On the semisimplicity of reductions and adelic openness for
E-rational compatible systems over global function fields
Gebhard Böckle, Wojciech Gajda and Sebastian Petersen
December 19, 2018

Contents
1 Introduction

22 Notation

7

3 Basic results on compatible systems over function fields

93.1

Compatible systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

3.2

The motivic group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

3.3

Reduction to the curve case and almost independence . . . . . . . . . . . . . . .

16

3.4

Geometric monodromy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

4 Absolute irreducibility

18

4.1

Basic results on absolute irreducibility . . . . . . . . . . . . . . . . . . . . . . . .

18

4.2

Chin’s theorem in the non-pure case . . . . . . . . . . . . . . . . . . . . . . . . .

20

4.3

G-compatible systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

5 Saturation

25

5.1

Reminders from [Nor87, Sect. 2] . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

5.2

Basics on k-saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

5.3

On saturation within any reductive group G . . . . . . . . . . . . . . . . . . . . .

33

5.4

Saturation and Lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

5.5

Saturatedness of some non-reductive groups . . . . . . . . . . . . . . . . . . . . .

40

5.6

Reductions of reductive groups and saturation . . . . . . . . . . . . . . . . . . . .

42

6 Residual saturation in the absolutely irreducible case

44

6.1

On central characters and lifting mod λ representations . . . . . . . . . . . . . .

44

6.2

Potentially tame mod λ compatible systems lift . . . . . . . . . . . . . . . . . . .

47

1

6.3

Mod λ absolute irreducibility for almost all λ. . . . . . . . . . . . . . . . . . . . .

49

6.4

Mod λ saturatedness for almost all λ. . . . . . . . . . . . . . . . . . . . . . . . .

50

7 Independence of lattices and main result on saturation
7.1

Saturation under changing representations . . . . . . . . . . . . . . . . . . . . . .

52

7.2

Geometric mod λ semisimplicity for almost all λ. . . . . . . . . . . . . . . . . . .

54

8 Adelic openness for some E-rational compatible systems

58

8.1

Summary of parts of [Pin98] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

8.2

Simple adjoint compatible systems . . . . . . . . . . . . . . . . . . . . . . . . . .

60

References

1

52

67

Introduction

Let κ be a finite field of characteristic p with an algebraic closure κ and absolute Galois groupΓκ = Gal(κ/κ). Let X be a normal geometrically connected variety over κ with arithmetic
fundamental group π1 (X), omitting a base point in the notation. Its base change under κ → κ
will be Xκ , and we write π1geo (X) := π1 (Xκ ) for the geometric fundamental group of X, so that
one has the short exact sequence 1 → π1geo (X) → π1 (X) → Γκ → 1.
0
Let further E be a number field and let PE
denote its set of finite places not above p. In thefollowing we denote by ρ• = (ρλ )λ∈PE0 an E-rational compatible system consisting of a continuous0, where Eλ denotes the completion ofhomomorphism ρλ : π1 (X) → GLn (Eλ ) for each λ ∈ PE
E at λ, subject to the usual compatibility condition, fully recalled in Section 2. Throughout
the introduction, we assume that all ρλ are semisimple, possibly by semisimplifying an initiallygiven system. Then the Zariski closure Gλ of ρλ (π1 (X)) in GLn,Eλ is a reductive subgroup. We
geo
geo
also denote by Ggeo
λ the Zariski closure of ρλ (π1 (X)) in GLn,Eλ . The group Gλ is semisimple
cf. Proposition 4.6. In the pure case this is a result of Deligne cf. Theorem 3.36.

There are two main sources of such systems. First, let f : Y → X be any smooth proper morphi
ism. Then by Deligne, [Del80], for any i ∈ Z, the family or higher direct images (Retf∗ Q` )`∈PQ0 ,provides a Q-rational compatible system (pure of weight i). Second, suppose X is a curve over
κ and Π is an cuspidal automorphic representations for GLn over the adele ring of the function
field κ(X) of X. Then by L. Lafforgue, [Laf02], and by Drinfeld if n = 2, [Dri87], to Π one canattach an E-rational compatible system, where E depends on Π. We shall refer to systems of
the former kind as cohomological and of the latter kind as automorphic.
Roughly, the motivation for the present work is to study under what conditions on ρ• thehomomorphism
Y
Y0
ρA :=
ρλ : π1 (X) −→ G(ApE ) :=Gλ (Eλ )
0
λ∈PE

0
λ∈PE

Q0
has open image, where
denotes the restricted product for a suitable choice of compact open
subgroups of Gλ (Eλ ). If there would be a Mumford-Tate like group, as is often the case if π1 (X)2is replaced with the absolute Galois group of a number field, then one would expect the Goλto arise from a single group over E. Therefore we think of the above as an adelic openness
question. Unlike in the number field case just mentioned, there is a well-known obstruction toadelic openness if the groups Gλ have non-trivial torus quotients, that stems from the smallnessof Gal(κ/κ) ∼= Ẑ. Therefore one has to study the above question either under the hypothesis thatare semisimple. One can show thatall Gλ are semisimple, or for π1geo (X) when the groups Ggeo
λthe geometric case does imply the arithmetic case, and so, for simplicity of exposition, for the
rest of the introduction we shall focus on the geometric case and always have superscripts geo.
Unless Ggeo
is simply connected for almost all λ, it has been well-known for a while that theλabove is not the right question, even for E = Q. A good formulation of adelic openness ise geo the universal cover of Ggeo , andgiven in [HL15] by Hui and Larsen. For this, denote by G
λ
λ
geo
geo
by hρA (π1 (X)), ρA (π1 (X))i the group generated by the commutators of ρA (π1geo (X)). TheQ
e geo
e geo (Ap ) := 0 0 G
latter can, in a natural way, be regarded as a subgroup of GE
λ∈P E λ (Eλ ), wheree λ defined over Oλ so that the restricted
we assume that for almost all λ we have a model of G
product makes sense. Then it is suggested in [HL15] to study the openness ofe geo (Ap ).
hρA (π1geo (X)), ρA (π1geo (X))i ⊂ G
E(1)For cohomological Q-rational compatible systems, the openness of the inclusion (1) is an immediate consequence of the recent article [CHT17] by Cadoret, Hui, Tamagawa. In the present
work, in Corollary 7.4 and Corollary 7.6, we shall give a proof for arbitrary Q-rational compatible
systems that is independent of [CHT17]. We also indicate a second proof of this fact based on[CHT17] and on the work of L. Lafforgue, using some reduction techniques of the present work.
For general E there are obvious further conditions, as can be deduced from [Pin98]. Namely
one needs that E is the subfield of Q that is generated over Q by {Tr(ρλ (Frobx )) | x ∈ X}. We
shall deduce explicit conditions from [Pin98], under which this holds. We shall also single out
one class of E-rational compatible system where we can prove fully the openness of the inclusion(1); see Theorem 8.14.
A main problem in studying adelic openness is to understand the image of the reduction ρ̄λ of0, and in fact most of the present work concerns precisely this question.
ρλ for almost all λ ∈ PE
Let us choose an Oλ -lattice Λλ in Eλn that is stable under the action of π1 (X) via ρλ .1 Following[LP95, Prop. 1.3], the groups Gλgeo , defined as the Zariski closure of Ggeo
λ in AutOλ (Λλ ) endowedwith the unique structure of reduced closed subscheme, are smooth group schemes over Oλ foralmost all λ. Let Gkgeo
denote the special fiber of Gλgeo , regard it as a subgroup of GLn,kλ via a
λchoice of basis of Λλ , and let ρ̄λ be the reduction of ρλ with respect to the lattices Λλ . Then we
have the following inclusions of groups:ρ̄λ (π1geo (X)) ⊂ Gkgeo(kλ ) ⊂ GLn (kλ ).
λgeo
Unlike Ggeo
λ , the group Gkλ is a priori not known to be reductive. Now for subgroups of GLnover finite fields Nori has defined an algebraic hull in [Nor87], often called the Nori envelope.
We shall write ρ̄λ (π1geo (X))satkλ for the closed subgroup of GLn,kλ that is the Nori envelope ofρ̄λ (π1geo (X)). Nori’s construction has been generalized by Serre to the concept of saturation,
which is the notion we will in fact use. We shall prove the following main result:
1 In Corollary 7.6 we show that it suffices to choose Λgeo
λ stable under the action of π1 (X).

3

Theorem 1.1 (Theorem 5.52, Theorem 6.11 and Proposition 5.18, Theorem 6.14 and Theo0
rem 7.3). Suppose Ggeo
is connected for all λ. Then for almost all λ ∈ PE
the following hold:λ(a) The group Gkgeo
is saturated.
λ(b) The group ρ̄λ (π1geo (X))satkλ is semisimple, and thus independent of the choice of Λλ .
geo
geo
geo
(c) The inclusion ρ̄λ (π1 (X))sat
is semisimple.
kλ ⊂ Gkλ is an equality and Gλ
The first major result in this direction is [Lar95] due to Larsen, who proved the above for Qrational compatible systems not for almost all `, but for all ` in a set of density one.2 Veryrecently the result as stated was proved for cohomological Q-rational compatible systems in theimportant work [CHT17, Thm. 7.3 and Cor. 7.5]. We shall give proofs of the three statements
above with methods independent of those of either [Lar95] or [CHT17].
As an immediate consequence of Theorem 1.1 (c) and results of Serre, we obtain the following
result, which for geometric Q-rational systems is [CHT17, Thm. 1.1].
0Corollary 1.2. Suppose Ggeo
, the represenis connected for all λ. Then for almost all λ ∈ PE
λtation ρ̄λ |π1geo (X) is semisimple and independent of the choice of Λλ .

Several of our key steps rely on automorphic methods, mainly [Laf02] and [Chi04], and only indirectly on [Del80]. In fact we shall first prove the theorem for automorphic E-rational compatible
systems. Moreover several of our arguments benefit from the freedom of enlarging an initiallygiven coefficient field.
Part (a) of Theorem 1.1 is obtained from combining the work of Larsen-Pink on the groups Gλwith work of Nori, Serre and others on saturation. We exploit the fact that the root system
of Gkgeo
looks like that of a reductive group, even when it is not known to be the case. The
λresult of (a) is stated in Theorem 5.52. Parts (b) and (c) are first proved in the case where ρ• isabsolutely irreducible, and using a result of Drinfeld, we also reduce to the case of a curve X,
see Corollary 3.33. In that case, using automorphic results, we use a further result of Drinfeld
who showed that ρ̄λ is absolutely irreducible for almost all λ, see Theorem 6.11. By standard
results on saturation, given in Proposition 5.18, (b) is then an immediate consequence.
From (a) and (b) it follows that one has the inclusion indicated in (c). We then use that theinclusion ρ̄λ (π1geo (X))satkλ ⊂ GLn,kλ is of low `λ -height (for `λ  0), and that the smaller groupis saturated, to lift this representation over the Witt vectors, still in the absolutely irreducible
case. Whenever the inclusion of connected groups in (c) is proper, and char kλ is large enough,this leads to a congruence of cuspidal automorphic forms for GLn with trivial determinant
and uniformly bounded conductor. An argument similar to (b) implies that only finitely many
geo
congruences are possible, i.e., that ρ̄λ (π1geo (X))satkλ ⊂ Gkλ can be proper only for finitely manyλ, and this completes (c) in the absolutely irreducible case; see Theorem 6.14. We note that we
include a proof for (b) also to stress that the method used here allows one to prove the following
result – for the notion of (tame) mod λ compatible system and for the notation ρ̄• ⊗OE OE 0 usedbelow we refer to Definition 6.6.
Theorem 1.3. Suppose that X is curve and that ρ̄• is a potentially tame E-rational semisimple
mod λ compatible system. Then there exists a finite extension E 0 of E and an E 0 -rational
compatible system ρ• , unique up to isomorphism, whose semisimplified reduction is ρ̄• ⊗OE OE 0 .
2 The formulation in [Lar95] is somewhat different, but it is not difficult to carry out the transition.

4

The general case of Theorem 1.1 is obtained by a reduction procedure to the absolutely irreducible case, explained in the proof of Theorem 7.3. Given any compatible system ρ• , one canin a systematic way find an absolutely irreducible compatible system ρ0• , of possible different
0,geo
dimension, such that there is a map Ggeo
with finite kernel, for all λ. We show in the
λ → Gλrather technical Lemma 7.1 that this can be done, by the use of an intermediate system thatallows one to move integral properties also, in such a way that if (b) and (c) hold for ρ0• , thenthey hold for ρ• . Lemma 7.1 is at the heart of the independence statement in Theorem 1.1(b).
We find the above an interesting reduction method in its own right. It relies on work of Chin who
in turn builds on work of L. Lafforgue. If E is sufficiently large and if the Gλ are semisimple, thenChin essentially shows that the compatible system ρ• in fact arises from an M -compatible systemr• = (rλ : π1 (X) −→ M (Eλ ))λ∈PE0 , with Zariski dense image, where M is split semisimple overE, by composing r• with a representation α : M → GLn , defined over E that is independentof λ. It is stated in Corollary 4.13, following [Chi04] and [BHKT16, Prop.6.6]. Choosing anirreducible (almost faithful) representation α0 instead, and using α ⊕ α0 as an intermediate step,
makes our argument possible. For instance one can take for α0 the tensor product of the adjoint
representations of the simple quotients of M ; see also Remark 4.15. We call the group M the
split motivic group of ρ• over E and α the split motivic representation, following [Ser94a]; see
Definition 3.22. The motivic group satisfies M ⊗E Eλ ∼= Gλ (if ρ• itself is E-rational).
This decomposition into tensor factors also allows one to reduce the original question about theadelic openness of (1) to the case where M is absolutely simple, cf. Lemma 8.12 and Proposition 8.13. For such M and for α the adjoint representation, we shall prove adelic openness.
Corollary 1.4 (Corollary 8.15; see also Theorem 8.14). Let ρ• be an E-rational semisimple
compatible system such that all Ggeo
are connected. Suppose either that (i) E = Q or that (ii)λthe split motivic group M is simple of adjoint type, the split motivic representation α is the adjoint
representation3 and E is the subfield of Q that is generated over Q by {Tr(ρλ (Frobx )) | x ∈ X}.4Thene geo (Ap )hρA (π1geo (X)), ρA (π1geo (X))i ⊂ G
Eis an inclusion of an open subgroup.
Furthermore, in our setting we answer Conjecture 5.4 of [LP95] in the affirmative.
Corollary 1.5 (see Corollary 7.9). Suppose that ρ• is an E-rational compatible system. Thenthe groups Ggeo,o
are unramified for almost all λ.
λFor cohomological Q-rational compatible systems the above two corollaries are also straightforward consequences of [CHT17]. Corollary 1.5 was proved in [LP95] for compatible systems
attached to abelian varieties over global function fields.
Let us end the introduction with a brief survey of the individual sections, noting that eachsection will begin with an outline of its contents. Section 2 collects the main notations usedthroughout the article. Section 3 recalls some standard results on compatible systems of Galois
representations, tailored to the setting of this work, where the Galois group is the arithmetic
fundamental group π1 (X) of a normal variety X over a finite field κ, or its geometric counterpart
3 The pair (M, α) is possibly defined only over some finite extension of E.
4 Given the stated properties on (M, α), by [Pin98] the representation ρ

5•

may always be defined over this E.

π1geo (X). In doing so, we also fix some further notation, and we present extensions of existing
results or give proofs of presumably known results that we could not locate in the literature.
Themes are the Weil restriction for compatible systems, Serre’s result on the independence of λof Gλ /Goλ , the motivic group and representation, the reduction of general X to the curve case,an extension of the independence of images result of [BGP18], and a reminder on Deligne’s resulton the semisimplicity of the representations ρλ |π1geo (X) .
The subsequent Section 4 provides various results on absolutely irreducible compatible systems.
We begin by collecting a number of basic and mostly elementary results. In Proposition 4.6, we
generalize a result of Chin from pure to arbitrary semisimple compatible systems. In Subsection 4.3, building on [Chi04] and [BHKT16], we explain how to use tools from representation
theory to study compatible systems. We recall the notion of a G-compatible system for G a
reductive group. Then we show that any semisimple compatible system with motivic group M
can be recovered from an M -compatible system (with Zariski dense image) together with the
motivic representation α – if E is large. By passing to other (almost faithful) representationsof M , one can obtain the same group M , up to a uniform finite kernel, from various choices ofabsolutely irreducible compatible systems; see Corollary 4.14. This corollary serves as a crucial
reduction step in the later Theorem 7.3, but is also important in Subsection 8.2.
Section 5 focusses on saturation. After recalling the foundational correspondence of Nori between
exponentially generated subgroups and nilpotently generated Lie algebras, the second subsection
provides many results on saturation in the sense of Serre that we could not locate in the literature,
and we link saturation with Nori’s results. Our main interest is in saturation within GLn but
in the third subsection, in relation to the Weil restriction and representations of low `-height,
we also recall the notion of saturation within a general reductive group. In Subsection 5.5 weshow for a new class of groups that they are saturated. In the last subsection, this is applied
to the special fibers of the smooth group schemes Gλ introduced in [LP95] and recalled above
Theorem 1.1. This will prove Theorem 1.1(a). We also discuss that the relevant results from
[LP95, § 1] hold for E- and not only Q-rational compatible systems.
In Section 6, we prove three main results on E-rational (mod λ) compatible systems ρ• . Aftersome preliminary observations in Subsection 6.1, in Theorem 6.8 in Subsection 6.2 we shall provethat any (suitably defined) potentially tame E-rational mod λ compatible system (over an infinite
index set) is the reduction of a compatible system. From this we shall deduce Theorem 6.11
in Subsection 6.3, which says that any absolutely irreducible compatible system has absolutely
irreducible reduction for almost all λ. The proof of this was first published by Drinfeld in [Dri15],
and we basically give a more detailed version of his proof. The final Subsection 6.4 contains theproof of Theorem 6.14, which is part (c) of Theorem 1.1 in the absolutely irreducible case.
Section 7 is concerned with the proof of Theorem 1.1(c) in general. In the first subsection we shall
explain that the geometric monodromy groups Ggeoof a compatible system ρ• , even integrally,
λarise from an absolutely irreducible compatible system, up to a uniform finite kernel. In thesecond subsection, we will deduce from this Theorem 1.1(c), and moreover Corollary 1.5 as well
as a precursor of Corollary 1.4 for Q-rational compatible systems.
Section 8 begins by recalling important results of Pink from [Pin98] in Subsection 8.1 on the
structure of compact subgroups of G(F ), where G is a reductive group and F is a product of`-adic fields. From this we shall in Subsection 8.2 deduce Corollary 1.4 – mainly the assertion
under hypothesis (ii). In Remark 8.16 we shall explain with a sketch of proof how far one

6could weaken the hypothesis in (ii) of Corollary 1.4 to still have an adelic openness result. The
subsection also explains how, up to some left-open details in Remark 6.16, one can directly from
[CHT17] obtain Theorem 1.1(c) without using our results from Section 6.
Acknowledgements: Part of this work was done during a stay of the three authors at Adam
Mickiewicz university in Poznań financed by NCN grant no. UMO-2014/15/B/ST1/00128. G.B.
was supported by the DFG in the FG1920 and within the SPP1489. W.G. was partially supported by the NCN grant no. UMO-2014/15/B/ST1/00128 and the Alexander von Humboldt
Foundation. The authors thank Prof. G. McNinch, for valuable feedback, in particular on Section 5, and the anonymous referee for a careful reading and many useful suggestions.

