following much more ambitious conjecture:e  H be a surjection, with central
Conjecture 4.3.1. Let F and E be number fields, and let H
torus kernel, of linear algebraic groups over E. Suppose we are given a homomorphism ρ : GF,E →eE 0
H. Then if F is imaginary, there is a finite extension E 0 /E and a homomorphism ρ̃ : GF,E 0 → H
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lifting ρ ⊗E E 0 . If F is totally real, then such a lift exists if and only if the Hodge number parity
obstruction of Corollary 3.2.8 vanishes.
To indicate the scope of this conjecture, let X/F be any smooth projective variety, and consider for any k ≤ dim X the motive H 2k (X)(k) (or Prim2k (X)(k), having chosen an ample line
bundle). This gives rise to an orthogonal representation of GF , and the conjecture in this case (forGSpin → SO, or the variant with the full orthogonal group in place of SO) amounts to a generalization of the Kuga-Satake construction to arbitrary orthogonally-polarized motivic (over F)
e and H (for instance, GLn → PGLn , where necessarily we
Hodge structures. For other choices of H
will have coefficient field larger than Q), the conjectured generalization is even more mysterious(compare Lemma 4.1.33).
e → H beNote the role of Lemma 4.1.31 in building our confidence in this conjecture. Let H
a morphism of groups, say over Q, as in the conjecture. One way of formulating the FontaineMazur conjecture is that GF ⊗Q Q` should be isomorphic to the Tannakian group for the Tannakian
category Repg,ss (ΓF ) of semi-simple geometric Q` -representations of ΓF . In particular, a geometricQ`e ` ) of ρ` : ΓF → H(Q` ) should arise from an algebraic homomorphism of Q` lift ρ̃` : ΓF → H(Qe . Lemma 4.1.31 then tells us that if ρ arises from some GF ⊗ Q → H, thengroups GF ⊗ Q` → H
Q`e of homomorphisms of Q-groups.
we can find a lift ρ̃ : GF ⊗ Q → H
Now that we know to look for such a thing, we conclude by giving one more example in which
it is easy to construct a ‘generalized Kuga-Satake motive.’
4.3.2. Motivic lifting: abelian varieties. If A/C is an abelian surface, then H 2 (A, Q) 
∧ H 1 (A, Q) has Hodge numbers h2,0 = 1 and h1,−1 = 4. The classical Kuga-Satake construction then associates to A another abelian variety, KS (A), with rational cohomology C + (H 2 (A, Q))(or the analogue with Prim2 (A, Q), if we have fixed a polarization), and a theorem of Morrison([Mor85]) asserts that KS (A) is isogenous to A8 (or A4 if we work with Prim2 (A, Q)). We now
show that this construction can be generalized to abelian varieties of any dimension.
Let F be any subfield of C, and let A/F be an abelian variety of dimension g with a fixedpolarization. Consider the algebraic representation2

ρ : GF → GSp(H 1 (AC , Q))  GSp2g ,or, for the theory over C, its restriction to GC . We will compose ρ with a homomorphism GSp2g →GSpinN for suitable N to produce the generalized Kuga-Satake motive. Let W = H 1 (AC , Q), andlet re1 +e2 : Sp(V) → GL(Ve1 +e2 ), in the weight notation of §2.8, denote the irreducible representation of Sp(V) obtained as the complement of the trivial representation in ∧2 (W). This absolutely irreducible representation, defined over Q, has image in SO(Ve1 +e2 ), where the quadratic
form is induced from the pairing canonically induced on ∧2 (W), and which coincides with theinduced polarization on Prim2 (AC , Q). Since Sp(W) is simply-connected, there is a lift to an algebraic homomorphism r̃ : Sp(W) → Spin(Ve1 +e2 ). Over C this follows from standard Lie theory,and such a homomorphism descends to Q (see Lemma 4.1.31); since the map of Lie algebrassp(W) → so(Ve1 +e2 ) is defined over Q, we see that theover Q is ΓQ -invariant, hence descends2gmap
to a morphism over Q. The dimension of Ve1 +e2 is 2 − 1, which is odd (= 2n + 1) if g is even,133

