GLOBALLY REALIZABLE COMPONENTS OF LOCALDEFORMATION RINGSFRANK CALEGARI, MATTHEW EMERTON, AND TOBY GEE

Abstract. Let n be either 2, or an odd integer greater than 1, and fix aprime p > 2(n + 1). Under standard “adequate image” assumptions, we showthat the set of components of n-dimensional p-adic potentially semistable local
Galois deformation rings that are seen by potentially automorphic compatible
systems of polarizable Galois representations over some CM field is independent of the particular global situation. We also (under the same assumption
on n) improve on the main potential automorphy result of [BLGGT14b], replacing “potentially diagonalizable” by “potentially globally realizable.”

Contents
1. Introduction
1.1. Our results
1.2. History and motivation
1.3. A sketch of the proofs
1.4. Acknowledgements
1.5. Notation, conventions, and background material
2. Globally realizable representations
2.1. Global realizability
3. Compatible systems
3.1. Component groups
3.2. Representation theory
3.3. A Lemma on 4-dimensional polarizable automorphic compatible
systems
3.4. Factorization of compatible systems
3.5. A technical lemma
4. Building Lifts
4.1. Deformations of a ⊗ b and strong irreducibility
4.2. Local Swapping
References

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The first author was supported in part by NSF Grants DMS-1404620 and DMS-1701703.
The second author was supported in part by the NSF grants DMS-1303450, DMS-1601871, andDMS-1902307. The third author was supported in part by a Leverhulme Prize, EPSRC grant
EP/L025485/1, Marie Curie Career Integration Grant 303605, ERC Starting Grant 306326, anda Royal Society Wolfson Research Merit Award.
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F. CALEGARI, M. EMERTON, AND T. GEE

1. Introduction
1.1. Our results. The results of this paper are motivated by the questions of levelraising/lowering for automorphic forms, and the weight part of Serre’s conjecture,
as well as a question about “automorphic components” related to the Fontaine–
Mazur conjecture and to global approaches to the p-adic local Langlands program.
However, our main results are stated purely in terms of Galois representations.
We need to introduce a few definitions before stating our main theorem. Let F
be a CM field with maximal totally real subfield F + , and write GF for the absolute
Galois group Gal(F /F ). (In this paper, all CM fields will be imaginary and so F/F +is a quadratic extension.)
We say that a representation s : GF → GLn (Fp ) is reasonable if it satisfies
the hypotheses needed to apply the automorphy lifting theorems of [BLGGT14b];that is, p > 2(n + 1), F does not contain a primitive pth root of unity, and s
is polarizable, odd, and is irreducible when restricted to GF (ζp ) . We say that a
representation s : GF → GLn (Qp ) is reasonable if its reduction mod p is reasonable.
We say that a compatible system of l-adic representations of GF is weakly
irreducible if for a positive density set of primes l, its l-adic Galois representations are irreducible. Conjecturally, this is equivalent to all of the l-adic representations being irreducible, but this seems to be very hard to prove; weak irreducibility is a well-behaved substitute for that stronger condition. By the resultsof [BLGGT14b, PT15], if a compatible system is odd, regular, and polarizable, thenit is weakly irreducible if and only if it is potentially automorphic (in the sense thatit potentially corresponds to a cuspidal automorphic representation).
The condition that the representation s is polarizable is best expressed in termsof the group Gn introduced in [CHT08]; it is a reductive group with connected component GLn × GL1 and component group of order 2, and as explained in [BG14,§8.3], it is very closely related to the C-group of an n-dimensional unitary groupover F + which splits over F . Then the polarizability of s is equivalent to the existence of a prolongation of s to a representation ρ : GF + → Gn (Fp ). In particular,
this implies that sc ∼= s∨ µ|GF + for some character µ of GF + .
Suppose that ρ : GF + → Gn (Fp ) is a prolongation of s, and let v be a finite placeof F + . A polarized component for ρ|GF + is, by definition, an irreducible componentvof a deformation ring for lifts of ρ|GF + which are of some fixed inertial type, andv
of a fixed regular Hodge type if v|p. If v|p, then we say that such a component isglobally realizable if it occurs globally, in the sense there is some totally real field L+∼ +with a CM quadratic extension L, a place w|p of L+ for which L+
w = Fv , and a(polarizable, odd, reasonable) representation r : GL → GLn (Qp ) which is part ofa (polarizable, odd, weakly irreducible) compatible system, which prolongs to arepresentation ρ0 : GL+ → Gn (Qp ) whose restriction ρ0 |GL+ gives rise to a point onw
this component (so that in particular, ρ0 |G ∼= ρ|G ). If the place v splits in F
+
Lw

+
Fv

as wwc , then the deformation ring for ρ|GF + can be identified with a deformationv
ring for s|GFw , as in [CHT08]. Conjecturally, every component is expected to be
globally realizable (and the analogous statement for places v - p is known), but
proving this seems to be a very hard problem.

GLOBALLY REALIZABLE COMPONENTS OF DEFORMATION RINGS

3

Our main theorem is the following (see Theorem 4.2.11 for a more precise statement, and see the notation and conventions section at the end of this introduction
for any unfamiliar terminology).
Theorem A. Assume that either n is odd, or that n = 2. Let F be a CM field,
and let s : GF → GLn (Fp ) be a reasonable representation, with prolongation ρ.
Let S be a finite set of finite places of F + , such that S contains all of the places
at which ρ is ramified and all of the places lying over p. For each place v ∈ S,
let Cv be a component for ρ|GF + , which is globally realizable if v|p.
v
Then there exists an odd, regular, polarized, weakly irreducible compatible system ({sλ }, {µλ }) of GF -representations with associated p-adic representation s, anda prolongation ρ of s, which satisfies:(1) ρ lifts ρ, and for each place v ∈ S, the representation ρ|GF + lies on Cv .
v(2) ρ is unramified outside S.
Note that by the very definition of global realizability, the hypothesis thateach Cv is globally realizable is a necessary condition for the conclusion of thetheorem to hold.
The hypothesis that s is reasonable is needed in order to apply the theoremsof [BLGGT14b], and some restriction on p, n and the size of the image of s iscertainly necessary; for example, the results of [LL15] show that the analogous
result fails for modular forms of weight 2 if p = 2 (in fact there are also dihedral
counterexamples due to Serre with p = 3, see [Car89, §4.4]). More generally,
calculations in Galois cohomology suggest that if p ≤ n+1, and s|GF (ζp ) is reducible,
then it unreasonable to hope for a global lifting result with control of the local
representations at all places. Thus the only unnaturally restrictive hypothesis in
Theorem A is the exclusion of even integers n > 2, which is a byproduct of our
methods; this is because given a compatible systems of n-dimensional polarizablel-adic representations, we cannot deduce the oddness of all the representations in
the compatible system from the oddness of a single representation.
An almost immediate corollary of our results is the following potential automorphy theorem, which may be of independent interest. Subject to the restrictionthat n = 2 or n is odd, it improves on [BLGGT14b, Thm. 4.5.1] by replacing “potentially diagonalizable” by “globally realizable”. Note that if v|p in F + splits in F ,then the global realizability of ρ|GF + only depends on s|GFw for w|v in F .
v

Theorem B (Corollary 4.2.12). Assume that either n is odd or n = 2. Let F be a
CM field, and let (s, µ) be a polarized representation, wheres : GF → GLn (Qp )is odd and ramified at only finitely many primes. Suppose that s is reasonable. Let ρ
be the corresponding prolongation of s, and assume that ρ|GF + is globally realizablev
for each v|p. Then (s, µ) is potentially automorphic.
In fact (as noted in the abstract), we could weaken the hypothesis that ρ|GF + isv
globally realizable to requiring only that it is potentially globally realizable, because
this is equivalent to global realizability by Corollary 4.2.13. Perhaps surprisingly,
if s is automorphic, then we cannot deduce that s is also automorphic; this isbecause our methods make considerable use of potential automorphy results forother representations in a compatible system containing s. On the other hand, if

4F. CALEGARI, M. EMERTON, AND T. GEE

we did know that weakly irreducible compatible systems are automorphic (rather
than just potentially automorphic), then a version of the Breuil–Mézard conjecture
for odd-dimensional globally realizable representations and a version of the weightpart of Serre’s conjecture for odd-dimensional unitary groups would both follow
from combining Theorem A with the methods of [GK14].
1.2. History and motivation. We now give a somewhat leisurely overview of our
motivations and of previous work on similar questions. Ultimately, the problems
that we are working on are motivated by congruences between modular forms;
more specifically, we are concerned with congruences between eigenforms. Such
congruences can often best be understood in terms of the corresponding Galois
representations, and in particular in terms of the restrictions of these (global) Galois
representations to (local) decomposition groups. It is therefore natural to wonder
whether there is a local to global principle for the existence of such congruences.
The first results in the literature that we know of that are explicitly formulated
in this way are those of [DT94], which we now recall. Let p > 3 be prime, andlet f be a newform of level prime to p and weight 2. We can (after choosing anembedding Q ,→ Qp ) associate a p-adic Galois representation ρf : GQ → GL2 (Qp )to f , and thus a mod p representation ρf : GQ → GL2 (Fp ). We say that an
irreducible representation ρ : GQ → GL2 (Fp ) or ρ : GQ → GL2 (Qp ) is modular of
weight 2 if it is isomorphic to some ρf (respectively ρf ).
Then the main result of [DT94] is as follows. Suppose we are given a modular
representation ρ : GQ → GL2 (Fp ) of weight 2, and that for each prime l 6= p,
we are given a lifting of ρ|GQl to a p-adic representation ρl : GQl → GL2 (Qp ).
Suppose also that all but finitely many of the ρl are unramified. Then there is alift ρ : GQ → GL2 (Qp ) of ρ which is modular of weight 2, with the property thatfor all l 6= p, we have an isomorphism of restrictions to inertia ρ|IQl ∼= ρl |IQl .
This result is best possible, in the sense that any modular representation isnecessarily only ramified at finitely many primes, and that (for example, because
spaces of modular forms are finite-dimensional) it is unreasonable to pin down ρ|GQl
more than specifying it on inertia. In particular, by local-global compatibility, theconductor of ρ|IQl determines the l-power part of the level of the modular form
associated to ρ.
It is natural to wonder whether this result can be extended to cover modular
forms of higher weight or with p dividing their level, and to allow some controlof ρ|GQp . Since the local behavior at p of ρf is highly dependent on the weight of
the newform f and the power of p dividing its level, these two questions are closely
related, and the most general local to global result of this kind is best formulated
in terms of components of local deformation rings. If we fix a weight k ≥ 2, and akfinite extension K/Qp , then we can consider the deformation ring RK
for liftingsof ρ|GQp which become semistable over K with Hodge–Tate weights 0, k − 1. If ρ
is modular of weight k and some level, then for some sufficiently large K, ρ|GQpkcorresponds to a point of RK. (For example, if the corresponding modular form
has level prime to p, then we can take K = Qp .)k
The spectrum of RK
has finitely many irreducible components, and given such acomponent, we can ask whether there exists a lift ρ with the property that ρ|GQp lieson this component. This is the correct analogue of what we are demanding at theplaces l 6= p; indeed, it turns out that specifying ρ|IQl is equivalent to demanding

GLOBALLY REALIZABLE COMPONENTS OF DEFORMATION RINGS

5

that ρ|GQl lies on a particular component of a deformation ring for ρ|GQl . Provingthat the lifting problem still admits a solution when we specify a component at p ismuch harder then the case in which we only specify components away from p, but
it follows from the results of [Kis09a] that such a lift exists under a mild (“Taylor–Wiles”) condition on ρ. In particular, this again gives a complete understanding of
the possibilities for the weight and level of the corresponding modular form.
When we restrict to the case that 2 ≤ k ≤ p + 1 and K = Qp , the lifting problem
is closely related to Serre’s conjectures [Ser87] on the weight and level of modular
Galois representations. For example, while not formulated in this way, Serre’s
conjecture on the minimal weight and level is equivalent to asking that ρ admits amodular lift whose ramification away from p is as small as possible (as measured
by the Swan conductor), and whose weight is as small as possible, compatible withthe property of locally having a crystalline lift of the corresponding Hodge–Tateweights.
It is relatively straightforward to formulate conjectural generalizations of theseresults. For example, a detailed formulation of a generalization of Serre’s conjectures to Hilbert modular forms was made in [BDJ10], and the weights are describedin terms of the existence of local crystalline lifts in a similar fashion to that described above. One can make similar conjectures for automorphic representations
on unitary groups over CM fields (or equivalently, for conjugate self dual automorphic representations of GLn over CM fields), and it is these generalizations that willconcern us below. (However, we don’t expect any straightforward generalizationof these results to hold outside of settings which are discrete series at the infinite
places.)
While some of the arguments of [DT94] and of the related papers on the weight
and level parts of Serre’s conjecture can be generalized to Hilbert modular forms,
it seems hard to adapt them to prove the conjectures of [BDJ10] completely, andmuch harder still to study congruences between forms on GLn in this way. However,in the mid 2000s a new approach to these problems was discovered by Khare–
Wintenberger [KW09] and the third author [Gee07], which we will now describe in
the setting of modular forms.
The approach is via the deformation theory of global Galois representations andautomorphy lifting theorems. Suppose that as above we are given components
of deformation rings for ρ|GQl for each prime l, which are unramified for all but
finitely many l. Then there is a corresponding deformation ring Runiv for the global
representation ρ, and the Qp -points of its spectrum precisely correspond to the
Galois representations that we are hoping to construct. If we can show that the setof Qp -points is nonempty, then we can hope to show that the Galois representations
are modular using modularity lifting theorems (the Taylor–Wiles method).
The tangent space to Runiv can be computed by Galois cohomology, and itturns out that Runiv always has dimension at least 1. (This computation relies on
the weight k being at least 2, and more generally on us being in a discrete series
context.) Heuristic arguments lead us to expect that Runiv is a finite Zp -algebra,
and if this is the case, the lower bound on the dimension guarantees the existenceof Qp -points.

6

F. CALEGARI, M. EMERTON, AND T. GEEThere is no known purely Galois-theoretic argument guaranteeing this finitenessin general (although it can sometimes be arranged at the cost of allowing additional ramification away from p by an argument of Ramakrishna [Ram02]). However, modularity lifting theorems are proved by identifying deformation rings suchas Runiv with Hecke algebras, which are finite over Zp by definition, so in principle
it is enough to prove an appropriate modularity lifting theorem (which can thenbe used in the final step of the argument to deduce that the Galois representations
that we have constructed are actually modular).
Unfortunately, this argument is circular as written, because what the Taylor–Wiles method allows us to prove is that if some Qp -point of Runiv is modular,then Runiv may be identified with a Hecke algebra; but it gives us no assistance withproducing a Qp -point in the first place. A key insight of Khare–Wintenberger isthat this argument can be combined with base change and/or potential modularity
to avoid the circularity. Suppose that F is a totally real finite extension of Q, andthat ρ|GF is irreducible. Then we may consider the deformation problem for ρ|GF
given by the restrictions to places of F of the conditions we imposed over Q, andthe corresponding deformation ring RFuniv is again of dimension at least one.
Now, by definition Runiv is an RFuniv -algebra, and it is in fact a finite RFuniv algebra. It is therefore enough to prove a modularity lifting theorem for RFuniv .
This allows us to reprove many cases of the theorem of [DT94], in the following
way. Suppose for simplicity that for each prime l 6= p, both the original modular
representation ρf |GQl and the local representation ρl are finitely ramified (that is,they become unramified after restriction to a finite extension of Ql ). Then we may
choose a finite solvable totally real extension F/Q so that the restrictions to thefinite places of F of these representations are actually unramified. In particular, thecorresponding restrictions of ρf |GQl and the local representation ρl lie on the same
component of the corresponding local deformation ring (indeed, the unramified
local deformation ring is formally smooth).
By solvable base change, ρf |GF is modular, and by the choice of F , it gives a pointof RFuniv . A modularity lifting theorem then shows that RFuniv is a finite Zp -algebra.
Thus the same is true of Runiv , and so Runiv has Qp -points, which give the soughtafter Galois representations; the modularity of these representations again follows
from solvable base change. The more general case, in which the ramification canbe potentially unipotent, can be handled in the same way when given some levelraising and level-lowering results over F ; by choosing F appropriately, one reduces
to a relatively straightforward case (see [SW01]).
A variant on this argument makes it possible to state and prove results about
Galois representations that make no reference to automorphic forms (although theproofs make heavy use of automorphic techniques). To this end, rather than assuming that ρ is modular, assume only that it is irreducible and odd, in the sensethat ρ(c) is non-scalar, where c is a complex conjugation. (Of course, since Serre’s
conjecture is a theorem, this implies that ρ is modular, but we can and will makean analogous assumption in more general contexts where the analogue of Serre’s
conjecture is open.) Then the same Galois cohomology calculations go through,and if we want to produce lifts of ρ with specified local properties, it is enough by
the above arguments to find a finite (not necessarily solvable) extension of totallyreal fields F/Q for which ρ|GF is modular (that is, it comes from a Hilbert modularform).

GLOBALLY REALIZABLE COMPONENTS OF DEFORMATION RINGS

7

An argument of Taylor [Tay02, Tay06] can be used to prove such “potential modularity” theorems. The idea is as follows: one can find a moduli space whose F points correspond to abelian varieties, part of whose p-torsion is isomorphic to ρ|GF ,and the corresponding part of whose l-torsion, for some fixed prime l 6= p, is isomorphic to an induction of a character. Since inductions of characters are always
modular, in favourable circumstances one can use modularity lifting theorems toprove that (part of) the l-adic Tate module of the corresponding abelian variety
is modular, and thus that (part of) the p-adic Tate module is modular, and finally that ρ|GF is modular. That F -points exist for F sufficiently large follows
from a theorem of Moret–Bailly, which also allows one to impose the kinds of local
conditions that are needed in order to apply modularity lifting theorems.
As well as producing lifts of ρ with specified local properties, it turns out that
potential modularity allows one to prove that that each p-adic representation ρthat is constructed in this way is part of a compatible system of l-adic representations. Indeed, this property is automatic for Galois representations associated toautomorphic forms, so that ρ|GF is part of a compatible system. By solvable basechange, the same is true for ρ|GF 0 whenever F/F 0 is a solvable extension of totallyreal fields, and an argument with Brauer’s theorem (see [Die04]) makes it possible
to put these together to give the required compatible system.
We now digress briefly to discuss another aspect of compatible systems that will
be of fundamental importance throughout this paper. In general, it seems to behopeless to understand the components of local potentially semistable deformation
rings of mod p representations in any concrete way, and this in turn places serious
restrictions on automorphy lifting theorems. However, it is possible to understandthem in the Fontaine–Laffaille case, which by definition is case that p is unramified
in the base field, and the weight is small relative to p. It is also very hard toprove automorphy lifting theorems when the global mod p Galois representation
has small image (in particular, when the image is reducible). This can make itvery difficult to prove the potential automorphy of a given Galois representation(or its automorphy, even if we know that it is residually automorphic). However,
if instead we are given a weakly irreducible compatible system of l-adic Galois
representations, then for all large l, the l-adic representation will be Fontaine–
Laffaille, and its residual image will be suitably large. Accordingly, one can hopeto prove the potential automorphy of the l-adic representation for some l, andimmediately deduce the potential automorphy for all l. As we now explain, this has
been carried out in considerable generality. In our arguments, this will allow us to
use compatible systems of Galois representations as a kind of proxy for automorphic
representations, without assuming the Fontaine–Mazur conjecture.
We briefly review the history of higher-dimensional potential automorphy theorems. Many of the arguments discussed above were generalized to polarizable(that is, essentially conjugate self-dual) n-dimensional Galois representations of
CM fields in the papers [CHT08, Tay08, HSBT10] (the corresponding automorphic
representations being those on general unitary groups). In particular, [Tay08] uses
automorphy lifting techniques to prove the kind of level raising/lowering results that
we applied after base change above, and the third author’s paper [Gee11] deduced
an n-dimensional version of the theorem of Diamond–Taylor in low weight.
Potential modularity, while powerful, has its limitations, the chief of which isthat the method explained above only works for modular forms of weight 2 (that8

F. CALEGARI, M. EMERTON, AND T. GEE

is, for Galois representations with Hodge–Tate weights 0, 1), because for reasons of
Griffiths transversality, the required moduli spaces only exist in this case. Allied tothis is the difficulty mentioned above that the deformation rings for ρ|GQp are much
more complicated than those for l 6= p, and much less well understood (for example,
it is certainly no longer the case that deformations are potentially unipotent oninertia). Indeed, when k is large compared to p, very little is known about thecomponents of these deformation rings, and we do not know whether we can makebase changes to make representations lie on the same component in any generality,
which limits our ability to change components at p (recall that for l 6= p, in thediscussion above we used that any finitely ramified representation can be madeto lie on the unique unramified component after a suitable base change). (It is
however worth noting that when k = 2 arguments of this kind are possible, evenfor Hilbert modular forms over totally real fields in which p is highly ramified, by
the results of [Kis09b] and [Gee06]. These results were a crucial part of the proofof the weight part of Serre’s conjecture for Hilbert modular forms [GK14], and thelack of anything similar for higher-dimensional representations is one of the reasonsthat less is known about the weight part of Serre’s conjecture in dimension greaterthan 2.)
In higher dimensions the situation is worse; the potential automorphy theoremsof [HSBT10] apply only in weight 0 (that is, the lowest discrete series; the corresponding Galois representations have Hodge–Tate weights 0, 1, . . . , n − 1), and infact only to ordinary Galois representations. While this was enough to prove theSato–Tate conjecture (by proving the potential automorphy of the symmetric powers of the 2-dimensional Galois representations attached to elliptic curves, which
are ordinary at most primes), it falls well short of proving the potential modularity
of compatible systems of Galois representations in any generality. This shortcoming was resolved in [BLGGT14b], which also introduced a way of systematicallychanging the weight of the representations, or more generally of moving between
components of deformation rings at p. The argument involves a refinement of amethod of Harris [Har09] (the “tensor product trick”), the basic idea of which is
as follows: given a global Galois representation with regular Hodge–Tate weights,
by taking the tensor product with another representation, one can produce a representation (of much higher dimension) of weight 0. It is then possible to apply
potential modularity techniques to this representation.
Of course, to be useful, one has to have a way of “undoing” the tensor product again (on both the Galois and automorphic sides). In general, this is a hard
problem, but if the representation we tensor with is the induction of a character,it turns out to be relatively straightforward, essentially because the tensor productof an n-dimensional representation with the induction of a character is itself theinduction of an n-dimensional representation.
We also obtain a way of moving between weights and between components ofdeformation rings at places dividing p, in the following way. Let ρ : GF → GLn (Fp )be our original mod p representation, and let r : GF → GLn (Qp ) be induced froma character. Fix a deformation ring R1univ for ρ corresponding to certain given local
conditions. Given a deformation ρ of ρ, we obtain a deformation ρ⊗r of ρ⊗r, and we
let R2univ be the corresponding deformation ring. This procedure makes Runiv into
an R2univ -algebra, and an argument analogous to the one of Khare–Wintenberger
that we mentioned above shows that it is even a finite R2univ -algebra.