2

Notation

Notation and conventions.
◦ We fix a prime number p throughout and write ` for some prime number different from p.
◦ κ is a finite field with prime field Fp and cardinality q.
◦ X is a normal geometrically irreducible variety over κ with function field κ(X) of dimension
at least 1; its set of closed points is denoted by |X| and the residue field of x ∈ |X| is κx .
◦ Xreg denotes the set of regular points of X; the subset Xreg is dense open in X.
◦ For any field K, K denotes an algebraic closure of K, K sep ⊂ K the separable closure of
K in K, and ΓK = Gal(K sep /K) its absolute Galois group.
◦ By π1 (X) we denote the étale fundamental group of X; its geometric fundamental group is
π1geo (X) := π1 (X ×κ κ); one has a short exact sequence 1 → π1geo (X) → π1 (X) → Γκ → 1.
◦ If K is a finite field, we denote by FrobK : K → K, α 7→ α1/#K the geometric Frobenius
of ΓK . For x ∈ |X| and a choice ϕ : π1 (x) → π1 (X) we write Frobx for ϕ(Frobκx ).
◦ For a (pro-)finite group Γ denote by Γ+` the (topological closure of the) normal subgroupof Γ generated by all (pro-)`-Sylow subgroups of Γ. If ` is clear from context, we simply
write Γ+ .
◦ For a linear algebraic group G over a field K, let Go denote its identity component, Gderthe derived group of Go , and Ru (G) the unipotent radical of G. The adjoint representation
will be AdG : G → Aut(Lie G).
◦ In this text semisimple and reductive groups need not be connected.
◦ E denotes a number field; any number field will carry a fixed embedding E ,→ Q into a00
fixed algebraic closure of Q; the set of finite places of E not above p is PE. For λ ∈ PE
wedenote by `λ the rational prime under λ, by Eλ the completion at λ, by $λ a uniformizer,
by Oλ its ring of integers and by kλ its residue field; for E 0 a finite extension field of E and
0
0
0
0
0
λ0 ∈ PE
0 , we write Eλ0 , Oλ0 and kλ0 , and we write λ |E for the restriction of λ to E. We
also use this notation for E = Q, so that for λ ∈ PQ0 we have the objects Qλ , Zλ and kλ .
7◦ If K is a local field or a number field, by OK we denote its ring of integers.
◦ We call α ∈ Q plain of characteristic p, cf. [Chi04, Sect. 2], if α ∈ (Z[1/p])× , and we callP ∈ Q[T ] plain of characteristic p if its roots are all plain of characteristic p.
◦ For Q ∈ pN and w ∈ Z, a Q-Weil number of weight w is an element α ∈ Q such that (i) αis plain of characteristic p and (ii) for any embedding ι : Q → C one has |ια|C = Qw/2 . We
call P ∈ Q[T ] pure of weight w for Q if every root of P is a Q-Weil number of weight w.
◦ If ρ is a representation of a group G and if ϕ : H → G is a (natural) group homomorphism(for instance the inclusion of a subgroup), we write ρ|H for the restriction ρ ◦ ϕ of ρ to H.
We use the same notation for the restriction of families of representations.
◦ An n-dimensional E-rational compatible system ((ρλ )λ∈PE0 , (Px )x∈|X| ), or for short ρ• ,0
consists of a homomorphism ρλ : π1 (X) → GLn (Eλ ) for every λ ∈ PE
and a monic degree0n polynomial Px ∈ E[T ], such that for each (λ, x) ∈ PE × |X| one has
charpolρλ (Frobx ) (T ) = Px (T )with respect to the embedding E ,→ Eλ . An E-rational compatible system is n-dimensional
for some n. The system is called pure of weight w, if for every x ∈ |X| the polynomial Px
is pure of weight w for #κx . The system is called tame, if for any discrete rank 1 valuation0
the extension (K sep )Ker ρλ |ΓK is tame at v. It is called
v of K = κ(X) and for any λ ∈ PE
potentially tame if for some finite cover X 0 → X the restriction ρ• |π1 (X 0 ) is tame.
0
we denote by Gρ• ,λ ,◦ For an E-rational n-dimensional compatible system ρ• , and λ in PE(o)or simply Gλ , the Zariski closure of ρλ (π1 (X)) in GLn/Eλ , and we write αλ , respectively
(o)

(o)

αρ• ,λ , for the inclusion of Gλ into GLn/Eλ . If we consider π1geo (X) in place of π1 (X), we
add a superscript geo to the notation, e.g. Ggeo
instead of Gλ etc.
λ◦ Mρ• (or simply M if ρ• is clear from context) is a motivic group for ρ• defined over a finite
extension F/E; it is a connected reductive algebraic group, that is usually also split; itcomes with a split motivic representation α : M ,→ GLn/F ; see Subsection 3.2.
◦ For α ∈ Q plain of characteristic p, and L = Q(α), we define the L-rational compatible
system ρα,• = (ρα,λ : Γκ → GL1 (Lλ ))λ∈PL0 by ρα,λ (Frobκ ) = α. If α is a #κ-Weil number
of weight w, then ρα,• is pure of weight w. Via π1 (X) → Γκ we regard ρα,• as a compatible
system of representations of π1 (X).
◦ If τ : π1 (X) → GLn (E) is a continuous representation (with the discrete topology onE), we denote by ρτ,• the compatible E-rational system (pure of weight 0) defined byρτ,λ = ((GLn (E) ,→ GLn (Eλ )) ◦ τ )λ∈PE0 .
Remark 2.1. It is an elementary observation, cf. [Chi04, last parag. in Sect. 2], that for acompatible system ρ• , for all x ∈ |X| the polynomials Px are plain of characteristic p.
Convention 2.2. From Section 4 on, all compatible systems are assumed to be semisimple.

8

3

Basic results on compatible systems over function fields

This section recalls some standard results on compatible systems of Galois representations, tailored to the setting relevant to this work, where the domain is π1 (X) or π1geo (X), respectively.
In doing so, we fix some further notation, and we present extensions of existing results or give
proofs of presumably known results that we could not locate in the literature.
Subsection 3.1 recalls change of coefficients and a kind of Weil restriction for compatible systems,
and it recalls Serre’s results on independence of λ of π0 of the monodromy groups. It alsoclassifies 1-dimensional compatible systems. Subsection 3.2 recalls a result of Chin that gives an
independence of λ of the identity component of the monodromy group after a finite coefficientchange. Subsection 3.3 explains a reduction from X to a curve that preserves the monodromy
groups (and the Galois images up to a uniform finite amount). Moreover it extends results from
[BGP18] on independence of images in the sense of Serre to arbitrary compatible systems. The
last subsection recalls a semisimplicity result of Deligne and reduces investigations on monodromy
groups attached to π1geo (X) to the case of semisimple compatible systems ρ• of π1 (X).

3.1

Compatible systems

We collect some basic results for later use.
Lemma 3.1 ([Ser68, I.2.3, Theorem]). Suppose that ρλ and ρ0λ are semisimple representations
π1 (X) → GLn (Eλ ) with charpolρλ (Frobx ) = charpolρλ (Frobx ) for all x ∈ |X|. Then ρλ ∼= ρ0λ .
In particular, if two semisimple compatible systems ρ• and ρ0• have the same Frobenius polyno0
one has ρλ ∼mials Px (T ), Px0 (T ) ∈ E[T ] for all x ∈ |X|, then ρ• ∼= ρ0λ .
= ρ0• , i.e, for all λ ∈ PE
Definition 3.2. If E 0 is a finite extension of E, we define the coefficient extension ρ• ⊗E E 0 ofρ• to E 0 by(ρ• ⊗E E 0 )λ0 := (GLn (Eλ ) ,→ GLn (Eλ0 0 )) ◦ ρλ00
for all λ0 ∈ PE
0 with contraction λ to PE .
00
Lemma 3.3. For a finite extension E 0 /E and any λ0 ∈ PE
0 above λ ∈ PE one hasGρ• ⊗E E 0 ,λ0 = Gρ• ⊗Eλ Eλ0 0 .
Proof. The assertion of the lemma follows from the following well-known result: Let L ⊃ K bean extension of fields field, let T be a K-variety and TL be its base change to L. Then the Zariski
closure in TL of any subset S ⊂ T (K) is the base change under K → L of the Zariski closure of
S in T , i.e., the operations base change and Zariski closure commute for varieties.
Let E 0 be a subfield of E. Then the Weil restriction ResE/E 0 GLn/E can be regarded as a closed
subgroup of GLn[E:E 0 ]/E 0 , via the natural transformation functor from E 0 -algebras R0 given by
ResE/E 0 GLn/E (R0 ) = GLn (E ⊗E 0 R0 ) ,→ GLn[E:E 0 ] (R0 ),where the equality on the right is given by choosing a basis for E over E 0 .

9Definition 3.4. The Weil restriction ResE/E 0 ρ• of ρ• to E 0 is defined by assigning to any0λ0 ∈ PE
0 the representationYMGLn (Eλ ),(ResE/E 0 ρ• )λ0 :=ρλ : π1 (X) → ResE/E 0 GLn/E (Eλ0 0 ) =
λ|λ0

λ|λ0

0
where the product is over all λ ∈ PE
above λ0 . It is an E 0 -rational compatible system.

An important example of a Weil restriction is ResE/Q ρ• ; cf. Theorem 3.7. For fields E, E 0 ⊃ E0 ,the set of field homomorphism σ : E → E 0 with σ|E0 = idE0 is HomE0 (E, E 0 ).
Lemma 3.5. Suppose that ρ• is E-rational. Let E 0 ⊂ E be a subfield, and F/E an extension
such that F/E 0 is finite Galois. Then the following hold:(a) If we denote by⊗σE F the tensor product of F over E with respect to σ : E → F , thenMρ• ⊗σE F.
ResE/E 0 ρ• ⊗E 0 F ∼=σ∈HomE 0 (E,F )(b) If ρ• is semisimple, then so is ResE/E 0 ρ• .
00(c) If Gρ• ,λ is semisimple for all λ ∈ PE
, then GResE/E0 ρ• ,λ0 is semisimple for all λ0 ∈ PE
0.

Proof. The proof of (a) is immediate from the definitions. We deduce (b) from (a). For thisrecall that a representation V of a profinite group Γ over E 0 is semisimple, if V ⊗E 0 F is semisimple as a representation of Γ over F : To see this, consider a Γ-stable submodule W ⊂ V
and aPΓ-equivariant splitting s : V ⊗E 0 F → W ⊗E 0 F of W ⊗E 0 F ,→ V ⊗E 0 F . Then1
τ ∈Gal(F/E 0 ) τ (s) : W → V is a Γ-equivariant splitting of W ,→ V . Thus it suffices to[F :E 0 ]show the semisimplicity of the right hand side of (a). For this note that since ρ• is semisimple,
then so is ρ• ⊗σE F (cf. [CR62, Cor. 69.8]), and thus also the direct sum over all σ ∈ HomE 0 (E, F ).
0For (c) let µ be in PF0 , let λ0 ∈ PE
0 be below µ, and denote for σ ∈ HomE 0 (E, F ) by µ|σ the0
under µ with respect to σ. By (a) the group H 0 := GResE/E0 ρ• ,λ ⊗Eλ0 Fµ is a closed
place of PE
Q
subgroup of H := σ∈HomE0 (E,F ) Gρ• ,µ|σ ⊗σEµ|σ Fµ . Let prσ : H → Hσ := Gρ• ,µ|σ ⊗σEµ|σ Fµ be
the canonical projection. Let R be the radical of H 0 . By Lemma 3.3, it will suffice to show that
R is trivial. By our hypothesis and Lemma 3.3, the groups Hσ are semisimple. Since prσ (H 0 )is a closed subgroup of Hσ , see [Spr09, 2.2.5], it must be all of Hσ , by the definition of H 0 andHσ . Now because prσ : H 0 → Hσ is surjective and Q
Hσ is semisimple, we have R ⊂ Ker prσ forall σ. But then R lies in the kernel of the injection σ prσ , and hence R must be trivial.

Remark 3.6. For H and H 0 as in the previous proof, without any further hypotheses there
is not more we can say than what we explained above. Proposition 8.13 gives examples with
H 0 = H. But from Proposition 8.13 and its proof one can also construct examples where in fact
H0 ∼= Gρ• ,µ|σ0 ⊗Eµ|σ0 Fµ for any σ0 ∈ HomE 0 (E, F ). Section 8 will give further results about
possible H 0 .
Theorem 3.7 ([Ser67, Prop. 2 and its Cor.]). Let Γ be a compact subgroup of GLn (Q` ) with
Zariski closure G. If G◦ is semisimple, then Γ is open in G(Q` ). In particular, if ρ• is a Qrational compatible system such that Gρ• ,` is semisimple, then ρ` (π1 (X)) is open in Gρ• ,` (Q` ).
10

Example 3.8. Clearly, in Theorem 3.7 one cannot replace Q` by a proper finite extension L: LetΓ and G be as in Theorem 3.7 and suppose that Go is semisimple and non-trivial. Then Γ is
Zariski dense in G ⊗Q` L by Lemma 3.3. But G(Q` ) is not open in G(L), and so Γ is not openin G(L). For positive results, see Theorem 8.14(e) and Corollary 8.15.
The following apparently well-known result we learned from [Chi04, Thm. 1.6]:
Lemma 3.9. Let ρλ : π1 (X) → GLn (Eλ ) be semisimple. Denote by G ⊂ GLn,Eλ its Zariski
closure and by Ggeo ⊂ G that of ρλ (π1geo (X)). Then Gder = Ggeo,o and Gder is semisimple.
Proof. For lack of a reference, we include a proof. Recall that by our conventions from Section 2,
we denote by Gder the derived group of the identity component of G, hence it is connected. The
group π1geo (X) is normal in π1 (X), and thus so is Ggeo ⊂ G, and then also Ggeo,o in Go . The
latter groups are connected reductive, because ρλ is semisimple by hypothesis, and ρλ |π1geo (X)is so because of a result of Deligne – see Theorem 3.36. Hence the quotient Ḡ := Go /Ggeo,o isreductive and connected. It is also abelian because the image of π1 (X)/π1geo (X) ∼= Ẑ is dense in
it. It follows that Ḡ is a torus. This proves the first assertion; cf. [LP95, Sec. 2]. The second
follows again from Theorem 3.36.
For later use, we record the following consequences of Lemma 3.9 and Theorem 3.7.
0Corollary 3.10. If ρ• is Q-rational, then ρ` (π1geo (X)) ⊂ Ggeoρ• ,` (Q` ) is open for all ` ∈ PQ .

Corollary 3.11. Suppose ρ• is E-rational semisimple
the groups Q
Gρ• ,λ areandgeo semisimple for
Q
0. Then for all ` ∈ PQ0 , the subgroupρ(πall λ ∈ PE
(X))
⊂
ρ
1
λ|` λ
λ|` λ (π1 (X)) is open.
Proof. Using Lemma 3.5(c) and the Weil restriction ResE/Q ρ• , we may assume E = Q. BecauseGρ• ,` is semisimple, we deduce Ggeoρ• ,` = Gρ• ,` from Lemma 3.9. The result now follows from
Theorem 3.7.
Remark 3.12. In Corollary 7.5 we shall give an adelic refinement of Corollary 3.11.
Regarding the discrepancy between Gλ and Goλ one has the following result due to Serre:
Theorem 3.13 ([Ser13a, p. 15–20] or [LP92, Prop. 6.14]5 ). The kernel ofρλ mod Goλ : π1 (X) → Gλ /Goλ0
is finite and independent of λ ∈ PE.

Definition 3.14. We say that ρ• is connected or has connected monodromy if Goλ = Gλ for one,0
and hence, by Theorem 3.13, for all λ ∈ PE.
By Theorem 3.13 for any compatible system ρ• there exists a finite étale cover X 0 → X such thatρ• |π1 (X 0 ) has connected monodromy. By the following result, whose proof is a simple exercise,
restriction to π1 (X 0 ) leaves Goλ invariant:Lemma 3.15. Let X 0 → X be any finite cover. Then Goρ• |π1 (X 0 ) ,λ0
= Goρ• ,λ for all λ ∈ PE.