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and even (= 2n) if g is odd. In either case, we have the representation r$n of Spin(Ve1 +e2 )Q having
highest weight $n =i=1 χiPn

2.23

Lemma 4.3.2. Let c denote the non-trivial (central) element of the kernel of Spin(Ve1 +e2 ) →SO(Ve1 +e2 ). If g ≡ 2, 3 (mod 4), then r̃(−1) = c, and if g ≡ 0, 1 (mod 4), then r̃(−1) = 1.
Proof. In all cases, re1 +e2 (−1) = 1, so r̃(−1) equals either 1 or c. Since c acts as −1 in any of thespin representations (c is the element −1 of the Clifford algebra), it suffices to compute r$n ◦ r̃(−1).
The weights of Ve1 +e2 are{±(ei + e j )}1≤i< j≤g ∪ {ei − e j }i, j ,except with one copy of the weight zero deleted (so that zero has multiplicity g − 1 rather than g).
It follows that r$n ◦ r̃ has a weight equal to


g
X
 X
1  X
(ei + e j ) +(ei − e j ) =(g − i)ei .
2
1≤i< j≤g

1≤i< j≤g

In particular, r$n ◦ r̃(−1) is multiplication by (−1)

g(g−1)/2

i=1, and the lemma follows.



Corollary 4.3.3. r̃ : Sp(W) → Spin(Ve1 +e2 ) extends to an algebraic homomorphism GSp(W) →GSpin(Ve1 +e2 ). If g ≡ 2, 3 (mod 4), then this map can be chosen so the Clifford norm coincides
with the symplectic multiplier; if g ≡ 0, 1 (mod 4), then this map can be chosen to factor throughSpin(Ve1 +e2 ). The composition
ρ

r̃

GF →
− GSp(W) →− GSpin(Ve1 +e2 ) ,→ GL(C + (Ve1 +e2 ))defines the generalized Kuga-Satake lift of A.
Proof. This follows immediately from Lemma 4.3.2 and the identifications:
Gm × Sp(W) ∼→− GSp(W)h(−1, −1)i
Gm × Spin(Ve1 +e2 ) ∼→− GSpin(Ve1 +e2 ).
h(−1, c)i
When g ≡ 2, 3 (mod 4), we take the map Gm → Gm to be the identity, and when g ≡ 0, 1 (mod 4),
we take it to be trivial.
Remark 4.3.4.
• Repeating the above arguments with ∧2 (W) in place of Ve1 +e2 , we cansimilarly construct lifts GF → GSpin(∧2 (W)).
• When g = 2, this recovers the classical construction. In that case, the composition r$n ◦ r̃is the identity, and, decomposing C + (Ve1 +e2 ) as 4 copies of r$n (as GSpin(Ve1 +e2 ) representation), the identification (up to isogeny) KS (A) ∼ A4 is nearly a tautology.
The motivic formalism now tells us that r$n ◦ r̃ ◦ ρ is (the Betti realization of) an object of
MF . Since the `-adic realizations of ρ form a weakly (in fact, strictly) compatible system, thesame is true for the `-adic realizations of this Kuga-Satake motive. In this case, however, we cansay more, and will realize this explicitly (and unconditionally) as a Grothendieck motive. The firststep is to compute the plethysm r$n ◦ r̃; we will do this, but first mention an equivalent, structurally
23

We use the common fundamental weight notation here. This is the spin representation in the odd case; one of
the half-spin representations in the even case.
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appealing plethysm. First we treat the Dn case, that is when d = 2g = dim H 1 (X) satisfies 2n = d2 ;this amounts to dim X being even. Let V = H 2 (X) = ∧2 W. We use the following (common)notation for fundamental weights of Dn :• $i = χ1 + . . . + χi , and r$i P= ∧i V, for i = 1, . . . , n − 2;nχ
• $n−1 = χ1 +...+χ2 n−1 −χn , $n = 12 i , and r$n−1 , r$n are the two half-spin representations.
As representations of so2n , we have the following identity:2nM∧i (V) = r$n−1 ⊕ r$n ⊗2 .