GLOBALLY REALIZABLE COMPONENTS OF DEFORMATION RINGS

9

It is not immediately obvious that this buys us anything, as to apply the techniques we used above we will need to prove an automorphy lifting theorem for R2univ .
To this end, suppose that we also have another deformation ring R3univ for ρ, andanother lift r0 of r, and that the deformation problem corresponding to the tensor product of a deformation for R3univ with r0 again corresponds to R2univ . If r
and r0 are both inductions of characters, and if we know that R3univ has automorphic points, then we can prove an automorphy lifting theorem for R2univ , deduce its
finiteness over Zp , and then prove the existence of lifts (and automorphy lifting)for R1univ , the original problem of interest.
It may not be obvious that this is generally applicable, but in fact in combination
with base change techniques it gives enough flexibility to prove the potential automorphy of compatible families, by moving between Fontaine–Laffaille and ordinary
weight 0 deformation problems. A little thought shows that this argument allows us
to move freely between components of local deformation rings at places dividing p,
provided that the corresponding representations are potentially diagonalizable, inthe sense that after some base change, the components contain a point which isa direct sum of crystalline characters. It turns out to be straightforward to showthat Fontaine–Laffaille representations and ordinary representations are potentiallydiagonalizable, giving the claimed potential automorphy result. (The most general
result about the existence of Galois representations proved in [BLGGT14b] is essentially Theorem A above, but with “globally realizable” replaced by “potentiallydiagonalizable”.)
While this gives a general potential automorphy result for compatible systems, it
is unsatisfactory as an answer to our original lifting question, due to the restriction
to potential diagonalizable representations. This restriction is problematic for two
reasons: firstly, beyond the Fontaine–Laffaille and ordinary cases, nothing is known
about the potential diagonalizability or otherwise of n-dimensional representations.
(It is quite plausible that all potentially crystalline representations are potentiallydiagonalizable, but this seems to be very hard to establish.) In addition, potentially diagonalizable representations are by definition potentially crystalline, so they
do not (unlike Theorem A) tell us anything about general potentially semistable
representations.
1.3. A sketch of the proofs. We now briefly sketch the proof of Theorem A,
omitting many of the trickier technical details; in particular, we completely ignore
places away from p in our discussions, as the difficulties they present are similar to,
but simpler than, the difficulties for places dividing p. We also suppress all mentionof choices of polarization. In essence, our idea is to go beyond the potentiallydiagonalizable case, by allowing ourselves to tensor with representations that are
not necessarily induced from characters. There are some obvious difficulties withthis approach, chief among them that on both the automorphic and Galois sides it
is hard to “undo” a tensor product. We overcome this by using compatible systems,
rather than individual Galois representations.
Our first technique is a variant of the argument of [BLGGT14b] explained above,that of tensoring with auxiliary representations in order to move between different
components. However, for our purposes it is insufficient to tensor with a fixed global
representation. Instead, we put ourselves in the following situation: suppose given
representations r, s : GF → GLn (Qp ) which belong to compatible systems, andsuppose for simplicity that for each place v|p, we have r|GFv ∼= s|GFv . Let Av be

10

F. CALEGARI, M. EMERTON, AND T. GEE

the component corresponding to r|GFv , and let Bv be the component corresponding to s|GFv . For each v, let Cv be one of Av and Bv , and let Dv be the other.
Let RC and RD be the corresponding global deformation rings for r and s respectively. (We should really be considering prolongations of these representations to
Gn -valued representations of GF + , but we ignore this point for the purposes of thisintroduction.)
We would like to produce representations r0 , s0 corresponding to points of RC
and RD , which belong to compatible systems; in this way we will be able to swap
components between different residual representations. We initially accomplish
this under very restrictive hypotheses, which we will later relax; note that we
certainly need to assume at first that r ⊗ s is irreducible, and that r ⊗ s has regular
Hodge–Tate weights. Now, the deformation ring Runiv corresponding to r ⊗ s isfinite over Zp , because r ⊗ s is part of a compatible system, and thus potentiallyautomorphic. Taking the tensor product of representations coming from RC and RD
b D into an Runiv -algebra, and we are able to show that it is in fact amakes RC ⊗R
univ
finite R-algebra, so that in the same way as before, we can see that RC , RD bothhave Qp -points, which will correspond to the lifts r0 , s0 that we want to produce.
However, we need to show that these lifts belong to compatible systems. The
tensor product r0 ⊗ s0 does belong to a compatible system of representations {tl }(because it corresponds to a point of Runiv , and is thus potentially automorphic),
so we would like to have a result ensuring that if one representation in a compatible
system is a tensor product, they all are. This should be true quite generally, but
it seems hard to prove, and we only establish it under rather restrictive hypotheses(which are ultimately sufficient for our needs).
In outline, we argue as follows. We assume that the Zariski closure of the imageof s0 contains SLn (Qp ). Under a strong irreducibility hypothesis (which can be arranged at infinitely many primes l by imposing local conditions), we can show thatthe representations tl decompose accordingly as tensor products tl = rl0 ⊗ s0l , where
the Zariski closure of the image of s0l contains SLn (Ql ). Here we make crucial use
of a result of Larsen and Pink [LP92]. Then, if l is sufficiently large, the representations rl0 and s0l belong to compatible families by the results of [BLGGT14b]; this is
the only place that we need our assumption on n, as we have to know that rl0 and s0l
are odd in the sense of [BC11]. This oddness is automatic if n is odd, and can beproved if n = 2 by the methods of [Cal11, Cal12], but it seems to be beyond thereach of current technology if n > 2 is even. Our assumptions then show that (after possibly twisting) the p-adic representations in these compatible systems are r0
and s0 , as required.
With this component swapping result in hand, the basic outline of the proof of
Theorem A is as follows. Suppose for simplicity that there is only a single place v|pof F , let r be some lift of r, and let C be the given globally realizable component.
By definition, this means that there is another CM field L, a place w|p and a
representation s (which is part of a compatible system) such that s|GLw lies on C.
Then if we apply our swapping result to r|GF L and s|GF L , we can produce a liftof r|GF L which lies on the correct component at some place over v. If we could
produce a lift with this property at all places above v, then we would be done by
the usual Khare–Wintenberger method; in order to do this, we replace F L with its
Galois closure, and s with its various Galois conjugates, and then inductively apply
the swapping result to each of these conjugates in turn.

GLOBALLY REALIZABLE COMPONENTS OF DEFORMATION RINGS

11

Unfortunately, the actual argument is much more complicated than this straightforward outline. The problem is that all of the results that we are applying have
hypotheses that we have been ignoring; for example, we need r ⊗s to be irreducible,
we need r ⊗ s to have regular Hodge–Tate weights, and we need to satisfy the restrictive hypotheses of our main swapping result, which are a mixture of local andglobal assumptions.
We are able to handle the various local assumptions away from p by more basechange tricks, but these cannot help with the global problems. Since the fields F, L
and the representations r, s are arbitrary, we cannot hope to arrange that theirrestrictions to the Galois closure of F L are irreducible. Instead, we make use of an
idea introduced in [Cal12], and use the theorem of Moret–Bailly mentioned above
to construct auxiliary global representations with large image, which locally admit
potentially diagonalizable lifts of arbitrarily large weights.
These representations are constructed over extensions of F, L that we have little
control over, and we have to go to some lengths to ensure that we can arrange all of
the properties we need. Rather than swapping directly between r and s, we instead
make a long chain of swaps, going via many auxiliary representations, and making
many base changes and descents by the Khare–Wintenberger argument.
1.4. Acknowledgements. The authors are grateful to Florian Herzig and theanonymous referee for their close readings of an earlier version of the paper, andtheir many helpful remarks and suggestions, which have improved its readability.
We particularly thank the referee for pointing out a gap in the proof of Theorem 3.2.17 in the case m = n = 2, which led us to add Lemma 3.3.1.
1.5. Notation, conventions, and background material. All representations
considered in this paper are assumed to be continuous with respect to the natural
topologies, and we will never draw attention to this. If M/F is an extension of number fields, then we will write M gal for the Galois closure of M over Q, and M F -gal for
the Galois closure of M over F . As usual, if K is a field of characteristic zero, thenwe write GK = Gal(K/K) for its absolute Galois group, and if K is furthermore a
local field, we write IK for the inertia subgroup of GK .
1.5.1. Polarizable representations. We begin by recalling some definitions and results from [CHT08, BLGGT14b] concerning polarizable representations.
Recall from [CHT08] that the reductive group Gn over Z is given by the semidirect product of Gn0 = GLn × GL1 by the group {1, } where (g, a)−1 = (a · t g −1 , a).
We let ν : Gn → GL1 be the character which sends (g, a) to a and sends  to −1.
Let Γ be a group, with an index 2 subgroup ∆. Fix an element γ0 ∈ Γ \ ∆.
Let R be a (commutative) ring. Then by [CHT08, Lem. 2.1.1], there is a naturalbijection between:• the set of homomorphisms ρ : Γ → Gn (R) which induce isomorphisms∼Γ/∆ −→ Gn /Gn0 , and• the set of triples (r, µ, h , i) consisting of homomorphisms r : ∆ → GLn (R)and µ : Γ → R× , and a perfect R-linear pairing
h , i : Rn × Rn → R
which for all x, y ∈ Rn and δ ∈ ∆ satisfies

12

F. CALEGARI, M. EMERTON, AND T. GEE– hx, r(γ02 )yi = −µ(γ0 )hy, xi, and– hr(δ)x, r(γ0 δγ0−1 )yi = µ(δ)hx, yi.
This correspondence is given by taking r = proj. onto first factor of Gn◦ of ρ|∆ , andµ = ν ◦ ρ, and setting
hx, yi = t xA−1 y,where ρ(γ0 ) = (A, −µ(γ0 )). We say that the pair (r, µ) is polarized 1 and that r
is polarizable, and is µ-polarized. If we are given a polarized pair (r, µ), then we will
sometimes refer to a corresponding homomorphism ρ : Γ → Gn (R) (which depends
on the choice of γ0 , as well as on a choice of pairing h , i witnessing the polarizabilityof (r, µ)) as a prolongation of the pair (r, µ).
Given a polarized pair (r, µ), we call µ the multiplier character of the pair (r, µ).
Given two polarized representations (r1 , µ1 ) and (r2 , µ2 ), there is a polarized representation (r1 ⊗ r2 , δΓ/∆ µ1 µ2 ), where δΓ/∆ denotes the unique non-trivial characterof Γ/∆ (see [BLGGT14b, §1.1] for the explicit description of this construction as
an operation on Gn -valued representations).
Suppose now that R is a complete local Noetherian ring, that r : ∆ → GLn (R)is such that r mod mR is absolutely irreducible, and that µ : Γ → R× is a charactersuch that∼ r∨ ⊗ µ|∆ .
(1.5.2)r γ0 =
Giving such an isomorphism is equivalent to giving a pairing
h , i : Rn × Rn → R
which for all x, y ∈ Rn and δ ∈ ∆ satisfies hr(δ)x, r(γ0 δγ0−1 )yi = µ(δ)hx, yi for
all x, y ∈ Rn and δ ∈ ∆. Since the isomorphism (1.5.2) is unique up to scaling by
elements of R× (because of our assumption that r mod mR is absolutely irreducible),
we see that the corresponding pairing h , i is also unique up to scaling. In particular,
if (r, µ) is polarized, then the pairing h , i that yields a prolongation of (r, µ) is unique
up to scaling.
If ρ is one particular prolongation, corresponding to a pairing h , i, then we seethat conjugating ρ by the element (1, λ−1 ) ∈ GLn (R) × GL1 (R) = Gn◦ (R) ⊂ Gn (R)scales h , i by λ; the relevant computation is that(1, λ−1 )(g, a)(1, λ) = (1, λ−1 )(g, a)(1, λ)−1 = (1, λ−1 )(g, a)(λ, λ)
= (λg, a).
Thus we see that all possible prolongations of (r, µ) are obtained from the given
prolongation ρ by such conjugations. We also see that the possible pairings arisingfrom the choice of a prolongation are independent of the choice of the element
γ0 ∈ Γ \ ∆ used to construction the bijection described above between (certain)
homomorphisms ρ : Γ → Gn (R) and (certain) triples (r, µ, h , i).
We now consider the particular case that Γ = GF + , ∆ = GF , where F is a
(totally complex) CM field with maximal totally real subfield F + . We say that the
1This is established terminology, and so we use it here. Note though that in general the pair(r, µ) may not determine the pairing h , i uniquely, even up to a scalar multiple. However, if R isa field and r is absolutely irreducible, then, as we will observe below, the pairing h , i is uniquelydetermined up to a scalar.

GLOBALLY REALIZABLE COMPONENTS OF DEFORMATION RINGS

13

pair (r, µ) is polarized and odd if it is polarized, and for all complex conjugations c ∈GF + , we have µ(c) = −1. In particular, we have the following standard lemma.
Lemma 1.5.3. Suppose that the characteristic of R/mR is not 2. If (r, µ) is polarized, n is odd, and r mod mR is absolutely irreducible, then (r, µ) is automaticallyodd.
Proof. This follows from the fact that any odd-dimensional perfect pairing thatis preserved up to scaling by a residually absolutely irreducible group action (incharacteristics other than 2) is necessarily symmetric. Indeed, let c be any complex
conjugation, take γ0 equal to c, and let h , i denote the pairing arising from a choiceof prolongation. Since c2 = 1, we find (using the first of the properties satisfied
by the pairing arising from a prolongation) that hx, yi = −µ(c)hy, xi. On the other
hand, as we already remarked, the pairing is necessarily symmetric. Thus we findthat µ(c) = −1, as required.

We can restrict a global representation ρ : GF + → Gn (R) to the decompositiongroup GFv+ of any finite place v of F + . Note that if v is inert or ramified in F ,then GFv+ is not contained in GF , so we are in the situation above with Γ =GFv+ and ∆ = GFv . If however v splits in F then GFv+ is contained in GF , sothat ρ(GFv+ ) ⊂ Gn0 (R) = GLn (R) × R× , so that (µ being fixed) the data of therepresentation ρ is the same as the data of the corresponding representation r :GFv+ → GLn (R).
1.5.4. Compatible systems. Let F be a number field. We recall some definitions
from [BLGGT14b, §5]. Note that what we call a “compatible system” is a “weakly
compatible system” in [BLGGT14b].
By a compatible system R of n-dimensional representations of GF defined overM we shall mean a 5-tuple(M, S, {Qv (X)}, {rλ }, {Hτ })where(1) M is a number field.
(2) S is a finite set of primes of F .
(3) for each prime v 6∈ S of F , Qv (X) is a monic degree n polynomial in M [X].
(4) for each prime λ of M (with residue characteristic l, say)rλ : GF → GLn (M λ )is a semisimple representation such that
• if v 6∈ S is a prime of F and v - l, then rλ is unramified at v andrλ (Frobv ) has characteristic polynomial Qv (X).
• if v|l then rλ |GFv is de Rham and in the case v 6∈ S crystalline.
(5) for τ : F ,→ M , Hτ is a multiset of n integers such that for any M ,→ M λover M , the τ -labelled Hodge–Tate weights of rλ are Hτ .
We will call R regular if for each τ : F ,→ M every element of Hτ has multiplicity 1.
We will refer to a rank 1 compatible system of representations as a compatible
system of characters.
Remark 1.5.5. By abuse of terminology, we refer to a collection of Galois representations {rλ } as a compatible system if it extends to a 5-tuple R as above. In thiscase, we say that the compatible system {rλ } is unramified outside S if it extends to14

F. CALEGARI, M. EMERTON, AND T. GEEsuch a 5-tuple with the given finite set S. Note that if {rλ } is unramified outside S,
then the individual representations rλ are unramified outside S ∪ {v|l} where λ has
residue characteristic l.
Remark 1.5.6. By a slight abuse of terminology, if F 0 /F is a finite extension of
number fields, and S is a finite set of places of F , then we will sometimes saythat a compatible system of representations of GF 0 is unramified outside S if it isunramified outside of the set of places of F 0 lying over S. Similarly, we will saythat an extension F 00 /F 0 is unramified outside of S if it is unramified at all placesof F 0 not lying over a place of S, and so on. In particular, we will apply frequently
apply this convention to quadratic extensions F/F + where F is CM with maximal
totally real field F + .
Note that if M 0 /M is a finite extension, then a compatible system defined over M
naturally determines a compatible system with M 0 -coefficients. We regard these twocompatible systems as equivalent. Similarly one can enlarge S, and we also regard
compatible systems associated in this way as equivalent. We may then consider theequivalence classes of the equivalence relation generated by these equivalences, andit follows easily from [PT15, Lem. 1.1] that to each equivalence class of compatible
systems is associated a minimal choice of M , namely the field generated by thecoefficients of the polynomials Qv (X).
For this reason, we generally suppress M in the below. Somewhat abusively,
we shall often assume that M comes with a fixed embedding M ,→ Qp for eachprime p, and hence talk of the p-adic representations : GF → GLn (Qp )associated to {sλ }.
We also introduce the following convenient short-hand terminology.
Definition 1.5.7. Let F be a number field, and let r : GF → GLn (Ql ) be a
representation. Then we say that r is Fontaine–Laffaille, or Fontaine–Laffaille atall primes dividing l (for emphasis), if l is unramified in F , and for all τ : F ,→ Ql ,the τ -labelled Hodge–Tate weights of r are contained in an interval of length (l − 2)(the precise interval possibly depending on τ ).
Note that in particular if {rλ } is a compatible system, then all but finitely manyof the rλ are Fontaine–Laffaille.
If F is CM (in this paper all CM fields are imaginary), we denote its maximal
totally real subfield by F + . If F is CM, and if M = (M, SF + , {X − αv }, {µλ }, {w})is a compatible system of characters of GF + , then we will call (R, M) a polarized(and odd) compatible system if for all primes λ of M the pair (rλ , µλ ) is polarized(and odd). (Here SF + denotes the set of places of F + lying below an element of
S.) We will call R polarizable (and odd) if there exists an M such that (R, M) isa polarized (and odd) compatible system. Note that µλ (cv ) is independent of λ, so
oddness of a polarized compatible system can be checked at a single λ.
Recall from [BLGGT14b, §2.1] that a polarized automorphic representation ofGLn (AF ) is a pair (π, χ) consisting of an automorphic representation π of GLn (AF ),+ ×
and a character χ : A×
→ C× with χv (−1) = (−1)n for all v|∞, such
F + /(F )c ∼ ∨that π = π ⊗ (χ ◦ NF/F + ◦ det). We say that an automorphic representation πof GLn (AF ) is polarizable if there exists a χ such that (π, χ) is polarized.

GLOBALLY REALIZABLE COMPONENTS OF DEFORMATION RINGS

15

If (π, χ) is a regular algebraic cuspidal polarized automorphic representationof GLn (AF ), then there is an associated polarized and odd compatible system({rλ (π)}, {ε1−n rλ (χ)}), as explained in [BLGGT14b, §5.1]. (See also [BLGGT14b,
Thm. 2.1.1]; note that ε denotes the cyclotomic character, and rλ (χ) is the compatible system associated to χ, regarded as an automorphic representation of GL1 (AF + ).
The assumption that χv (−1) = (−1)n ensures the oddness of this compatible system.) We say that the pair of compatible systems ({rλ (π)}, {ε1−n rλ (χ)}) is automorphic. We write rp (π) for the associated p-adic representation, and we say thata representation r : GF → GLn (Qp ) is automorphic if it is isomorphic to rp (π)
for some π; note that a compatible system is automorphic if and only if for someprime p, its associated p-adic representation is automorphic.
The following definition is one of several closely related definitions that one could
make of what it means for a compatible system to be potentially automorphic; conjecturally, all of these definitions are equivalent, and are equivalent to automorphy,
but this seems to be very difficult to prove.
Definition 1.5.8. If F is CM, then we say that a pair of compatible systems ({sλ }, {ψλ }),with the sλ being n-dimensional and the ψλ being characters, is potentially automorphic if for every finite Galois extension F (avoid) /F , there is a finite Galois
extension of CM fields L/F , which is linearly disjoint from F (avoid) /F , and is suchthat ({sλ |GL }, {ψλ |GL }) is automorphic.
Similarly, we say that a compatible system {sλ } is potentially automorphic if it
may be extended to a potentially automorphic pair of compatible systems ({sλ }, {ψλ }).
Definition 1.5.9. We will call R pure (of weight w ∈ Z) if• for each v 6∈ S, each root α of Qv (X) in M and each ı : M ,→ C we have|ıα|2 = qvw ;• and for each τ : F ,→ M and each complex conjugation c in Gal(M /Q) wehave
Hcτ = {w − h : h ∈ Hτ }.
In the following definition, and throughout the body of the paper, “density”
means “Dirichlet density”.
Definition 1.5.10. We say that a compatible system {sλ } is weakly irreducible if
there is a positive density set of rational primes l so that for all primes λ|l, therepresentation sλ is irreducible.
One expects that the irreducibility of a single Galois representation in a compatible system should imply the irreducibility of all representations, but this is
unknown in general. On the other hand, the notion of weak irreducibility turns out
to be easy to work with in light of the following results.
Lemma 1.5.11. Let F be CM. Then a regular, odd, polarizable compatible system of representations of GF is weakly irreducible if and only if it is potentiallyautomorphic.
Proof. Any automorphic compatible system is weakly irreducible by [PT15, Thm.
1.7]; it follows immediately that potentially automorphic compatible systems arealso weakly irreducible. Conversely, a weakly irreducible compatible system is potentially automorphic by the results of [BLGGT14b]; see Theorem 2.1.16 below. 

16

F. CALEGARI, M. EMERTON, AND T. GEE

Lemma 1.5.12. Let F be CM, and let {rλ } be a weakly irreducible regular, odd,
polarizable compatible system of representations of GF . Then {rλ } is pure in thesense of [BLGGT14b, §5.1].
Proof. Since {rλ } is potentially automorphic by Lemma 1.5.11, it is pure by [BLGGT14b,
Cor. 5.4.3].
Lemma 1.5.13. Let F be CM, and let {rλ } be a regular, odd, polarizable compatible
system of representations of GF . Suppose furthermore that {rλ } is pure. Then we
may write {rλ } = ⊕si=1 {ri,λ }, where each {ri,λ } is a weakly irreducible, regular,
odd, polarizable compatible system of representations of GF .
Proof. This is immediate from [PT15, Thm. 2.1] and Lemma 1.5.11.



We will occasionally need to make use of compatibility at ramified places. While
we have not built this into our definition of a compatible system, it follows frompotential automorphy, as in the following result.
Proposition 1.5.14. Let F be CM, and let {rλ } be a weakly irreducible regular,
odd, polarizable compatible system of representations of GF with field of coefficients M .
Let v be a finite place of F , and suppose that v - Nλ (respectively, that v|Nλ).
Then:(1) For each finite extension K/Fv , rλ |GK is unramified (respectively, crystalline) for some λ if and only if it is so for all λ.
(2) Suppose that (1) holds. Then there is a representation rv of IK/Fv over M
such that for each λ, rλ |IK/Fv ∼= rv ).
= rv (respectively, WD(rλ |GFv )|IK/Fv ∼Proof. By Lemma 1.5.11, {rλ } is potentially automorphic, so it is strictly compatible by [BLGGT14b, Cor. 5.4.3]. Strict compatibility means by definition that theWeil–Deligne representation corresponding to rλ |GFv is independent of v, so theconsequences follow immediately.
We next establish some results describing how the property of weak irreducibilityof a compatible system behaves under restriction. To begin with, suppose that F
is a number field and that {rλ } is a compatible system of representations of GF ,and that the Zariski closure of the image of rλ is Gλ . Let G◦λ ⊂ Gλ denote theconnected component of the identity. The following is a theorem of Serre.
Theorem 1.5.15. The pre-image of G◦λ in GF is independent of λ.
Proof. See [LP92, Prop. 6.14].



As a corollary, we have the following result, that allows us to ensure that certain
restrictions of weakly irreducible compatible systems remain weakly irreducible.
Lemma 1.5.16. Let F be a number field, and let {rλ } be a compatible system
of GF -representations. Then there exists a finite extension F (avoid) /F with thefollowing property: if L/F is a finite extension linearly disjoint from F (avoid) ,and if r = rλ is any representation in the compatible system which is irreducible,
then r|GL is irreducible. In particular, if {rλ } is weakly irreducible then {rλ |GL } isweakly irreducible.

GLOBALLY REALIZABLE COMPONENTS OF DEFORMATION RINGS

17

Proof. Suppose that r = rλ is irreducible. If L is any finite degree extension of F ,then the Zariski closure of the image of r|GL will also contain G◦λ , because G◦λ isconnected and hence does not contain any finite index subgroups.
By Theorem 1.5.15, the pre-image of G◦λ in GF is independent of λ. Let F (avoid)
denote the corresponding fixed field. If L is disjoint from F (avoid) , then the component group of r|GL will be isomorphic to the component group of r, and hence
the Zariski closures of r and r|GL will coincide. In particular, r|GL will be irreducible.

We also have the following variant of the preceding result, which shows that
arbitrary restrictions (to CM extensions)) of weakly irreducible compatible systems(satisfying the appropriate hypotheses) remain weakly irreducible, provided that atleast one member of the restricted compatible system is irreducible.
Lemma 1.5.17. Let F be CM, and let {rλ } be a weakly irreducible regular, odd,
polarizable compatible system of representations of GF . Let M be a CM extensionof F . If some rµ |GM is irreducible, then {rλ |GM } (which is a priori a regular, odd,
polarizable compatible system of representations of GM ), is again weakly irreducible.
Proof. By Lemma 1.5.12, the weakly compatible system {rλ } is odd, polarized,
regular and pure. These properties are inherited by the system {rλ |GM }, which istherefore a direct sum of weakly irreducible compatible systems by Lemma 1.5.13.
Since rµ |GM is irreducible, there can only be one compatible system in this direct
sum, and {rλ |GM } is weakly irreducible, as required.