5 The references only consider E-rational compatible systems for E = Q. The adaptation to general E is minor.

11Below we need the following immediate consequence of [KL81, Thm. 1] by Katz and Lang:
Theorem 3.16. Denote by π1 (X)ab,p the maximal abelian quotient of π1 (X) modulo its pro-p
subgroup. Then the following hold:(a) For any x ∈ X, the cokernel of the canonical map π1 (x) → π1 (X)ab,p is finite.
(b) There exists a finite extension X 0 → X, say with field of constants κ0 , such that the induced
homomorphism π1 (X 0 ) → π1 (X)ab,p factors via the canonical map π1 (X 0 ) → Γκ0 .
The following results cover 1-dimensional representations. We first need some notation. For×α ∈ Z` and can : π1 (X) → Γκ the canonical map, define ρα : π1 (X) → GL1 (Q` ) as the composite
continuous homomorphism
can

r

α
π1 (X) −→ Γκ −→GL1 (Q` ) determined by rα (Frobκ ) = α.

(2)Lemma 3.17. Let ρ : π1 (X) → GL1 (Q` ) be a continuous representation and define αx =ρ(Frobx ) for all x ∈ |X|. Then the following hold:×(a) There exists α ∈ Z` such that ρ0 := ρ ⊗ (ρα )−1 has finite image.
(b) If all αx lie in Q and are plain of characteristic p, then α in (a) lies in Q and is plain ofcharacteristic p. In this case, if L = Q(α) and µ ∈ PL0 is the place induced from Q ,→ Q` ,then ρα = (ρα,µ ) ⊗Lµ Q` is a member of a compatible system.
(c) If all αx are #κx -Weil numbers of weight w, then α in (a) is a q-Weil number of weight w.
(d) If all αx lie in a number field E, then α in (a) can be chosen to lie in E.
Proof. For x ∈ |X| let dx be the degree of κx over κ. Because X is geometrically irreducible overthe finite field κ, we can find x1 , x2 ∈ |X| such that gcd(dx1 , dx2 ) = 1.6 Let m1 , m2 ∈ Z be suchthat 1 = m1 dx1 + m2 dx2 and set α = αxm11 αxm22 . Since ρ is continuous and π1 (X) is compact all
αx lie in Z×` , and hence so does α. This choice also proves (d) once the proof of (a) is complete.
To complete (a), note first that the natural map Γκx → Γκ is given by Frobx 7→ Frobdκx . By
Ddxfor any x, x0 ∈ |X|. A direct
Theorem 3.16 there exists D ∈ N such that αx x0 = αxDd
00 D
computation now shows that (ρ ) is the trivial character (on all Frobx ), and hence the image
is finite, proving (a) (and (d)).
For (b) and (c) we only have to show that, independently of the construction of α in (a), it will
have the properties asserted there. The remaining part of (b) is clear from the definition of ρα,• .
Let therefore D denote the order of ρ0 . Evaluating (ρ0 )D at Frobx shows αdx D = αxD , i.e., that α
is one of the Ddx -th roots of αx . This implies (b) and (c) since the properties considered there
are stable under taking roots.
Remark 3.18. The proof also shows that if one αx is plain of characteristic p, or a #κx -Weilnumber pure of weight w, then the same holds for all αx .
6 Lacking a reference, we give a short proof: Let κ be the finite extension of κ of degree r. Fix two rationalrprime numbers `1 6= `2 . Because X is geometrically irreducible, suitable estimates based on the Weil conjectures,see [Zyw16, Thm. 2.1], imply limj→∞ X(κ`j ) = ∞ for i ∈ {1, 2}. Thus we can find xi ∈ |X| with dxi a power of `i .
i

12

Corollary 3.19 (cf. [Chi04, 4.2, 4.3]). Let ρ• be a 1-dimensional E-rational compatible system.
Then the following hold:(a) There exists α ∈ E × plain of characteristic p and a continuous representation τ : π1 (X) →GL1 (E) such that ρ• = ρα,• ⊗ ρτ,• .
(b) If ρ• is pure of weight w, then α in (a) is a q-Weil number of weight w.
(c) There exists a finite étale cover X 0 → X and some α ∈ E × which is plain of characteristicp (and a #κ0 -Weil number of weight w if ρ• is pure of weight w), such that ρ• |π1 (X 0 ) is
isomorphic to ρα,• |π1 (X 0 ) , and in particular factors via Γκ0 for κ0 the constant field of X 0 .
Proof. Because ρ• is compatible 1-dimensional, ρλ (Frobx ) can be identified with some αx ∈ E ×
0that is independent of λ. In particular, αx lies in Oλ for all λ ∈ PE
and is thus plain ofcharacteristic p; if ρ• is pure of weight w, then moreover αx is a #κx -Weil number of weight0
and an embedding Eλ0 ,→ Q`λ . Let α ∈ E be as in Lemma 3.17 for
w. Fix one λ0 ∈ PE
ρλ0 ⊗Eλ0 Q`λ0 . It follows that ρ−1α,• ⊗ ρ• is a compatible system that is of finite order at λ0 .
Then by Theorem 3.16 (or by Theorem 3.13) there exists a finite cover X 00 → X such thatρ0• |π1 (X 00 ) is trivial. Now one can simply apply complex representation theory of finite groups
to deduce the existence of τ : π1 (X) → GL1 (E) such that (a) holds. Part (b) follows from
Lemma 3.17(c). Moreover (c) follows by taking for X 0 the cover of X that corresponds to thekernel of τ from (a).

3.2

The motivic group

For repeated later use, we first recall two basic facts on the representation theory of split semisimple connected groups in characteristic zero, and state an immediate corollary:
Theorem 3.20 ([DG70, IV.3.3.3], [Tit71, Thm. 2.5]). Suppose that M is a split reductive
connected group over a field F of characteristic 0. Then the following hold.
(a) The category of finite dimensional representations of M is semisimple.
(b) The irreducible representations of M are in bijection with highest weight representations,parameterized by the dominant weights of M .
Corollary 3.21. Let M and F be as in Theorem 3.20, and let F 0 be any extension of F .
Then V 7→ V ⊗F F 0 sets up a bijection between the isomorphism classes of irreducible, finitedimensional representations of M and those of M ⊗F F 0 .
Definition 3.22. Let ρ• be a semisimple E-rational n-dimensional compatible system. We calla pair (M, α) consisting of(i) a connected reductive algebraic group M over E,(ii) and a faithful E-rational representation α : M → GLna motivic pair for ρ• , M the motivic group of ρ• and α the motivic representation of ρ• , if the0
following conditions hold: For any λ ∈ PE
one has an isomorphism of algebraic groups'

ιλ : M ⊗E Eλ −→ Goρ• ,λ ,13

(3)such that the following diagram is commutative up to conjugationiµM ⊗E E λ

/ G◦

ρ• ,λ

α⊗E Eλ(4)&z inclGLn,Eλ ;i.e., there is an isomorphism of representations M → GLn under the identification in (3).
We call (F, M, α) a motivic triple for ρ• if F is a finite extension of E and if (M, α) is a motivic
pair for ρ• ⊗E F . The motivic triple is called split if M is split.
Remark 3.23. Let ρ• be a semisimple E-rational n-dimensional compatible system. Then for split
motivic triples (F, M, α) for ρ• , the root datum and the weights defining (M, α) are independentof F . Therefore we shall sometimes simply speak of the split motivic group and the split motivic
representation, assuming that a sufficiently large F has been chosen.
For the same reason, if M is split, then the pair (M, α) is uniquely determined by the properties(3) and (4) for a single µ.
Remark 3.24. (a) The definition of the motivic group M should also include conditions at theplaces of E above p; this is possible using the work [Abe] of Abe that attaches certain
isocrystals at places above p and thus extends the work of L. Lafforgue; cf. also [Dri15].
For the present work these places are irrelevant, and so we omit them.
(b) It is expected that the motivic group exists (over E). The existence of α over E depends
on a Brauer obstruction; cf. [Dri15, E.9]. But ignoring the places above p, it might well bepossible that α can be realized over E also. See [Hui18] for recent progress.
We shall use of the following result of Chin, generalizing [LP95, Thm. 2.4], at various places:
Theorem 3.25 ([Chi04, Thm. 1.4 and Thm. 1.6]). For any pure semisimple E-rational ndimensional compatible system ρ• there exists a split motivic triple (F, M, α) for ρ• . Moreover0
the isomorphism (3) (for ρ• ⊗E F )(cf. Lemma 3.9) for any µ ∈ PF0 with contraction λ ∈ PE
induces an isomorphism of algebraic groupsGoρ• | geo
π1(X) ,λ⊗Eλ Fµ ∼= M der ⊗F Fµ ,(5)and the diagram (4) an isomorphisms of representations M der → GLn
α|M der ⊗F Fµ ∼= αρo• | geo
π1(X) ,λ⊗Eλ Fµ .

(6)

Remark 3.26. It cannot be expected that in Theorem 3.25 there is a smallest or optimal choicefor F , since M is split over F . This is analogous to the well-known fact that for representationsof finite groups over Q there is in general no smallest field of definition – because of the Schurindex. A simple example in the present case would be a system ρ• of dimension 4 with Gλ0 a0quaternion division algebra for some λ0 ∈ PE
and Gλ ∼= GL2,Eλ for all other λ ∈ PE . Then
there could be many quadratic extensions of E over which a split motivic group exists, but overE this is not possible.
Lemma 3.27. Suppose ρ• is E-rational and semisimple with motivic pair (M, α). Then:14

(a) If E 0 /E is a finite extension, then (M ⊗E E 0 , α ⊗E E 0 ) is a motivic pair for ρ• ⊗E E 0 .
(b) If E 0 ⊂ E is a subfield and if ρ• is pure, then ResE/E 0 ρ• has a motivic triple (F 0 , M 0 , α0 ),and if M is semisimple, then so is M 0 .
(c) If X 0 → X is a finite cover, then (M, α) is a motivic pair for ρ• |π1 (X 0 ) .
Proof. Part (a) is immediate from Lemma 3.3, and Part (c) is immediate from Lemma 3.15. We
shall now prove Part (b). ResE/E 0 ρ• is a semisimple compatible system by Lemma 3.5(b), and
it is easy to see that ResE/E 0 ρ• is pure (of the same weight as ρ• ). By Theorem 3.25 it follows
that ResE/E 0 ρ• has a motivic triple (F 0 , M 0 , α0 ). If M is semisimple in addition, then all Gρ• ,λare semisimple, thus all GResE/E0 ρ• ,λ0 are semisimple as well by Lemma 3.5(c), and this implies
that M 0 is semisimple.
Remark 3.28. In Proposition 4.6 and Corollary 4.8 we shall remove the purity hypothesis in
Theorem 3.25 and Lemma 3.27(b), respectively.
00Remark 3.29. In Lemma 3.27(b), for λ ∈ PE
and λ0 ∈ PE
0 the contraction of λ, the group
GResE/E0 ρ• ,λ0 is a closed subgroup of ResEλ /E 0 0 Gρ• ,λ . In general the containment is proper; forλinstance for ρ• = ρ0• ⊗E 0 E with ρ0• an E 0 -rational compatible system.

The proof of Theorem 3.25 crucially relies on the work of L. Lafforgue on the global Langlands
correspondence. Since in later parts we shall need to directly apply his work, we now state it ina form to be used later; our formulation also requires [Chi04, Thm. 4.1].
Theorem 3.30 ([Laf02, Thme. (p. 2), Thme. VI.9, Prop. VII.4]). Let X be a smooth projective
curve over κ. Then the following holds:(a) Suppose Π is a cuspidal automorphic representation of GLn/Aκ(X) with finite order central
character τ : π1 (X) → GL1 (Q), conductor N which is a divisor on X, and Hecke field E.
For x ∈ |X| r Supp N let pΠ,x ∈ E[T ] be the Hecke polynomial at x. Then there exists a
finite extension field E 0 of E and an E 0 -rational compatible system(ρΠ,λ : π1 (X r Supp N ) → GLn (Eλ0 ))λ∈PE0 0 ,0pure of weight 0 with determinant ρτ,• , such that for all λ ∈ PE
0 and x ∈ |X| r Supp N the
Frobenius polynomial of ρΠ,• is equal to pΠ,x . Moreover each ρΠ,λ is absolutely irreducible.

(b) If ρ : π1 (X rS) → GLn (Q` ) for some finite S ⊂ |X| is continuous and absolutely irreducible
and if det ρ(π1 (X)) is finite, then there exists a cuspidal automorphic representation Π of
GLn/Aκ(X) with finite central character such that one hasρ = ι ◦ ρΠ,λ00
00for some λ0 ∈ PE
0 and embedding Eλ0 → Q` , with E from part (a). In particular, ρΠ,λ0 isa member of a compatible system that is pure of weight 0.

Moreover the bijection set up above is also compatible with the local Langlands correspondence
at all places dividing the conductor. In particular, the conductors of ρΠ,λ and of Π coincide.

15

3.3

Reduction to the curve case and almost independenceThis subsection collects two deeper results on compatible systems needed in the later parts ofthis work. The first result explains why for a given compatible system ρ• , after a finite basechange X 0 → X all ramification of ρ• |π1 (X 0 ) is tame, and in fact pro-`λ for any ρλ , and why for0
tame systems one can find a curve C embedded into X, say via ι, such that for all λ ∈ PEone
has ρλ (π1 (X)) = ρλ (ι∗ (π1 (C))). The second result is an extension of the main result of [BGP18].
We show the almost independence in the sense of [Ser13b] of arbitrary Q-rational compatible
systems of certain representations of π1geo (X).
We say that a profinite Galois extension L of K = κ(X) has at most pro-` ramification if for any
discrete rank 1 valuation v of K with extension w to L and completions Kv and Lw , the inertia
subgroup in Gal(Lw /Kv ) is a pro-` group.
Lemma 3.31. Let ρ• be a compatible system. Then there exists a finite étale cover X 0 → X
0the ramification of ρλ |π1 (X 0 ) is pro-`λ . In particular, ρ• restricted tosuch that for any λ ∈ PE
0
π1 (X ) is tame.
Proof. The argument is a slight modification of the argument given in [BGP18, § 4 and § 6]. Letρ• be the E-rational compatible system. Choose a finite cover X 0 → X, such that for two places0with `λ0 6= `λ1 the image of ρλ0 (π1 (X 0 )) is pro-`λ0 and that of ρλ1 (π1 (X 0 )) is proλ0 , λ1 ∈ PE
`λ1 . Consider any embedding ι : C → X 0 of a curve into X 0 , and let ι∗ : π1 (C) → π1 (X) be thecorresponding homomorphism of fundamental groups. Then ρ• ◦ ι∗ is an E-rational compatible
system of representations of π1 (C). For any place c of K = κ(C) denote by (rλ,c , Nλ,c ) the
associated Weil-Deligne representation, and by Ic the inertia subgroup of ΓKc . By [Del73,
Thm. 9.8] the family of representations rλ,c |Ic is independent of λ. By our hypotheses, theimage is an `λ0 and an `λ1 -group, and hence trivial. It follows that all ramification of ρλ ◦ ι∗
is unipotent, and hence at most pro-`λ . Then [BGP18, Prop. 4.6], which builds on work of
Kerz-Schmidt-Wiesend, cf. [KS10, Thm. 4.4], completes the proof of the lemma.
We also need the following result due to Drinfeld:
Theorem 3.32 ([Dri12, Prop. 2.17 and 2.18]). Let Y be a normal geometrically connected variety
over κ and let U ⊂ Y be a dense regular open subvariety. Then there exists a smooth irreducible
curve C on U such that the canonical map π1 (C) → π1 (U )/H is surjective, where H ⊂ π1 (U ) isdefined as follows:(a) H is a closed normal subgroup such that π1 (U )/H contains an open pro-` subgroup, or(b) Y is projective and H is the kernel of π1 (U ) → π1tame (U ), where π1tame (U ) is the quotientof π1 (U ) that classifies covers of U which are at most tamely ramified at codimension 1
divisors of Y r U .
Moreover one can assume that C passes through any finite number of points of U , and in particular one may, cf. Footnote 6, assume that C is geometrically irreducible.
The following corollary is an immediate consequence of Lemma 3.31 and Theorem 3.32(b).

16Corollary 3.33. For any compatible system ρ• , there exists a finite étale cover π : X 0 → X and0
a curve C embedding into X 0 via some ι such that ρλ (π1 (X 0 )) = ρλ (ι∗ (π1 (C))) for all λ ∈ PE.
If ρ• is tame, then one can take π = idX .
As an application of [BGP18] and Lemma 3.31, we also record the following result that is apartial strengthening of the main result of [BGP18].
Theorem 3.34. Let ρ• be a Q-rational compatible system for π1 (X). Then ρ• |π1geo (X) is almost
independent in the sense of Serre, cf. [Ser13b, § 1]. Moreover there is a finite cover X 00 → Xsuch that for almost all ` ∈ PQ0 the group ρ• (π1geo (X 00 )) is `-generated.
Proof. By Lemma 3.31 there exists a finite (possibly ramified) Galois cover X 0 → X such thatρ` is `-tame for each ` ∈ PQ0 . In particular ρ• satisfies Condition S(X, κ) of [BGP18, Def. 5.1].
Using the notation from [BGP18, § 3], by [BGP18, Thm. 3.6], for each ` ∈ PQ0 there exists a
short exact sequenceg`
1 → M` → ρ` (π1 (X)) → H` → 1of profinite groups with H` finite in Jor` (J 0 (n)) and M` in Σ` (2n ). Now apply [BGP18, Prop. 5.7]
to the family (g` ◦ ρ` : π1 (X) → H` )`∈PQ0 (compatibility makes no sense here), to find a finite(possibly ramified) cover X 00 → X 0 such that X 00 → X is Galois and ρ` (π1geo (X 00 )) ⊂ M` (2n ) forall ` ∈ PQ0 . Now apply [BGP18, Thm. 5.8] to ρ• |π1geo (X 00 ) to deduce that this family, and hence
also ρ• |π1geo (X) is almost independent.
Remark 3.35. For an E-rational compatible system ρ• , Theorem 3.34 can be applied to ResE/Q ρ• .