i=0

The plethysm problem that we expect to solve, then, is to describe a representation rKS (X) of Sp(W)such that2nM∧i (∧2 W) = (rKS (X) )⊗2 .
i=0(More ambitiously, we could attempt this with ∧2k (W) instead.) Similarly,
in the Bn case, we havePPiχifundamental weights $i = 1 χi for i = 1, . . . , n − 1, and $n = 2 , corresponding, respectively
to the wedge powers ∧i V of the standard representation (r$1 ) and the spin representation. As
representations of so2n+1 , we have the identitynM⊗2∧i (V) = r$,ni=0so in this case we want a representation rKS of Sp(W) satisfyingnM⊗2∧i (∧2 (W)) = rKS.
i=0

Proposition 4.3.5. Writing ωi (i = 1, . . . , g) for the usual fundamental weights24 of Cg , then in
all cases (g odd or even) we have∧• (∧2 (W))  2g (Vω1 +...+ωg−1 )⊗2 ,as Sp(W)-representations. Here ∧• denotes the full exterior algebra. This is deduced from theabove discussion and the two calculations:• (even) As Sp(W)-representations,(r$n−1 ⊕ r$n ) ◦ r̃  2g/2 Vω1 +...+ωg−1 .
• (odd) As Sp(W)-representations,r$n ◦ r̃  2(g−1)/2 Vω1 +...+ωg−1 .
Proof. We treat the even case, the odd case being essentially identical. There are g2 −g non-zeroweights in ∧2 (W), and the weight zero occurs with multiplicity g. The weights of (r$n−1 ⊕ r$n ) ◦ r̃
are sums of plus or minus any n = g2 − 2g weights of ∧2 (W), then the total divided by two. It follows
P
that the highest weight of (r$n−1 ⊕ r$n ) ◦ r̃ is, as previously computed, g−1
i=1 ωi , but moreover thatgg/2
2
it occurs with multiplicity 2 (here 2 = n − (g − g); we can choose the weight +0 or −0 thisωi = e1 + . . . + ei in the standard coordinate system

24

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many times). It follows that (r$n−1 ⊕ r$n ) ◦ r̃ contains 2g/2 Vω1 +...+ωg−1 ; by dimension count (see thefollowing Lemma 4.3.6), they are isomorphic.
Lemma 4.3.6. With the above notation, the dimension of the irreducible Sp(W)-representationVω1 +...+ωg−1 is 2g(g−1) .
Proof. Simplifying the Weyl dimension formula, we findY 2g + 1 − (i + j)+,
dim Vω1 +...+ωg−1 = 2|Φ |2g + 2 − (i + j)1≤i≤ j≤gwhere the number |Φ+ | of positive roots is 2g . The product telescopes: fix an i, and the corresponding product over j is equal to 12 . The lemma follows.