We close the present discussion of weak irreducibility with the following result,which establishes the weak irreducibility of certain tensor products of compatible
systems.
Lemma 1.5.18. Let F be CM, and let {sλ }, {tλ } be regular, odd, weakly irreduciblepolarizable compatible systems of representations of GF . Assume that {sλ ⊗ tλ } isregular, and that at least one representation sµ ⊗ tµ is irreducible. Then {sλ ⊗ tλ }is weakly irreducible.
Proof. By Lemma 1.5.12 each of {sλ }, {tλ } is pure, so that {sλ ⊗ tλ } is regular,pure, odd, and polarizable. By Lemma 1.5.13 it is therefore a direct sum of weakly
irreducible compatible systems; since sµ ⊗ tµ is irreducible, this direct sum can only
consist of a single compatible system, as required.

1.5.19. Deformation rings. When we consider deformation rings and automorphy
lifting theorems, we can no longer use algebraically closed coefficient fields. To thisend, we adopt the convention that O will denote the ring of integers in a finite
extension E/Qp with residue field F, and that E will be chosen large enough suchthat all representations under consideration are defined over E (and all mod p representations are defined over F); as always, the precise choice of E is unimportant.
As usual, let F be a CM field with maximal totally real subfield F + . Following [BG17], we work in a slightly more general context than [CHT08, BLGGT14b],
allowing ramification at primes of F + which are inert or ramified in F . This allows
us to make cleaner statements, and is also necessary for some of our argumentswith auxiliary primes.
Fix a prime p > 2 and a polarized residual representation (s, µ) of GF + with sabsolutely irreducible and (as always in the body of this paper) p > 2. We choose

18

F. CALEGARI, M. EMERTON, AND T. GEE

once and for all an element γ0 ∈ GF+ \GF (e.g. we could choose γ0 to be the complex
conjugation at one of the real places of F + ), and we let ρ : GF + → Gn (F) be a
fixed prolongation of (s, µ), following the procedure discussed in Subsection 1.5.1.
Let µ : GF + → O× be a lift of µ. Let A be a complete local Noetherian Oalgebra. Then a µ-polarized lifting of ρ to A is a representation ρ : GF + → Gn (A)with ν ◦ ρ = µ and ρ (mod mA ) = ρ. A µ-polarized deformation of ρ to A is
a 1 + Mn (mA )-conjugacy class of liftings. As in [Che11, Lem. 1.5]), µ-polarized
liftings and deformations ρ in this sense are equivalent to the data of a lifting or
deformation s of s which satisfies sc ∼= µs∨ (where the equivalence arises fromtaking s = ρ|GF ).
We also need to consider the corresponding local deformation problems. We
refer to [BG17, §3] for the definitions of deformations of fixed inertial and Hodge
types. Let v be a finite place of F , let ρv : GFv+ → Gn (F) be a representation
with multiplier µ, and let µ be a lift of µ. Then a µ-polarized lifting of ρv to A
is a representation ρv : GFv+ → Gn (A) with ν ◦ ρv = µ and ρv (mod mA ) =ρv . If we fix an inertial type IFv+ → Gn (E), then in the case l 6= p, we may
consider the universal framed deformation O-algebra R,τ of inertial type τ ; thisring is non-zero for a finite and nonempty set of inertial types τ . We refer toan irreducible component of any R,τ [1/p] as simply “a µ-polarized componentfor ρv ”; such a component uniquely determines τ . By [BG17, Lem. 3.4.1], each
irreducible component is invariant under conjugation, in the sense that conjugation
by elements of ker(Gn (R,τ ) → Gn (F)) preserves each irreducible component. We
will sometimes speak of polarized components, rather than µ-polarized components,
when the choice of µ is clear from the context.
If v|p, then in the same way we let R,τ,v denote the universal framed deformation O-algebra of ρ for µ-polarized potentially semistable lifts of inertial type τand Hodge type v. We again refer to an irreducible component of any R,τ,v [1/p]as simply “a µ-polarized component for ρv ”. Note that such a component again
uniquely determines τ and v; we say that a component is regular if v is regular(that is, the labelled Hodge–Tate weights are distinct); we will always assume thisin our main results.
Return now to the global situation of a polarized residual representation (s, µ)with prolongation ρ. Suppose that v splits in F as wwc . A choice of embedding F + → Fv+ gives rise to a choice of w|v in F . With respect to this choice,the representation ρ|GF + has image in Gn◦ (F) = GLn (F) × GL1 (F), and the projecv
tion to the first factor is the representation s|GFw . (A difference choice of embedding F + → Fv+ corresponding to wc |v in F would have the effect of replacing s|GFw
by µ ⊗ s∨ |GFw ' s|GFwc .) If ρv : GFv+ → Gn (A) is a µ-polarized lifting of ρ|GF + ,v
then the projection to GLn (A) gives rise to a lift sw : GFw = GFv+ → GLn (A)of s|GFw .
Lemma 1.5.20. If v splits in F , then the assignment ρv 7→ sw is an equivalenceof categories between the µ-polarized liftings of ρ|GF + and the liftings of s|GFw .
v

Proof. Under this identification, the representation ρv is simply the representation ρv = (sw , µ) : GFv+ → Gn◦ (A) ⊂ Gn (A), from which the result is clear. (cf. thediscussion in [CHT08, §2.3].)


GLOBALLY REALIZABLE COMPONENTS OF DEFORMATION RINGS

19

By Lemma 1.5.20, if v splits in F , then we can identify µ-polarized components with components of the corresponding lifting rings for s|GFw after choosing
a prime w|v (or after choosing an embedding F + → Fv+ which gives a canonicalchoice of w|v). We will sometimes do this without comment later in the paper.
Convention 1.5.21. Let F/F + be a CM extension. Given a prime v in F + , wechoose an embedding F + → Fv+ which in turn determines a choice of prime in F
above v which we denote by w.
Convention 1.5.22. If s : GF → GLn (F) is as above, with prolongation ρ : GF + →Gn (F), and if (following Convention 1.5.21) w denotes a prime of F lying over theprime v of F + , then we will often write “µ-polarized component for s|GFw ” rather
than “µ-polarized component for ρ|GF + ”.
v

0Given another representation ρ : GFv+ → Gm (F), and polarized components C,
D for ρ, ρ0 respectively, then there is a well-defined component C ⊗ D for ρ ⊗ρ0 . Similarly, if L/Fv+ is a finite extension, there is a well-defined µ|GL -polarizedcomponent C|L for ρ|GL . (In the case that v is a split prime, this is [BG17, Lem.
3.5.1], and the general case is proved in exactly the same way.)
We will frequently make use of the following lemma without further comment.
Lemma 1.5.23. Let F be a CM field, and let {rλ } be a regular, odd, polarizable,
weakly irreducible compatible system of representations of GF . Then for each λ andeach finite place w of F , the representation rλ |GFw lies on a unique component of
the corresponding deformation ring.
Proof. It suffices to prove that rλ |GFw defines a smooth point of the corresponding
deformation ring. By [BG17, Cor. 3.3.5], it is enough to prove that there is a finite
extension L/Fw such that rλ |GL is pure in the sense of [TY07, §1]. Since {rλ }is potentially automorphic by Lemma 1.5.11, this follows from the main theoremsof [Car12, Car14] (which show that automorphic Galois representations are pure).

We now return to global deformation problems.
Proposition 1.5.24. Let F be a CM field, and let p > 2 be prime. Let (r, µ) bean absolutely irreducible polarized representation of GF . Suppose that r is odd, andthat r|GF (ζp ) is absolutely irreducible, and let µ be a de Rham lift of µ. Let S bea finite set of finite places of F + such that r and µ are unramified outside S. For
each place v ∈ S, let Cv be a µ-polarized component for r|GFv , which is regular
if v|p.
Let Runiv be the universal deformation ring for µ-polarized deformations of rwhich are unramified outside S, and lie on the component Cv for each v ∈ S.
Then Runiv has Krull dimension at least one.
Proof. This is [BG17, Cor. 5.1.1] (note that the condition there of being “discrete
series and odd” is by definition the same as being odd in the sense of this paper). 
1.5.25. Automorphy lifting and adequate representations. We end this section byrecalling some results concerning automorphy lifting theorems. Let F be a CM
field with maximal totally real subfield F + . We say that a finite place v of F
is split if v|F + splits in F . In order to apply the (potential) automorphy results

20

F. CALEGARI, M. EMERTON, AND T. GEEof [BLGGT14b], we need to assume that all of the places v|p, and all of the places
at which our Galois representations are ramified, are split places; we will avoid
making such assumptions in our main results by making base changes.
We have the following theorem; the notion of adequacy is recalled in Definition 1.5.29 below.
Theorem 1.5.26. Let F be a CM field, and let p > 2 be prime. Suppose that p - n
and that ζp ∈/ F . Let (r, µ) be a polarized automorphic Galois representation, wherer : GF → GLn (Qp ), and assume that r(GF (ζp ) ) is adequate.
Let S be a finite set of places of F + which includes all places at which (r, µ)
is ramified, and all places dividing p, and for each v ∈ S let Cv denote the local
component at v on which ρ|GFv lies, where ρ : GF + → Gn (Qp ) is the prolongationof (r, µ). Assume that every place in S is a split place.
Let RC denote the global deformation O-algebra for r which parameterizes deformations of ρ that are µ-polarized, that are unramified outside S, and that for eachv ∈ S, lie on the corresponding component Cv . Then RC is a finite O-algebra, andany representation corresponding to a Qp -point of RC is automorphic.
Proof. The automorphy statement is essentially [Cal12, Thm. 7.1], and the finiteness statement follows easily from the proof of loc. cit. (cf. [Tho12, Thm. 10.1]).
We only need to justify the slightly weaker hypotheses that we are making here,
in comparison to assumptions 4(c) and 4(d) in the statement of [Cal12, Thm. 7.1].
Assumption 4(d) was only made because local-global compatibility at places dividing p was unknown at the time that [Cal12] was written, but it is now available
in the required generality thanks to [BLGGT14a]. Assumption 4(c) is satisfied
by Lemma 1.5.23.

As a consequence, we have the following useful finiteness result.
Lemma 1.5.27. Let p be an odd prime, and let F be a CM field with ζp 6∈ F .
Let {(sλ , µλ )} be a weakly irreducible, odd, regular, polarized compatible systemof n-dimensional representations of GF . Assume that p - n, let (s, µ) be the padic representation coming from the given compatible system, with corresponding
prolongation ρ, and assume that s(GF (ζp ) ) is adequate.
Let S be a finite set of finite places of F + which contains all of the places dividing p, and is such that ρ is unramified outside S. For each v ∈ S, let Cv denotethe local component at v on which ρ|GFv lies. Let RC denote the global deformation
O-algebra which parameterizes deformations of ρ that are µ-polarized, that are unramified outside S, and that, for each v ∈ S, lie on the corresponding component Cv .
Then RC is a finite O-algebra.
Proof. By Lemma 1.5.11, {(sλ , µλ )} is potentially automorphic. By [BLGGT14b,
Lem. 1.2.3], we can reduce to the case that {(sλ , µλ )} is in fact automorphic, andall the places in S are split places, in which case it follows from Theorem 1.5.26
that RC is a finite O-algebra.

1.5.28. Adequate representations. Let k be a field of characteristic p. We always
assume it is sufficiently large to contain all the eigenvalues of any representation
under consideration. Let V be a vector space over k and let G ⊂ GL(V ) be a groupwhich acts absolutely irreducibly. We firstly recall from [Tho12] what it means
for G to be adequate.

GLOBALLY REALIZABLE COMPONENTS OF DEFORMATION RINGS

21

Definition 1.5.29. G is adequate if the following conditions hold.
(1) H 0 (G, ad0 (V )) = 0.
(2) H 1 (G, k) = 0.
(3) H 1 (G, ad0 (V )) = 0.
(4) For every irreducible G-submodule W ⊂ ad0 V , there is an element g ∈ Gwith an eigenvalue α such that tr eg,α W 6= 0 (where eg,α is the projection
to the generalized α-eigenspace of g).
Lemma 1.5.30. Suppose that G acts absolutely irreducibly on V . Then the following are equivalent.
(1) Condition (4) of Definition 1.5.29.
(2) The set of semisimple elements of G spans ad(V ) ⊗k k as a k-vector space.
Proof. This follows from Lemma A.1 of the appendix to [Tho12], namely the equivalence between (i) and (iii).
Lemma 1.5.31. Suppose that V and V 0 are absolutely irreducible representationsof a group Γ. Suppose that the projective images of Γ on V and V 0 are disjoint,that is, the group Γ surjects onto the product of the projective images of Γ on V
and V 0 , and denote the projective images by PG and PG0 . Then the images of Γon ad(V ) and ad(V 0 ) are PG and PG0 respectively, and the image of Γ on ad(V ⊗V 0 ) is PG × PG0 .
Proof. The fact that Γ acts on ad(V ) as PG is completely formal. Hence under theassumption that the projective images PG and PG0 are disjoint, Γ acts on ad(V ) ⊕ad(V 0 ) via PG × PG0 . The kernel of the map
GL(ad(V )) × GL(ad(V 0 )) → GL(ad(V ⊗ V 0 ))consists precisely of pairs of scalar matrices (λ, λ−1 ). But it is not possible forany g ∈ G (or PG) to act on ad(V ) as a non-trivial scalar — this would force theaction of g on V itself to be scalar and then to be trivial on ad(V ).

Remark 1.5.32. The proof of Lemma 1.5.31 is just another way of saying that themap
PGL(V ) × PGL(V 0 ) → GL(ad(V ⊗ V 0 ))is injective.
The following lemma is similar to Lemma A.2 of the appendix to [Tho12].
Lemma 1.5.33. Suppose that V and V 0 are absolutely irreducible representationsof a group Γ. Suppose that the projective images of Γ on V and V 0 are disjoint,
and that the images of Γ on V and V 0 satisfy condition (4) of Definition 1.5.29.
Then the image of Γ on V ⊗ V 0 satisfies condition (4) of Definition 1.5.29.
Proof. Let G and G0 denote the images of Γ in V and V 0 respectively. By Lemma 1.5.30,
the semisimple elements g of G and g 0 of G0 span ad(V ) and ad(V 0 ) respectively. In
particular, the elements g ⊗ g 0 , which are also semisimple, span ad(V ) ⊗ ad(V 0 ) =ad(V ⊗ V 0 ).
Let g and g 0 be any pair of semisimple elements in G and G0 respectively. By
Lemma 1.5.31, there is a γ ∈ Γ which acts projectively on V and V 0 by g and g 0
respectively. Hence it acts on V and V 0 by λg and λ0 g 0 respectively for scalars λand λ0 . Hence it acts on V ⊗ V 0 by a scalar multiple of g ⊗ g 0 . In particular, it

22

F. CALEGARI, M. EMERTON, AND T. GEEspans the same line in ad(V ⊗ V 0 ) as g ⊗ g 0 . Hence these elements span ad(V ⊗ V 0 ),as required.

Lemma 1.5.34. Suppose that V and V 0 are absolutely irreducible representationsof a group Γ of dimensions n, n0 > 2(p + 1) respectively, whose projective images
are disjoint. Then the image of Γ acting on V ⊗ V 0 is adequate.
Proof. By Theorem A.9 of the appendix to [Tho12], the images of Γ acting on V
and V 0 are both adequate. That condition (4) of Definition 1.5.29 holds for theimage H of Γ acting on V ⊗ V 0 then follows from Lemma 1.5.33.
The adjoint representation of H has image PG × PG0 by Lemma 1.5.31, so there
is a surjective map H → PG × PG0 whose kernel Z is central in H (and acts by×scalars on V ⊗ V 0 ). Certainly Z injects into k and so has order prime to p. Let M
and M 0 be PG- and PG0 - modules respectively. Since Z has order prime to p,
inflation–restriction gives
H 1 (H, M ⊗ M 0 ) = H 1 (PG × PG0 , M ⊗ M 0 ).
Another application of inflation–restriction gives an exact sequence0

H 1 (PG, M ) ⊗ (M 0 )PG → H 1 (PG × PG0 , M ⊗ M 0 ) → M PG ⊗ H 1 (PG0 , M 0 ).
Letting M = M 0 = k or M = ad(V ) and M 0 = ad(V 0 ), we see the two exteriorgroups vanish by the adequacy of the images of Γ on V and V 0 , and hence so does
the middle group. Absolutely irreducibility is easy, and the lemma follows.

Lemma 1.5.35. Let G ⊂ GL(V ), and let H ⊂ G be a normal subgroup with G/Hof order prime to p, such that H is adequate. Then G is adequate.
Proof. If condition (4) of Definition 1.5.29 holds for H, it obviously holds for G.
So it suffices to check the cohomological conditions. We have (since G/H has orderprime to p) that
H 1 (G, M ) = H 1 (H, M )G/H .
Hence if the right hand side vanishes, then so does the left hand side. Similarly, if a
representation of G is absolutely irreducible after restriction to H, it is absolutely
irreducible.
Lemma 1.5.36. Let A ⊂ GLn (k) be absolutely irreducible with n > 2(p + 1).
If B ⊂ GLn (k) has projective image containing PSLn (l) for some sufficiently large
extension l/k (depending on A), then the image of A ⊗ B is adequate.
Proof. By using Lemma 1.5.35, we may assume that B has projective image exactly PSLn (l) for some l. This is a simple group. By taking l large enough sothat the projective image of A has order less than that of PSLn (l), we deduce thatthe projective images of A and B have no nontrivial common quotients. It follows
by Goursat’s Lemma that the image of A ⊗ B surjects onto to the product of the
projective images of A and B. We now finish by invoking Lemma 1.5.34.
The following is an immediate consequence of Lemma 1.5.36.
Lemma 1.5.37. Suppose that p > 2(n + 1), that L is a number field, and thata : GL → GLn (Fp ) is an irreducible representation. Then if q is a sufficiently
large power of p (depending on a), and b : GL → GLn (Fp ) has projective image
containing PSLn (Fq ), then (a ⊗ b)(GL ) is adequate.

GLOBALLY REALIZABLE COMPONENTS OF DEFORMATION RINGS

23

2. Globally realizable representations
2.1. Global realizability. Let E/Qp be a finite extension with ring of integers O
and residue field F. Let K/Qp be a finite extension, and letr : GK → GLn (F)be a representation.
Definition 2.1.1. Say the representation r admits many diagonalizable lifts if
the following holds: for any C ≥ 0, r admits a potentially diagonalizable lift withthe property that, for each embedding σ : K ,→ Qp , the σ-labelled Hodge–Tateweights of the lift all differ by at least C.
Remark 2.1.2. We expect that every representation r admits many diagonalizable lifts. In this paper, we will use a base change trick (based on Lemmas 2.1.3
and 4.1.8) to avoid needing to know this.
Lemma 2.1.3. For any representation r : GK → GLn (F), there is a finite extension L/K such that r|GL admits many diagonalizable lifts. Moreover, any L/Ksuch that r and the mod p cyclotomic character ε become trivial over GL has thisproperty.
Proof. Choose L so that each r|GL is trivial, and the mod p cyclotomic characterof GL is also trivial. For each integer C > 0, 1 ⊕ εC ⊕ · · · ⊕ ε(n−1)C is a potentiallydiagonalizable (in fact, diagonal) crystalline lift of r|GL , all of whose σ-labelled
Hodge–Tate weights differ by at least C.
Convention 2.1.4. We will frequently consider potentially diagonalizable lifts ofan r which admits many diagonalizable lifts. Whenever we do so, we will always
choose the lifts to have Hodge–Tate weights that are sufficiently spread out (inthe sense that the condition of Definition 2.1.1 holds for some sufficiently large C)
that all representations formed in the arguments that we make (which will involve
tensoring representations together) have regular Hodge–Tate weights. In order tostreamline the paper, we will not make this explicit in any of our arguments.
Definition 2.1.5 (Polarized Local Isomorphisms). Let F be a CM field. Supposethat a : GF → GLn (F) and b : GF → GLn (F) are absolutely irreducible polarizable
representations with respect to a character µ, so (in particular) they both prolong
to representations
ρ(a), ρ(b) : GF + → Gn (F),each of which is uniquely determined up to conjugation by an element in Gn0 (F).
Let v be a prime in F + , and let w be a prime above v in F . We define a polarized
isomorphism a|GFw ' b|GFw to be an isomorphism of representations which extends
to an isomorphism of polarized representations:
ρ(a)|GF + ' ρ(b)|GF + .
v

v+

If v ∈ F splits in F , then any isomorphism between a|GFw and b|GFw extends
to such an isomorphism, because GFv+ = GFw ⊂ GF , andρ(a)|GF = a × µ|GF : GF → GLn (F) × GL1 (F) = Gn◦ (F) ⊂ Gn (F)(and similarly for ρ(b)|GF ), so that ρ(a)|GF + = ρ(a)|GFw = a|GFw × µ|GFw (resp.
v

ρ(b)|GF + = ρ(b)|GFw = b|GFw × µ|GFw ). On the other hand, if v is inert or ramifiedv

24

F. CALEGARI, M. EMERTON, AND T. GEE

in F/F + and a|GFw = b|GFw is reducible, then this restriction may admit morethan one polarization, and so the requirement that the representations ρ(a)|GF +v

and ρ(b)|GF + be isomorphic may be a non-trivial condition.
v

Definition 2.1.6. Let F be a CM field. We say that a representation s : GF →GLn (Fp ) is reasonable if:• ζp ∈/ F , and s|GF (ζp ) is irreducible.
• s is polarizable and odd.
• p > 2(n + 1).
Definition 2.1.7. Let F be a CM field. We say that a representation s : GF →GLn (Fp ) is pleasant if:• ζp ∈/ F , and s|GF (ζp ) is irreducible.
• s is polarizable and odd.
• p > 2(n + 1).
• All the primes v|p in F + split in F .
• For each place w|p of F , s|GFw admits many diagonalizable lifts.
Lemma 2.1.8. Let s be a reasonable representation of GF . Let F (avoid) /F be a
finite extension. Then there is a finite extension L/F of CM fields, which is linearly
disjoint from F (avoid) over F , such that s|GL is pleasant.
ker r

Proof. Replace F (avoid) with F (avoid) · F(ζp ). Let E = E + F where E + /F +is any totally real extension linearly disjoint from F (avoid) /F + with the property
that for each place wE |p of E, both s|GEw and ε|GEw are trivial, where ε is theEE
mod-p cyclotomic character. It follows from Lemma 2.1.3 that s|GEw then admitsEmany diagonalizable lifts, and moreover, the same is true if one replaces E + by any
further finite extension. It now suffices to ensure the primes v|p in E + split in E.
To achieve this, we cross with a quadratic extension. Namely, let L+ = M + E + ,where M + /F + is a quadratic extension with the property that Mv+ ' Fv for v|p
in F + , and such that L = L+ F is linearly disjoint from F (avoid) over F .
Definition 2.1.9. Let K/Qp be a finite extension, and let ρ : GK → Gn (F) be a
representation with multiplier µK . Let µK be a de Rham lift of µK . A µK -polarizedcomponent C for ρ is globally realizable if there exists a CM number field F
and an odd, regular, polarized, weakly irreducible compatible system ({sλ }, {µλ })over F , with corresponding p-adic representation (s, µ), with the following properties:(1) The residual representation s is reasonable.
(2) There exists a prolongation ξ : GF + → Gn (F) of (s, µ), and a place v of F + ,such that Fv+ ' K, µ|GF + = µK , ξ|GF + ∼= ρ, and the representation ξ|GF +v
vv
lies on C.
We say that a de Rham lift ρ : GK → Gn (O) of ρ is globally realizable if it lies on
a globally realizable component.
Remark 2.1.10. In Definition 2.1.9, if the condition holds for one prolongation ξ,
then it holds for any prolongation. Indeed, we saw in Section 1.5 that any two
prolongations are conjugate by some element of 1 × GL1 ⊂ Gn◦ , and the componentsof the local deformation ring are invariant under conjugation by [BG17, Lem. 3.4.1].