3.4

Geometric monodromy

Our final result here shows that the geometric monodromy is unaffected by semisimplification.
We begin with an important result due to Deligne:
Theorem 3.36 ([Del80, Cor. 1.3.9, Thm. 3.4.1(iii)]). Let K be a finite extension of Q` for some` ∈ PQ0 , and let ρ : π1 (X) → GLn (K) be pure. Then ρ|π1geo (X) is semisimple, and the Zariski
closure of ρ|π1geo (X) is an extension of a finite group by a semisimple group.
Corollary 3.37. Let K and ρ be as in Theorem 3.36, and denote by ρss the semisimplificationof ρ as a representation of π1 (X). Then one has an isomorphism of representationsgeo
ρss |π1geo (X) ∼= ρ|π1geo (X) : π1 (X) → GLn (K).

Corollary 3.37 is an immediate consequence of Theorem 3.36 and the following lemma, whose
simple proof we leave as an exercise.
Lemma 3.38. Let V be a finite dimensional representation of a group G and let H be a subgroupof G. Denote by V Γ-ss the semisimplification of V as a representation of Γ ∈ {G, H}. If theaction of H on all simple G-representations is semisimple, then as H-representations one has
V G-ss ∼= V H-ss .
If in addition V is semisimple as an H-representation, then V G-ss ∼= V as H-representations.
17

4

Absolute irreducibility

Subsection 4.1 provides various results on absolutely irreducible compatible systems. Lemma 4.1
shows that if one member of a compatible system is absolutely irreducible, then all of its members have this property, and also that connected absolutely irreducible compatible systems canbe twisted by a character so that their monodromy is semisimple; then π1 (X) and π1geo (X)have the same monodromy groups. If the monodromy is connected we show that absolute irreducibility is equivalent to the irreducibility of its split motivic representation. Proposition 4.6
in Subsection 4.2 generalizes Theorem 3.25 of Chin. It shows the existence of a motivic triple
for arbitrary and not only pure compatible systems. In Subsection 4.3 we explain how to use
tools from representation theory to study compatible systems. We recall the notion of a Gcompatible system for G a reductive group from [BHKT16]. Then we show, using [Chi04] and[BHKT16], that any compatible system with split motivic group M can be recovered from a
M -compatible system (with Zariski dense image) together with its split motivic representation.
We show in Corollary 4.14 that any connected compatible system with semisimple split motivic
group has, up to a (uniform) finite kernel, the same monodromy groups as a suitable connected
absolutely irreducible compatible system. This corollary serves as a crucial reduction step in thelater Theorem 7.3.
In accordance with Convention 2.2, from now on all compatible systems are semisimple.

4.1

Basic results on absolute irreducibility

Recall that a representation ρ : Γ → GLn (K) for a group Γ and a field K is called irreducible if
the vector space K n underlying the representation contains no non-trivial Γ-stable subspace. It
is called absolutely irreducible, if ρ ⊗K K is irreducible. The latter is equivalent to ρ ⊗K L being
irreducible for all finite field extensions K → L, or for all field extensions K → L.
Lemma 4.1. Let ρ0 : π1 (X) → GLn (Q` ) be absolutely irreducible. Then:0−1(a) There exists α ∈ Z×
` such that det(ρ ⊗ ρα ) has finite order for ρα defined as in (2).

(b) If all αx := det(ρ0 (Frobx )) lie in Q and are plain of characteristic p, then α from (a) has
this property. In this case ρα = ρα,• ⊗Lµ Q` where L = Q(α) and µ ∈ PL0 is the placedefined from Q(α) ,→ Q ,→ Q` . If furthermore all αx are #κx -Weil numbers of weight w,
then α is q-Weil number of weight w.
(c) Suppose that all αx are plain of characteristic p. Then through any finite set of points of
Xreg there is a smooth geometrically irreducible curve C ⊂ Xreg , a number field E 0 and an
E 0 -rational compatible system ρ0• for π1 (C) such that(i) each ρ0λ is absolutely irreducible, and
0
0
0
Q` ∼
(ii) for some λ0 ∈ PE
= ρ0 |π1 (C) .
0 above ` and embedding Eλ → Q` one has ρλ ⊗E 0
0
0
λ
0

0
(d) Suppose that ρ• is E-rational compatible system of π1 (X) such that for some λ0 ∈ PE
0
above ` and embedding Eλ0 → Q` one has ρλ0 ⊗Eλ0 Q` ∼= ρ . Let α be as in (a). Then:(i) ρλ is absolutely irreducible for all λ ∈ PE .

18

(ii) α ∈ Q, and, assuming α ∈ E, ρ• ⊗ ρ−1α,• has finite order determinant and is pure ofgeo
weight 0 when restricted to π1 (Xreg ), and one has Gρ• |π1geo (X),λ = Gρ• ⊗ρ−1(X),λ .
α,• |π1

×Proof. For (a) we apply Lemma 3.17(a) to find β ∈ Z` such that (det ρ0 ) ⊗ ρ−1
β has finite image,where ρβ is the representation π1 (X) → GL1 (Q` ) that factors via Γκ and sends Frobκ to β. Let−10
α be an n-th root of β. Then (a) is clear since det(ρ0 ⊗ ρ−1α ) = (det ρ ) ⊗ ρβ . Part (b) follows
now from Lemma 3.17(b) and (c).
00
To prove (c), let ρ00 := ρ0 ⊗ ρ−1
α for α from (b). Clearly ρ is absolutely irreducible and by(a) it’s determinant has finite order. Note also that by the Čebotarov density theorem ρ00 andρ00 |π1 (Xreg ) have the same image. Let H := Ker ρ00 . From the compactness of π1 (X) it follows
that π1 (X)/H = ρ00 (π1 (X)) contains an open pro-` subgroup; see for instance [KLR05, page 1].
By Theorem 3.32(a) we can find a smooth geometrically irreducible curve C on Xreg , throughany finite number of points on Xreg , such that under the natural map π1 (C) → π1 (X) we have
ρ00 (π1 (X)) = ρ00 (π1 (C)). From Theorem 3.30 by L. Lafforgue, we see that ρ00 |π1 (C) is a memberof a compatible system ρ00• (which amounts to (ii) for ρ00 ) and such that all ρ00λ are absolutely
irreducible. By taking for E 0 a sufficiently large number field, we may assume that ρ00• is E 0 rational and L ⊂ E 0 for L from (b). Part (c) now follow by defining ρ0• := ρ00• ⊗ ρα,• .

Finally, we prove (d). We apply (c) to ρλ0 ⊗Eλ0 Q` to obtain a curve C on Xreg and an E 0 compatible system ρ0• for π1 (C) with the properties in (c). Let us replace both E and E 0 by
EE 0 and call this field E again. Observe that now ρ• |π1 (C) ∼= ρ0• by Lemma 3.1. Hence (d)(i)−1
is implied by (c)(i). The assertion on det(ρ• ⊗ ρα,• ) follows from (a) and the classification in
Corollary 3.19. The purity of ρ• ⊗ ρ−1α,• of weight 0 on π1 (Xreg ) follows by varying the curves Cin (c) and using that all ρ0• ⊗ ρ−1|α,• π1 (C) are pure of weight zero from Theorem 3.30. The lastassertion is obvious since ρα,• |π1geo (X) is trivial.
0, orDefinition 4.2. We call an E-compatible system ρ• absolutely irreducible if for all λ ∈ PE0equivalently, by Lemma 4.1(c), for one λ ∈ PE , the representation ρλ is absolutely irreducible.
L
We call ρ• absolutely completely reducible if it is isomorphic to the (finite) direct sum i ρi,• ofabsolutely irreducible compatible systems ρi,• .

The following result is well-known for single representations.
Proposition 4.3. For every compatible system ρ• there exists a finite extension E 0 of E suchthat ρ• ⊗E E 0 is absolutely completely reducible.
00
Proof. Fix a place λ ∈ PE
and choose a finite extension E 00 /E and a place λ00 ∈ PE
00 above λ
00 00such that (ρ• ⊗E E )λ is a direct sum ⊕i∈I ρi,λ00 of absolutely irreducible representations. By
Lemma 4.1(a), (b) there exists a finite extension E 0 /E 00 such that ρi,λ00 ⊗Eλ00 Eλ0 is a member
000
of an E 0 -rational absolutely irreducible compatible system ρi,• for some λ0 ∈ PE(for
0 above λ00
all i we can arrange for the same λ ). Then ρ• ⊗E E and ⊕i∈I ρi,• are isomorphic compatible
systems by Lemma 3.1, and, by construction, ⊕i∈I ρi,• is absolutely completely reducible.

In order to investigate the geometric monodromy, the following two results will be useful:Lemma 4.4. Let ρ• be absolutely irreducible and connected. Then the following are equivalent:19(a) det ρ• has finite order,(b) det ρ• is trivial,
0(c) Gρ• ,λ = Gρ• |πgeo (X) ,λ for all λ ∈ PE,
10(d) Gρ• ,λ is semisimple for all λ ∈ PE,
0(e) Gρ• ,λ has finite center for all λ ∈ PE.

Proof. Since by the absolute irreducibility of the ρλ and the connectivity of ρ• the group Gλ0
is connected reductive for all λ ∈ PE
, the equivalence (d)⇐⇒(e) is clear. The equivalence(c)⇐⇒(d) follows from Lemma 3.9. Also, if ρ• is connected then so is det ρ• , and this proves(a)⇐⇒(b). By (d) and Galois cohomology, the maximal abelian quotient of the abstract group
Gλ (Eλ ) is finite, and this implies (a). Finally, because ρ• is absolutely irreducible, the theoremof Burnside shows that the Eλ -linear hull of ρλ (π1 (X)) is Mn×n (Eλ ), and hence the center of
Gλ is equal to the center of GLn intersected with Gλ . This gives (b)⇒(e).
Proposition 4.5. Suppose ρ• is connected and a direct sum of absolutely irreducible compatible
systems ρi,• each of which has finite order determinant. Then for all λ ∈ PE the group Gρ• ,λ issemisimple connected, and one has an equality of monodromy groups Gρ• ,λ = Gρ• |πgeo (X) ,λ .
1

Proof. By our hypotheses the Gλ are connected reductive. Since Gλ surjects onto each Gρi,• ,λ ,the latter are connected as well.7 By Lemma
Q 4.4 the image of the center Zλ of Gλ in Gρi,• ,λ isfinite for all i. Since Gλ is a subgroup of i Gρi,• ,λ , we deduce that Zλ is finite, and thus Gρ• ,λis semisimple. The asserted equality of monodromy groups now follows from Lemma 3.9.

4.2

Chin’s theorem in the non-pure caseIn this subsection we shall prove that the purity assumption in Theorem 3.25 is unnecessary.
Below we write diag(a1 , · · · , at ) for the diagonal matrix in GLt with diagonal entries a1 , · · · , at .
We shall also use this notation when P
a1 ,. . . ,at are themselves diagonal matrices with ai of size ri
to obtain a matrix in GLr with r = i ri . For the following result, recall the notion of a (split)
motivic triple from Definition 3.22.
Proposition 4.6. Let ρ• be an E-rational semisimple compatible system. Then there exists a
split motivic triple (F, M, α) for ρ• , and it satisfies (5) and (6) of Theorem 3.25.
Proof. By Lemma 3.27(a) and (c), in the following we may and will (multiple times) replace X
by a connected finite étale cover and E by a finite extension.
After replacing E by a finite extension we may assume that ρ• is absolutely completely reducible(cf. Proposition 4.3) and that there exists a 1-dimensional E-rational compatible system δi,•and an absolutely irreducible E-rational compatible systems σi,• pure of weight zero and withfinite order determinant such that
ρ• =

t
M

σi,• ⊗ δi,•i=1
7 The converse is not true as can be seen by considering SL (K) × {±1} → SL (K) × SL (K), (g, ε) 7→ (g, g · ε),222
where we regard {±1} as the center of SL2 (K) for a field K of characteristic zero.

20

(cf. Lemma 4.1). After replacing X by a finite étale cover we can assume that all the compatible
systems involved so far are connected and that there exists for every i an element βi ∈LE × suchtthat δi,• = ρβi ,• . In particular the system δi,• is trivial on π1geo (X). We define σ• := i=1 σi,• ,geo
geo
so that σ• is pure of weight 0 and ρ• |π1geo (X) = σ• |π1geo (X) . Denote by (F, M , α ) a split
motivic triple for σ• that exists by Theorem 3.25. Note that for every λ ∈ PE0 the group Gσ• ,λis semisimple and we have Gσ• ,λ = Gσ• |πgeo (X) ,λ by Proposition 4.5. If ri is the rank of σi,• and1Pt
r = i=1 ri , then it is also clear that αgeo (M geo ) ⊂ GLr1 ,F × · · · × GLrt ,F . After enlarging E
again we can also assume E = F .
0iDefine δ•0 = ⊕ti=1 ρ⊕rβi ,• . Then δ• is a compatible system of rank r with values in the centerof GLr1 ,F × · · · × GLrt ,F and ρλ (g) = σλ (g)δλ0 (g) for every g ∈ π1 (X). By possibly replacing
X again by a finite cover X 0 → X (see Theorem 3.13), we can assume that the groups Gδ•0 ,λ ,0
λ ∈ PE, are all connected. Next we construct a split motivic triple for δ•0 : Let

Bi = diag(βi , · · · , βi ),| {z }
ri times

B = diag(B1 , · · · , Bt ) ∈ GLr (E), and let hBi be the cyclic group of the integral powers of B.
Note that B is a diagonal matrix which lies in the center of GLr1 ×· · ·×GLrt . Let Z be the Zariski
closure of hBi in GLr,E . Let Zλ ⊂ GLr,Eλ be the Zariski closure of δλ0 (π1 (X)). Because hBi isdense in δλ0 (π1 (X)) in the profinite topology, using Lemma 3.3 it follows that Zλ ∼= Z ⊗E Eλ .
Since the groups Gδ•0 ,λ = Zλ are connected, they are tori, and hence Z is a torus. This showsthat (E, Z, incl) is a split motivic triple for δ•0 .
We now show that Gρ• ,λ = Gσ• ,λ Gδ•0 ,λ . Clearly the left side is contained in the right side. To
show the opposite inclusion, we claim that some open subgroup of δλ0 (π1 (X)) lies in ρλ (π1 (X)).
By Corollary 3.11, there exists a connected finite étale cover X 0 → X, a pure constant field
extension that depends on λ, such that σλ (π1 (X 0 )) = σλ (π1geo (X 0 )). Let d be the degree of
X 0 → X. Choose a section s : Γκ → π1 (X) of the canonical map p : π1 (X) → Γκ andchoose g ∈ π1 (X) such that p(g) is a topological generator of Γκ . Then g d ∈ π1 (X 0 ) and there
exists h ∈ π1geo (X 0 ) such that σλ (h) = σλ (g d ). We define a := g d h−1 . Then σλ (a) = 1 andp(a) (topologically) generates an open subgroup of Γκ , and thus ρλ (a) = δλ0 (a) (topologically)
generates an open subgroup of δλ0 (π1 (X)). This finishes the proof of the claim.
Because Gδ•0 ,λ is connected, it follows, that the groups ρλ (π1 (X)) and σλ (π1 (X))δλ0 (π1 (X)) havethe same Zariski closure, i.e., thatGρ• ,λ = Gσ• ,λ Gδ•0 ,λ ,where Gσ• ,λ is semisimple and Gδ•0 ,λ is a torus commuting with Gσ• ,λ .
The torus Z commutes with the semisimple algebraic group αgeo (M geo ) and thus we may define
M := αgeo (M geo ) · Z
inside GLr,E . It follows that (E, M, incl) is a split motivic triple for ρ• , and that (5) and (6) of
Theorem 3.25 hold.
We have the following immediate consequence (of the above proposition and notion of Chin21

triple). 8
Corollary 4.7. Let ρ• be an E-rational compatible system. Then0(a) the root datum of the groups Gρ• ,λ is independent of λ ∈ PE,0(b) the formal character of the representation Gρ• ,λ ⊂ GLn,Eλ is independent of λ ∈ PE.

Corollary 4.8. The statement of Lemma 3.27(b) remains true without the purity assumption.
Proposition 4.9. Suppose ρ• is E-rational connected, and let (F, M, α) be a split motivic triplefor ρ• . Then α is irreducible if and only if ρ• ⊗E F is absolutely irreducible.
Proof. It is clear that if ρ• ⊗E F is absolutely irreducible, then α is irreducible. For the converse,
suppose that α is irreducible. Let F 0 /F be a finite extension and µ0 ∈ PF0 0 . Now (F 0 , M ⊗F
F 0 , α ⊗F F 0 ) is a split motivic triple for ρ• ⊗E F 0 by Lemma 3.27(a). As M is connected split
reductive we can conclude that α ⊗F Fµ0 0 is irreducible (cf. Corollary 3.21). From Definition 3.22
we now see that the action of G◦ρ• ⊗E F 0 ,µ0 is irreducible, and hence (ρ• ⊗E F 0 )µ0 is irreducible, as
desired.

4.3

G-compatible systems

Let G be a reductive group over Q. Recall that a representation ρ : Γ → G(Q) is called Gcompletely reducible if whenever P ⊂ G is a parabolic subgroup with ρ(Γ) ⊂ P (Q), then there
exists a Levi subgroup L ⊂ P such that ρ(Γ) ⊂ L(Q). It is called G-completely irreducible if
there is no parabolic P ⊂ G such that ρ(Γ) ⊂ P (Q). For L ⊃ Q an algebraically closed field andg ∈ G(L), we write gs ∈ G(L) for the semisimple part of g in its Jordan decomposition g = gs gu .
Definition 4.10 ([BHKT16, Def. 6.1]). A G-compatible system of π1 (X) is a datum((ρµ )µ∈P 0 , [gx ]x∈|X| )Q

consisting of◦ a continuous G-completely reducible representation ρµ : π1 (X) → G(Qµ ) for every µ ∈ PQ0 ,◦ a semisimple G(Q)-conjugacy class [gx ] for every x ∈ |X|
such that for any x ∈ |X| and any µ ∈ PQ0 , we have[gx ] = [ρµ (Frobx )s ]as an equality of semisimple conjugacy classes in G(Qµ ).
We call homomorphisms ρ, ρ0 : π1 (X) → G(Q` ) equivalent, and write ρ ∼= ρ0 , if they are conjugate.
We call G-compatible systems ρ• , ρ0• equivalent, and write ρ• ∼= ρ0• , if we have ρµ ∼= ρ0µ for all0µ ∈ PQ .
8 [LP92] proves the first part for a density 1 set, Serre proves the second part in general; both for Q-rational
systems. See [LP92], the first two pages of the introduction.