2Corollary 4.3.7. Let F be any subfield of C, and let X/F be an abelian variety of any dimension g, giving rise to a representation
ρH1 (X) : GF → GSp(H 1 (X, Q)).
Then the Kuga-Satake lift (Corollary 4.3.3 and remark following) of the representation
GF → SO(H 2 (X, Q)(1))
gcan be explicitly realized in the spin (or sum of half-spin) representation as 2b 2 c copies of the composition rω1 +...+ωg−1 ◦ ρH1 (X) with the highest weight ω1 + . . . + ωg−1 representation of Sp(H 1 (X, Q)).25
Let us call this object of MF the Kuga-Satake motive KS (X). Then KS (X) is in fact a Grothendieckmotive, for either numerical or homological equivalence.
Proof. It remains to check that KS (X) can be cut out by algebraic cycles. We start with theexplicit description (due to Weyl; see §17.3 of [FH91]) of the representation Vω1 +···+ωg−1 . From nowP
Pg−1
on, abbreviate λ = g−1
1 ωi , and r =1 i (the length of λ); in fact, what follows applies to anarbitrary partition λ. ThenVλ = W hri ∩ Sλ (W)as Sp(W)-representation. We have to explain this notation: Sλ (W) denotes the Schur functor associated to the partition λ, which explicitly is equal to the image of the Young symmetrizer cλ acting
on W ⊗r ; and W hri is the subspace of W ⊗r given by intersecting the kernels of all the contractions(1 ≤ p < q ≤ r)
c p,q : W ⊗r →− W ⊗(r−2)
v1 ⊗ . . . ⊗ vr 7→ hv p , vq i · v1 ⊗ . . . ⊗ v̂ p ⊗ . . . ⊗ v̂q ⊗ . . . ⊗ vr ,where h·, ·i represents the symplectic form on W. The result will now follow from some basic facts
about algebraic cycles and, crucially, the Lefschetz and Künneth Standard Conjectures for abelianvarieties (due to Lieberman; for a proof, see [Kle68]). We will write πiX for the algebraic cycle
on X × X inducing H(X)  H i (X) ,→ H(X). Fix a polarization of X, giving rise to a Lefschetz
operator LX . Recall Jannsen’s fundamental result ([Jan92]) that numerical motives form a semisimple abelian category. This and the Künneth Standard Conjecture for abelian varieties imply thatthe category of numerical motives generated by abelian varieties over F is a semi-simple Tannakian
category. We deduce that the following are (numerical) sub-motives of H(X r ):25

Extended to GSp on the center by the prescription of Corollary 4.3.3. Throughout this argument we will ignore
these extra scalars to simplify the notation.
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Version 4 - Submitted to MEMO• M = (X, π1X , 0), i.e. the object corresponding to H 1 (X);• The kernel of each contraction H 1 (X)⊗r → H 1 (X)⊗(r−2) . For notational simplicity, takep = 1, q = 2. If we were working in cohomology, we would compute the kernel of c1,2 as


h·,·i
ker H 1 (X) ⊗ H 1 (X) −−→ Q(−1) ⊗ H 1 (X)⊗r−2 ,⊗2where recall that the polarization h·, ·i is defined by hw1 , w2 i = Ld−1=
X (w1 ∪w2 ). Now, W2
2
22
Sym (W)⊕∧ (W), and cup-product kills Sym (W), while mapping ∧ (W) isomorphicallyto H 2 (X). The kernel of c1,2 is therefore isomorphic to
Sym2 (H 1 (X)) ⊕ Prim2 (X) ⊗ H 1 (X)⊗r−2 .