GLOBALLY REALIZABLE COMPONENTS OF DEFORMATION RINGS

25

Remark 2.1.11. If v splits in F , then by Lemma 1.5.20, we can identify the µpolarized deformation ring for ρ with the lifting ring for s|GFw : GFw → GLn (K)(which is independent of µK ). Then, in the setting of Definition 2.1.9, it follows
from [CHT08, Lem. 4.1.6] that the condition of a component being globally realizable is independent of the choice of µK (as we can twist compatible systems byalgebraic characters).
Remark 2.1.12. Note that by definition if a component is globally realizable thenit is regular.
Remark 2.1.13. While it is not obvious from the definition, as a consequence of ourmain results we can show that if n = 2 or n is odd, then any potentially globallyrealizable component is globally realizable. More precisely, a component C for ρ :GK → Gn (F) is globally realizable if and only if there exists a finite extension L/Ksuch that C|L is globally realizable. See Corollary 4.2.13.
Remark 2.1.14. Any (regular) potentially diagonalizable representation is globallyrealizable; this is easily proved using the methods of [EG14, App. A], and in particular if n = 2 or n is odd, it is a simple consequence of Corollary 4.2.13 below (whichshows that it is enough to prove this after an arbitrary base change), together
with [BLGGT14b, Lem. A.2.5] (which shows how to globalize local representations
which are induced from characters).
The notion of being globally realizable can also be formulated in automorphic
terms:
Lemma 2.1.15. A µK -polarized component C for ρ is globally realizable if andonly if there exists a CM number field F , a regular algebraic cuspidal polarized
automorphic representation (π, χ) of GLn /F , and a prolongation ρp (π) of rp (π)
such that:(1) There is a prime v in F + such that Fv+ ' K, ρp (π)|GFv ∼= ρ, (ε1−n rp (χ))|GK =µK , and the representation ρp (π)|GF + lies on C.
v(2) The residual representation rp (π) is reasonable.
Proof. For the “if” direction, note that if these conditions are satisfied, then we may
take ({sλ }, {µλ }) in the definition of global realizability to be ({rλ (π)}, {ε1−n rλ (χ)}).
Conversely, if C is globally realizable, then we apply Theorem 2.1.16 below, takker s
ing F (avoid) to be F
(ζp ), and S to be the set of places of F which lie over p.
Then the conditions in the Lemma are satisfied by the automorphic representation
corresponding to the compatible system ({sλ |GL }, {µλ |GL+ }).

(i)

(i)

Theorem 2.1.16. Let ({sλ }, {µλ }), i = 1, . . . , r be compatible systems of odd,
regular, weakly irreducible polarized Galois representations over a CM field F . Let
S be a finite set of finite places of F + , and let F (avoid) /F be a finite extension.
Then there is a finite Galois extension L/F of CM fields with the properties that:• L is linearly disjoint from F (avoid) over F .
• Every place in S splits completely in L+ .
(i)(i)
• Each ({sλ |GL }, {µλ |GL+ }) is automorphic.
(i)

(i)

Proof. By [BLGGT14b, Thm. 5.4.1], each compatible system ({sλ }, {µλ }) is potentially automorphic over some finite extension Li /F , which is linearly disjoint

26

F. CALEGARI, M. EMERTON, AND T. GEE(i)from F (avoid) over F . (Strictly speaking, that result assumes that all of the sλare irreducible, but as explained in the introduction to [PT15], all that is actuallyneeded is that there is a positive density set of rational primes l such that for each(i)
λ|l, sλ is irreducible.)It suffices to show that L/F can be chosen simultaneously for all i, in such a
way that all places of F above S split completely. This can be arranged by aslight refinement of the arguments of [BLGGT14b]; we explain the main idea here,
referring the interested reader to the proof of [EG14, Prop. A.6] for a more detailed
treatment of a similar result.
(i)
As a first step, note that since each {sλ } is potentially automorphic, it follows
from [PT15, Lem. 1.5, Thm. 1.7] that there is a positive density set of rational(i)primes l such that if λ|l, then each {sλ } is irreducible. Therefore by [BLGGT14b,(i)
Prop. 5.3.2] we can choose l and λ|l such that each sλ is Fontaine–Laffaille at all(i)(i)
primes dividing l, each sλ is irreducible, and indeed sλ |GF (ζl ) is irreducible. We
also assume that l > 2(max dim siλ + 1)
In the main argument of [BLGGT14b], the field L/F is constructed by a (finite)
number of applications of the theorem of Moret–Bailly, applied to a particular
moduli space T over F + (see the proof of [BLGGT14b, Thm. 3.1.2]). By the versionof the theorem of Moret–Bailly given in [BLGGT14b, Prop. 3.1.1], we can arrangethat the places in S all split completely in L+ provided that T has Fv+ -rational
points for all places v ∈ S. This need not be the case, but we can in any casechoose a finite solvable extension of totally real fields M + /F + so that T has Mw+ rational points for each place w of M + lying over a place in S. We then replace T
by the Weil restriction ResM + /F + T , and running the arguments of [BLGGT14b],
we find a finite Galois extension L+ /F + , linearly disjoint from F (avoid) over F + , inwhich all places in S split completely, with the property that if we set L = L+ F
(i)then the restrictions {sλ |GLM + } are automorphic. Since the extension LM + /L is(i)solvable, it follows that each {sλ |GL } is automorphic, as required.



3. Compatible systems
Our aim in this section is to prove results showing that if one representation in
a compatible system is a tensor product, then the compatible system is a tensor
product. We do this under somewhat restrictive hypotheses (see Theorem 3.4.3
below), which we will suffice for the results of the the following section due to somebase change tricks and arguments with auxiliary places. We also prove a number ofother results about compatible systems that we will use in Section 4. Our results aremostly Lie-theoretic, and in particular we make crucial use of the results of [LP92].
3.1. Component groups. Recall that by Theorem 1.5.15, any compatible system
has a well-defined component group. We have the following technical lemma.
Lemma 3.1.1. Let F be a number field, let r : GF → GLn (Fp ) be irreducible, andsuppose that p > max(n, 3). Let L/F denote the field F (ker(r)). Let {rλ } be any
compatible system of Galois representations such that rp = r and such that det(rp )has infinite image. Let F 0 /F be a finite Galois extension which is linearly disjoint
from L/F . Then the component group of {rλ |GF 0 } is independent of F 0 /F .
Proof. Let G denote the Zariski closure of im(rp ), and let G0 denote the connected component of G. Let im◦ (rp ) denote the intersection of im(rp ) with G0 .

GLOBALLY REALIZABLE COMPONENTS OF DEFORMATION RINGS

27

Let F ◦ denote the fixed field of im◦ (rp ). By Theorem 1.5.15, F ◦ is independentof p, and G/G0 ' Gal(F ◦ /F ). It suffices to show that F ◦ is completely contained in L. The image im(rp ) of rp inside GLn (Zp ) naturally admits a surjection
onto Gal(F ◦ /F ). If Gal(F ◦ /F ) has order prime to p, then this surjection factors
through im(rp ), since the kernel of im(rp ) → im(rp ) is a pro-p group. But this
implies that F ◦ is contained in L. Thus we may assume that Gal(F ◦ /F ) has orderdivisible by p, and hence that the component group G/G◦ has order divisible by p.
Note that G acts irreducibly because r is irreducible. By assumption, an elementof order p in G/G◦ induces an outer automorphism of G◦ of order p. If this automorphism is trivial, then, by Schur’s lemma, this automorphism may be modified
by an inner automorphism to be a scalar which does not lie in G◦ . The assumptionthat the determinant has infinite image, however, implies that (since G is reductive
and irreducible) the center Z is infinite, and hence that Z ◦ = Z. This is a contradiction, and hence this order p element induces a non-trivial outer automorphismof G◦ , and hence also of its Lie algebra g. The automorphism group of any simple
Lie algebra has order at most 3 < p. Thus this order p automorphism must act
by permuting the simple factors. Yet G acts on a space of dimension n, and hence
there are at most n simple factors. Hence we obtain a non-trivial element of order p
in Sn , which is impossible for p > n.

3.2. Representation theory. In this section, we begin by proving some basic
representation theoretic lemmas for reductive groups G. All the representations we
consider below are assumed to be finite dimensional.
3.2.1. Reductive linear algebraic groups. Let k be an algebraically closed field of
characteristic zero. If G is a connected reductive linear algebraic group over k,then we let Gder denote the derived subgroup of G — it is a connected semisimple
linear algebraic group — and let Z denote the centre of G. The natural morphismof connected reductive linear algebraic groups
Gder × Z ◦ → G(where as usual Z ◦ denotes the connected component of the identity in Z) is surjective, and its kernel is contained in (the anti-diagonally embedded copy of) theintersection Gder ∩ Z ◦ , and thus is contained in the centre of Gder ; in particular, it
is finite.
e der denote the simply connected cover of Gder ; it is again a connected
We let G
e der →
semisimple linear algebraic group, and the kernel of the natural surjection G
der
der
◦
e
e
G
is finite and central. We write G := G × Z (and note that the possible
e der is ameliorated by the fact that Ge der isambiguity in our use of the notation Genaturally identified with the derived subgroup of G). The composite morphisme=G
e der × Z ◦ → Gder × Z ◦ → G
Gis again surjective, and its kernel is finite and central.
e der is semisimple and simply connected, it may be written as a direct
Since Ge is a direct product of
product of almost simple linear algebraic groups. Thus G
such groups and a torus.
If H and J are linear algebraic groups, then any irreducible representation Wof the product H × J may be factored (uniquely, up to isomorphism) as a tensor
product W ∼= U ⊗k V, where U (resp. V ) is an irreducible representation of H (resp.

28

F. CALEGARI, M. EMERTON, AND T. GEEJ). Applying this remark in the context of the preceding discussion (thinking of arepresentation of G as a representation of Gder × Z ◦ via inflation), we find that theirreducible representations of G are obtained from the irreducible representations
V of Gder by choosing a character of Z ◦ which coincides with the given actionof Gder ∩ Z ◦ on V (Schur’s Lemma ensures that this action is indeed given by acharacter) and extending the Gder -action on V to an action of G (thought of as aquotient of Gder × Z ◦ ) via having Z ◦ act through this choice of character.
e so that gder is the Lie
If g denotes the Lie algebra of G (or equivalently of G),
e der ), then passing to the induced gder -action
algebra of Gder (or equivalently of Ge der induces an equivalence of categories between the category of finite-dimensional G
derrepresentations over k, and the category of finite-dimensional g -representations
over k. In particular, this equivalence induces a bijection between the isomorphisme der and the isomorphism classes of irreclasses of irreducible representations of G
derducible representations of g .
The following lemma (and its proof) is a very special case of a theorem of Rajan [Raj04].
Lemma 3.2.2. Let U and V be two non-trivial representations of a simple Liealgebra g over an algebraically closed field of characteristic zero. Then U ⊗ V is
reducible.
Proof. Without loss of generality we may assume that U and V are irreducible.
Let the highest weights of U and V be λ and µ respectively. Then, if U ⊗ V isirreducible, it must be the irreducible representation of highest weight λ + µ. By
the Weyl character formula, this implies thatdim(U ) dim(V ) Y hρ + λ, αihρ + µ, αi=.
1=dim(U ⊗ V )hρ, αihρ + λ + µ, αi+ΦEach individual factor has the form:hρ, αi2 + hρ, αi(hλ, αi + hµ, αi) + hλ, αihµ, αi≥ 1.
hρ, αi2 + hρ, αi(hλ, αi + hµ, αi)
Since the pairing is non-negative, we obtain a contradiction unless hλ, αihµ, αi = 0
for each root α ∈ Φ+ . Because g is simple, there exists a maximal root β ∈Φ+ such that, for any dominant weight ν, one has hν, βi ≥ hν, αi for any α ∈Φ+ . In particular, assuming without loss of generality that hλ, βi = 0, we deducethat hλ, αi = 0 for all roots in Φ+ , which implies that λ = 0 and the corresponding
representation is trivial.

We say that an irreducible representation W of the connected reductive linear
algebraic group G is tensor indecomposable if, for any isomorphism W ' U ⊗ Vof G-representations, either U or V is a character.
Lemma 3.2.3. Let G = G◦ be a connected reductive Lie group over an algebraicallye denote the finite cover of G constructed
closed field of characteristic zero, and let G
in the preceding discussion. If W is an irreducible representation of G of dimene via inflation, has a factorization
sion > 1, thought of a representation of GOW 'Vie where the Vi areas a tensor product of tensor indecomposable representations of G,
unique up to reordering and twisting and have dimension > 1.

GLOBALLY REALIZABLE COMPONENTS OF DEFORMATION RINGS

29

e is a direct product of almost simple
Proof. As noted above, the algebraic group G
linear algebraic groups Gi and a torus T . Any irreducible representation of G is thena tensor product of irreducible representations of the Gi up to twist by a characterof T . It then suffices to show that the tensor indecomposable representations are
precisely the representations of a single simple factor Gi up to twist. This follows
immediately from Lemma 3.2.2 (which shows that an irreducible representation of
any Gi is automatically tensor indecomposable).
We next establish some purely representation-theoretic results.
Lemma 3.2.4. Let V and W be finite dimensional linear representations of agroup G over an algebraically closed field. Suppose that V ⊗ W decomposes as adirect sum of characters. Suppose, in addition, there are at most three isomorphismclasses of characters which occur in direct sum. Then V and W also admit such a
decomposition.
Proof. First consider the case when both V and W are irreducible. Note that or
any character χ,
dim HomG (V ⊗ W, χ) = dim HomG (V, W ∨ χ) ≤ 1,
where the latter inequality follows from the irreducibility of V and W together with
Schur’s Lemma. Since the number of distinct characters is at most three, it follows
that dim(V ⊗ W ) ≤ 3, and thus at least one of V or W is a character, and theresult follows.
In the general case, when V and W are not necessarily irreducible, choose irreducible subrepresentations V 0 ⊂ V and W 0 ⊂ W . Then V 0 ⊗ W 0 ⊂ V ⊗ W , and sofrom what we’ve already proved, we find that each of V 0 and W 0 is necessarily acharacter. We then see that
V = V ⊗ W 0 ⊗ (W 0 )−1 ⊂ (V ⊗ W ) ⊗ (W 0 )−1
is a direct sum of at most three characters (possibly with multiplicities), and similarly for
W = (V 0 )−1 ⊗ V 0 ⊗ W ⊂ (V 0 )−1 ⊗ V ⊗ W.
Remark 3.2.5. The preceding result is false when there are four distinct characters.
Indeed, one can take V = W to be the irreducible 2-dimensional representation(over Q, say) of the quaternion group Q of order 8.
We recall the definition of a strongly irreducible representation.
Definition 3.2.6. A representation of a group G (either a Lie group or a Galois
group) is strongly irreducible if it remains irreducible after restriction to any finite
index subgroup H ⊂ G.
Remark 3.2.7. A representation of a reductive Lie group G is strongly irreducible
if and only if it remains irreducible after restriction to the connected component G◦ ⊂ G. If a topological group G acts continuously on W and a subgroup H ⊂ G fixes a closed subspace 0 ( V ( W , then the closure of H in G
also fixes V . Hence continuous representations of a Galois group G are strongly
irreducible if and only if they remain irreducible after restriction to any closed finite
index subgroup H.

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F. CALEGARI, M. EMERTON, AND T. GEE

Lemma 3.2.8. Let V and W be irreducible representations of a group G over analgebraically closed field k of characteristic zero, and suppose that the action of G
on V is strongly irreducible. Then, up to scaling, there is at most one non-trivial
bilinear G-equivariant pairing:
W × V → k(µ),
where k(µ) is the twist of the trivial representation by a character µ of G, and µ isallowed to range over all characters of G.
Proof. Any such pairing gives rise to an isomorphism V ' W ∨ (µ). If there existed
two such pairings with the same µ which were not the same up to scalar, then there
would be two corresponding isomorphisms in HomG (V, W ∨ (µ)). Using either of theidentifications of V with W ∨ (µ), we deduce that2 ≤ dim HomG (V, W ∨ (µ)) = dim HomG (V, V ),which contradicts the irreducibility of V by Schur’s Lemma. If there were two suchpairings with different µ, then, denoting one character by µ and the other by µ ⊗ χfor a non-trivial character χ, we deduce that V ' W ∨ (µ) and V ' W ∨ (µ ⊗ χ)and thus V ' V (χ). Taking determinants of both sides, it follows that χn = 1.
Hence χ defines a map: G → µn ⊂ k × , and in particular the image of χ is finite.
Let H denote the kernel of this map, which is of finite index. Then we deducethat dim HomH (V, V ) ≥ 2. By Schur’s Lemma, this implies that V |H is reducible,
contradicting our assumptions.

3.2.9. Galois representations. Throughout this subsection we let F be a (not necessarily CM) number field.
Lemma 3.2.10. Let r : GF → GLm (Qp ) be a representation with image Γ :=r(GF ), and suppose that the (semisimple) residual representation r has image r(GF ) =Γ ⊂ GLm (Fp ) containing SLm (Fq ) for q a sufficiently large power of p (we cantake q = p if p > 5, and q = 25 if p = 5). Then the Zariski closure G of Γ
contains SLm (Qp ).
Proof. After conjugation, the image is a closed subgroup of GLm (O) for O thering of integers in some finite extension of Qp . By the main result of [Man15], theimage therefore contains a conjugate of SLm (W (Fq )), from which the result follows
immediately (since SLm is unirational, by [Bor91, Thm. 18.2]).
Lemma 3.2.11. Let {rλ } be a weakly compatible system of GF -representations ofdimension m = n2 . Suppose that r = rp = a ⊗ b where a and b are n-dimensional
representations which have images whose Zariski closure contains SLn , and thecorresponding Lie algebra of rp contains sln × sln . Then the component group of
the compatible system {ad(rλ )} is trivial.
Proof. By Theorem 1.5.15, the pre-image of G◦λ in GF is independent of λ, andso the component group is independent of λ, and thus we may choose λ = p. We
have ad(r) = ad(a) ⊗ ad(b). By assumption, the Zariski closures of the imagesof ad(a) and ad(b) are both PGLn , and hence the Zariski closure of ad(r) is G =PGLn × PGLn . Since G = G◦ , the component group is trivial.
The following result relates irreducibility and strong irreducibility of Galois representations.

GLOBALLY REALIZABLE COMPONENTS OF DEFORMATION RINGS

31

Lemma 3.2.12. Let r : GF → GLn (Qp ) be an irreducible Galois representation
which is Hodge–Tate with regular Hodge–Tate weights. If r is not strongly irreducible, then r is induced from a strongly irreducible representation over some finite
extension M/F .
Proof. This is proved in the course of the proof of [CG13, Cor. 4.4]; we recall theargument. For any finite extension E/F , either r|GE is irreducible or it decomposes
into a sum of distinct irreducible representations. This follows immediately fromthe fact that r|GE has distinct Hodge–Tate weights at any prime w|l. (Note thatr|GE is necessarily semisimple: if V denotes any irreducible subrepresentation, thenthe various translates of V by elements of GF are stable under the corresponding
conjugates of GE , and so we see that r|GE becomes completely decomposable under restriction to a finite index subgroup, so must already have been semisimple.)
Suppose then that r|GE is reducible for some finite extension E/F . Replacing E by
its normal closure over F , we may assume that the extension E/F is Galois, andF
the claim that r is induced is immediate from [CG13, Lem. 4.3]. If r = IndG
GM s,then s is also Hodge–Tate with regular Hodge–Tate weights, so by induction on n
we may assume that r is induced from a strongly irreducible representation.

The following lemma will prove useful for lifting Galois representations along
central extensions:Lemma 3.2.13. Suppose that r : GF → GLmn (Qp ) is a Galois representation
whose image has Zariski closure inside the image of the map GLn × GLm → GLnm .
Then there exist Galois representations rA and rB of dimension n and m respectively such that r ' rA ⊗ rB .
Proof. There is a central extension
0 → Z → GLn (Qp ) × GLm (Qp ) → Γ(Qp ) → 0,
where Γ denotes the image of GLn × GLm in GLnm , and Z is the Qp -points ofa torus embedded anti-diagonally. The result then follows directly from [Con18,
Proposition 5.3].
Lemma 3.2.14. Fix n ≥ m, and let r : GF → GLmn (Qp ) be a strongly irreducible
representation such that the Lie algebra of the Zariski closure of the image of r isisomorphic to t × h × sln , where:• t is a torus of rank at most 1,• h is semisimple, and• the corresponding mn-dimensional representation of h × sln is the tensor
product of an m-dimensional representation of h with the standard representation of sln .
Then either:(1) r decomposes as a tensor product a ⊗ b where a is of dimension m and b
is of dimension n, and furthermore the Zariski closure of the image of b
contains SLn ; or(2) m = n, and r is the tensor induction of an n-dimensional representationof GL for some quadratic extension L of F .
Moreover, in case (1), the representations a and b are unique up to twisting by acharacter, or possibly permuting a and b if m = n and h = sln .

32

F. CALEGARI, M. EMERTON, AND T. GEE

Proof. Let G denote the Zariski closure of the image of r, and let G◦ denote theconnected component of G. For now, let us regard r as a (faithful) representationof G. We shall exhibit a factorization of r as a tensor product of representations ofsome cover of G (possibly after passing to a quadratic extension) and then promote
this to an actual factorization of Galois representations by Lemma 3.2.13.
By our assumptions, together with the general discussion of (3.2.1), we may finda finite cover of G◦ by a group of the form T × H × SLn , where H is a product ofalmost simple groups, and T is a torus. If we let r◦ denote the restriction of r to G◦ ,regarded as a representation of T × H × SLn by inflation, then r◦ is irreducible (asr is strongly irreducible by assumption), and so we may write r◦ = a◦ ⊗ b◦ ⊗ c◦ ,where a◦ is an irreducible representation of H, b◦ is the standard n-dimensional
representation of SLn , and c◦ is an irreducible (and hence one-dimensional) representation of T . (We do not assert that any of a◦ , b◦ , or c◦ are representationsof G◦ .)
After twisting by a character, we may assume (for example by [Pat12, Lem.
2.3.15]) that the determinant of r has finite image, and hence that the determinantof r◦ is trivial. Thus we may in fact assume that c◦ , and hence t and T , are trivial,
and we do so from now on. Thus we assume that G◦ admits a finite cover by H×SLn ,and that r◦ admits a corresponding tensor factorization a◦ ⊗ b◦ with b◦ being thestandard representation; we now attempt to extend this tensor factorization to acorresponding tensor factorization of r.
Since G◦ is normal in G, we obtain a conjugation action of G on G◦ , and hence
on its universal cover H × SLn . (Recall that the formation of universal covers isfunctorial in pointed spaces, and note that the conjugation action of G on G◦ acts
via automorphisms of the pointed space (G◦ , 1).) Suppose firstly that h 6= sln ; thenthere are no non-trivial morphisms sln → h (since h has a faithful representationof dimension m ≤ n), so that any automorphism of h × sln must fix the sln factor,
and correspondingly any automorphism of H × SLn must fix the SLn -factor.
The component group G/G◦ is then endowed with a homomorphism(3.2.15)

G/G◦ → Out(SLn )to the group of outer automorphisms of SLn , which we claim is trivial. To see this,note first that if n = 2, then Out(SL2 ) is trivial, and so we are done. If n ≥ 3, thenthe outer automorphism group of SLn is cyclic of order 2. If G/G◦ surjects onto
this outer automorphism group, then the restriction of r◦ to the SLn -factor, which
is a direct sum of m copies of b◦ , is isomorphic to its outer twist, which is a direct
sum of m copies of (b◦ )∨ . Since n ≥ 3, the standard representation of SLn is not
self-dual, and hence this is not possible. Thus (3.2.15) is trivial, as claimed.
The action of G on the SLn -factor by conjugation is thus an inner action, andso induces a homomorphism G → PGLn , compatible with the given map SLn →G◦ . Let K ⊆ G denote the kernel of this homomorphism; the compatibility justremarked upon shows that K contains the image of H in G◦ . Then we obtain asurjection K × SLn → G compatible with the given surjection H × SLn → G◦ .
Correspondingly, we obtain a tensor factorization r = a ⊗ b of r inflated to thecover K × SLn of G compatible with the factorization r◦ = a◦ ⊗ b◦ of G◦ . This now
induces a factorization of Galois representations by Lemma 3.2.13. Note that b◦
is the standard representation of SLn , so the Zariski closure of the image of b
contains SLn , as claimed.