22

0
The system ρ• is called E-rational if G is defined over E and if for all λ ∈ PE
there is a repre000sentation ρλ : π1 (X) → G(Eλ ), such that for all µ ∈ PQ one has ρµ = ρµ|E ⊗Eλ Qµ .

We say that a representation ρ : π1 (X) → G(Q` ) is a member of a G-compatible system ρ• , ifthere exists µ ∈ PQ0 , and an isomorphism ι : Qµ → Q` such that ρ ∼= ι ◦ ρµ .
We say that ρ• has Zariski dense image if ρµ (π1 (X)) ⊂ G(Qµ ) is Zariski dense for all µ ∈ PQ0 .
Remark 4.11. Since for GLn there is a bijection between semisimple conjugacy classes and monicpolynomials of degree n given by the characteristic polynomial (and defined over any field), GLn compatible systems are the compatible systems we consider in the rest of this work.
From now on, for the remainder of this section, we assume that X is a curve. The following two
results from [BHKT16] make strong use of results from [Chi04].
Theorem 4.12 ([BHKT16, Thm. 6.5, Prop. 6.6]). Suppose ρ : π1 (X) → G(Q` ) is a continuous
homomorphism with Zariski dense image for some prime ` 6= p and G is semisimple. Then:(a) The homomorphism ρ is a member of a G-compatible system ρ• .
(b) The system ρ• from (a) has Zariski dense image.
(c) The system ρ• from (a) is unique up to equivalence.
(d) Suppose ρ• is a G-compatible system with Zariski dense image. Then there exists a number
field E and an E-rational G-compatible system ρ0• that is equivalent to ρ• .
Corollary 4.13. Let ρ• be a GLn -compatible connected system with semisimple groups Gλ .
Then there exists a finite extension F of E such that ρ• ⊗E F has a motivic pair (M, α), andwith Zariski dense image, uniquesuch that there exists an F -rational M -compatible system ρM
•
∼
ρ
⊗
F
.
up to equivalence, such that α ◦ ρM=•
E•
Proof. By passing from E to a finite extension, we may assume that ρ• itself possesses a motivic
pair (M, α). Write MF and αF for M ⊗E F and α ×E F for any field extension F of E.
Choose λ ∈ PE . Then, by the definition of motivic pair, there is a continuous homomorphism
ρ0 : π1 (X) → M (Eλ ) with Zariski dense image, such that ρλ = αEλ ◦ρ0 . By Theorem 4.12 there isa finite extension F of E and an F -rational MF -compatible system ρ0• , which contains ρλ ×Eλ Fµ
as a member for some µ ∈ PF0 above λ. Then αF ◦ ρ0• is an F -rational GLn -compatible system(with semisimple conjugacy classes [α(gx )]x∈|X| ). From αF ◦ ρ0µ ∼= ρλ ×Eλ Fµ and Lemma 3.1,
the corollary follows.
Recall that a representation β : G → GLn of a reductive connected group G is called almost
faithful if Ker β is a finite subgroup scheme.
Corollary 4.14. Let ρ• be a GLn -compatible connected system with split motivic triple (F, M, α)such that M is semisimple, and suppose that ρMfrom Corollary 4.13 is defined over F . Then•the following hold:(a) For any representation β of M (defined over F ), we have an F -rational compatible connected system ρβ• := β ◦ ρM• .
i(b) If α = ⊕i αi , then ρ• = ⊕i ρα• . In particular, ρ• is absolutely completely reducible.

23

(c) If ϕ : M → M 0 is a surjective homomorphism, then ϕ ◦ ρM
is an M 0 -compatible system•0with Zariski dense image, and if α is a faithful representation of M 0 , then (F, M 0 , α0 ) is a0
motivic triple of the compatible system ρ0• := α0 ◦ ϕ ◦ ρMand ρ0,M= ϕ ◦ ρM••• .
(d) There exist ϕ, α0 in (c) such that ϕ is a central isogeny and α0 is faithful and irreducible(and so α0 ◦ϕ is almost faithful), and then the compatible system ρ0• is absolutely irreducible.
Q(e) Suppose that M ∼= i Mi for simple groups Mi and with projections pri : M → Mi (which
always exists
Then ρM
is isomorphic to the•
Q if MMis simply connected or of adjoint type).
M
product i (pri ◦ρ• ) of the Mi -compatible systems pri ◦ρ• .
N(f ) Suppose that M is as in (e) and α is irreducible. Then α = i αi for faithful irreducible
representations αi of Mi (considered as representations of M via pri ), the systems ρi,• :=M
αNi ◦ pri ◦ρ• are absolutely irreducible with motivic triple (F, Mi , αi ), and we have ρ• =i ρi,• .
It is sometimes useful to depict (d) in the following diagram, where d = dim α0 :/4 GLn (Fµ )ρµπ1 (X)ρMµ/ M (Fµ )α⊗F Fµα0 ⊗F Fµ*
ρ0µ

/ GLd (Fµ ),Proof. Part (a) is immediate from Corollary 4.13. The first part of (b) is clear from the definitions
αii
areof the ρα• . For the second part, let the αi be irreducible summands of α. Then the ρ•
absolutely irreducible by Proposition 4.9. Part (c) is straightforward from the previous results.
For the construction of α0 and ϕ in (d), see Remark 4.15. Part (e) is obvious. The assertionin (f) is a well-known fact in the representation theory of reductive groups. For some further
details see Lemma 5.37(c).
Remark 4.15. Either of the following two constructions show that in Corollary 4.14(d), one may
realize ρ0• as a direct summand of compatible system constructed out of ρ• .
(a) Q
Let ϕ be the central isogeny M → M ad to the adjoint quotient of M and write M ad =0i∈I Mi with Mi simple. Let α be the representation ⊗i∈I AdMi . Because each AdMi is0irreducible, so is α . Now α defines a homomorphism M → GLd for d = dim α, and hencea monomorphism AdM ,→ AdGLd ∼= α ⊗ α∨ of M -representations. It follows that α0 occurs∨ ⊗#I
as a direct summand of (α ⊗ α ).
L⊗#I(b) Write α as a sum
contains ai∈I αi with
N irreducible representations αi . Then αsummand isomorphic to γ := i∈I αi . Let β be an irreducible summand of γ whose highest
weight is the sum of the highest weights of the αi ; cf. Theorem 3.20. The restriction of β tothe simply connected cover of M is of the form ⊗j βj corresponding to the simple factors ofthat cover, cf. Lemma 5.37(c), and it can easily be verified that the highest weight of eachβj is non-trivial. Hence each βj , and hence also β are almost faithful. Take now ϕ as the
canonical map M → M 0 := M/ Ker(β) and α0 as the representation on M 0 induced from β.

24Remark 4.16. Suppose the split motivic group M of a connected compatible system is of adjoint
type. If M is also simple, then we shall prove in Section 8, using [Pin98] by Pink, that there
is a natural field of definition Eλ0 ⊂ Eλ for each ρλ over which ρλ has open image. If M is not
simple, this cannot be expected. Take for instance representations ρi , i = 1, 2, with open image
in SL2 (Z` ) and SL2 (Z`2 ), respectively, and form their tensor product.

5

Saturation

This section begins by recalling the foundational correspondence of Nori between exponentiallygenerated subgroups and nilpotently generated Lie algebras. The second subsection concerns themore general notion of saturation introduced by Serre. The literature seems to focus mainly on
algebraically closed base fields while our main interest is in finite base fields. The adaptations areminor, and we also investigate the relation between being exponentially generated and between
being saturated. One more aspect of saturation is that it can be developed within general
reductive groups and not just within GLn , which was Nori’s original setting. Our treatment
focusses on the GLn case but in Subsection 5.3 we also present some results in the general case,useful when combining Weil restriction and saturation. Subsection 5.4 provides a result on liftinga mod `-representation of small `-height of a saturated group to a representation over the Witt
vectors of a lift of that group. In Subsection 5.5 we show for a new class of groups that they are
saturated. For k = Fp our findings can be deduced from results in [CHT17, § 7]. Our motivation
for this class of groups stems from some results of Larsen-Pink. Their results attach smoothgroup schemes over Z` to the monodromy at ` of a Q-rational compatible systems, if ` is large.
In the last subsection, Subsection 5.6, we first extend their results to E-rational systems, andthen we show that the smooth reductions of these subschemes of GLn are saturated for `λ  0.
We remind the reader once more that from Section 4 on all compatible systems are semisimple.

5.1

Reminders from [Nor87, Sect. 2]

We fix a natural number M ≥ 3. By ` we denote a prime and by n a natural number such thatn ≤ M < `. We let k be a field of characteristic ` with an algebraic closure k ⊃ k. As in [Nor87,
Sec. 1], we define
`
nil
`
GLuni
n (k) := {u ∈ GLn (k) | u = 1} and Mn (k) := {X ∈ Mn (k) | X = 0},

and further
expn : Mnnil (k) → GLunin (k), X 7→

`−1
X
1

i!
i=0

nil
X i , logn : GLunin (k) → Mn (k), u 7→ −

`−1
X
1

i
i=1

(1 − u)i . (7)Then expn and logn are well-defined mutually inverse bijections. For X ∈ Mnnil (k) let fX : Ga →GLn denote the morphism t 7→ expn (tX) of algebraic groups; it is defined over k.
Definition 5.1 ([Nor87, § 1]). A Lie subalgebra L of Mn (k) is called nilpotently generated if L
is the k-span of L ∩ Mnnil (k).

25

If L is nilpotently generated, the algebraic subgroup of GLn,k generated by the one-parameter
subgroups fX for all X ∈ L ∩ Mnnil (k) will be denoted by expn L. It is a geometrically connected
subgroup scheme.
If A is a closed subgroup-scheme of GLn,k , one defines hlogn A(k)i as the F` -span of {logn u |nilu ∈ A(k) ∩ GLunin (k)} ⊂ Mn (k). It is an F` -sub Lie algebra of Mn (k).
Definition 5.2 ([Nor87, Def. 2.3, Rem. 2.14]). Let L be a k-Lie subalgebra of Mn (k) and let
A be a closed subgroup-scheme of GLn,k . Then (L, A) is an acceptable pair if and only if A issmooth and geometrically connected, Lie A = L, and for all X ∈ Mnnil (k) one has
X ∈ L ⊗k k ⇐⇒ expn (X) ∈ A(k).
One calls L acceptable if there exists A so that (L, A) is an acceptable pair.9
Definition 5.3 ([Nor87, Def. 2.4]). A closed subgroup-scheme A of GLn,k is exponentiallygenerated if there exists S ⊂ Mnnil (k) such that A is generated by the groups fX for all X ∈ S.10
Theorem 5.4 ([Nor87, Thm. A]). There is a natural number c1 (M ) ≥ 2M − 1 so that whenever
k is a field of characteristic ` > c1 (M ), and if 1 ≤ n ≤ M , then (1) and (2) below are true:(1) If L is a nilpotently generated Lie subalgebra of Mn (k), then (L, expn L) is an acceptablepair, and in particular L = Lie(expn L).
(2) If A ⊂ GLn,k is exponentially generated, then A is smooth and Lie A = hlogn A(k)i andfurthermore A = expn (Lie A).
Note that if A is exponentially generated, then by (2) the F` Lie subalgebra hlogn A(k)i ofMn (k) is in fact a k Lie subalgebra. By definition it is nilpotently generated, and thus from A =expn (Lie A) and Lie A = hlogn A(k)i it follows that (Lie A, A) is an acceptable pair. As stressedin [Nor87], the above theorem sets up a bijection L → exp L between nilpotently generated Lie
subalgebras of Mn (k) and exponentially generated subgroup schemes of GLn,k .

5.2

Basics on k-saturation

Let 1 ≤ n ≤ M and k be as in the previous subsection. We further assume that ` > c1 (M ) ≥2M − 1. This subsection contains a detailed treatment of some aspects of saturation over perfectfields, building on [Ser94b, Ser98, Sei00, McN02, Ser05, Del14, BDP16], that mostly focus onk = k. Throughout this subsection we assume that k is perfect and we let G ⊂ GLn,k be a closed
smooth k-subgroup such that Go /Ru (G) is quasi-split, i.e., G contains closed subgroup B suchthat the base change Bk of B to k is a Borel subgroup of Gk . The latter condition seems natural
since the groups we are interested in should have an ample supply of unipotent subgroups. But
also, our main case of interest is that of finite fields, and, by a result of Lang, Go /Ru (G) is
quasi-split if k is finite; see [Spr09, Exer. 16.2.9].
By Gu we denote the closed subscheme of G of unipotent elements. By the Jordan decompositionin linear algebraic groups, it is equal to GLn,u ∩ G. Since GLn,u is defined over k – it is defined
9 Nori develops this notion more generally over any commutative ring and not only over a field.
10 Thereis a related notion by Suslin, Friedlander and Bendel of subgroup schemes of exponential types,
cf. [SFB97] or [McN02].

26

by the closed condition that the characteristic polynomial be (T −1)n –, so is Gu . Because n < `,uni
we have u` = 1 or equivalently (u − 1)` = 0 for u ∈ GLunin (k). For u ∈ GLn (k) and t ∈ k, wedefine

`−1
X
t
i t
u :=
(u − 1)= expn (t logn u) ∈ GLn (k).
(8)
i
i=0
Definition 5.5 ([Ser94b, § 4.2]). (a) One calls G saturated (in GLn,k ) if for any u ∈ Gu (k)and t ∈ k, the element ut ∈ GLn (k) lies in G(k).
(b) One defines the saturation of G (over k) as the intersection of all closed saturated subgroupsof GLn,k that contain G, and denotes it by Gsat
k .
Lemma 5.6. The intersection of all saturated subgroups of GLn,k that contain G is defined overk and hence equal to Gsatk ⊗k k.
Proof. Let H ⊂ GLn,k be a saturated subgroup that contains G. Let Ak be the Hopf algebra ofGLn,k , Ak its base change to k, IH,k the Hopf ideal that defines H, and IG,k the Hopf ideal thatdefines the k-subgroup G. Note that k = k sep since k is perfect. Let σ ∈ Gal(k/k). Then σ(IH,k )is an ideal of Ak , and because the Hopf structure of Ak is defined over k, σIH,k is a Hopf ideal.11
The ideal σ(IH,k ) defines the subgroup σH ⊂ GLn,k . The group σH is saturated, because Gu−1

is defined over k and for u ∈ G(k) and t ∈ k one has σ(u)t = σ(uσ (t) ) ∈ σH.S The intersectionof the Gal(k/k)-orbit of H is defined by the ideal of Ak spanned by JH := σ∈Gal(k/k) σIH,k .
Since G is contained in H we have IG,k ⊗k k ⊂ JH . Moreover J
TH is a Hopf ideal, and by itsvery definition JH is invariant under Gal(k/k). It follows that σ∈Gal(k/k) σ(H) is saturated,
contained in H, defined over k and contains G. The lemma follows; cf. [Bor91, I.14].
00 ∼Corollary 5.7. For any algebraic extension k → k 0 , the base change (Gsatk )⊗k k ⊂ GLn,k ⊗k k =0GLn,k0 is the saturation of G ⊗k k in GLn,k0 .

Proof. By the previous lemma, the group Gsat
k is the intersection of all saturated subgroups H
of GLn,k that contain G – because this intersection is defined over k. But the same intersection
also defines (G ⊗k k 0 )sat
k0 . Hence the result is clear.
Corollary 5.8. Suppose H is a closed subgroup of GLn,k such that H ⊗k k 0 is saturated forsome finite extension k → k 0 . Then H is saturated.
0∼ sat
Proof. Consider H ⊂ Hksat . By base change this becomes H ⊗k k 0 ∼= (H ⊗k k 0 )sat
k 0 = Hk ⊗k k .
sat∼
But then we must have H = Hk .

The following result and its proof are inspired by [Ser94b], and in particular § 4 of op.cit.
Proposition 5.9. Define ⊗0 : GLn1 ,k × GLn2 ,k → GLn,k , (A, B) 7→ A ⊗ B for ni ∈ N suchthat n = n1 n2 . Let Γ be a finite group with homomorphisms ri : Γ → GLni ,k , i = 1, 2, and set
r = ⊗0 (r1 × r2 ) : Γ → GLn . Let Gi ⊂ GLni ,k be closed saturated subgroups for i = 1, 2. Then
11 A Hopf ideal of A is an ideal I such that ∆(I) ⊂ A ⊗ I + I ⊗ A for the comultiplication ∆ : A → A ⊗ A, suchthat S(I) ⊂ I for the inversion S : A → A, and such that ε(I) = 0 for the counit ε : A → k.

27

(a) ⊗0 (G1 × G2 ) is saturated in GLn,k .
0sat(b) If Gi contains ri (Γ), i = 1, 2, and r(Γ)satk = ⊗ (G1 × G2 ), then ri (Γ)k = Gi for i = 1, 2.