To realize the analogous numerical motive, we can define Sym2 (M) as
1(X × X, (1 + (12)) · (π1X × π1X ), 0).
2
For the projector, we have taken the product of two commuting idempotents, the firstone being the Young symmetrizer associated to the Schur functor Sym2 . There is also a
projector p2 onto primitive cohomology (see 1.4.4, 2.3, and 2A11 of [Kle68]), so (X, p2 , 0)realizes Prim2 (X) as a numerical motive. Finally, since direct sums and tensor products
exist in our category, we can therefore describe the kernel of any c p,q as a numericalmotive M p,q .
• In general, the Young symmetrizer cλ is not quite an idempotent. Write c2λ = x−1 cλ (xdepends on λ, but we have fixed a λ), making xcλ an idempotent algebraic correspondence
on X r . cλ commutes with the r-fold Künneth projector (π1X )r (this is easily checked at thelevel of cohomological correspondences, so a fortiori holds for numerical equivalence),so(X r , xcλ · (π1X )r , 0)defines a numerical motive, denoted Sλ (H 1 (X)).
• We would like to conclude the proof by intersecting the objects M p,q (p < q) and Sλ (H 1 (X)).
This is possible for numerical motives since the category is abelian. Finally, since all of
the cycles considered in the proof are cycles on (disjoint unions of) abelian varieties incharacteristic zero, where numerical and homological equivalence coincide,26 we also deduce the existence of a homological motive corresponding to KS (X).
Remark 4.3.8. A weight calculation yields the Hodge numbers of KS (X).
4.3.3. Coda. Identifying among all rational Hodge structures those that are motivic is one of
the fundamental problems of complex algebraic geometry, and this generalized Kuga-Satake theory
would systematically construct new motivic Hodge structures from old, in a way not achievable bysimply playing the Tannakian game. I hope that investigation of these phenomena will provide a
stimulating testing-ground for thinking about Hodge theory in non-classical weights.
It is also tempting to ask what should be true if we replace F by other fields, especially finitelygenerated subfields of C. Our motivic descent in the hyperkähler case works as written, except
In the presence of the Standard Conjecture of Hodge type, the Lefschetz conjecture implies ‘num = hom’: see
Proposition 5.1 of [Kle94].
26

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Version 4 - Submitted to MEMOfor the critical absence of Tate’s basic vanishing result; in other contexts, it might be hoped thata similar descent works, conditional on the relevant cases of the Tate conjecture. I suspect only
a qualitative potential lifting result will hold in this generality–that would suffice to imply theanalogous lifting conjecture for C itself. But another clearly important question to ask is: what, if
any, is the analogue of Tate’s vanishing theorem when the number field F is replaced by any field
finitely-generated over Q?

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Index of symbols∗
Chom(X, Y), 101
C + , 117
C + (VZ ), 117•
Cmot(X, Y)E , 104
C F , 15
DdR , 23
DHT , 23
Fcm , 16G, 61
e 61, 71G,
G, 39, 61, 71
GFM , 105
GSpin, 55
G∨ , LG, 15
G∨sc , 72
e 50
H,
H, 50
HTτ , 25, 36
HTτ (ρ), 34
HB∗ , 101
KS (X), 118
KS (X), for an abelian variety X, 136
L(λ), 89
LF , 70
LZ , 117
L` , 118
MF , 104
MT (A), 106
mv , 27
Prim2k (X)(k), 115
Qcm , 16Q(π f ), 38
r0 , 127
recv , 15rspin,V , rspin,Alg , rspin,T : homomorphisms giving thespin representations associated to the various
orthogonal spaces V, Alg, T , 125rspin , 117
Sλ , 136
S , 71
Spin, 55
Te, 61(·)D , with argument a topological group, 16
e with argument a field, 16(·),∼ s , 78∼w , 15, 78∼ew , 78
∼w,∞ , 16, 78
ΓF , 15Θρ,ι , 26θρ,ι , 73
ι` , 15ι∗∞,` (τ), ι∗ (τ), 15
ι∞ , 15µv , νv , 2µιv , νιv , 2
πiX , 102σπ, 38
ρA , 118ρ`M , 117ρV , 118
ρ M , for a motive M: the associated motivic Galois
representation, 117τ∗`,∞ (ι), τ∗ (ι), 15
ω̃, 62
A: short-hand for the abelian variety AF 0 , 119
A(T ), 124A∗hom (X), 101
Ak , 115
AF 0 : André’s descent of the classical Kuga-Satake
abelian variety, 118
Alg: space of algebraic cycle classes as an object ofMF , 124
B: ‘spin representation’ abelian variety, 119
B(T ), 127
BdR , 23
BHT , 23
B+ , 119B− , 119
B0 , 127
Bk , 115
B+k , 115cλ , 136
C(VZ ), 117
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T : transcendental lattice as an object of MF , 124
TF , 70
TF,E , 112
tv , 27
VE , 116
VQ , 116
VZ , 117
V` , 116
W hri , 136
WE , 117
WV , WAlg , WT : spin representations associated to thevarious orthogonal spaces V, Alg, T , 125
Wn , for some odd integer n, 131
W+,E , 117
W−,E , 117
Wn,+ , Wn,− , for some even integer n, 131Z, 61
e 61, 71
Z,