GLOBALLY REALIZABLE COMPONENTS OF DEFORMATION RINGS

33

We now prove that this factorization is unique. Suppose that r ' a0 ⊗ b0 ' a ⊗ b.
We already have uniqueness of these representations over G◦ by Lemma 3.2.3.
Hence it follows that Hom(a, a0 ) as a G-representation has a summand which becomes trivial when restricted to G◦ , and hence has a summand on which G acts
through the finite quotient G/G◦ . If this factor is one dimensional, then a and a0
are isomorphic up to twist. If this factor has dimension > 1, then, over G◦ , we seethat Hom(a, a0 )|G◦ = Hom(a, a)|G◦ has at least two trivial factors, which implies
that a is reducible over G◦ , contradicting the strong irreducibility of r. The same
logic applies to b, as required.
Suppose finally that h = sln . The argument proceeds as above, except now wehave to allow the possibility that G/G◦ also swaps the factors. Assuming we are inthis case, replacing G by G0 where G0 is the kernel of the map G → G/G◦ → S2 , weobtain a tensor factorization of Galois representations as above over some quadratic
extension. But then the image of r must coincide (up to twist) with the tensor
induction of the corresponding n-dimensional representation (of a or b) from thisquadratic extension.
Lemma 3.2.16. Consider p-adic representations a and b of GF of dimensions mand n with m ≤ n. Suppose that
(1) The representation a ⊗ b is irreducible.
(2) The residual representation b has image containing SLn (Fq ) for q a sufficiently large power of p (in the sense of Lemma 3.2.10).
Let A and B denote the Zariski closures of the images of a and b respectively.
Let Ader and B der denote the corresponding derived subgroups, and Ader,◦ and B der,◦the connected components of these groups. Let G denote the Zariski closure of theimage of a ⊗ b. Then the corresponding representation of Gder,◦ is the natural representation of Ader,◦ × B der,◦ corresponding to the tensor product of the two naturalrepresentations. This identifies Gder,◦ with the image of Ader,◦ × B der,◦ in the automorphism group of the exterior tensor product of the two natural representations.
Proof. By Lemma 3.2.10, we have B der,◦ = SLn . Certainly the connected subgroup Gder,◦ ⊂ SLn2 lies inside the image Γ of Ader,◦ × B der,◦ under the exterior tensor product. Since Γ is connected (since it is the image of a connectedgroup under an isogeny), it contains no proper finite index subgroups. Thus toprove Gder,◦ ,→ Γ is an isomorphism, it suffices to prove it is an isogeny, which we
can do on the level of Lie algebras.
Denote the Lie algebras of A and B by a ⊕ tA and b ⊕ tB where a and b aresemisimple, and tA and tB are tori of rank at most one. Note that b = sln .
After twisting, we may assume that tB is trivial, and that the Lie algebra of G
is g ⊕ tG where tG ' tA . Let d ⊕ tD denote the Lie algebra (decomposed as asemisimple part d and a torus tD ) of the Zariski closure of the image of a ⊕ b.
There is an inclusion d ⊕ tD ⊂ a ⊕ b ⊕ tA , which induces an isomorphism tD ' tA .
Hence d ,→ a ⊕ b. On the other hand, since (a ⊕ b)⊗2 contains a ⊗ b, there is asurjection d  g. There is an isomorphismad0 (a ⊗ b) ' ad0 (a) ⊕ ad0 (b) ⊕ ad0 (a) ⊗ ad0 (b) .
The corresponding Lie algebras of the images of ad0 (a) and ad0 (b) are a and b
respectively, and the Lie algebra of the image of ad0 (a ⊗ b) is g. Hence there are

34

F. CALEGARI, M. EMERTON, AND T. GEEmapsa ⊕ b ⊇ d  g → a ⊕ b.
The corresponding maps d → a and d → b (either coming from the inclusion
into a ⊕ b or via the map to g) may be identified with each other, because thesemisimple part of the Lie algebra of a is canonically identified with the Lie algebraof ad0 (a). Moreover, these maps are both surjective, and thus g is identified with d.
By Goursat’s Lemma, the inclusion g ⊆ a ⊕ b is the pullback to a ⊕ b of the graphof an isomorphism
a/nA ' b/nB ,for some ideals nA or nB which may be identified with the kernels of the projections from g → b and g → a respectively. Since b = sln is simple, either both sides
are trivial, in which case d ' g ' a ⊕ b (and we are done), or there is a surjection a  b. In this latter case, by rank considerations (since A acts faithfully on a
space of dimension m ≤ n), we deduce that a ' sln and that the map above induces
an isomorphism of Lie algebras a ' b, and thus g ' a ' b is diagonally embedded
in a ⊕ b. This implies that, still on the Lie algebra level, the representation a ⊗ bmust come from the tensor product of an n-dimensional representation of sln with asecond n-dimensional representation of the same sln . In either case (standard tensor
standard or standard tensor dual), the corresponding representation would be reducible (as also follows from a special case of Lemma 3.2.2). This implies (returning
to the Lie group level) that, over some finite extension, a ⊗ b is either isomorphic
to the direct sum of a 1-dimensional representation
and an irreducible n2 − 1dimensional representation or an irreducible n2 -dimensional representation and anirreducible n+1
2 -dimensional representation (depending on whether the representations of sln are the same or dual to each other). But since a⊗b itself is irreducible
by hypothesis, it can only decompose over a finite extension into irreducible representations of the same dimension. The claim follows.

Theorem 3.2.17. Let {rλ } be a compatible system of GF -representations of dimension mn. Suppose there exists a prime p with r = rp satisfying the following:(1) There exist p-adic representations a and b of GF of dimensions m and nwith m ≤ n such that r ' a ⊗ b.
(2) The representation a ⊗ b is irreducible.
(3) The residual representation b has image containing SLn (Fq ) for q a sufficiently large power of p (in the sense of Lemma 3.2.10).
Suppose firstly that (m, n) 6= (2, 2). Let λ be a place with residue characteristic lsuch that rλ is strongly irreducible. Then there exist representations aλ and bλ , ofdimensions m and n respectively, such that rλ = aλ ⊗bλ . Moreover, the image of bλhas Zariski closure containing SLn (Ql ), and aλ and bλ are unique up to twist by acharacter and up to permutation — the latter being possible only when n = m.
Suppose now that m = n = 2. In this case, assume also that {rλ } is odd, regular,
polarizable, and weakly irreducible. Then there exists a set of primes l of density one
such that for each λ|l, rλ is strongly irreducible and admits a decomposition rλ =aλ ⊗ bλ satisfying the conditions in the previous paragraph.
Proof. Let A and B denote the Zariski closures of the images of a and b respectively. By Lemma 3.2.16, if G denotes the Zariski closure of the image of a ⊗ b,then we may identify the corresponding representation of Gder,◦ with the natural

GLOBALLY REALIZABLE COMPONENTS OF DEFORMATION RINGS

35

representation of Ader,◦ × B der,◦ corresponding to the tensor product of the twonatural representations.
Let λ denote a prime for which rλ is strongly irreducible. Let Gλ denote theZariski closure of the image of rλ , and let Gder
denote the corresponding derived
λsubgroup, and Gder,◦theconnected
componentofthis group. By the strong irreλducibility assumption, the corresponding representation of Gλder,◦ is irreducible. By
a Theorem of Serre [LP92, Prop. 6.12], the formal character of a compatible system
of Galois representations is independent of λ. In particular, the formal charactersof the corresponding representations of Ader,◦ × B der,◦ = Ader,◦ × SLn and Gλder,◦coincide. It is possible for the formal characters of irreducible representations ofconnected groups to coincide even when the groups differ (for example, there exist 27- dimensional irreducible representations of G2 and SL3 with the same formal
character). However, what is true ([LP92, Thm. 5.6, Prop. 5.7]) is that every such
equality arises from taking the tensor product of a list of (explicitly given) basic
similarity relations, described explicitly in §5.3.1–5.3.4 of [LP92]. In particular,
note that that the standard representation of SLn for any n > 2 does not admit
any basic similarity relations (which one can also deduce by a consideration ofranks), while in the case n = 2, the only similarity relation which intervenes in oursituation is that given by the coincidence of the formal character of the standard
representation of Sp4 with that of the external tensor product of two copies of thestandard representation of SL2 . Applied to our situation, it follows that if n > 2(which we assume for the time being, returning to the case n = 2 at the end of theproof) there exists a connected semisimple group Hλder,◦ such that the representation of Gder,◦is the tensor product of an irreducible n-dimensional representation
λ
der,◦
of Hλ
with the standard representation of SLn . Note that the Lie algebra hof Hλder,◦ need not a priori be equal to the Lie algebra a of Ader,◦ , but this does
not concern us. The result then follows from Lemma 3.2.14, once we show that the
representation is not a tensor induction from a quadratic extension.
We now prove that this case can not occur. If rλ is a tensor induction with
Lie algebra containing h × sln , then we must have h = sln acting via the standard
representation. Hence, once more by [LP92, Thm. 5.6, Prop. 5.7] (and the fact thatthe standard representation of SLn does not admit any basic similarity relations),
we deduce that the Lie algebra of a ⊗ b also contains sln × sln , and thus that the
Zariski closure of a contains SLn . (To see that the Zariski closure of the imageof a contains SLn rather than a quotient of SLn by some finite group, we use
the tautological fact that a has a faithful representation in dimension n.) By
Lemma 3.2.11, we deduce that the component group of ad(rλ ) is trivial. Yet supposethat G is the Zariski closure of the image of rλ , and let H be the index two subgroupfrom which rλ is tensor induced. ThenG0G0ad(TensorIndG
H V ) ' 1 ⊕ IndH ad (V ) ⊕ TensorIndH ad (V ).

In particular, we see (looking at the second factor) that the image of G acting
on ad(rλ ) surjects onto G/H, and so the component group is non-trivial, a contradiction.
We now return to the case n = 2, where we have the additional assumption thatthat the 4-dimensional compatible system {rλ } is odd, regular, polarizable, andweakly irreducible, and thus potentially automorphic by Lemma 1.5.11. By [Xia19,
Thm. 2], for a density one set of l, rλ is irreducible for all λ|l. We claim that for

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F. CALEGARI, M. EMERTON, AND T. GEE

any λ for which rλ is irreducible, rλ is also strongly irreducible. If this fails to be
the case, then since rλ is regular, it follows (as in the proof of [CG13, Cor. 4.4])that rλ is induced from a quadratic extension of F , and hence rλ ' rλ ⊗ χ forsome non-trivial quadratic character χ. Since χ lives in a compatible system, we
see that every rλ is induced from a common quadratic extension, and in particularno rλ is strongly irreducible, contradicting our assumptions. In particular, since r
is assumed irreducible, it is strongly irreducible.
Since rλ is strongly irreducible for a set of primes l of density 1, it suffices toshow (given the argument above in the case of general m, n) that the set of primes lfor which there exists λ|l with Gλder,◦ having Lie algebra sp4 is a set of density
zero. Suppose not; then by Lemma 3.3.1 below, we deduce that the compatible
system {∧2 rλ } decomposes as a direct sum of a 1-dimensional and a 5-dimensional
compatible system. Since ∧2 (a⊗b) = det(b)⊗Sym2 (a)⊕det(a)⊗Sym2 (b), it follows
that at least one of Sym2 (a) and Sym2 (b) must have a one-dimensional factor. But
then either a or b is induced from a quadratic extension, so r is induced from aquadratic extension, contradicting the strong irreducibility of r which we proved in
the previous paragraph.

3.3. A Lemma on 4-dimensional polarizable automorphic compatible systems. The following lemma was used in the proof of Theorem 3.2.17. Recall thatfor a Galois representation rλ , we denote the Zariski closure of the image of rλby Gλ .
Lemma 3.3.1. Let F be a CM field, and let {rλ } be a 4-dimensional compatible
system of representations which is odd, regular, polarizable, and weakly irreducible.
Suppose that there exists a set of primes l of positive upper density with the property that for some λ|l we have Gλder,◦ = Sp4 . Then the compatible system of 6dimensional representations {∧2 rλ } decomposes as a direct sum of two compatible
systems of dimensions 5 and 1 respectively.
Proof. Let T be a set of primes l as in the statement of the lemma. If l ∈ Tthen we let λ denote the choice of a particular λ|l with the property that Gλder,◦ =Sp4 . Then the Zariski closure of the image of rλ is a subgroup of GSp4 (M λ )containing Sp4 (M λ ). By [BLGGT14b, Prop. 5.3.2], after possibly replacing T witha smaller set of primes (still of positive upper density), we also may assume that,for any fixed finite extension H/F ,rλ |GH(ζl ) → GSp4 (Fl )is irreducible for l ∈ T . We apply this with H equal to the compositum of all thequadratic extensions of F unramified outside the set of primes of bad reduction forthe compatible system.
Consider the (semisimple) Galois representations:∧2 rλ |GF (ζl ) → GL6 (Fl ).
For each l ∈ T , the representation ∧2 rλ admits a 1-dimensional summand. We claimthat for all but finitely many l ∈ T , the complementary 5-dimensional summandis also irreducible. To see this, consider the various possible images of rλ : GF →GSp4 (k) with k = Fl under the additional assumption that they act irreducibly.
The classification of such maximal subgroups (as first computed in [Mit14]) showsthat, for l > 2, either:

GLOBALLY REALIZABLE COMPONENTS OF DEFORMATION RINGS

37(1) The image contains Sp4 (Fl ).
(2) The image stabilizes a decomposition k 4 = k 2 ⊕ k 2 , and rλ is thus induced
from a quadratic extension of F .
(3) The projective image is contained within the group PGL2 (k) acting via thesymmetric cube representation.
(4) The projective image has absolutely bounded order.
For a more modern reference, one could also consult [BHRD13], in particular tables 8.12 and 8.13, which list the maximal subgroups of Sp4 (q) := Sp4 (Fq ) andfrom which one can read off the maximal subgroups of the almost simple extension PGSp4 (Fq ) of Sp4 (Fq )/Z(Sp4 (Fq )) by hδi = Z/2Z using the rightmost column.
For the convenience of the reader, we note that groups listed there of type C1 correspond to reducible representations, those of type C2 , C3 , and C5 correspond togroups of type (2), and the remaining groups of type C6 or of class S have absolutely bounded image (type (4)) with the exception of SL2 (Fq ) which corresponds
to (3).
In case (1), the representation ∧2 rλ decomposes as a direct sum of an irreducible 5-dimensional representation and a 1-dimensional representation. Moreover, because Sp4 (Fl ) is quasisimple (a perfect central extension of a simple group),
the projective image of the representation does not change after restriction to thesolvable extension F (ζl ).
We claim that cases (2) and (4) can only hold for finitely many l. In the case (2),it follows, similarly to the proof of [CG13, Lemma 2.6], that, for l sufficiently large,
the representation is induced from a quadratic extension unramified at l. But
the quadratic extension must also be unramified outside the fixed finite set S ofprimes of bad reduction for the compatible system, and so must be contained in H,
contradicting our assumption on the irreducibility of rλ restricted to H(ζl ). For
case (4), note that since rλ is regular, the order of the projective image of rλ (evenafter restriction to inertia at primes above λ) tends to infinity with l, as follows
from Fontaine–Laffaille theory.
Finally, in case (3), ∧2 of the symmetric cube representation of a subgroupof GL2 (k) is the direct sum of a character plus a (twist of) the symmetric fifth
power. If l is large, the only subgroups of GL2 (k) for which the 4-dimensional
representation is irreducible but the 5-dimensional representation is not are those
whose projective image is S4 ; since the image of the symmetric cube representationof such a subgroup then has bounded projective order, arguing again by Fontaine–
Laffaille theory, as in case (4), we see that this case may also be ruled out if l
is sufficiently large. In conclusion, we see that if l is sufficiently large, then the
5-dimensional summand is irreducible, as required.
Shrinking T further, we see that we may assume that for all l ∈ T , the representation ∧2 rλ decomposes as the sum of a character and a 5-dimensional representation sλ with the properties that sλ is Fontaine–Laffaille, and sλ |GF (ζl ) is irreducible.
The representation sλ is regular and essentially conjugate self-dual (since rλ is), andis also odd (by Lemma 1.5.3). We can certainly assume that l ≥ 11, so it follows
that sλ is potentially automorphic by [BLGGT14b, Theorem C], and hence extends
to the desired compatible system.

3.4. Factorization of compatible systems. Our main result in this section is
Theorem 3.4.3. We begin with two preparatory lemmas. The following is a variant
on the main results of [Cal11, Cal12].

38

F. CALEGARI, M. EMERTON, AND T. GEE

Lemma 3.4.1. Let F be a CM field, and let r : GF → GLn (Ql ) be an irreducible
regular polarizable representation of GF . Suppose either that n is odd, or that n = 2.
If n = 2, suppose further that l ≥ 11, that Sym2 r is irreducible and is Fontaine–
Laffaille at all places dividing l, and that Sym2 r|GF (ζl ) is irreducible. Then r is
odd.
Proof. In the case that n is odd, this is immediate from Lemma 1.5.3. Suppose now
that n = 2. Let Pr denote the projective representation Pr : GF → PGL2 (Qp )associated to r. The polarizability of r implies that Prc ' Pr∨ ' Pr, and hence
that Pr extends to a representation Pr : GF + → PGL2 (Qp ). Thus Ad0 (r) =Ad0 (Pr) also extends to a representation Ad0 (r) : GF + → GL3 (Qp ). It follows
from [BLGGT14b, Cor. 4.5.2] (and the case n = 3 of the result being proved)that Ad0 (r) is potentially automorphic, and thus (by [Tay12, Prop. A]) the image of any complex conjugation is non-scalar, and thus the image of any complex
conjugation under Pr is non-scalar.
We show that this implies that r is odd. Because we are in dimension 2, there is
2certainly a non-degenerate pairing on Qp and a character µ of GF + which satisfies
hr(σ)x, rc (σ)yi = µ(σ)hx, yi
for any complex conjugation c. Since Pr is not dihedral, the pairing is unique, andto prove oddness it suffices to show that hx, yi = hy, xi. If not, then the pairing is
symplectic, and hx, xi = 0 for all x. This implies thathr(σ)x, rc (σ)xi = 0
for all σ. Because the dimension is 2, we have hx, yi = 0 for x 6= 0 only when y is amultiple of x. Since r is irreducible, it follows that Pr(σ) = Prc (σ) for all σ, which,by Schur’s lemma, implies that Pr(c) is scalar, contradicting the result above. 
The following lemma gives a local condition for a representation not to be of theform ρ ⊗ (ρc )∨ up to twist for some representation ρ.
Lemma 3.4.2. Let F be a CM field. Let v be a prime in F + which is inert
in F , and denote by w the corresponding prime in F . Denote by c the non-trivial
element of Gal(F/F + ) = Gal(Fw /Fv+ ). Suppose that l is a prime distinct from thecharacteristic of v and w. Let×ψ : GFw → Ql

be a non-trivial ramified character such that ψ c |IFw = ψ −1 |IFw . Suppose thatsw : GFw → GLn2 (Ql )is a representation such that2⊕(n −n)sw |IFw ∼.
= ψ|⊕nIF w ⊕ 1

Then, if n > 2, we cannot write sw in the formsw ' θ ⊗ (ρ ⊗ (ρc )∨ )×

where ρ : GFw → GLn (Ql ) and θ : GFw → Ql is a character. If additionally ψ|IFw
has order > 2, then there is no such decomposition when n = 2 either.

GLOBALLY REALIZABLE COMPONENTS OF DEFORMATION RINGS

39

Proof. Suppose for the sake of contradiction that we can write sw ' θ ⊗ (ρ ⊗ (ρc )∨ ).
Consider the representations ρ and (ρc )∨ restricted to IFw . We are assuming forthe sake of contradiction that2⊕n⊕(n −n)).
ρ ⊗ (ρc )∨ |IFw ' θ|−1IFw ⊗ (ψ|IFw ⊕ 1

By Lemma 3.2.4, we deduce that ρ and ρc restricted to IFw are both given by directsums of characters.
Suppose that, after restriction to IFw , the representation ρ decomposes as the
direct sum of n copies of a single character φ. The inertia group IFw is normal
in GFw and in GFv+ . For a representation % of IFw and for σ ∈ GFv+ , it thus
makes sense to define a representation %σ of IFw by %σ (g) = %(σgσ −1 ). Since ρ isa representation of GFw , we see that ρσ ' ρ for any σ ∈ GFw . Hence if σ ∈ GFw ,then we haveφ⊕n ' ρ|IFw ' ρσ |IFw ' (ρ|IFw )σ ' (φσ )⊕n ,and thus φ = φσ . It follows that if c ∈ GFv+ is any lift of the non-trivial elementof GFv+ /GFw = Gal(Fw /Fv+ ), then φc does not depend on this lift. In particular,
still assuming that ρ|IFw decomposes as a direct sum of n copies of φ, we haveρc |IFw ' (φc )⊕n .
This would imply that ρ ⊗ (ρc )∨ is a direct sum of n2 equal characters of IFw , contradicting the fact that ψ is ramified, and thus non-trivial on inertia. In particular,
ρ (and by symmetry ρc ) contains at least 2 distinct characters.
Let ϕ be a character of IFw occurring in ρc , so that ϕ−1 occurs in (ρc )∨ . Then
ρ|IFw = (ρ|IFw ⊗ ϕ−1 ) ⊗ ϕ ⊂ (ρ ⊗ (ρc )∨ )|IFw ⊗ ϕ = θ−1 ϕ ⊗ sw |IFw ,2

so ρ|IFw ⊂ θ−1 ϕ ⊗ (ψ ⊕n ⊕ 1⊕(n −n) ) and it follows that ρ restricted to IFw is adirect sum of at most two distinct characters with some multiplicity. From theprevious discussion, precisely two such characters occur. The same argument gives
the same result for ρc . Thus we may write
ρ|IFw = φr1 ⊕ φn−r,2(ρc )∨ |IFw = ϕs1 ⊕ ϕn−s,2

for some 0 < r < n and 0 < s < n. Since each pair of characters are distinct,
we must have φi ϕj 6= φi ϕk and φj ϕi 6= φk ϕi when j 6= k. In order for the tensor
product to contain precisely two distinct characters, this forces the equalities φi ϕj =φj ϕi and φi ϕi = φj ϕj . Since one character occurs in ρ ⊗ (ρc )∨ exactly n times (andthe other n2 − n times), we deduce, after some appropriate re-ordering, thatrs + (n − r)(n − s) = n.
This has no solutions in integers 0 < r < n and 0 < s < n when n > 2, acontradiction. (Indeed, we have 2n = rs + rs + (n − r)(n − s) + (n − r)(n − s) ≥r+s+(n−r)+(n−s) = 2n with equality if and only if r = s = (n−r) = (n−s) = 1.)
Now let us consider the case n = 2. In this case, the required character identities
imply that φi /φj = ϕi /ϕj is a character of order dividing 2. On the other hand,
the ratio of the two distinct characters occurring in ρ ⊗ (ρc )∨ |IFw may be identified
both with ψ and with φi /φj with i 6= j, which implies that ψ has order 2, acontradiction.
Combining the two previous lemmas (as well as lemmas from previous sections),
we are now able to prove a theorem which allows us to factor a compatible system

40

F. CALEGARI, M. EMERTON, AND T. GEE

into the tensor product of two other compatible systems given such a factorization
at one prime p.
Theorem 3.4.3. Let F be a CM field and let p > 2 be prime. Suppose that n is
odd or that n = 2. Let ({sλ }, {µλ }) be an odd, polarized, regular, weakly irreducible
compatible system of representations with associated p-adic representation (s, µ).
Suppose that we can write(s, µ) = (a, µ1 ) ⊗ (b, µ2 )where a and b are n-dimensional representations which are de Rham at all places v|p.
Suppose also that:• There is a positive density set of places l such that for each λ|l, the representation sλ is strongly irreducible.
• s is strongly irreducible.
• The residual representation b has image containing SLn (Fq ) for q a sufficiently large power of p in the sense of Lemma 3.2.10.
• There is a finite place x - p of F + which is inert in F , and a character×ψ : GFx → Qp , such that:c– ψ |IFx = ψ −1 |IFx .
– ψ|IFx has (finite) order greater than 2.
– a|GFx is unramified, and– b|IFx ∼= ψ|IFx ⊕ 1⊕(n−1) .
Then there are odd, polarized, regular, weakly irreducible compatible systems ({aλ }, {µ1,λ }),({bλ }, {µ2,λ }) whose associated p-adic representations are respectively (a, µ1 ) and (b, µ2 );so in particular {sλ } = {aλ ⊗ bλ }.
Proof. Under our first assumption, we deduce from [BLGGT14b, Prop. 5.3.2] that
there exists a set of primes l of density 1 with l > 2(n + 1) and x - l, such thatfor each λ|l, sλ is Fontaine–Laffaille at all places dividing l, sλ is strongly irreducible, and sλ |GF (ζl ) is irreducible. If n = 2, we furthermore can and do assume
that Sym2 sλ is irreducible and Fontaine–Laffaille, and l ≥ 11. (The irreducibilityof Sym2 sλ is an easy consequence of the assumption that sλ is strongly irreducible— for example, one can deduce it from Lemma 3.2.8.)
By Theorem 3.2.17, we may choose l such that we can write sλ = al ⊗ bl ,where al and bl are both n-dimensional representations of GF , the Zariski closureof bl contains SLn (Ql ), and the unordered pair {al , bl } is unique up to twist. It
follows from [Pat12, Thm. 3.2.10] that we can choose al and bl to be unramified at
all but finitely many places, and de Rham at all places dividing l. (We apply the
result to the surjection from GLn × GLn to its image in GLn2 ; [Pat12, Hyp. 3.2.4] issatisfied because F is CM and sλ is polarizable.) It then follows from [LY16, Prop.
3.3.4] that we can furthermore ensure that al and bl are in fact crystalline at allplaces dividing l. Moreover, the regularity of sλ immediately implies the regularityof a and b.
Since sλ is Fontaine–Laffaille at all places dividing l and sλ |GF (ζl ) is irreducible,
we see that each of al and bl is Fontaine–Laffaille at all places dividing l, andthat al |GF (ζl ) and bl |GF (ζl ) are irreducible. Since sλ is strongly irreducible, al and bl
are strongly irreducible.