Proof. Note first that ⊗0 (G1 × {1}) and ⊗0 ({1} × G2 ) are commuting subgroups in GLn . Hence⊗0 (G1 ×G2 ) is a subgroup of GLn,k . It is easy to see that it is closed. Part (a) now follows from thefollowing two simple facts whose proof we leave to the reader: (i) If u = u1 ⊗ u2 ∈ ⊗0 (G1 × G2 ) ⊂GLn is unipotent in GLn , then so are ui ∈ GLni ; (ii) one has expn (A ⊗ 1n2 ) = expn1 (A) ⊗ 1n2(k) and B ∈ Mnniland expn (1n1 ⊗ B) = 1n1 ⊗ expn2 (B) for A ∈ Mnnil(k).
1
2
To see (b), note that by (a) and our hypotheses we have inclusions
0
sat
sat
0
⊗0 (G1 × G2 ) = r(Γ)satk ⊂ ⊗ (r1 (Γ)k × r2 (Γ)k ) ⊂ ⊗ (G1 × G2 ).

It follows that we have equality everywhere, and (b) follows from the injectivity of ⊗0 .
e for Ru (G)Gder  G and G+ for the subgroup of G generated by Gu ;In the following we write G
recall that by our convention Gder = (Go )der . As observed above, Gu is defined over k, and,e and G+ are defined
because k is perfect, Ru (G) is defined over k (see. [Spr09, 12.1.7]). Thus Ge and G+ .
over k. We now clarify some further properties of G
Lemma 5.10.

e contains a Borel subgroup B
e defined over k.
(a) The group Ge as in (a). Then G
e is generated by the subgroup schemes(b) Let T be a maximal torus of a B
e defined over k and containing T .12
Ru (B), where B ranges over the Borel subgroups of Ge and defined over the perfect field k,
Proof. Since Ru (G) is closed normal connected unipotent in G
e → G/Re u (G) is a kand since the inverse image of a k-Borel subgroup under the canonical map G
e u (G)
Borel subgroup, it suffices to prove the lemma for the connected semisimple group H = G/Rewhich satisfies H = H. By hypothesis H is quasi-split, and hence (a) is clear.
e be a k-Borel subgroup of H and T ⊂ B
e a maximal k-torus. Let H 0 be the closed
Let B(reduced) smooth subgroup scheme of H generated by the unipotent subgroups Ru (B) where B
runs through all k-Borel subgroups of H containing T . By [Con11, 5.2.12], there exists a uniquee ∩ B 0 = T , and over k the group B 0 is the opposite Borel
k-Borel subgroup B 0 of H such that B
0e It follows that Lie H ⊗k k contains all subspaces of Lie H ⊗k k on which T acts
subgroup to B.
non-trivially. In particular, it implies that H 0 (k) contains the root subgroups of H(k). Since H
is semisimple, we deduce H 0 = H which proves (b).
e is normal in G. It is contained in G+ .
(a) The group G
e and Go /G
e commute in G/G.
e(b) The groups G+ /GCorollary 5.11.

e → G/Go is well-defined, its image is the subgroup generated by(c) The homomorphism G+ /G
e u → (G/Go )u .
all element of G/Go of order `, and it sets up a bijection (G+ /G)
12 The result only needs ` > max{3, m + 1 | Go /R (G) has a quotient of type A }.
um28

Proof. To see (a) note that Ru (G) and Gder are characteristic subgroups of G and hence so
e = Ru (G)Gder . The assertion G
e ⊆ G+ follows from Lemma 5.10. To prove (b), observe
is G
ee It will
first that G/G is an extension of the finite group G/Go by the torus T := Go /G.
+ esuffice to show that any element ū of G /G acts trivially on T . If not, passing to k we geta non-trivial homomorphisms Z/(`) → Aut(T k ) ∼= GLm (Z) for m = rank T ≤ n. The imageof a generator of Z/(`) is a matrix A ∈ GLm (Z) of exact order `. The cyclotomic polynomialΦ` = (x` − 1)/(x − 1) is then irreducible and by the Cayley-Hamilton Theorem it must divide thecharacteristic polynomial of A. This implies ` − 1 ≤ m ≤ n ≤ M , contradicting our hypothesis` ≥ 2M − 1(≥ M + 2) since M ≥ 3. This completes (b) and shows the first part of (c).
For the second part of (c) recall first that under homomorphisms of algebraic groups unipotent
elements are mapped again to such (see [Spr09, Thm. 2.4.8]). To complete the second assertionof (c), let ū ∈ G/Go be unipotent, i.e., of order `m for some m ≥ 1. Let u ∈ G be a preimage,and split u = us uu in its (commuting) unipotent and semisimple parts. We have u`u = 1 because` > n and the image of us is semisimple and hence of order prime to `. It follows that ū is theimage of uu , and that its order is ` (cf. [Ser98, Lect. 3, Prop. 3]).
e u → (G/Go )u is injective. So suppose that u, u0 are elements
It now remains to show that (G+ /G)+ e
in (G /G)u that have the same image in (G/Go )u . It follows that there exists t in the torus
e such that u0 = ut. Clearly the elements of the torus are semisimple, and we deduce u = u0
Go /Gfrom the uniqueness in the Jordan decomposition of linear algebraic groups.
e → G/Go is always finite, or if (G+ ∩Go )/Ge
Question 5.12. We do not know if the kernel of G+ /G
can contain a torus.
Lemma 5.13. (a) If T is a torus, ϕ : G → T is a homomorphism of algebraic groups, then G
is saturated if and only if Ker ϕ is so.
(b) G is saturated if and only if Go is saturated and G/Go has order prime to `; cf. [Ser94b,
Prop. 11].
e is saturated and G
e = G+ .
(c) G is saturated, if and only if Ge(d) If G ⊂ GLn,k is exponentially generated, then G is saturated and G = G+ = G.
Proof. Concerning (a) and (b) let us first show that if G is saturated, then so are Ker ϕ and
Go . This follows from the fact that any homomorphism from Ga to a finite group scheme or amultiplicative group scheme is trivial. In (b) note also that the subgroup of G/Go generated byelements of order ` coincides with the image of G+ by Corollary 5.11(c). However because G
is saturated, every element of G+ is contained in a 1-parameter subgroup and hence G+ ⊂ Go
which implies that G/Go is of order prime to `.
For the ‘if part’ of (a), suppose that Ker ϕ is saturated. Now note that the inverse image ofT r {1T } under ϕ is disjoint from Gu , and hence G is saturated if and only if Ker ϕ is saturated.
For the ‘if part’ of (b) note that if G/Go is of order prime to `, then by Corollary 5.11(c), we
have G+ ⊂ Go , and hence Gu ⊂ Go , so that G is saturated if and only if Go is so.
We now prove (c) and begin with the only if part. If G is saturated, then by (b) G+ ⊆ Go ande
Go is saturated. Let ψ : Go → T be the maximal torus quotient of Go . It is easy to see that G
e is saturated by (a), and G+ ⊂ G
e because G+ maps to the identity
is the kernel of ψ. Hence G
e ⊆ G+ by Corollary 5.11, we obtain G
e = G+ .
under ψ. Since we also have G
29

e = G+ and that G
e is saturated. The first condition together
For the if part of (c) assume that G
with part(b) and Corollary 5.11(c) shows that it suffices to prove that Go is saturated. For this
we apply the ‘if part’ of (a), which allows us to deduce the saturatedness of Go from that ofe = Ker ψ with ψ : Go → T as in the previous paragraph.
G
The first assertion of (d), i.e., that G is saturated, follows from ` > c1 (M ), Theorem 5.4 andDefinitions 5.3 and 5.2: Let u be in Gu (k). Then logn u ∈ Mnnil (k), expn (logn u) = u, and by
Definition 5.2(c), we have logn u ∈ Lie G ⊗k k. But then t logn u ∈ Lie G ⊗k k for all t ∈ k,e now
and again by Definition 5.2(c) we have ut = expn (t logn u) ∈ G(k). The equality G+ = G
follows from (c), and G = G+ holds because G is exponentially generated.
e = G+ and G
e is exponentially generated.
Proposition 5.14. G is saturated if and only if G
e is saturated, and we need to show
Proof. By Lemma 5.13(c) and (d), we may assume that Gethat it is exponentially generated. By Lemma 5.10, G is generated by its subgroup schemes
e defined over k. Let now U = Ru (B) forRu (B), where B ranges over the Borel subgroups of G
such a B. It suffices to show that any such U is exponentially generated.
Because k is perfect, the connected unipotent group U is k-split, and hence trigonalizable inside
GLn,k , see [Spr09, Ex. 12.3.5(3), Rem. 14.1.5, Thm. 14.1.4]. This means that there existsg ∈ GLn (k) such that gU g −1 is a closed subgroup of the subscheme Un,k ⊂ GLn,k of upperetriangular unipotent matrices. Thus below we shall assume U = Un,k ∩ G.
(c)

For 1 ≤ c ≤ n define Un,k as the normal closed subscheme of Un,k of elements u such thatthe matrix (ai,j )i,j∈{1,...,n} := u − 1 satisfies ai,j = 0 for all i ≥ j − c. Clearly the quotient
(c)

(c+1)

Un,k /Un,k

is isomorphic to Gca . From [Spr09, Thm. 14.2.6] one deduces an isomorphisms
U∼
=

n
Y

(U ∩ Un,k )/(U ∩ Un,k ) ∼
=
(c)

(c+1)

n
Y

GdaU,c

c=1

c=1

d
(c)
(c+1) ∼
of k-schemes with dU,c defined by any isomorphism (U ∩ Un,k )/(U ∩ Un,k ) =GaU,c . It follows
that the k-span of hlogn U (k)i ⊂ Lie Un (k) has dimension at least dim U . If we take a basis(Xi )i∈I of this k-span it follows that the subgroup U 0 of Un,k generated by the fXi , i ∈ I, hase these subgroups lie in G,e anddimension at least #I ≥ dim U . By the saturation property of G
0hence for dimension reasons U = U . This proves that U is exponentially generated.

Remark 5.15. Suppose k is finite and G ⊂ GLn,k satisfies G = G+ . This holds for instance if G
is a finite group generated by unipotent elements. By Proposition 5.14 the group Gsat
k coincides
with what is often called the Nori envelope of G ⊂ GLn,k , i.e. the smallest exponentially closed
subgroup that contains G (which only exists if G is generated by Gu , i.e., if G = G+ ).
Corollary 5.16. Suppose H is a normal subgroup of G. Then(a) The group G normalizes Hksat , and Hksat is a normal subgroup of Hksat G.
(b) The group G+ is a normal subgroup of G, and Gsat = (G+ )sat
k G.

30

Proof. For (a) note that by the normality of H in G, the adjoint action of G on GLn permutes thesaturated subschemes of GLn,k that contain H, and so by Lemma 5.6, the group G normalizes
H sat . The second part of (a) is immediate from the first.
For the first part of (b) note that the adjoint action of G on GLn,k clearly preserves Gu and thus
G+ , and hence G+ is normal in G. Regarding the second part of (b), observe that ⊇ is clear
from the definitions. To show ⊆ it thus suffices to show that (G+ )satk G is saturated. Now fromthe first paragraph it follows that (G+ )sat
isanormalsubgroupof(G+ )satkk G. Hence we obtaincanonical epimorphisms
ϕ

ψ

+ sat
+ sat ∼+ sat+(G+ )satk G −→ Ḡ := (G )k G/(G )k = G/((G )k ∩ G) ←− G/G .

From ψ, the definition of G+ and Corollary 5.11(c), it follows that Ḡo is a torus and Ḡ/Ḡo isof order prime to `. From Lemma 5.13(a) and (b) applied to ϕ, and from the saturatedness of+ sat(G+ )satk we deduce that (G )k G is saturated.
Corollary 5.17. The group (G+ )sat
is connected. In particular, if G is connected, then so is
k
sat
Gsat
k , and if G has no torus quotient, then neither does Gk .
Proof. Suppose first that G = G+ ; i.e., Go /Ru (G) is connected semisimple and G/Go is a
finite and generated by the elements of (G/Go )u which have order dividing ` (see the proof ofCorollary 5.11). It will suffice to show that Gsat
k is exponentially generated.
In a first step, we choose for any unipotent element ū ∈ G/Go (k) a unipotent element u ∈ G(k)mapping to it; see the proof of Corollary 5.11(c). Then Gsat has to contain the 1-parameter subgroups t 7→ ut for any such t. The group H generated by these and G is invariant under Gal(k/k)
e and by construction H must be contained inand hence a k-group. Moreover H = H + = H,e
Gsat . Hence it suffices to show that Gsat is exponentially generated, assuming that G+ = G.
kke is generated by
For this we proceed as in the proof of Proposition 5.14. The group G = G
the groups UB = Ru (B) where B ranges over the Borel subgroups of G defined over k and−1
containing a fixed maximal torus T , and for a fixed UB let gB ∈ GLn (k) be such that gB UB gB
=
−1
−1
0
gB GgB ∩Un,k . Let UB be the subgroup of gB Un,k gB generated by UB together with the imagesof the fX for X ∈ hlogn UB (k)i. If UB0 = UB then as in the proof of Proposition 5.14 we deducethat UB and hence G are exponentially generated.
Suppose now that some UB0 properly contains UB , so that dim UB0 > dim UB . Let G0 be thesubgroup of GLn,k generated by the UB0 . Because the UB0 are connected unipotent, the group G0
e 0 = G0 . Moreover we have dim G0 > dim G. We conclude the proof
is connected and satisfies G
by induction on dim G.
+ sat
For the remaining assertions, note that by Corollary 5.16 we have Gsatk = (G )k G. By what we+ satjust proved (G )k is connected and has no non-trivial torus quotient; cf. Lemma 5.13(d). Thus
if G is connected or has no-non-trivial torus quotient, the same property will hold for Gsatk .

The following result is straightforward from [Ser94b, 4.3], [Ser98, Thm. 8] and Corollary 5.17.
We omit the proof.
Proposition 5.18 (Serre). Suppose the representation of G on V = k n afforded by G ⊆ GLn,kis a direct sum V = ⊕i Vi of absolutely irreducible representations. Then31

(a) Gsat
k preserves each Vi and acts absolutely irreducibly on it,o(b) (Gsatk ) is reductive,(c) if G = G+ , then Gsat
k is semisimple connected.
o
13
Conversely, if (Gsatk ) is reductive, then the action of G on V is semisimple.

Proposition 5.19. Suppose the representation of G on VQ= k n afforded by G ⊆ GLn,k is a direct
sum V = ⊕i Vi of representations, and denote by pri : i Autk (Vi ) → Autk (Vi ) the projectionsatonto the i-th factor. Then pri maps Gsat
k onto the saturation pri (G)k formed inside Autk (Vi ).
Q
Q
Proof. Observe first that i Autk (Vi ) is saturated. Also, since G lies in i Autk (Vi ), for elements
expdim Vi .
of G we have logn =logdim Vi and for elements of Lie G we have expn =ii
Thus (i): if H ⊆ Autk (Vi ) is saturatedQand contains pri (G), then pr−1
(H)issaturatedandicontains G. Moreover (ii): if G ⊆ N ⊆ i Autk (Vi ) is saturated, then pri (N ) is saturated andsatcontains pri (G). Now from (i) we deduce the inclusion Gsat
⊆ pr−1i (pri (G)k ), and from (ii)
k
sat
sat
that pri (G)k ⊆ pri (Gk ), and the assertion follows.

××

The following result will be useful when comparing geometric and arithmetic mod λ monodromy.
Proposition 5.20. Suppose that : (i) G ⊂ SLn , (ii) N is a closed normal subgroup of G withfinite cyclic quotient, (iii) N (k) acts absolutely irreducibly on V := k n , (iv) ` > n. Then
o
sat
sat
+
sat o
sat
+ sat
(a) (Nksat )o = (Gsatk ) and Gk = GNk , and if N = N , then also (Nk ) = Nk = (G )k ;sat(b) the group Gsat
is finite cyclic, and its order divides n! and is bounded by n(n − 1).
k /NkProof. Let us first prove the proposition under the further hypothesis N = N + . By Corollary 5.16, the group Nksat is normal in Nksat G, and from (ii) it follows that Ḡ := Nksat G/Nksat
is finite cyclic. Consider the adjoint homomorphism c : Nksat G → Aut(Nksat ), g 7→ cg definedby conjugation, and denote by cout : Nksat G → Out(Nksat ) the induced homomorphism to theouter automorphisms. By [Con11, 7.1], the group Out(Nksat ) injects into the group of automorphisms of a based root datum attached to N sat . Because of N = N + and (iii), Proposition 5.18
implies that Nksat is connected semisimple, and it follows that Out(Nksat ) is a subgroup of theautomorphisms of the Dynkin diagram of Nksat . The Dynkin diagram (of Nksat over k) is theunion of the diagrams of the simple factors of Nksat ⊗k k, and the diagram contains of at mostn − 1 = rank SLn ≥ rank Nksat vertices. It follows that the maximal cyclic subgroup in Out(Nksat )has order bounded by n − 1. Define H as the kernel of the restriction cout |G . Then N ⊂ H ⊂ G,
the group G/H is cyclic of order bounded by n − 1, and Ker cout = Nksat H.
Let now h ∈ H be an element whose image is a generator of the cyclic group H/N , and let
h0 ∈ Nksat (k) be an element such that ch0 = ch . Then h0 h−1 maps to a generator of Nksat H/Nksat ,and ch0 h−1 is trivial, so that h0 h−1 commutes with Nksat and hence by (iii) lies in the scalars in
SLn . It follows that the order of h0 h−1 divides n, and hence that Nksat H/Nksat is cyclic of orderdividing n. We deduce that the order of GNksat /Nksat divides n! and is bounded by n(n − 1).
13 The condition n = h
GLn ≤ p needed in [Ser98, Thm. 8] follows from our standing assumption ` > 2M − 1;

see Remark 5.26 for the meaning and value of hGLn .