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Index of terms and concepts
Hodge symmetry, 109
Hodge-Tate Galois representation, 23
Hodge-Tate-Sen weights, 25
hyperkähler variety, 115

L-packetarchimedean, 65
unramified, 65
W-algebraic automorphic representation, 3, 39
W-arithmetic, 40
Q-curves, 121`-adic Hodge theory property P, 5

image of a functorial transfer, 33
cyclic automorphic induction, 45
Tate’s lifting problem, 63
infinity-type of a Hecke character, 27
infinity-type of an automorphic representation, 38
infinity-types of automorphic representations:questions about, 50
infinity-types of Hecke characters, all possible, 32

algebraic correspondence, 101
archimedean purity lemma of Clozel, 41
Artin motive, 105
automorphic Langlands group, 70
C-algebraic, 37
C-algebraic automorphic representation, 3central character of an automorphic representation, 29
central character of an automorphic
representation–how to compute, 62
CM coefficients, 10
CM descent prototype, 28
CM field, 16
compatible system of `-adic representations valued inlinear algebraic group, 22
correspondence between automorphic representationsand Galois representations, 15K3 surface, 115Künneth Standard Conjecture, 102
Kuga-Satake construction, 118
Kuga-Satake motive associated to an abelian variety,136
L-algebraic, 37
L-algebraic automorphic representation, 3labeled Hodge-Tate weights, 24, 34, 36
labeled Hodge-Tate weights of a Hodge-Taterepresentation, 25
Lie irreducible, 86
Lie multiplicity free, 88
Lie-multiplictiy free, or LMF, 10
LMF, 88
local reciprocity map, 15

de Rham Galois representation, 23
equivalence of automorphic representations and`-adic representations, 78

mixed-parity Hilbert modular form, 40
mixed-parity Hilbert modular representation, 46
modified commutativity constraint for Mhom
F , 102
motivated correspondences, 104
motivated cycles, 103
motive for homological equivalence, 101
motives for motivated cycles, 104
motivic Galois group of an object M of MF , 105
Mumford-Tate conjecture for hyperkähler varieties,
129
Mumford-Tate group of a Q-Hodge structure, 106

fiber functor, 99
fiber of a functorial transfer
Tate’s lifting problem, 65
fibers of a functorial transfer
GL2 × GL2 tensor product, 44
Fontaine-Mazur conjecture, 12
Fontaine-Mazur-Langlands conjecture, 13
Fontaine-Mazur-Tate conjecture, 12Galois lifting problem, 4, 50
geometric Galois representation, 12
Grunwald-Wang special case, 63

neutral Tannakian category, 99
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polarized variety, 115
potential automorphy theorem, 53
primitive cohomology, as a motive, 115
pure weakly compatible system, 14
Ramanujan conjecture (archimedean), 40
regular automorphic representation for the group GLn ,38
regular filtered module, 37
rigid tensor category, 99
root data for Spin and GSpin groups, 56
Sen operator, 25, 26
solvable base-change, 33Standard Conjectures, 12
Taniyama group, 70, 112
Tate conjecture for hyperkähler varieties, 124
Tate’s theorem, 17
tensor category, 98transcendental lattice, or subspace, of the Hodge
structure Prim2k (XC , Q)(k), 116
type A Hecke character, 3, 28
type A0 Hecke character, 2, 28
weak transfer of automorphic representations, 65
weakly compatible system of λ-adic representations
valued in a reductive group, 14
weakly compatible system of λ-adic, or `-adic,
representations, 13

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Bibliography[AC89]

James Arthur and Laurent Clozel, Simple algebras, base change, and the advanced theory of the trace
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