GLOBALLY REALIZABLE COMPONENTS OF DEFORMATION RINGS

41

We now show that al and bl are both polarizable. Since scλ ∼= µλ s∨λ , it follows
that
al ⊗ bl ∼= µλ (acl )∨ ⊗ (bcl )∨ ,and by the uniqueness of al and bl up to twist, we see that there are characters×ψl , ϕl : GF → Ql such that either al ∼= ψl (acl )∨ ,= ϕl (acl )∨ , or that al ∼= ψl (bcl )∨ , bl ∼
c
∨
c
∨
bl ∼
= ϕl (bl ) . In either case we have sλ ∼= ψl ϕl sλ , and it follows from Lemma 3.2.8
that ψl ϕl = µcλ = µλ .
In the first case, we have sλ ∼= al ⊗ bl ∼= ϕl al ⊗ (acl )∨ . This contradicts
Lemma 3.4.2 (the hypotheses of which are satisfied by our assumptions on a|GFx
and b|GFx and by Proposition 1.5.14).
We are therefore in the second case, and we need to show that ψl , ϕl both extend
to characters of GF + . Taking the conjugate dual of the isomorphism bl ∼= ϕl (bcl )∨ ,
c
c c ∨
∼
∼
we see that also bl = ϕl (bl ) , so that bl = (ϕl /ϕl )bl . Since bl is strongly irreducible,
we have ϕl = ϕcl , and ϕl extends to GF + . Similarly, ψl also extends, as required.
Since al , bl are polarizable, it follows from Lemma 3.4.1 that they are both odd.
We can now apply [BLGGT14b, Thm. 5.5.1] to the polarized representations (al , ψl )and (bl , ϕl ), obtaining compatible systems ({aλ }, {ψλ }) and ({bλ }, {ϕλ }) whose corresponding l-adic representations are (al , ψl ), (bl , ϕl ) respectively. Since (sλ , µλ ) =(al , ψl ) ⊗ (bl , ϕl ), we have (s, µ) = (ap , ψp ) ⊗ (bp , φp ), where (ap , ψp ) and (bp , ϕp )are the p-adic representations corresponding to ({aλ }, {ψλ }) and ({bλ }, {ϕλ }) respectively.
Since we also have (s, µ) = (a, µ1 ) ⊗ (b, µ2 ), it follows from Lemma 3.2.14 (since
we are assuming that s is strongly irreducible) that the pairs {ap , bp } and {a, b}
agree up to twist. After possibly exchanging {aλ } and {bλ }, we may supposethat bp is a twist of b. Since b is strongly irreducible, and both b and bp areboth polarizable and de Rham, the twist must be by an algebraic character of GF .
Replacing {bλ } by the twist by the corresponding compatible system of characters,and {aλ } by the inverse twist, we have constructed the sought-after compatible
systems (note that we must have µ2 = ϕp by another application of Lemma 3.2.8,
so that also µ1 = ψp ).

3.5. A technical lemma. We end this section with a technical lemma that will
be used in Section 4.1. We begin with some equally technical preliminaries.
Lemma 3.5.1. Let H be a reductive linear algebraic group with a faithful irreducible
linear representation V , such that the restriction of V to H ◦ remains irreducible.
Let ZH denote the center of H. Then there is an injective map
H/ZH H ◦ → Out(H ◦,der ).
In particular, if ZH is connected, then the component group of H injects into Out(H ◦,der ).
Proof. If ZH ◦ denotes the centre of H ◦ , then the natural morphism H ◦,der ×ZH ◦ →H ◦ is surjective (since H ◦ is a connected reductive linear algebraic group). Thus we
see that the irreducible representation V of H ◦ remains irreducible when restricted
to H ◦,der ; and we also see that the conjugation action of H ◦ on H ◦,der is via
inner automorphisms, so that there is indeed a well-defined morphism H/ZH H ◦ →Out(H ◦,der ).
Suppose now that some element h ∈ H acts on H ◦,der via an inner automorphism.
We must show that h ∈ ZH H ◦ . Multiplying h by an element of H ◦,der , we may
assume that h actually centralizes H ◦,der . Since V is an irreducible representation

42

F. CALEGARI, M. EMERTON, AND T. GEEof H ◦,der , we see that h acts on V by scalars, and thus commutes with the actionof H. Since H acts faithfully on V , we see that h ∈ ZH , as required.

Lemma 3.5.2. Let H be a connected semisimple algebraic group with a faithful
irreducible representation of dimension ≤ 2d . If p > max(d, 3), then p does notdivide the order of Out(H ◦,der ).
Proof. Suppose that we have an element of Out(H ◦,der ) of order divisible by p.
Any such outer automorphism induces a non-trivial outer automorphism of thecorresponding Lie algebra. Since the simple Lie algebras only have automorphismsof order at most 3, any such automorphism can must consist of a permutation of
the simple factors. But if there is a faithful representation of dimension at most 2d ,then there can be at most d simple factors, and Sd has no elements of order p
if p > d.

The following lemma is the main result of this subsection. Recall that if M/F
is a finite extension, we let M F -gal denote the Galois closure of M over F .
Lemma 3.5.3. Let F be a number field, and let
a, b : GF → GLn (Fp )be such that a ⊗ b is irreducible. Let {rλ } denote a weakly compatible system of n2 dimensional regular representations of GF such that rp = a ⊗ b, where a is adeformation of a, b is a deformation of b, and p > max(n, 3). Assume that theimage of b contains SLn (Fq ) for q sufficiently large, in the sense of Lemma 3.2.10.
If for some λ, rλ is induced from an extension M/F , then [M F -gal : F ] has orderprime to p.
Proof. We may (and so) assume, after tensoring {rλ } by a compatible system ofcharacters if necessary, that the image of det(rλ ) is infinite. (Note that any twistof rλ is also induced from M/F .) Let G◦λ ⊂ Gλ denote the connected componentof the Zariski closure Gλ of rλ . Let N/F denote the fixed field of the inverse
image of G◦λ . Since the connected component G◦λ is a normal subgroup of Gλ , itfollows that N/F is Galois. On the other hand, if rλ is induced from M , thencertainly M ⊂ N , and hence M F -gal ⊂ N . Hence it suffices to show that thecomponent group of Gλ has order prime to p. For this, we work at the prime p(since the order of the component group is independent of λ by Theorem 1.5.15).
To this end, we now let G denote the Zariski closure of rp and G◦ the connectedcomponent of the identity in G. By Lemma 3.2.16, we may assume that Gder,◦ is thenatural representation of Ader,◦ × B der,◦ , where A and B are the Zariski closuresof the images of a and b respectively, and B der,◦ = SLn (Qp ). Let Z denote thecenter of G. Because det(G) is infinite, it follows that the Lie algebra g contains
a non-trivial torus t, and hence exp(t) ⊂ Z is infinite. Because G acts irreducibly,×by Schur’s Lemma, it follows that Z consists of scalars, and thus Z = Qp = Z ◦ isconnected.
Because rp is regular, it follows from Lemma 3.2.12 that rp is induced from astrongly irreducible representation sp over some extension Mp /F . By comparison
with the decomposition of Gder,◦ above, this strongly irreducible representation
has dimension mn for some m|n with m[Mp : F ] = n. In particular, the Galois
closure MpF -gal /F of Mp over F has the property that Gal(MpF -gal /F ) is a subgroupof Sn , and thus has order prime to p.

GLOBALLY REALIZABLE COMPONENTS OF DEFORMATION RINGS

43

Let H denote the Zariski closure of the image of sp |G F -gal . Since [MpF -gal : F ]Mp

has order prime to p, it is enough to prove that the component group of H has orderprime to p. There is an inclusion G◦ ⊂ H ⊂ G, and ZH = Z = Z ◦ is connected.
Moreover, since sp is strongly irreducible, it follows that H has a faithful irreducible
representation of dimension mn ≤ n2 ≤ 2n such that the restriction to H ◦ is also
irreducible. The result now follows from Lemmas 3.5.1 and 3.5.2.

4. Building Lifts
4.1. Deformations of a ⊗ b and strong irreducibility. In this section, we showthat strong irreducibility of compatible systems can be ensured by imposing ramification conditions at finite places. These ramification conditions become trivial aftera finite base change, and we will use the Khare–Wintenberger method to remove
them from our final results.
We begin with the following preparatory lemma:
Lemma 4.1.1. Let K/Qh be a finite extension of degree prime to p for someprime h ≡ 1 mod p. Suppose that l 6= h, and let
ρ : GK → GLm (Ql )be a representation such that ρ|IK has finite p-power order. If p - m, then ρ admits
a one dimensional sub-quotient which is the restriction of a character of GQh .
Note that the lemma applies to both l = p and l 6= p.
Proof. Since not every irreducible constituent of ρ can be of degree a multiple of p,
we may reduce to the case that ρ is irreducible. The image of inertia has p-power
order and h 6= p. Hence the representation is tamely ramified. Any such irreducibleK
representation is of the form ρ = IndG
GL ψ for a character ψ of GL , where L/K is
unramified (although we don’t use this fact) and the degree [L : K] = m. It follows
that
n := [L : Qh ] = [L : K][K : Qh ]is prime to p. Also, since ρ|IK has p-power order, we see that ψ|IL has p-power
order.
It is elementary to see that, since p - n and p|(h − 1), the largest power of p×
dividing |kL| is the same power of p which divides |F×
h | = h − 1. (Indeed, (h − 1)×
divides |kL |, which in turn divides hn − 1, and we have (hn − 1)/(h − 1) ≡ n(mod p).) By local class field theory, it follows that the character ψ of GL (whose
restriction to inertia is, as we have observed, of p-power order) is the restriction ofa character of GQh , which we also denote by ψ. Yet thenGKK
ρ = IndG
GL ψ = ψ ⊗ IndGL 1.
K
The representation IndG
GL 1 contains a copy of the trivial representation. Since ρ
is irreducible, it follows that L = K, and ρ = ψ, as claimed.
Let F be a CM field. Consider a pair of irreducible representations:a, b : GF → GLn (Fp )44

F. CALEGARI, M. EMERTON, AND T. GEEsuch that a ⊗ b is irreducible and a and b are polarizable. We now build on
Lemma 3.5.3, showing how we can control the extensions M/F from which representations inside a compatible system containing a lift of a ⊗ b could possibly be
induced.
We will shortly prove Lemma 4.1.3, which enables us to show that, for certain
deformation problems, any deformation of a ⊗ b which is induced must be induced
from one of a finite number of possible fields, independent of certain classes ofauxiliary ramification sets Σ. We start with the following preparatory lemma:
Lemma 4.1.2. Let [Fv : Qv ] be a finite extension, and letr : GFv → GLn (Ql )be either potentially unramified if l 6= v or potentially crystalline if l = v. Assume that the restriction of the Weil–Deligne representation WD(r) to the inertia
group IFv factors through a group of p-power order. Suppose that r is induced froma finite extension Mv /Fv whose Galois closure MvFv -gal over Fv has order prime
to p. Then Mv /Fv is unramified.
Proof. The formation of Weil–Deligne representations is compatible with inducW
G
tions. Hence, if r = IndGFMvv s, then WD(r) = IndWFMvv WD(s). It follows thatthe kernel of WFv acting on WD(r) is contained in WMv and thus also containedin WMvFv -gal . But the image of WD(r) restricted to IFv has p-power order by assumption, and thus the inertia subgroup of Gal(MvFv -gal /Fv ) has p-power order.
But Gal(MvFv -gal /Fv ) has order prime to p by assumption, and thus Mv /Fv is unramified.
Lemma 4.1.3. Fix a finite set S of places of F + containing all primes above p andall the primes at which a or b are ramified. Let {rλ } denote a weakly irreducible
compatible system of odd, polarizable, regular n2 -dimensional representations of GF
such that rp = c ⊗ d, where c is a deformation of a, and d is a deformation of b,and p > max(n, 3). Assume that the image of b contains SLn (Fq ) for q somesufficiently large power of p, in the sense of Lemma 3.2.10.
Suppose that {rλ } is unramified outside a finite set S ∪ Σ of places of F + (inthe sense of Remark 1.5.5), and that for primes v ∈ Σ, rp (IFv ) is finite. Supposethat λ is a prime such that the representation rλ is induced from M/F . Then:(1) [M F -gal : F ] has order prime to p,(2) [M : F ] ≤ n2 , and(3) M/F is unramified outside S.
In particular, if S is fixed, there are only finitely many such M independently of
the choice of λ, the compatible system {rλ }, and the set Σ.
Proof. By shrinking Σ if necessary, we may (and do) assume that Σ is disjoint
from S; in particular, the representation a ⊗ b is unramified at primes of Σ. In
particular, since the primes above p lie in S, we may assume that Σ does notcontain any such prime.
The fact that [M F -gal : F ] has order prime to p is an immediate consequence ofLemma 3.5.3. Let us consider the possible M from which rλ may be induced. The
degree [M : F ] is certainly bounded by n2 . If we prove that M/F is unramified atprimes not dividing S, then there are only finitely many such M/F as an immediate

GLOBALLY REALIZABLE COMPONENTS OF DEFORMATION RINGS

45

consequence of a theorem of Hermite (following Minkowski). Consider the representation rp locally at primes v ∈ Σ. By assumption, the image rp (IFv ) of the inertia
subgroup IFv for v ∈ Σ factors through a finite quotient. On the other hand, therepresentation a ⊗ b is unramified at v. Hence the image of inertia factors througha finite quotient of p-power order. By Proposition 1.5.14, the image of IFv factors
through a finite quotient of p-power order for all representations WD(rλ |GFv ).
Suppose that rλ is induced from M . Let v be a finite place of F . Then rλ |GFv isinduced from Mv . Moreover, the Galois closure MvFv -gal of Mv over Fv is containedin the localization of the Galois closure M F -gal of F at v, and thus Gal(MvFv -gal /Fv )also has order prime to p. If v ∈/ S, then either v ∈ Σ and the image of IFv
in WD(rλ |GFv ) factors through a finite quotient of p-power order, or v 6∈ Σ andthe image of IFv in WD(rλ |GFv ) is trivial. In either case, the claim that Mv /Fv isunramified follows immediately from Lemma 4.1.2.
Given an integer n and a finite set S of places of F + , we now introduce certain
auxiliary primes v for fields M/F of the kind which arose in Lemma 4.1.3.
Lemma 4.1.4. For each nontrivial M/F unramified outside S with Gal(M F -gal /F )of order prime to p, either M ⊂ F (ζp ), or there exist infinitely many finite places vof F + and w|v in F with the following properties:(1) NF/Q (w) = h is prime, and h ≡ 1 (mod pm ) for some pm > n2 .
(2) h is unramified both in M and in the fixed fields of a and b over F . In
particular, w is unramified in M F -gal /F .
(3) The images of a(Frobw ) and b(Frobw ) have orders prime to p.
(4) The decomposition group of M F -gal /F at w is non-trivial.
(5) v (resp. w) lies outside any given finite set of places of F + (resp. F ).
Proof. A choice of w in F determines a unique prime v of F + with w|v. Conditions (2) and (5) exclude only finitely many places. (By definition, any prime which
ramifies in M/Q is either ramified in F/Q, which is fixed, or is ramified in M/F ,which, by definition, is unramified outside the fixed set S.) The remaining three
conditions are a Chebotarev condition relative to the image of GF in
Gal(F (ζpm )/F ) × Gal(M F -gal /F ) × GLn (Fp ) × GLn (Fp ),where the maps to the two copies of GLn (Fp ) factor via a and b respectively.
It suffices to show that the image of GF contains an element whose projection
is trivial in the first group (for (1)), non-trivial in the second group (for (4)),
and semisimple in the last two groups (for (3)). The first two conditions can besatisfied simultaneously unless M F -gal ⊂ F (ζpm ). Since [M F -gal : F ] is prime to p,
this is equivalent to M ⊂ M F -gal ⊂ F (ζp ). Thus if M 6⊂ F (ζp ), then there existsa σ ∈ GF which is trivial in Gal(F (ζpm )/F ) and non-trivial in Gal(M F -gal /F ). It
kfollows that σ p also has this property for any k, because Gal(M F -gal /F ) has orderprime to p. On the other hand, the image of any sufficiently large p-power of anykelement of GLn (Fp ) has order prime to p. Thus the image of σ p for sufficiently
large k in the product above has the desired shape.
Definition 4.1.5. (A suitable choice) Continuing on with our running assumptions
on a and b, suppose in addition that the assumptions of Lemma 4.1.3 are in effect.
In particular, both a and b are n-dimensional residual representations of GF which
are polarizable, the representation a ⊗ b is absolutely irreducible, and there exists

46

F. CALEGARI, M. EMERTON, AND T. GEEa weakly irreducible compatible system {rλ } such that rp = c ⊗ d is a deformationof a⊗b. Let S be a set containing all primes above p and all primes where a and b are
ramified. Fix characters µ1 , µ2 unramified outside S such that a and b are µ1 and µ2 polarizable respectively. For v ∈ S, consider any collection of local µ1 -polarizedcomponents Cv and µ2 -polarized components Dv for a and b respectively. Suppose,
furthermore, that Cv ⊗ Dv is regular for all v|p. We now choose an auxiliary set Σ
disjoint from S and components Cv , Dv for v ∈ Σ as follows. (Ultimately, we
shall apply Lemma 4.1.3 with precisely this set Σ.) For each of the finitely many
fields M/F not contained in F (ζp ) and satisfying conditions (1), (2), and (3) in the
conclusion of Lemma 4.1.3, we choose two primes v in F + with w|v in F according
to Lemma 4.1.4, where the auxiliary set of places given by (5) is the set of primes
which ramify in any of the finitely many fields M/F . Moreover, for each M/F , wechoose this pair of places v so that they have distinct residue characteristics. We
now let Σ be the union of these pairs of places v of F + for all M/F . Our choiceguarantees that, for any prime l, and for any given M/F , there exists a v ∈ Σcorresponding to M/F with residue characteristic prime to l. We now choose Cv
×and Dv as follows. Recall that Fv+ ' Fw ' Qh . Let ψ : GQh → Qp be a charactersuch that ψ|IFw has order pm , which exists because h − 1 ≡ 0 mod pm . We consider
deformations c and d at v such that
n−1
n−1
M
M
c|IFw =
ψ i |IFw ,d|IFw =
ψ ni |IFw .
i=0

i=0

Such deformations exist (locally) because a(Frobw ) and b(Frobw ) are semisimple.
Since the characters ψ i |IFw (and similarly the characters ψ ni |IFw ) are pairwise distinct, this defines unions of components of the corresponding local deformation
rings. We choose Cv and Dv to be any of the resulting components. The corresponding inertial type of the n2 -dimensional compatible system (which is welldefined across the entire compatible system, by Proposition 1.5.14) at each v ∈ Σ
is
2
2
nM
−1
nM
−1
i
ψ |IFw =ψ i |IF + .
i=0

i=0

v

(Note that IFw = IFv+ .) In particular, it is a direct sum of distinct characters of
p-power order — here we use the assumption that pm > n2 . As usual, the corresponding polarized deformation rings of a and b are denoted by RC,Σ and RD,Σrespectively. We refer to the choice of auxiliary set Σ and corresponding components Cv and Dv as a suitable choice.
Equipped with the notion of a suitable choice, we now prove the following:
Lemma 4.1.6. Let {rλ } denote a weakly irreducible, polarizable, regular, odd compatible system of n2 -dimensional representations of GF such that rp = c⊗d, where cand d are deformations of a and b of types RC,Σ and RD,Σ respectively for a suitablechoice of Σ, Cv and Dv as in Definition 4.1.5. Assume that p > max(n, 3), andthat the image of b contains SLn (Fq ) for q some sufficiently large power of p, inthe sense of Lemma 3.2.10.
Suppose that some rλ is induced from M/F . Then M ⊂ F (ζp ).
Proof. Assume that rλ is induced from M/F . The choice of Cv and Dv for v ∈ Σensure that, for primes v ∈ Σ, rp (IFv ) is finite. Hence, by Lemma 4.1.3, M/F

GLOBALLY REALIZABLE COMPONENTS OF DEFORMATION RINGS

47

is one of finitely many fields which is unramified outside S, has degree boundedby n2 , and M F -gal /F has order prime to p. Suppose that M 6⊂ F (ζp ). By thesuitable choice of Σ as in Definition 4.1.5, there exists a v ∈ Σ corresponding to M
which satisfies the conditions of Lemma 4.1.4 and has residue characteristic notdividing N (λ). Let us writeFrλ = IndG
GM s : GF → GLn2 (Ql ).

Since the inertial type of rλ at v consists of distinct characters of IQh , it suffices toshow that this is incompatible with being an induction.
Let x be a place of M F -gal lying over w (note that Fw = Qh ). Since Gal(M F -gal /F )has order prime to p, it follows that [MxF -gal : Fw ] = [MxF -gal : Qh ] has order prime
to p. We have a representation
s|G F -gal : GMxF -gal → GLm (Ql ),Mx

2where m[M : F ] = n , and so p - m (since p - n). It follows from Lemma 4.1.1
that s|MxF -gal contains at least one subquotient ω which is the restriction of a character of GQh (note that s(IMxF -gal ) has finite p-power order, because rλ (IFw ) does,by the definition of a suitable choice of Σ). By the definition of an induction, there
is an identification
M
rλ |GM F -gal =sσ |GM F -gal ,σ∈Gal(M F -gal /F )/ Gal(M F -gal /M )σ

−1where s (g) = s(σgσ ).
The decomposition group of w in Gal(M F -gal /F ) is non-trivial by the choice of v
and w (more precisely, by condition (4) of Lemma 4.1.4). Moreover, because M F -galis the Galois closure of M , the intersection of the conjugates of Gal(M F -gal /M ) inside Gal(M F -gal /F ) is trivial. Hence, for a suitable choice of x|w in M F -gal , we
may ensure that there exists an element σ in the decomposition group of x above w
in Gal(M F -gal /F ) that does not lie in Gal(M F -gal /M ). It follows that s|GM F -gal ⊕sσ |GM F -gal is a summand of rλ |GM F -gal . But σ lies inside the decomposition groupof x, and hence σx = x, and sσ |G F -gal is the conjugate by σ of s|G F -gal . Since ω
Mx

Mxoccurs as a subquotient as s|G F -gal , it follows that ω σ is a subquotient of sσ |G F -gal ,Mx

Mx

and hence ω ⊕ ω σ is a subquotient of rλ |G F -gal . But ω is the restriction of a charMx

acter of GQh , and thus ω σ = ω, and ω ⊕ ω is a subquotient of rλ |G F -gal . By
Mx

assumption, the restriction rλ |IFw is a direct sum of distinct characters. Since w
is unramified in M F -gal /F by construction of Σ (cf. Lemma 4.1.4 (2)), the restriction rλ |G F -gal must also be a direct sum of distinct characters, and hence we haveMx
a contradiction.

Lemma 4.1.7. In the context of Lemma 4.1.6, assume also that rp |GF (ζp ) is absolutely irreducible. Then rp is strongly irreducible, and for a positive density ofprimes l, the representations rλ are strongly irreducible for all λ|l (that is, thecompatible system {rλ } is “weakly strongly irreducible”).
Proof. By Lemma 3.2.12, the representations rλ which are absolutely irreducible
are strongly irreducible unless they are induced. Since {rλ } is weakly irreducibleby assumption, for a positive density of primes l, the representations rλ are irreducible for all λ|l, and certainly rp is irreducible. Hence it suffices to show that the

48

F. CALEGARI, M. EMERTON, AND T. GEE

representations rλ and rp are not induced from some finite extension M/F . By theprevious lemma (Lemma 4.1.6), this can happen only if M ⊂ F (ζp ). Thus we will
be done if we can show that rp |GF (ζp ) is absolutely irreducible, and that {rλ |GF (ζp ) }is weakly irreducible.
Since rp |GF (ζp ) is absolutely irreducible by assumption, rp |GF (ζp ) is absolutely
irreducible. Since F (ζp )/F is CM, it follows from Lemma 1.5.17 that {rλ |GF (ζp ) }is weakly irreducible, as required.