32

From (iv) it now follows that GNksat /Nksat is finite of order prime to `. Then Lemma 5.13(b)implies that GNksat is saturated, and hence equal to Gsat
k . Thus (b) is proved. The equalityosat
Nksat = (Gsat)in(a)
followsfrom(b)becauseNisconnected and saturated. The firstkkequality in (a) follows by applying the proposition to N ⊆ G+ , in which case one deduces theequality from (b).
Let now N be arbitrary. Choose g ∈ G such that g maps to a generator of G/N , and let G0
be the subgroup of G generated by N + and g. Observe that N + is characteristic in G, andin particular a normal subgroup. Hence we can use (a) and (b) for the pair (N + , G0 ). From
GNksat = G0 Nksat = G0 (N + )Nksat we deduce that GNksat /Nksat ∼= G0 /(Nksat ∩ (G0 N + )satk ) is a0+ sat
quotient of G /(N )k , and this proves (b). From (b) it follows that GNksat is saturated since
the quotient GNksat /Nksat is of order prime to `; cf. the argument proving Lemma 5.13(b). From
o
sat
sat
(b) we also deduce (Gsatk ) ⊂ Nk ⊂ Gk , and this completes the proof of (a).

5.3

On saturation within any reductive group G

In this subsection, G will denote a connected reductive group over k. We shall recall some resultsfrom [Ser98] on saturation within arbitrary reductive groups G. Our presentation is also inspiredby [BDP16].
We begin with the definition of Dynkin height and Coxeter number.
Definition 5.21 ([Ser98, II.2] or [BDP16, Def. 4.3]). Let G be a reductive group over k, T amaximal torus of Gk , and R+ the set of positive roots for some Borel subgroup containing T .
The Dynkin height of a representation V of G isX
htG (V ) := max{hλ, α∨ i | λ a weight for the action of T on V ⊗k k}.
α∈R+

A representation ρ : G → GL(V ) with ` > htG (V ) is said to be of low `-height for G.
Remark 5.22. Suppose G has a root base (αi )i∈I , and V is an irreducible
representation withP
dominant weightP
λ+ and smallest weight λ− , and write λ+ − λ− = i mi αi for (mi ) ∈ QI≥0 .
Then htG (V ) = i mi . For instance, if G = GLn and V is the standard representation, thenVihtG ( V ) = i(n − i) (cf. [Ser98, Part II, Lect. 2]).
Example 5.23. Let E be a number field with the ring of integers OE and n ∈ N. Let G be a
reductive group scheme over OE [1/n] and ρ : G → GL(V) be representation over OE [1/n] withgeneric fiber G → GL(V ). Then for all primes p of OE [1/n] with residue field kp such thatchar kp > htG (V ), the base change Gkp → GL(Vkp ) is of low char kp -height.
Q
N
Lemma 5.24. (a) If G =Gi for reductive groups Gi , andi Vi is an (external)
PV =tensor product of representations Vi of Gi , then htG (V ) = htGi (Vi ).
(b) If π : G → G0 is a central isogeny of reductive k-groups, V 0 is a representation of G0 , andV the representation of G on V 0 via π, then htG (V ) = htG0 (V 0 ).
Proof. Part (a) is a simple exercise left to the reader. Concerning (b), we may assume that k = k.
Let T 0 be a maximal torus of G0 and T = pr−1 (T 0 ), denote by X, X 0 the character groups of T
and T 0 , by X ∨ and X 0,∨ their cocharacter groups, and let R and R0 be roots of G with respect
33

to T and G0 with respect to T 0 respectively. Let further f : X 0 → X be the homomorphismsof character groups induced by π|T : T → T 0 and f ∨ : X ∨ → X 0,∨ be the dual map. Then by[Spr09, 9.6, formula (44)] we have f ∨ (R+,∨ ) = R0+,∨ if we define R0 by a Borel subgroup B 0 ⊂ G0
containing T 0 and R by B = π −1 (B 0 ). Moreover a weight λ0 ∈ X 0 of V 0 becomes the weightλ0 ◦ π of V . By [Spr09, 9.6.1] for α ∈ R one has hλ0 , f ∨ (α)i = hλ ◦ π, αi. Summing over α ∈ R+
and maximizing over the weights λ0 of V 0 , part (b) of the lemma follows.
Definition 5.25. If the root system of Gk is irreducible, the Coxeter number hG is defined tobeX
hG = 1 +
ni ,
P
where
ni αi is the highest root of Lie Gk (with (αi )i∈I a root base), cf. [Bou68, VI.1.Prop. 31].
For a general reductive G one defines
hG = max{hH | H is a simple quotient of G}.
Remark 5.26. If the root system of Gk is irreducible, one has hG = hρ, β ∨ i + 1 where ρ is thehalf sum of the positive roots and β ∨ is the highest short coroot, see [Hum06, § 3.5].
Note Qalso that for G = GL(V ) one has hG = dim V for its Coxeter number and thus for
G = i GL(Vi ) one finds hG = max dim Vi .
Let gnil (resp. Gu ) be the reduced closed subscheme of g = Lie G (resp. G) with points thenilpotent (resp. unipotent) elements. Let U be the unipotent radical of a parabolic subgroup P
of G and u = Lie U . Suppose that ` ≥ hG , so that for the `-th step of the descending central seriesof u one has Z ` u = 0. Then the Campbell- Hausdorff group law ◦ makes sense in characteristic`, and it turns u into an algebraic group over k. Furthermore there is an unique isomorphism'

expU : (u, ◦) −→ U

(9)equivariant for the action of P and whose differential at the origin is the identity; see [Sei00,
Proof of Prop. 5.3] and [Ser98, Lect. 2]. Here k can be any field.
If ` > hG , there is a unique G-equivariant isomorphism'

expG : gknil −→ Gk,u ,(10)which induces (9) on the unipotent radical of each parabolic subgroup; see [Ser98, Thm. 3, p. 21],
[Sei00, Proof of Prop. 5.3] and [McN05, Prop. 11, p. 6] for details. Let logG : Gu → gnil denote
its inverse. The map expG is not necessarily defined over the field of definition k of G. Foru ∈ Gu (k), one defines the “t-power map”
Ga → G, t 7→ ut = expG (t logG u).

(11)

For G = GL(V ) the maps expG and logG are the maps in (7), and t → ut is the the map (8).
Definition 5.27 ([Ser98, Lect. 3]). A closed k-subgroup H ⊂ G is called saturated if for allu ∈ Hu (k) and t ∈ k the elements ut = expG (t logG u) lie in H(k).
For any closed subgroup Γ of G, its G-saturation ΓG-sat
is defined as the intersection of all closedksaturated k-subgroups of G that contain Γ.
34

Remark 5.28. For G = GLn and a closed subgroup H ⊂ G we clearly have Hksat = HkGLn -sat .
Lemma 5.29. Suppose G is quasi-split over k. Then expG is defined over k.
Proof. Let B be a closed subgroup of G, such that its base change Bk to k is a Borel subgroupof Gk . Let U be the unipotent radical of B, and so Uk is the unipotent radical of Bk , and let
expU : Lie U → U be the unique exponential map in (9) provided by [Sei00, § 5] and definedover k. Consider the subvariety
Y = {(g, AdLie G (g)n, AdG (g)(expU (n)) | g ∈ G, n ∈ Lie U } ⊂ G × Lie G × G}.
The graph of expU is closed in Lie G × G and defined over k, and since the adjoint action of G
on Lie G × G (by AdLie G × AdG ) is also defined over k, its orbit Y is defined over k, as well. Let
Z ⊂ Lie G × G be the image of Y under the projection onto the last two factors of G × Lie G × G.
Note that (Lie Gk )nil is the orbit of Lie Uk under the adjoint action. Hence after base change
from k to k, the uniqueness of expG in (10) implies that Zk is the graph of expG . It follows that
Z is the graph of a morphism defined over k, and hence that expG itself is defined over k.
Remark 5.30. The main issue about expG being defined over k is its domain. If, for instance, G
was the unit group of a quaternion algebra, then G has no non-trivial parabolics defined over k
and hence the result of Seitz (or Serre) is void.
Corollary 5.31. If G is quasi-split, then the intersection of all saturated subgroups of Gk that
contain H agrees with HkG-sat ⊗k k.
Proof. Because of Lemma 5.29 the isomorpism expG and its inverse logG are defined over k.
Now following the proof of Lemma 5.6 implies the result of the corollary.
The following are some important results from [Ser98, Lect. 4]:
Corollary 5.32 ([Ser98, Cor. 2 to Thm. 5]). The representation V is of low `-height for G if
and only if the following diagram is well-defined and it commutes
gnil

dρV

/ End(V )nilρV
/ GL(V )uni .

expGL(V )

expG


Gu

In other words, if V has small `-height for G, then the saturation inside G defined by Serreagrees with the saturation inside GLn defined by Nori.
Corollary 5.33 ([Ser98, Cor. 1 to Thm. 5]). If H is a closed, reductive and saturated subgroupof G, one has htH (V ) ≤ htG (V ).
The following corollary generalizes [EHK12, App. B, Prop. 25, 26] from F` to any finite field.

35

Corollary 5.34. Let H be a closed, connected reductive and saturated subgroup of GLn , and let
V be the standard representation of GLn . Denote by λ+ and λ− the highest and lowest weightof V regarded as a representationPof H with respect to a root base ∆. Let ε be in {±}. Then ελε
is dominant, and writing ελε = α∈∆ mεα α, one has max{mεα | α ∈ ∆, ε ∈ {±}} ≤ n − 1.
Proof. By [Bou75, VIII, Sec. 7.5, Prop. 11] the lowest weight of an irreducible representation of
G is the negative of a dominant weight. This shows that −λ− is dominant, and for λ+ this isclear. In particular the mεα lie in Q≥0 . Now from Corollary 5.33 and Remark 5.22 we deducethat max{mα + m−
α | α ∈ ∆} ≤ n − 1, and this completes the proof of the corollary.
The following result and its proof are inspired by [Fri10, §1, 2] where similar but not quite thesame results are developed.
Proposition 5.35. Let G be quasi-split with hG < `, let k 0 be a subfield of k such that k/k 0 is
finite separable. Then the following hold:(a) The Weil restriction G0 := Resk/k0 G is quasi-split, hG0 = hG , and expG0 is defined over k 0 .
(b) One has Resk/k0 expU = expG0 |Resk/k0 U for any parabolic P ⊂ G with unipotent radical U .
(c) Let H ⊂ G be a closed subgroup. If H is saturated in G, then so is Resk/k0 H in Resk/k0 G.
sat
For any H we have an inclusion (Resk/k0 H)sat
k0 ⊂ Resk/k0 (Hk ).
(d) Let V be a vector space over k and write W for V as a vector space over k 0 . Let
ι : Resk/k0 GL(Vk ) → GL(Wk0 ) be the embedding defined by the natural transformation
GLk⊗k0 R (V ⊗k0 R) → GLR (W ⊗k0 R) on k 0 -algebras R. Then:(i) We have htResk/k0 GL(V ) W = [k : k 0 ](dimk V − 1) = dimk0 W − [k : k 0 ].
(ii) If ` > dimk0 W − [k : k 0 ], then the following diagram commutes(Resk/k0 Endk (V ))nil
expResk/k0dι

/ Endk0 (W )nilι
/ GL(Wk0 )uni .

expGL(W )GL(V )


Resk/k0 GL(Vk )uni(iii) If H is closed and saturated in GL(V ) and if ` > dimk0 W − [k : k 0 ], then Resk/k0 H
is saturated inside GL(Wk0 ) (embedded via ι).
Proof. Note first that after base change to an algebraic closure k of k we have an isomorphism
Y
Resk/k0 G ⊗k k ∼G ⊗σk k,(12)=σ∈Homk0 (k,k)

where the map k → k in the factor for σ is given by σ. Using (12), it is now easy to see that
if P is a k-parabolic subgroup of G, then Resk/k0 P is a k 0 -parabolic subgroup of Resk/k0 G.
This implies the first claim of (a). The second claim of (a) also follows from (12) and thedefinition of hG . The last claim in (a) follows from the first two and Lemma 5.29. For (b) note
that, again by (12), Resk/k0 U is the unipotent radical of Resk/k0 P . Now (b) follows from the36characterization of expU given in the paragraph surrounding the equation (9) and the propertiesof the Weil restriction. By (b) the exponential expResk/k0 G after base change to k and restriction
to the factor σ in (12) is given by expG ⊗σk k. Since clearly H ⊗σk k is saturated in G ⊗σk k under
expG ⊗σk k, this implies the first assertion of (c); the second follows from the first. Part (d)(i)follows from (12) and Remark 5.22. Part (d)(ii) is a consequence of (i) and Corollary 5.32. Part(d)(iii) is a consequence of (c) and (d)(ii).
Example 5.36. It is in general not true that the operation of taking the saturation and oftaking a Weil restriction commute. For this, suppose k is a finite field with proper subfield k 0 ,let G = SLn,k and let H be the finite group SLn (k 0 ) considered as a reduced finite subgroupscheme. Then we have Hksat = G, because the Nori envelope has to contain all root groups forthe diagonal torus and so in particular the standard upper triangular Borel subgroup and its
opposite. This implies that Resk/k0 (Hksat ) = Resk/k0 G. However Resk/k0 H = H, and therefore(Resk/k0 H)sat= SLn,k0 inside Resk/k0 SLn,k0 via the the embedding SLn,k0 ,→ Resk/k0 SLn,k0kthat is the adjunction map of the Weil restriction. We leave further details to the reader.

5.4

Saturation and Lifting

In this subsection, in Proposition 5.40, we prove a result on lifting representations of low `-heightof a saturated semisimple subgroup G of GLn from a finite field of characteristic ` > 0 to its
ring of Witt vectors and thus to a finite extension of Q` .
We begin with some elementary preparations for the proof of Proposition 5.40.
Lemma 5.37. Let k be a field. Let H ⊂ GLn,k be a split semisimple connected closed subgroup.
Denote by V ∼= k n the resulting representation ρ̄ : H → Autk (V ) of the algebraic group H, and byee is semisimple connected and simply connected.
πk : H → H the universal cover of H, so that H
Suppose that V is irreducible. ThenQe ∼e i with He i split simple semisimple simply connected over k.
(a) H= i∈I He i . Then Te = Q Tei is a maximal split torus of He and(b) Let Tei be a maximal split torus of H
i
'
e
e
e
T := πk (T ) of H. Let ι : ⊕I X(Ti ) → X(T ) be the induced isomorphism on character groups.
e via ρ̄ ◦ πk one has V ∼(c) As a representation of H
= ⊗i∈I Vi (exterior tensor product) fore i → Autk (Vi ).
irreducible representations ρi : H(d) If µi ∈ X(Tei ) is the highest weight of Vi , then ι((µi )i∈I ) is the highest weight of V .
(e) The order of Ker πk is bounded by n = dim V and if char k = 0 or char k > n, then Ker πkis étale over k.
Proof. Part (a) follows from [Con11, parag. before 6.4.4 and Thm. 5.1.19], and (b) is clear. Parts(c) and (d) are standard facts. But we could not find a reference. Let µ be the highest weight of Ve i representation with highest weight µi .
and define (µi )i∈I := ι−1 (V ) and Vi as the irreducible HeThen (the external tensor product) W := ⊗i∈I Vi has highest weight µ as a representation of H.
e
Observe that by the Burnside theorem
We show that W is irreducible as a representation H(k):e i the homomorphism k[He i (k)] → Aut (Vi (k))
and the irreducibility of Vi as a representation of Hk37