We end this subsection with some results (versions of the Khare–Wintenberger
argument) that will allow us to remove the auxiliary conditions discussed abovefrom our final results. We remind the reader of the conventions introduced in
Definition 2.1.5 and Conventions 1.5.21 and 1.5.22, which will be in force throughout
the rest of this section; namely, we write w (possibly decorated by subscripts andsuperscripts) for a place of a CM field lying over the place v of its totally realsubfield, and we do not explicitly mention prolongations.
Lemma 4.1.8 (Descending the existence of a compatible system). Let S be a finite
set of finite places of F + which contains all the places dividing p, and let (a, µ) be×a polarized representation which is unramified outside of S. Let µ : GF + → Zp be
a de Rham lift of µ which is unramified outside of S. For each v ∈ S, let Av be aµ-polarized component for a|GFw . Assume that a : GF → GLn (Fp ) is reasonable.
Suppose that there is a finite Galois extension of CM fields L/F , linearly disjoint
ker afrom F(ζp ) over F , and an odd, polarized, regular, weakly irreducible compatible system of representations ({sλ }, {µ0λ }) of GL , with associated p-adic representation (s, µ0 ), with the properties that:(1) a|GL is reasonable.
(2) s ∼
= a|GL .
(3) µ0 = µ|GL .
(4) s is unramified outside of the places lying over S.
(5) For each place vL of L+ lying over a place v ∈ S, s|GLw lies on Av |L+
.
v
L

L

Then there is an odd, polarized, regular, weakly irreducible compatible system ofrepresentations ({aλ }, {µλ }) of GF , with associated p-adic representation (a, µ),with the properties that:(1) a lifts a.
(2) a is unramified outside of S.
(3) For each place v ∈ S, a|GFw lies on Av .
Proof. By Theorem 2.1.16, after replacing L by a finite extension, we can and do
assume that ({sλ }, {µ0λ }) is automorphic. Let RA be the universal O-deformation
algebra for a, for the deformations which are:• µ-polarized and odd.
• unramified outside of S.
• If v ∈ S, then the corresponding lift lies on Av .
Let RA0 be the universal O-deformation algebra for s, for the deformations whichare:• µ0 -polarized and odd.
• unramified outside of the primes lying over S.
• If v ∈ S, and vL |v is a place of L+ , then the corresponding lift lies on Av |L+.
v
LGLOBALLY REALIZABLE COMPONENTS OF DEFORMATION RINGS

49

By Proposition 1.5.24, RA has positive dimension, and by [BLGGT14b, Lem. 1.2.3]
it is finite over RA0 . Since RA0 is finite over O by Lemma 1.5.27, we deduce that RA
is finite over O. It follows that RA has Qp -points, and we let a be the representation
corresponding to such a point.
It remains to check that (a, µ) is part of a weakly irreducible compatible system. Since (by construction) (s, µ0 ) is automorphic, it follows from Theorem 1.5.26
that (a|GL , µ0 ) is automorphic. Then (a, µ) is part of a compatible system by
the usual argument with Brauer’s theorem; to be precise, it follows from the proofof [BLGGT14b, Thm. 5.5.1], as the appeal to [BLGGT14b, Thm. 4.5.1] in that proof
is only in order to prove potential automorphy, which we have already established.
(Note that the hypothesis on the component groups of the Galois representations
made in the proof of [BLGGT14b, Thm. 5.5.1] is guaranteed by Lemma 3.1.1; weare free to twist our representations by an algebraic character in order to guarantee that the determinant has infinite order.) This compatible system is weakly
irreducible by Lemma 1.5.11.
Corollary 4.1.9 (Level lowering for a compatible system). Let S be a finite setof finite places of F + , containing all of the places dividing p, and let (a, µ) be a×polarized representation which is unramified outside of S. Let µ : GF + → Zp be ade Rham lift of µ which is unramified outside of S. Assume that a : GF → GLn (Fp )is reasonable. For each v ∈ S, let Av be a µ-polarized component for a|GFw . Let Σ
be a finite set of finite places of F + , which is disjoint from S, and for each v ∈ Σ,
let Av be a µ-polarized component for a|GFw which is potentially unramified.
Suppose that there is an odd, polarized, regular, weakly irreducible system ofrepresentations ({sλ }, {µλ }) of GF , with associated p-adic representation (s, µ),with the properties that:(1) s ∼
= a.
(2) s is unramified outside of S ∪ Σ.
(3) For each place v ∈ S ∪ Σ, s|GFw lies on Av .
Then there is an odd, polarized, regular, weakly irreducible system of representations({aλ }, {µλ }) of GF , with associated p-adic representation (a, µ), with the properties
that:(1) a lifts a.
(2) a is unramified outside of S.
(3) For each place v ∈ S, a|GFw lies on Av .
ker a

Proof. We may choose a finite Galois extension L/F , linearly disjoint from F(ζp )over F , with the properties that a|GL remains reasonable, and for each v ∈ Σ,and each place vL |v of L, s|GLw is unramified. The result then follows fromL
Lemma 4.1.8, applied to ({sλ |GL }, {µλ |GL+ }).

4.2. Local Swapping. In this subsection we prove our main theorem (Theorem 4.2.11 below), building on a series of lemmas. We begin with the following
lemma, which will be used to move components between different mod p representations. Much of the rest of this section is devoted to relaxing the rather restrictive
hypotheses made in this lemma, culminating in Lemma 4.2.9 below.
We again remind the reader that we are using the conventions introduced in
Definition 2.1.5 and Conventions 1.5.21 and 1.5.22. We extend Convention 1.5.21

50

F. CALEGARI, M. EMERTON, AND T. GEEin the obvious way to subscripts and superscripts, so that w1 is a place of F over v1
in F + , wL is a place of L over vL in L+ , and so on.
Lemma 4.2.1 (Local swapping I). Suppose that either n > 1 is odd, or n = 2.
Let F be a CM field, and let ({aλ }, {µλ }) and ({bλ }, {νλ }) be two weakly irreducible,
odd, regular, polarized compatible systems of n-dimensional representations withcorresponding p-adic representations (a, µ) and (b, ν). Let S be a finite set of finite
places of F + containing all of the places lying over p, such that each of a, b, µ, or ν
is unramified outside of S. For each v ∈ S, write Av for the µ-polarized component
determined by a|GFw , and Bv for the ν-polarized component determined by b|GFw .
Assume that:(1) For each place v ∈ S with v|p, the component Av ⊗ Bv is regular.
(2) The representations a and b are reasonable, and (a⊗b)(GF (ζp ) ) is adequate.
(3) The image b(GF ) contains SLn (Fp ).
(4) There is a finite place x - p with x ∈ S of F + which is inert in F , and a×character ψ : GFx → Qp such that:• ψ c |IFx = ψ −1 |IFx .
• ψ|IFx has (finite) order greater than 2.
• a|GFx is unramified, and• b|IFx ∼= ψ|IFx ⊕ 1⊕(n−1) .
We let T ⊂ S \{x} be a set of places with the property that if v ∈ T , then there is anequality µ|GF + = ν|GF + and a polarized isomorphism a|GFw ∼= b|GFw . Furthermore,vv
if v ∈ T , then we set Cv = Bv and Dv = Av , while if v ∈ S \ T , then we set
Cv = Av and Dv = Bv .
Then there exist odd, regular, polarized, weakly irreducible compatible systems ({cλ }, {µλ })and ({dλ }, {νλ }) with corresponding p-adic representations (c, µ) and (d, ν), having
the following properties:• c∼= a and d ∼= b.
• For each place v ∈ S, c|GFw lies on Cv and d|GFw lies on Dv .
• c and d are unramified outside of S.
Proof. Note firstly that since b is reasonable, we have p > 2(n + 1) ≥ 6, so in particular b(GF ) is large enough that Lemma 3.2.10 applies. Note also that since a|GFx
is unramified but b|GFx is ramified, we have x ∈/ T . By Corollary 4.1.9, it suffices toconstruct the desired compatible systems after increasing the set S, provided thatthe additional components Av that we choose are potentially unramified. Note thatthe assumption that (a ⊗ b)(GF (ζp ) ) is adequate includes the assumption that a ⊗ b
is absolutely irreducible; accordingly, we allow ramification in S ∪ Σ, where Σ, andthe components Cv and Dv for v ∈ Σ, are a suitable choice as in Definition 4.1.5.
We set Av = Cv and Bv = Dv for v ∈ Σ.
We consider the following four global deformation O-algebras RA,Σ , RB,Σ , RC,Σ ,and RD,Σ , defined as follows (from this point onwards we drop the Σ from thenotation):(1) RA and RC are deformation rings for a; RB and RD are deformation ringsfor b.
(2) The deformations are polarized and odd; the multiplier characters of RA , RB , RC , RD
are respectively µ, ν, µ, ν.

GLOBALLY REALIZABLE COMPONENTS OF DEFORMATION RINGS

51(3) If v ∈ S ∪ Σ, then the restriction to GFw of the universal deformation
corresponding to RA , RB , RC , or RD , lies on the component Av , Bv , Cv ,or Dv respectively.
(4) The representations are unramified outside of S ∪ Σ.
We also consider a fifth deformation O-algebra RA⊗B for a ⊗ b, which is defined
to have the following properties:(1) The deformations are polarized and odd, with multiplier µνδF/F + .
(2) If v ∈ S ∪ Σ, then the corresponding lift lies on the component Av ⊗ Bv =Cv ⊗ Dv .
(3) The representations are unramified outside of S ∪ Σ.
Note that {aλ ⊗ bλ } is weakly irreducible by Lemma 1.5.18. It follows from
Lemma 1.5.27 (with S replaced by S ∪ Σ) that RA⊗B is a finite O-algebra. We
claim that RC and RD are also both finite over O.
To prove this, we first note that any deformation coming from RC tensored with
one from RD gives, functorially, a deformation of type RA⊗B . The representation a ⊗ b also gives such a point. By Yoneda’s Lemma, there exist corresponding
morphisms:b O RB ,b O RD .
RA⊗B → RA ⊗RA⊗B → RC ⊗Since RA⊗B is finite over O, it then suffices to show that the morphism RA⊗B →b O RD is finite.
RC ⊗By Nakayama’s lemma, we are reduced to showing that if A is an Artinian ka and eb to A (of types C and D
algebra such that a and b admit deformations eerespectively) so that ea ⊗ b is the trivial deformation of a ⊗ b, then the correspondingb O RD → A factors through some subalgebra A0 of A of uniformly bounded
map RC ⊗length. To show this, let M be the fixed field of the kernel of a ⊕ b. Then we findthatea|GM ⊗ eb|GM = (a ⊗k A)|GM ⊗ (b ⊗k A)|GM
is a free A-module with a trivial action of GM . Now, we claim that if V and W are
free A-modules with an action of GM such that the diagonal action on V ⊗A W is
trivial, then GM acts on V and W by scalars. Assume for the sake of contradictionthat GM does not act on V via a character. Then there exists an element v ∈ V
and g ∈ GM such that gv is not a multiple of v. Yet then (choosing w to be
any element of W \ mA W ), g(v ⊗ w) cannot possibly be a multiple of v ⊗ w, acontradiction. Hence the action of GM on V (and W ) is by scalars.
Let Rdet(C) , Rdet(D) , etc. denote the corresponding deformation rings for thedeterminants of our representations. We have the following diagram:Rdet(A⊗B)

b det(D)Rdet(C) ⊗R

RA⊗B

b D
RC ⊗R

We begin by showing that Rdet(C) and Rdet(D) are finite over O. The Hodgetheoretic conditions imply that any two characters of type det(C) (or det(D))
differ by a finite order character unramified outside S ∪ Σ, and with ramification
at v|p ∈ S ∪ Σ bounded purely by the corresponding type. The finiteness of Rdet(C)and Rdet(D) over O is now an immediate consequence of class field theory. (Fixing

52

F. CALEGARI, M. EMERTON, AND T. GEE

one such pair of characters τC and τD , the deformation rings over k are identified
with group rings k[Γ] for some finite ray class group Γ.)Hence we may additionally assume that the determinants of ea and eb are fixed.
From the argument above, we have also shown that the action of GM on the corresponding A-modules V and W is via a scalar. Since there are only finitely many
characters of GM of order n which are unramified outside any fixed finite set ofprimes, we deduce that there exists a finite Galois extension N/F such that theaction of GN on V and W is trivial. But this implies that the corresponding maps
from RC and RD to A factor through the quotient of the universal deformation
ring over k of a and b as representations of the finite group Gal(N/F ). But thefiniteness of these deformation rings follows exactly as in the proof of Lemma 1.2.3
of [BLGGT14b].
Since RC and RD are finite over O, and have dimension at least one by Proposition 1.5.24, this shows that RC and RD both have non-trivial Qp -valued points.
Make a choice of such points, and let c, d be the corresponding p-adic representations. Recall that {aλ ⊗ bλ } is weakly irreducible, and thus potentially automorphic
by Lemma 1.5.11. In particular, given any finite Galois extension F (avoid) /F , wecan find a CM Galois extension L/F which is linearly disjoint from F (avoid) /F
and is such that {(aλ ⊗ bλ )|GL } is automorphic. Furthermore, by replacing F (avoid)
ker a⊗b(ζp ), we may assume that (a⊗b)(GL(ζp ) ) is adequate, and ζp ∈/ L.
by F (avoid) F
Making a further quadratic base change if necessary, we can also assume that allplaces at which a ⊗ b is ramified, and all places lying over p, are split places. Then(c⊗d)|GL is automorphic by Theorem 1.5.26. As in the proof of Lemma 4.1.8, it follows from the proof of [BLGGT14b, Prop. 5.5.1] that c ⊗ d is part of a regular, odd,
polarizable weakly irreducible compatible system {tλ } (again, using Lemma 3.1.1
to guarantee the hypothesis on the component groups made in [BLGGT14b, Prop.
5.5.1]).
We will now apply Theorem 3.4.3 to the compatible system {tλ }, and so deducethat c, d are the p-adic representations corresponding to compatible systems {cλ },{dλ }, which (by construction) satisfy all the requirements of the lemma. To complete the proof, it suffices to verify that the hypotheses of Theorem 3.4.3 are satisfiedby {tλ }. The strong irreducibility assumptions are satisfied by Lemma 4.1.7 andthe conditions at the places in Σ, which imply that the hypotheses of Lemma 4.1.6
hold. The hypothesis on the image of b is satisfied by assumption. Finally, theplace x required in the hypotheses of Theorem 3.4.3 can be taken to be the place
x appearing in hypothesis (4) of the present lemma: indeed, the hypotheses on the
behavior of a and b at x involve just the restriction of these representations to theinertia group at x, and thus they are constant along the components Ax and Bx .
As noted above, x 6∈ T , and thus Cx = Ax and Dx = Bx , so that these same hypotheses are satisfied by c and d. This completes the verification of the hypothesesof Theorem 3.4.3, and so also completes the proof of the lemma.
Remark 4.2.2. The polarizability requirement in the previous lemma is essential,
even for n = 1. In particular, there are genuine global obstructions (arising fromunits) to producing characters with prescribed ramification properties at all primes.
However, the polarizable condition implies that (in our setting) the corresponding
characters are unitary and come from the −1-part of the corresponding ray classGLOBALLY REALIZABLE COMPONENTS OF DEFORMATION RINGS

53

groups; in particular, there are no unit obstructions provided that p 6= 2 (and in
this paper, p is never 2).
We now introduce potentially diagonalizable lifts into the picture. We remind
the reader of Convention 2.1.4 (that we will assume without explicit mention thatthe gaps between the Hodge–Tate weights of our potentially diagonalizable lifts are
sufficiently large); we will also sometimes assume without further comment thatin base change arguments the weights and types of the potentially diagonalizable
representations have been chosen compatibly.
Lemma 4.2.3 (Existence of potentially diagonalizable lifts). Let s : GF → GLn (Fp )be a pleasant representation in the sense of Definition 2.1.7, namely:• ζp ∈/ F , and s|GF (ζp ) is irreducible.
• s is polarizable and odd.
• p > 2(n + 1).
• All the primes v|p in F + split completely in F .
• For each place w|p of F , s|GFw admits many diagonalizable lifts.
Let µ be a character such that (s, µ) is polarized. Let µ be a de Rham lift of µ.
Let S be a finite set of finite places of F + , including all places at which s or µ is
ramified, and all places lying over p.
Then there is an odd, regular, polarized weakly irreducible compatible system ({sλ }, {µλ })of GF -representations whose associated p-adic representation is (s, µ), where:• s lifts s,• s is unramified outside of S, and• s is potentially diagonalizable at all places v|p.
Furthermore, if we fix a µ-polarized component Cv of s for each place v - p in S,
then we may assume that s|GFw lies on Cv for each v.
Proof. The existence of s follows immediately from [BG17, Thm. 5.2.1] (the hypotheses of which hold by [BLGGT14b, Thm. 4.3.1, 4.5.1]). That s is part of acompatible system follows from [BLGGT14b, Thm. 5.5.1].

We will also make use of the following variant on the previous result, where weno longer require that s|GFw admit potentially diagonalizable lifts for w|p in F , but
we allow a finite base change.
Lemma 4.2.4 (Existence of potentially diagonalizable lifts II). Let s : GF →GLn (Fp ) be a reasonable representation in the sense of Definition 2.1.6, that is:• ζp ∈/ F , and s|GF (ζp ) is irreducible.
• s is polarizable and odd.
• p > 2(n + 1).
Let µ be a character such that (s, µ) is polarized. Let µ be a de Rham lift of µ.
Let S be a finite set of finite places of F + , including all places at which s or µ is
ramified, and all places lying over p. Let F (avoid) /F be a finite extension.
Then there is a finite Galois extension of CM fields L/F , linearly disjoint fromthe field F (avoid) (ζp ) over F , and an odd, regular, polarized weakly irreducible compatible system ({sλ }, {µλ }) of GL -representations whose associated p-adic representation is (s, µ|GL ), where:• s|GL is pleasant,• s lifts s|GL ,54

F. CALEGARI, M. EMERTON, AND T. GEE• s is unramified outside of the places lying over SL , and• s is potentially diagonalizable at all places v|p.
Furthermore, if we fix a µ-polarized component Cv of s for each place v - p in S,
for each vL |v.
then we may assume that s|GFw lies on Cv |L+
v
L

L

Proof. By Lemma 2.1.8, we may find a finite extension L/F linear disjoint from F (avoid) (ζp )/F
such that s|GL is pleasant. The result then follows immediately from Lemma 4.2.3.

Remark 4.2.5 (Arguments both reasonable and pleasant). The notions of reasonable
and pleasant (Definitions 2.1.6 and 2.1.7, which were also just recalled in the statements of Lemmas 4.2.4 and 4.2.3 respectively) are very closely related. Pleasantrepresentations are automatically reasonable, and reasonable representations arepleasant after a finite extension (Lemma 2.1.8). Since, in the ultimate application,
we only assume our residual representations are reasonable, we have endeavored tostructure the arguments below to only require reasonable hypotheses on the relevant residual representations. However, at a number of points (including in the
conclusions of Lemmas 4.2.6 and 4.2.7 as well as during the proofs of Lemma 4.2.9
and Theorem 4.2.11), we deduce that certain residual representations satisfy thestronger condition of pleasantness, even though, almost all of the time, we only
ever use the fact that these resulting representations are reasonable. (One notable
exception is the proof of Lemma 4.2.6, where, to invoke Lemma 4.2.3, we requirethat r is reasonable.) Thus, to the close reader, it should be born in mind that when
it is stated that a certain residual representation is pleasant, the main implication
to take away is that is reasonable.
The following lemma and its corollaries will be used in our later arguments to
replace a given representation with one which behaves well under base change.
Lemma 4.2.6 (Auxiliary representations). Let (a, µ) be polarized and odd, wherea : GF → GLn (Fp ) is reasonable, and let S be a finite set of finite places of F + ,containing all the places lying over p, and all places at which (a, µ) is ramified. Let
F (avoid) /F be a finite extension, and let q be a power of p.
Then there is a finite Galois extension L/F with ζp ∈/ L, which is linearly disjoint
from F (avoid) over F , and a weakly irreducible, odd, polarized, regular compatible
system ({rλ }, {µλ |GL+ }) of GL -representations with associated p-adic representation (r, µ|GL+ ), which satisfies:(1) a|GL is pleasant.
(2) r and r|GLgal are pleasant, and r(GLgal ) ⊃ SLn (Fq ); here Lgal denotes the
Galois closure of L over Q.
(3) (r ⊗ a|GL )(GL(ζp ) ) is adequate.
(4) Each place in S splits completely in L+ . For each place vL of L+ lying overa place v ∈ S, we have polarized isomorphisms r|GLw ∼= a|GLwL = a|GFw .
L+(5) There is a finite place x of L which is inert in L and does not lie over any
×place in S, and a character ψ : GLx → Qp with ψ c |ILx = ψ −1 |ILx , suchthat µ|GL+ is unramified, ψ|ILx has finite order greater than 2, and r|ILx ∼=x

ψ|ILx ⊕ 1⊕(n−1) .
(6) There is an isomorphism r|ILx ∼= ψ|ILx ⊕ 1⊕(n−1) .

GLOBALLY REALIZABLE COMPONENTS OF DEFORMATION RINGS

55(7) For any place v of L+ not lying over S at which r|GLw is ramified, r(ILw )is finite.
(8) r|GLw is potentially diagonalizable for all places w|p of F .
Proof. By Lemma 4.2.3, it is enough to construct the odd, polarized pair (r, µ|GL+ )satisfying properties (1)–(5), since we may choose local components which force conker aditions (6)–(8). Replacing F (avoid) with F (avoid) F(ζp ), and applying Lemma 2.1.8,/ L.
we see that we can ignore the requirements that a|GL is pleasant, and that ζp ∈Choose a finite place y ∈/ S of F + which is inert in F and is such that µ|GF + isy×unramified, and choose a character ψ : GFy → Qp with ψ c |IFy = ψ −1 |IFy such that
ψ|IFy has finite order greater than 2. (Note that we can do this for every place ywhich is inert in F .)Replacing q with a power of q if necessary, we may assume that ψ, µ, and allof the representations a|GFw with v ∈ S are defined over Fq . We may also assume that PSLn (Fq ) is simple, and is not isomorphic to any Jordan–Hölder factor
of Gal(F gal /Q), and that q is large enough so that condition (3) follows automatically from condition (2) by Lemma 1.5.37. (Note that if r(GLgal ) ⊃ SLn (Fqk ) forsome k, then in particular r(GLgal ) ⊃ SLn (Fq ).)Arguing exactly as in the proof of [EG14, Prop. A2], by [Cal12, Prop. 3.2] (seealso [MB90, Thm. 1.2]) we can find a finite totally real extension L+ /F + , linearly
disjoint from F (avoid) over F + , in which all places above S ∪ {y} split completely,
and a surjective odd reasonable µ|GL+ -polarized representation r : GL → GLn (Fq )(where L = L+ F ) with the property that for each place vL of L lying over aplace v ∈ S, we have a polarized isomorphism r|GLw ∼= a|GLwL . In addition,Lfor each place x of L+ lying over y, we can ensure that r|ILx ∼= ψ|ILx ⊕ 1⊕(n−1) .
Furthermore, by the same restriction of scalars trick that we used in the proof of
Theorem 2.1.16, we can assume that L+ is of the form M + F + , where M + /Q is
Galois.
It remains to check property (2). Since M + /Q is Galois, we have Lgal =
+ gal
M F
= LF gal , and since we are assuming that PGLn (Fq ) is simple and is
not isomorphic to any Jordan–Hölder factor of Gal(F gal /Q), it follows that the
projective image of r|Lgal (ζl ) contains PGLn (Fq ), from which the required property
follows at once.
Corollary 4.2.7 (Auxiliary representations with specified components). Supposethat either n is odd or n = 2. Let F be a CM field, and let ({aλ }, {µλ }) be a weakly
irreducible, odd, regular, polarized compatible system of n-dimensional representations with corresponding p-adic representation (a, µ). Assume that a is reasonable.
Let S be a finite set of finite places of F + containing all of the places lying over p,
and chosen so that (a, µ) is unramified outside of S.
For each v ∈ S, write Av for the µ-polarized component determined by a|GFw .
Let F (avoid) /F be a finite extension, and let q be a power of p.
Then there is a finite Galois extension L/F with ζp ∈/ L, which is linearly disjoint
from F (avoid) over F , such that for each set T of places of L+ which divide p, there isa weakly irreducible, odd, polarized, regular compatible system ({sλ }, {µλ |GL+ }) of
GL -representations with associated p-adic representation (s, µ|GL+ ), which satisfies:(1) a|GL is pleasant.
(2) s is pleasant, and is independent of the choice of T (up to isomorphism).

56

F. CALEGARI, M. EMERTON, AND T. GEE(3) s|GLgal is pleasant, and s(GLgal ) ⊃ SLn (Fq ).
(4) (s ⊗ a|GL )(GL(ζp ) ) is adequate.
(5) for each place vL of L+ lying over a place v ∈ S, we have a polarized
isomorphism s|GLw ∼= a|GLwL .
L(6) for each place vL ∈/ T of L+ lying over a place v ∈ S, s|GLw lies on Av |GL+ .
LvL(7) s|GLw is potentially diagonalizable for all places wL lying over places in T .
L(8) There is a finite place x of L+ which is inert in L and does not lie over a×place in S, and a character ψ : GLx → Qp with ψ c |ILx = ψ −1 |ILx , suchthat µ|GL+ is unramified, ψ|ILx has finite order greater than 2, and s|ILx ∼=x

ψ|ILx ⊕ 1⊕(n−1) .
(9) For any place v of L+ not lying over S at which s|GLw is ramified, s(ILw )is finite.
(10) Both L/F , x, and ψ can be chosen independently of the choice of T .
Proof. By Lemma 4.2.6, we may construct a compatible system ({rλ }, {µλ |GL+ })which satisfies all the required conditions of the theorem except perhaps (6), and ispotentially diagonalizable at all places above p. We now apply Lemma 4.2.1 to thecompatible systems ({aλ |GL }, {µλ |GL+ }) and ({rλ }, {µλ |GL+ }) with respect to theset T . The result is a compatible system ({sλ }, {µλ |GL+ }) with s = r independentof T and such that (s, µ|GL+ ) lies on the component associated to a for all vL ∈/Tand the component associated to r (which is a potentially diagonalizable component) for all vL ∈ T , as required. Note that L is the field on whose absolute Galois
group s is defined and so can be chosen independently of T , and similarly x and ψ
can be chosen independently of T (as an examination of the relevant argumentsshows).

Corollary 4.2.8. Maintaining the notation and assumptions of Corollary 4.2.7, we
can instead produce L/F and {sλ } satisfying conclusions (1)–(7) of Corollary 4.2.7,
such that in addition s is unramified outside of S.
Proof. This follows from Corollary 4.2.7 by making a further base change to remove
the ramification at places (including x) not lying over S.
We can now establish the following improvement on Lemma 4.2.1, where we nolonger need to make assumptions (1)–(4) there; we even allow a = b.
Lemma 4.2.9 (Local swapping II). Suppose that either n is odd or n = 2. Let F
be a CM field, and let ({aλ }, {µλ }) and ({bλ }, {νλ }) be two weakly irreducible, odd,
regular, polarized compatible systems of n-dimensional representations of GF withcorresponding p-adic representations (a, µ) and (b, ν). Let S be a finite set of finite
places of F + , containing all of the places lying over p, and chosen such that (a, µ)and (b, ν) are unramified outside of S. For each v ∈ S, write Av for the µ-polarizedcomponent determined by a|GFv , and Bv for the ν-polarized component determinedby b|GFv . Assume that the representations a and b are reasonable.
Let T ⊂ S be a set of places with the property that if v ∈ T , then µ|GF + = ν|GF + ,vv
and there is a polarized isomorphism a|GFw ∼= b|GFw . If v ∈ T , then we set Cv = Bv
and Dv = Av , and if v ∈ S \ T , then we set Cv = Av and Dv = Bv .

GLOBALLY REALIZABLE COMPONENTS OF DEFORMATION RINGS

57

Then there exist odd, regular, polarized, weakly irreducible compatible systems({cλ }, {µλ }) and ({dλ }, {νλ }) with corresponding p-adic representations (c, µ) and (d, ν),with the properties that:• c∼= a and d ∼= b.
• For each place v ∈ S, c|GFw lies on Cv and d|GFw lies on Dv .
• c and d are unramified outside of S.
Proof. Since the statement is symmetric in ({aλ }, {µλ }) and ({bλ }, {νλ }), it isenough to prove that ({cλ }, {λµ }) exists. We apply Corollary 4.2.7 twice to thecompatible system ({bλ }, {νλ }). The point of this construction is to provide auxiliary compatible systems to which we can apply Lemma 4.2.1. In the first application, we take T in Corollary 4.2.7 to be the set of places of F + lying over p,
and in the second application, we take T to be the set of places in our S \ T
which lie over p. The extension L/F , the place x, and the character ψ can bechosen independently of T by Corollary 4.2.7 part (10), and hence we may makethe same choice on both cases. We deduce the existence of a finite Galois extensionof CM fields L/F , and odd, regular, polarizable, weakly irreducible compatible systems ({sλ }, {νλ |GL+ }) and ({tλ }, {νλ |GL+ }) of GL -representations, with associatedp-adic representations (s, ν|GL+ ) and (t, ν|GL+ ) respectively, such that:
• s∼
= t.
• a|GL , b|GL , and s = t are pleasant.
• s(GL ) contains SLn (Fp ).
• (s ⊗ a|GL )(GL(ζp ) ) is adequate.
• for each place vL of L+ lying over a place v ∈ S, we have polarized isomorphisms s|GLw = t|GLw ∼= a|GLwL .
L
L• for each place vL of L+ not dividing p and lying over a place v ∈ T , s|GLw
L.
and t|GLw lie on Bv |L+
vLL• s is potentially diagonalizable at every place dividing p.
• t is potentially diagonalizable at every place dividing p and lying over aplace in S \ T .
• at each place vL of L lying over a place v|p with v ∈ T , t|GLw liesL
on Bv |L+.
vL
• There is a finite place x of L+ which is inert in L and does not lie over a×place in S, and a character ψ : GL+→ Qp with ψ c |ILx = ψ −1 |ILx , suchxthat µ|GL+ is unramified, ψ|ILx has finite order greater than 2, andxt|ILx ∼
= s|ILx ∼= ψ|ILx ⊕ 1⊕(n−1) .
Applying Lemma 4.2.1 to ({aλ |GL }, {µλ |GL+ }) and ({sλ }, {νλ |GL+ }), we deduce
the existence of an odd, regular, polarizable, weakly irreducible compatible system ({uλ }, {µλ |GL+ }) of GL -representations with associated p-adic representation (u, µ|GL+ ),with the properties that:• u∼= a|GL .
• For each place vL of L+ lying over a place v ∈ S not dividing p, u|GLw
Llies on Av |GL+ .
vL• for every place vL |p of L+ lying over a place v ∈ S\T , u|GLw lies on Av |L+
.
v
L

L

58

F. CALEGARI, M. EMERTON, AND T. GEE• u is potentially diagonalizable at every place dividing p and lying over aplace in T .
• u is unramified outside of S.
Applying Lemma 4.2.1 to ({uλ }, {µλ |GL+ }) and ({tλ }, {νλ |GL+ }), we then obtainan odd, regular, polarized, weakly irreducible compatible system ({eλ }, {µλ |GL+ })of GL -representations with associated p-adic representation (e, µ|GL+ ), with theproperties that:• e∼= a|GL .
.
• For each place vL lying over a place v ∈ S \ T , e|GLw lies on Av |L+
vL
L• For each place vL lying over a place v ∈ T , e|GLw lies on Bv |L+.
vLL• e is unramified outside of S.
The existence of ({cλ }, {µλ }) now follows from Lemma 4.1.8, applied to ({eλ }, {µλ |GL+ }).

Corollary 4.2.10 (Merging components). Suppose that either n is odd, or n = 2.
Let F be a CM field, and let S be a finite set of finite places of F + , containing
all of the places lying
p. Let T be a set of places of F + which divide p. Let overv({aλ }, {µλ }) and ({rλ }, {νλv }) v∈T be weakly irreducible, odd, regular, polarizedcompatible systems of n-dimensional representations of GF with corresponding padic representations (a, µ) and (rv , ν v ).
For each v ∈ S, write Av for the µv -polarized component determined by a|GFw .
Assume that all of the representations a and {rv }v∈T are reasonable, and that the
representations (a, µ) and {(rv , ν v )}v∈T are unramified outside of S. Assume also,
for each v ∈ T , that µ|GF + = ν v |GF + , and that there is a polarized isomorphismv
v
a|GFw ∼= rv |GFw .
Then there exists an odd, regular, polarized, weakly irreducible compatible system ({cλ }, {µλ }) with corresponding p-adic representation (c, µ), with the properties
that:
• c∼
= a.
• For each place v ∈ S \ T , c|GFw lies on Av .
• For each place v ∈ T , c|GFw lies on the component determined by rv |GFw .
• c is unramified outside of S.
Proof. Let v1 , . . . , vm be the places in T . We claim that for each 0 ≤ i ≤ m, we canfind an odd, regular, polarized, weakly irreducible compatible system ({ciλ }, {µλ })with corresponding p-adic representation (ci , µ), with the properties that:• ci ∼= a.
• For each place v ∈ S \ T , ci |GFw lies on Av .
• For each j ≤ i, ci |GFw lies on the component determined by rvj |GFw .
jj• For each j > i, ci |GFw lies on the component determined by a|GFw .
jj• ci is unramified outside of S.
Assuming we can do this, we can take {cλ } := {cmλ }. We prove the existence ofthe {ciλ } by induction on i, taking {c0λ } := {aλ }. Then the existence of {ci+1λ } isvi+1
vi+1i
immediate from Lemma 4.2.9 applied to ({cλ }, {µλ }) and ({rλ }, {νλ }) (taking
the set T there to be {vi+1 }).
Our main theorem is the following (recall that the notion of a globally realizable
component was defined in Definition 2.1.9).

GLOBALLY REALIZABLE COMPONENTS OF DEFORMATION RINGS

59

Theorem 4.2.11. Assume that either n is odd or n = 2. Let F be a CM field,
and let (s, µ) be a polarized representation, where s : GF → GLn (Fp ) is reasonable.
Let S be a finite set of finite places of F + , such that S contains all of the places atwhich (s, µ) is ramified and all of the places lying over p. Let µ be a de Rham liftof µ which is unramified outside of S. For each place v ∈ S, let Cv be a µ-polarizedcomponent for s|GFw .
Assume that Cv is globally realizable for each v ∈ S which divides p. Then there
exists an odd, regular, polarized, weakly irreducible compatible system ({sλ }, {µλ })of GF -representations with associated p-adic representation (s, µ), which satisfies:(1) s lifts s, and for each place v ∈ S, the representation s|GFw lies on Cv .
(2) s is unramified outside S.
Proof. Fix a place v1 ∈ S which lies over p. By definition, the hypothesis thatCv1 is globally realizable means that we can find a CM field E, a place vE,2 |pof E + such that Ev+E,2 ∼= Fv+1 , a finite set SE of places of E + containing all of theplaces lying over p, and an odd, regular, polarized, weakly irreducible compatible system ({rλ }, {µE,λ }) of GE -representations with associated p-adic representation (r, µE ), such that r is reasonable, r is unramified outside of SE , and r|GEw
E,2
lies on Cv1 (which as usual implies in particular that µE |GE+ ' µ|GF + and that
vE,2
v1
∼
there is a polarized isomorphism r|G= s|G ). The reason we use the distinct
Ew
E,2

Fw
1

numbers 1 and 2 in the subscripts is that we shall ultimately construct a field M
containing both E and F , and there will be no a priori relation between the primesof M + over v1 in F + and vE,2 in E + .
Applying Lemma 4.2.4 to s, we may find a finite Galois extension L/F of
CM fields and an odd, regular, polarizable, weakly irreducible compatible system ({esλ }, {µλ |GL+ }) of GL -representations with associated p-adic representation se,such that s|GL is pleasant, se lifts s|GL , and se is potentially diagonalizable at allplaces over p. (Here {µλ } is the compatible system containing the character µ.)
This compatible system is our initial seed; we now begin constructing auxiliary compatible systems in order to apply our component swapping to ultimately construct
the desired representation.
Applying Corollaries 4.2.7 and 4.2.8 to ({esλ }, {µλ |GL+ }), we can, after possibly
further extending L, find an odd, regular, polarizable, weakly irreducible compatible
system ({tλ }, {µλ |GL+ }) of GL -representations with associated p-adic representation t, such that:• s|GL , t and t|G(LE)gal are pleasant. (The pleasantness of s|GL and t, and alsoof t|Lgal , is ensured by the very statements of Corollaries 4.2.7 and 4.2.8.
The pleasantness of G(LE)gal can then be achieved by taking q in Corollaries 4.2.7 and 4.2.8 to be sufficiently large, and in particular large enoughthat PGLn (Fq ) is simple and not isomorphic to any Jordan–Hölder factor
of Gal(E gal /Q).)• For each place vL of L+ lying over a place v ∈ S, we have a polarized
isomorphism t|GLw ∼= s|GLwL .
L• At each place vL of L+ lying over a place v ∈ S which does not divide p,t|GLw lies on Cv |L+.
vL
L• t|GLw is potentially diagonalizable for all places vL |p of L+ .
L
• t is unramified outside of the places lying over S.

60

F. CALEGARI, M. EMERTON, AND T. GEE

Now applying Corollaries 4.2.7 and 4.2.8 to ({rλ }, {µE,λ }), we may find a finite
Galois extension K/E of CM fields and an odd, regular, polarizable, weakly irreducible compatible system ({uλ }, {νλ |GK + }) of GK -representations with associatedp-adic representation (u, ν|GK + ), such that:• u|G(LK)gal and t|G(LK)gal are pleasant. (This can be achieved by noting that
we may choose K/E to be linearly disjoint not only from (LE)gal but also
from the Galois closure of the splitting field of t.)• For each place vK of K lying over a place in SE , we have a polarized
isomorphism u|GKv ∼= r|GKvK .
K• u|GKw is potentially diagonalizable for all places vK |p of K + lying overK
places in SE , except for the places vK,2 lying over vE,2 , for which u|GKw
K,2
lies on Cv1 |Kv+ .
K,2• ν|GK +vK,2' µE |GK +vK,2, and µE |GE+

vE,2

' µ|GF + . The first condition follows
v1

from the corollary we are invoking and the second condition was alreadyassumed to be true.
• u is unramified outside of the places lying over SE .
For ease of notation, we now write M = (LK)gal ; note then that M gal = M . We
also draw the following picture which represents the inclusions between various
fields which occur in this argument:

M+

M

K+

K

(LE)gal+(LE)gal

E+

E

L+

L

F+

F

Choose a place vM,2 above vK,2 and vE,2 in M + . We now conjugate {uλ } byelements of Gal(M/Q). Because Gal(M/Q) acts transitively on the primes above p
in M + , there exists in particular a σ ∈ Gal(M/Q) with σvM,2 = vM,1 for any choiceof prime vM,1 above our fixed place v1 of F + . Note that, by construction, there
will be a polarized isomorphismuσ |GMwM,1 ' u|GMwM,2 ' r|GMwM,2 ' s|GMwM,1 ' t|GMwM,1 ,GLOBALLY REALIZABLE COMPONENTS OF DEFORMATION RINGS

61

where the second isomorphism comes from the construction of {uλ }, and the third
isomorphism was one of defining properties of {rλ }. Similarly, there is an identification of charactersν σ |GMv+

M,1' ν|GMv+

M,2' µ|GMw+.

M,1

We now apply Lemma 4.2.9 to ({tλ |GM }, {µλ |GM + }) and ({uσλ |GM }, {νλ |GM + }) relative to the set T = {vM,1 }. We deduce the existence of an odd, regular, polarized,
weakly irreducible compatible system ({aλ }, {µλ |GM + }) of GM -representations withassociated p-adic Galois representation (a, µ|GM + ), with the properties that:• a = t|GM is pleasant.
• At the place vM,1 of M + lying over v1 in F + , the representation a|GMw
M,1lies on Cv1 |Mv+ .
M,1• a is potentially diagonalizable at all places dividing p other than the places
over vM,1 .
• At each place vM of M + lying over a place v ∈ S which does not divide p,
a|GMw lies on Cv |Mv+ .
MM• a is unramified outside of the places lying over S.
As above, we now conjugate {aλ } by elements of Gal(M/F ) = Gal(M + /F + ). Because Gal(M + /F + ) acts transitively on the primes above v1 in M + , for any suchprime veM,1 |p, there exists a σ ∈ Gal(M/F ) with σevM,1 = vM,1 , and thus we deduce that there exists an odd, regular polarized, weakly irreducible compatible
system ({aσλ }, {µσλ }) of GM -representations with associated p-adic Galois representation (aσ , µσ ), with the properties that:• veM,1 is a prime in M + above v1 in F + .
• aσ is pleasant.
• µσ |GM += µ|GM + , and there is a polarized isomorphismveM,1

aσ |GMwe

M,1

v
eM,1

σ

= t |GMwe

M,1= t|GMσwe

M,1= t|GMw

M,1∼= t|GMwe

M,1

for w
eM,1 |evM,1 in M . This is because the representations t|GMwe

are allM,1

isomorphic to the restrictions of s|GFw to GMw1 , and so do not depend on1
the choice of veM,1 above v1 in M .
lies on Cv1 |M + .
• aσ |GMwe
M,1

v
eM,1

• aσ is potentially diagonalizable at all places of M dividing p other thanthose lying over veM,1 .
We now apply Corollary 4.2.10 to the compatible systems ({tλ |GM }, {µλ |GM + })and ({aσλ }, {µσλ }), where we let σ range over a set of elements of Gal(M/F ) =Gal(M + /F + ) such that σevM,1 = vM,1 where veM,1 ranges exactly over the primesof M + above v1 in F + . We deduce the existence of an odd, regular, polarized,
weakly irreducible compatible system ({bλ }, {µλ |GM + }) of GM -representations withassociated p-adic Galois representation (b, µ|GM + ), with the properties that:• b = t|GM is reasonable.
• At every place veM,1 |p of M + which lies over v1 , b|GMwe

M,1lies on Cv1 |M +

.

v
eM,1

• b is potentially diagonalizable at all places of M dividing p and not lying
over v1 .

62

F. CALEGARI, M. EMERTON, AND T. GEE• At each place vM of M + lying over a place v ∈ S which does not divide p,b|GMw lies on Cv |Mv+ .
MM• b is unramified outside of the places lying over S.
We now use this deformation of t|GM to descend to a deformation of t over GL .
Namely, applying Lemma 4.1.8, we deduce the existence of an odd, regular, polarizable, weakly irreducible compatible system ({cλ }, {µλ |GL+ }) of GL -representations
with associated p-adic representation (c, µ|GL+ ), such that:
• c∼
= t.
• At each place vL of L+ lying over a place v ∈ S which does not divide p,.
c|GLw lies on Cv |L+
vL
L.
lies on Cv1 |L+• At each place vL,1 of L+ lying over v1 , c|GLw
v
L,1

L,1

• c|GLw is potentially diagonalizable for all places vL |p not lying over v1 .
L
• c is unramified outside of the places lying over S.
Hence we have succeeded in finding deformations of a representation t (which looks
locally like s|GL at each v|p but is globally different) which lies on the desired
component at the places lying over v1 and is potentially diagonalizable at all otherprimes.
We now use local swapping to find a corresponding deformation of s|GL which
has the correct behavior at the places lying over v1 and is potentially diagonalizable at all other primes. That is, by applying Lemma 4.2.9 to ({esλ }, {µλ |GL+ })and ({cλ }, {µλ |GL+ }) (taking the subset T of that lemma to be the set of all places
lying over p), we deduce the existence of an odd, regular, polarized, weakly irreducible compatible system ({dλ }, {µλ |GL+ }) of GL -representations with associatedp-adic Galois representation (d, µ|GL+ ), such that:• d∼= s|GL .
• At each place vL of L+ lying over a place v ∈ S which does not divide p,
d|GLw lies on Cv |L+.
vL
L• At each place vL,1 of L+ lying over v1 , d|GLwlies on Cv1 |L+
.
v
L,1

L,1

• d|GLw is potentially diagonalizable for all places vL |p not lying over v1 .
L
• d is unramified outside of the places lying over S.
We now descend from L to F : Applying Lemma 4.1.8, we construct an odd, regular,
polarized, weakly irreducible compatible system ({eλ }, {µλ }) of GF -representations
with associated p-adic representation (e, µ), such that:
• e∼
= s.
• At each place v ∈ S which does not divide p, e|GFw lies on Cv .
• e|GFw lies on Cv1 .
1• e|GFw is potentially diagonalizable for all places v|p of F + other than v1 .
• e is unramified outside of the places lying over S.
Since v1 was arbitrary, there exists a compatible system with these properties foreach choice of v1 . Applying Corollary 4.2.10 to these compatible systems, we obtain
the required compatible system ({sλ }, {µλ }).
As an immediate consequence, we have the following potential automorphy theorem.

GLOBALLY REALIZABLE COMPONENTS OF DEFORMATION RINGS

63

Corollary 4.2.12. Assume that either n is odd or n = 2. Let F be a CM field,
and let (s, µ) be a polarized representation, wheres : GF → GLn (Qp )is odd and ramified at only finitely many primes. Suppose that s is reasonable. Let ρ
be the corresponding prolongation of s, and assume that ρ|GF + is globally realizablev
for each v|p. Then (s, µ) is potentially automorphic.
Proof. Let F (avoid) /F be a finite Galois extension. By Theorems 4.2.11 and 2.1.16,
together with Lemma 2.1.8, there is a finite Galois extension of CM fields L/F ,linearly disjoint from F (avoid) /F and a polarizable regular algebraic cuspidal automorphic representation of GLn (AL ) such that rp (π) ∼= s|GL is pleasant, s|GL is only
ramified at split primes, and for each place v of L+ , rp (π)|GLv lies on the component
determined by s|GLv . Then s|GL is automorphic by Theorem 1.5.26.
Finally, as promised in Remark 2.1.13, we show that “potentially globally realizable” representations are globally realizable.
Corollary 4.2.13. Assume that either n is odd or n = 2. A component C for ρ :GK → Gn (F) is globally realizable if and only if there exists a finite extension L/Ksuch that C|L is globally realizable.
Proof. If C is globally realizable, then it is easy to see from the definitions that C|L
is globally realizable, by choosing an appropriate extension of CM fields E/F . Conversely, suppose that C|L is globally realizable. Let µρ be the multiplier character
for C, so that µρ |GL is the multiplier character for C|L . By the definition of global
realizability, µρ |GL is the restriction of a de Rham character of a totally real field,
so it is a power of the cyclotomic character times a finite order character; so thesame is true of µρ .
Exactly as in the proof of [EG14, Prop. A2], by [Cal12, Prop. 3.2] (see also [MB90,
Thm. 1.2]) we can find a CM field F with maximal totally real subfield F + , suchthat we have Fv+ ∼= K for all v|p, and a representation % : GF + → Gn (F) with multiplier µ, such that for each v|p, %|GF + ∼= ρ, and %|GF is reasonable. In particular,v
we have µ|GF + = µρ for all v|p.
v
We claim that we can find a (necessarily de Rham) lift µ of µ which satisfies µ|GF + = µρ for all v|p. Indeed, as explained above, we can write µρ = εr χρv
for some integer r and some finite order character χρ . By [CHT08, Lem. 4.1.1],
there exists a finite order global character χ such that χ|GF + = χρ for each v|p. Byv
construction, it follows that εr χ|GF + = µρ for all v|p, and thusv

εr χ = µ · ψ
for some finite order residual character ψ which is trivial for all v|p. But then µ =εr χψe−1 has the required property, where ψe is the Teichmüller lift of ψ.
ker %
(ζp )
Let E/F be a finite Galois extension of CM fields, linearly disjoint from F
such that for each place vE |p of E + we have Ev+E ∼= L. The result follows from
Theorem 4.2.11 (applied with F equal to our E, and s our %|GE ), together with
Lemma 4.1.8 (applied with a there being our %|GF , and L our E).
64

F. CALEGARI, M. EMERTON, AND T. GEE

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Email address: fcale@math.uchicago.edu
Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago,
IL 60637, USA
Email address: emerton@math.uchicago.edu

66

F. CALEGARI, M. EMERTON, AND T. GEE

Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago,
IL 60637, USA
Email address: toby.gee@imperial.ac.uk
Department of Mathematics, Imperial College London, London SW7 2AZ, UK