induced from the representation of Vi is surjective. But from our product situation it is thene→ Autk (V (k)) induced from the representation of V
clear that also the homomorphism k[H(k)]is surjective. Again from Burnside, we deduce that W (k) and hence W is irreducible. By theclassification of irreducible representations in terms of their highest weight for split reductive
groups, see [Jan03, II. Cor. 2.7], we find V ∼= W , and hence (c) and (d) are shown.
To prove (e), we may pass to an algebraic closure k of k. Since Ker πk is contained in the producte i , it suffices to show that the order of the center (as a group scheme) willof the centers of the HQ
be bounded by dim Vi . Then the order of Ker πk is bounded by i dim Vi = dim V . However
for G simply connected simple, over any field, the order of its center is always bounded by thesmallest dimension of a non-trivial representation, as follows from the classification: The kernel
has order at most m if G = SLm , 2 if G is of type Bm , Cm or E7 , 4 if G is of type Dm , 3 if G isof type E6 and 1 otherwise, e.g. [Bou68, Planches, 250ff., entry (VIII)]. The minimal dimensionsof non-trivial representations of G are given in [Bou75, VIII, table 2, p. 214]; they are m for
SLm , and 2m + 1, 2m, 2m, 27, 56, 248, 26, 7 for Bm , Cm , Dm , E6 , E7 , E8 , F4 and G2 , resp.
The étaleness of Ker πk now follows from char k > n or char k = 0.
e i is the kernel of H
e i → H,
Remark 5.38. The scheme theoretic intersection Zi := Ker πk ×He H
eso that Hi := Hi /Zi can be regarded as an almost simple subgroup of H. Moreover ZN
i acts00trivially on VQ
i , so that VQ
i can be regarded as a representation Vi of Hi . One can form
i Vi0over H := i Hi , and i Hi → H is a central isogeny whose kernel coincides with that of
N 0ei Vi . Thus in the above lemma, one can replace, as often done in the literature, the group H
by H 0 .
Q
0Moreover if H has trivial center, i.e. H = H ad , then HN= H. Then H = i Hi for simple connected adjoint groups, and the decomposition of V = i Vi can be carried out with irreducible
representations Vi of Hi . In this case (as in the case where H is simply connected), the Vi canalso be regarded as representations of H itself.
Remark 5.39. By reduction to the simply connected case, the argument for (e) shows that for any
connected semisimple group G with center Z over any field with an almost faithful representation
V one has dim V ≥ #Z.
The next result makes use of the notion of Weyl module. Weyl modules are natural representations V (λ)Z attached to a fixed dominant weight λ of a reductive group over Z. Via base change
they give rise to representations V (λ)A of the base changed group over any base ring A. If A
is a field of characteristic zero, then V (λ)A is irreducible of highest weight λ. If A is a field ofpositive characteristic, in general V (λ)A will be reducible. For more see [Jan03, II.8].
Proposition 5.40. Let the hypotheses and the notation be as in Lemma 5.37. Suppose further
that k is finite of characteristic ` > 2(n − 1) and that H ⊂ GLn is saturated, and denote by
W (k) the ring of Witt vectors of k. Thenei.
(a) Each Vi is a Weyl module for He Hei with special fiber H, H
e and Hei,(b) There exist semisimple W (k)-group schemes H, H,
e is simply connected, we have a universal cover π : H
e → H overrespectively; moreover H
Q ee∼
W (k) whose reduction is πk , and we have an isomorphism H
over
W (k) whose= iH
ireduction is the isomorphism in Lemma 5.37(b).
38ei over W (k) with the same highest weight than Vi . Then(c) Let Vi be the Weyl module for He which reduces to ρ̄ ◦ πk .
V := ⊗i∈I Vi is a representation ρ of H(d) For (V, ρ) from (c) we have Ker ρ = Ker π, and thus V is a representation of H over W (k)that reduces to the representation V of H.
e i given in
Proof. To prove (a) we shall verify the low alcove condition for Vi with respect to H
[McN98, Cor. 4.4.3] in order to deduce by loc.cit. that Vi is a Weyl module. Let Φi ⊂ X(Tei ) be
e i , choose a Borel subgroup Bi ⊂ H
e i and let ∆i be the corresponding root
the root system of H
base for Φi . Because H is saturated in GLn we deduce htH (V ) ≤ n − 1 from Corollary 5.34,
and from Lemma 5.24 we deduce htHe i (Vi ) ≤ htHe (V ) = htH (V ) ≤ n − 1, using for the htHe j (Vj ),j 6= i, that the Dynkin height is always positive.
We need to collect some results about the paring h·, ·i : X(Tei ) × X ∨ (Tei ) → Z. Denote by σi thee i , and by αi,0 the highest short root of He i , see [Car05, p. 251].
half sum of positive roots of H∨e i . For any
Then from Remark 5.26 we have hσi , αi,0 i = hHe i − 1 for the Coxeter number of H
simply connected simple group G the value of hG is at most as large as the dimension of thesmallest irreducible non-trivial representation of G (see the proof of Lemma 5.37(e)). Moreover∨from [Car05, p. 543ff.] one can check that hα, αi,0i ∈ {0, 1} for any α ∈ ∆i and that the value 1
is attained at least once and at most twice (this holds for any simply connected simple G).
P
Let µi be the highest weight of Vi with respect to Tei . Write µi =mi,α α for a uniqueα∈∆iituple (mi,α ) ∈ Q∆
e i (Vi ) ≤ n − 1 and from Corollary 5.34 it follows that
≥0 . Then from htHP
α∈∆i mi,α ≤ n − 1. Then by the above we findXX∨∨
hσi + µi , αi,0i ≤ hHe i − 1 +mi,α hα, αi,0i ≤ dim Vi − 1 +mi,α ≤ 2(n − 1).

α∈∆i

α∈∆i

Because ` > 2(n − 1), it follows now from [McN98, Cor. 4.4.3], that the Weyl module V̄ (µi ) fore i is simple, and hence isomorphic to Vi , which proves (a).
the weight µi of He Hei with special
Now we turn to (b). The existence of simply connected simple groups H, H,eefiber H, H and Hi , as well as the isogeny pr lifting prk follows from [Con11, Thm. 6.1.16].
∼ Q H
e i defines an isomorphism of root data whiche =Note also that the isomorphism Hi∈Ie and Q H
ei , and then again by [Con11,
yields an isomorphism between the root data for H
i
Q
∼
e=
e
Thm. 6.1.16] we obtain an isomorphism H
i Hi over W (k) whose reduction is the isomorphism in Lemma 5.37(b).
Regarding (c), note first that as explained in [Jan03, II.8] Weyl modules for split reductive
connected groups are defined over Z, and hence over any ring. Thus we have Weyl modules
ei defined over W (k) whose reduction to k is Vi . The highest weighs of Vi and of Vi
Vi for H
are the same. We remark that Weyl modules are irreducible over any field of characteristic
zero, and that their highest weight is independent of the characteristic. From their definitioninN
loc.cit. and arguing as in the proof of Lemma 5.37(c),(d), it follows that in fact V = i Vi is thee of highest weight ι((µi )i∈I ). Its reduction agrees with V = N Vi , and this
Weyl module for Hiproves (c).
e over W (k) whose image under π is a
Regarding (d), let Te denote a maximal split torus of H
maximal split torus T of H. Since the kernel of π is equal to the kernel of pr : Te → T , it suffices39

to consider the restriction of ρ to Te . This restrictionT
can be regarded as an n-tuple of characters(χi )i=1,...,n in Hom(T , Gm,W (k) ), and thus Ker π = i=1,...,n Ker χi . Their reductions form thecorresponding character tuple (χ̄i )i=1,...,n for ρ̄|Te . After identifying T with Grm,W k for somer ≥ 1, the tuples are represented by some r × n matrix over Z, and the order of the kernel isthe product of the elementary divisors of this matrix. By Lemma 5.37(e), over k, this order isbounded by n < ` (because n > 1 and ` > 2(n − 1)). In particular, this kernel is finite étale overW (k). Since pr from [Con11, Thm. 6.1.16] is constructed from the corresponding map of root
data over k, and since over the special fiber Ker ρ agrees with Ker πk , we deduce Ker π = Ker ρ,and this completes (d).

5.5Saturatedness of some non-reductive groups

We now assume that k is finite, that G is a connected smooth affine k-subgroup scheme of GLn,kbut not necessarily reductive, and that G contains a maximalLk-split torus T . Let V = k n
be the standard representation of GLn,k . Let Lie G ∼= Lie T ⊕ α∈Φ gα denote the eigenspacedecomposition under the adjoint action of T where Φ ⊂ X(T ) r {0} is the finite subset ofcharacters α of T whose weight space gα is non-zero. Throughout this subsection, we require
the following conditions to hold:(i) the weight spaces gα are 1-dimensional,(ii) for any α ∈ Φ we have Zα ∩ Φ ⊂ {±α} (i.e., Φ is reduced).
(iii) for any α ∈ Φ, we have{i ∈ N | αi is a weight of the action of T on End(V )} ⊆ {1, . . . , ` − 1}.
Note that if G ⊂ GLn,k is a reductive subgroup, then (i) and (ii) clearly hold and (iii) holds if
End(V ) is `-restricted.
Lemma 5.41. For α ∈ Φ, let Tα ⊂ T be the subtorus Ker α and define Gα := ZG (Tα ). In thefollowing we consider (gα , +) as a group scheme isomorphic to Ga . Then(a) Gα is a smooth subgroup scheme of G and Lie(Gα ) = Lie T ⊕ gα ⊕ g−α .
(b) If gβ 6= 0 for β ∈ {±α}, then there exists a unique k-homomorphism uβ : gβ → Gα , suchthat duβ = idgβ and uβ (λt) = β(λ)uβ (t) for all t ∈ k. Let Uα be the image of uα .
∼ Uα × U−α , and(c) If dim gα ⊕ g−α = 2, then Gα is either reductive of rank 1, or Ru (Gα ) =∼ T.
Gα /Ru (Gα ) =(d) If dim g−α = 0, then Gα is solvable and sits in an extension 1 → Uα → Gα → T → 1.
Proof. Part (a) is a special case of [Con11, Lem. 2.4.4], using condition (ii). If Gα is reductive,then assertions (b) and (c) are clear. In the other case Ru (Gα ) is non-trivial, and Gα /Ru (Gα )must be a reductive group of rank 0, i.e., a torus. For dimension reasons, the induced map fromT to that torus is an isomorphism and thus Gα sits in a short exact sequence 1 → Ru (Gα ) →Gα → T → 1. If dim Ru (Gα ) = 1, then we must have Lie Ru (Gα ) ∼= gα and hence the existenceof uα follows from [Spr09, 14.3.11], and (d) follows. Because of hypotheses (i) and (ii) in theremaining case Lie Ru (Gα ) ∼= gα ⊕ g−α has dimension 2. Since k is perfect, we can regard40

Ru (Gα ) as a 2-dimensional connected subgroup of Un . [Spr09, 14.3.11] guarantees the existenceof subgroups isomorphic to Ga on which T acts via α and −α, respectively. The existence of
the uα and hence Uα are now straightforward: One simply has to check that any action of Gm
on Ga is of the form Gm × Ga → Ga , (λ, t) 7→ λi t for some unique i ∈ Z (and this generalizes
in an obvious way to actions of T on Ga ). To see that Uα and U−α commute, note that theircommutator lies in Ru (ZG (Tα )) but has weight 0 for the action of T , and hence is trivial.
Lemma 5.42. Let α be in Φ and let uα : gα → Uα as in the previous lemma. Then for anyu ∈ Uα (k) and corresponding X = u−1
α (u) one has
expn (t logn (u)) = uα (tX).
Proof. The following argument is inspired by [Ser98, proof of Thm. 5 and its Cor. 2]. For
explicitness we may assume that the k-split solvable group T Uα is contained the the upper
triangular Borel subgroup of GLn,k , and Uα in Un,k . Consider the homomorphism
ϕα,t : Ga →P
Un,k , t 7→ uα (t). It can be given as a polynomial in t of the form 1 + i≥1 ai (α)ti , for suitableai (α) ∈ End(V ) (V = k n ). Consider the adjoint action of λ ∈ T (k):XX
1+ai (α)αi (λ)ti = 1 +ai (α)(α(λ)t)i = uα (α(λ)t) = λuα (t)λ−1
i≥1

i≥1



= λ 1+XX
ai (α)ti λ−1 = 1 +λai (α)λ−1 ti .

i≥1

i≥1

It yields ai (α)αi (λ) = λai (α)λ−1 for all i ∈ N, and thus either ai (α) = 0 or αi is a T -weightof End(V ). Since by hypothesis (iii) the T -weights of End(V ) are `-restricted, it follows thatai (α) = 0 for i ≥ `. Now consider
expn (t logn (u)) − uα (tX)
as a polynomial in t with coefficients in End(V ). By what we just proved and by the definitionof expn it follows that this polynomial has degree at most ` − 1. As it vanishes for the ` values
t = 0, . . . , ` − 1, it is identically zero.
Remark 5.43. It seems an interesting question whether one can use the uα above (and theirconjugates) together with the Baker-Campbell-Hausdorff formula to define a Springer type isomorphism between Gu (k) and (Lie Gk )nil , even if G is not semisimple or reductive. If G issemisimple such a Springer isomorphism was used by Serre in his approach to saturation forgeneral reductive groups. A related question where G is reductive but ` < hG is asked in[McN05, Rem. 27]: does there exist a Springer isomorphism whose restriction to the unipotent
radical of any parabolic is as in formula (9).
Theorem 5.44. The group G is saturated.
Proof. Let H be the subgroup of G generated by the Uα (for one fixed T ). Then by Lemma 5.10
e Because of Lemma 5.42 each Uα and hence also H is exponentially generated.
we have H = G.
e by Corollary 5.11(c), and hence G is saturated by Proposition 5.14.
Since G is connected G+ = G

41

5.6

Reductions of reductive groups and saturation

The first aim here is to generalize a result of Larsen-Pink from Q- to E-rational compatible
systems. In Theorem 5.52 we deduce from this the saturatedness of an interesting class of apriori not necessarily reductive groups over a finite field. In order to carry out the first part,which is a straightforward extension of results of Larsen and Pink, we recall some notions andresults from [LP92, Sects. 4,6,7]. We follow this reference rather closely.
Throughout this subsection, we let K be any field of characteristic zero, and we assume that all
algebraic groups and varieties will be defined over K. Denote bych : GLn → Gm × Gn−1athe morphism associating to a matrix the coefficients of its characteristic polynomial. Let Dn =Gnm be the diagonal split maximal torus of GLn . The Weyl group of GLn with respect to Dn isthe symmetric group Sn , acting on Dn by permutation of factors. The restriction of ch to Dn isa finite morphism which identifies Gm × Gn−1
with the scheme-theoretic quotient Dn /Sn .
a
Let T0 ⊂ Dn be any closed subtorus. Then T0 is also split over K, and as a map between
split tori, the inclusion T0 ⊂ Dn is defined over Q (or even Z), and hence the variety ch(T0 ) ⊂is defined over Q. The variety ch(T0 ) is geometrically connected, because T0 is so.
Gm × Gn−1aLet ρ : T0 → Dn denote the representation corresponding to T0 ,→ Dn , and consider on Dn
the action of Sn = NGLn (Dn )/CGLn (Dn ) that permutes its entries. Set Aut(T0 , ρ0 ) := {α ∈Aut(T0 ) | ∃σ ∈ Sn : ρ ◦ α = σ ◦ ρ} – the automorphism group of the pair (T0 , ρ) (cf. [LP92]).
'
One has an isomorphism NSn (T0 )/CSn (T0 ) → Aut(T0 , ρ0 ), and the pair (T0 , ρ0 ) is determinedby ch(T0 ) up to isomorphism. (cf. [LP92]).
Following [LP92, 4.4], for any σ ∈ Sn r CSn (T0 ), we define a proper subgroup Hσ of T0 : ifσ(T0 ) = T0 , then Hσ = {t ∈ T0 | σ(t) = t}; otherwise, set Hσ := T0 ∩ σ(T0 ). The intersectionof different tori of the same dimension is a subgroup of smaller dimension. Also, the condition
σ(t) = t can be expressed in the vanishing of at least one character of T0 . Thus dim Hσ < dim T0
for all Hσ . Since chSis finite on Dn one deduces dim ch(Hσ ) = dim Hσ < dim ch(T0 ) = dim T0 .
In particular Y := σ∈Sn rCS (T0 ) ch(Hσ ) is a proper closed subset of ch(T0 ).
n

For a connected reductive group G ⊂ GLn there exists a torus T0 ⊂ GLn as above such thatevery maximal torus of G is, over an algebraic closure K of K, conjugate to T0 . Over K, thesemisimple part of any g ∈ G can be conjugated into T0 , so that ch(T0 ) = ch(G) pointwise. It
follows that ch(G) = ch(T0 ) ⊂ Gm × Gn−1
is Zariski-closed and that T0 is, up to conjugation byaSn , uniquely determined by ch(G). Hence Y only depends on ch(G) = ch(T0 ) (cf. [LP92, 4.3]).
Definition 5.45 ([LP92, § 4]). Let G ⊂ GLn , be a connected reductive subgroup, and let
Y ⊂ ch(G) be as above. An element g ∈ G is Γ-regular if and only if ch(g) ∈/ Y.
The importance of Γ-regular elements g ∈ G is explained in [LP92, 4.4-4.7]: One has:(a) The element g is regular semisimple in G, and it lies in a unique maximal torus Tg of G;
denote by ρg : Tg ,→ GLn the induced representation of Tg on K n .
(b) The element g, via the eigenvalues of ρg (g) together with their multiplicities, determine
uniquely the formal character of the action of Tg on K n via ρg from (a).

42Let ρ• be an E-rational n-dimensional compatible system. Following the proof of [LP92, 6.11],
the following result is immediate.
Lemma 5.46 ([LP92, 6.11]). Let g ∈ π1 (X) and Γ ⊂ π1 (X) be an open subgroup. Then the0
Zariski closure of ch(ρλ (gΓ)) in (Gm × Gn−1
)Eλ is defined over E and independent of λ ∈ PE.
a
Remark 5.47. Lemma 5.46 also applies to compatible systems over number fields.
Convention 5.48. In the remainder of this subsection, ρ• is assumed to be connected.
Definition 5.49 ([LP95, § 1]). A place x ∈ |X| is called good (for ρ• (and λ)) if ρλ (Frobx ) is
Γ-regular.
The following result is proved in [LP92] for Q-rational compatible systems. The adaptations to
the general case are minor, and we omit them.
0
Proposition 5.50 ([LP92, 7.2, 7.4]). For any λ ∈ PE
, the set of all g ∈ π1 (X) such that ρλ (g) isΓ-regular is open and dense in π1 (X). In particular, the set of good places x ∈ |X| has Čebotarov
density 1. Moreover for x ∈ |X| the condition of Frobx being Γ-regular is independent of λ.
0
we choose an Oλ -lattice Λλ ⊂ Eλn that is invariant under the action
Suppose that for every λ ∈ PEof π1 (X) via ρλ . Denote by Gλ the Zariski closure of Gλ in AutOλ (Λλ ), endowed with the uniquestructure of reduced closed subscheme. This is a flat group scheme over Oλ . We have thefollowing immediate generalization of a result of Larsen and Pink:

Proposition 5.51 ([LP95, Prop. 1.3]). For all `λ  0, the following hold: The group Gλ issmooth. It contains a closed subtorus Tλ defined over Oλ whose generic fiber is a maximal torus
Tλ of Gλ , and whose special fiber Tkλ is a maximal torus of Gkλ = Gλ ×Oλ kλ . Moreover(a) the set of roots (and their multiplicities) of Gkλ with respect to Tkλ is the same as the setof roots of Gλ with respect to Tλ , and(b) the formal character for the action of Tλ on Eλn and for the action of Tkλ on kλn agree.
Proof. There are some explanations in order. The proof in [LP95] for Q-compatible systems
consists of two parts. First a nice model Tλ (in the sense of [CHT17]) in Gλ of a maximal torusof Gλ is constructed. This argument uses Γ-regular elements recalled earlier in this subsection,and it works for E-compatible systems with any number field E; see also [CHT17, 9.2.1].
In aQ
second step, for the proof of smoothness, an Oλ -scheme map ϕλ from Xλ := Tλ ×Oλ α Uα formodels Uα of the root spaces of Gλ to Gλ is considered. According to [Win02, footnote, p. 3]
the argument in [LP95] for ϕλ being an isomorphism near zero and Xλ being Oλ -smooth seems
to be incomplete. This is remedied in [Win02, Thme. 1] for Gλ reductive, and in [CHT17, 9.1.1]
for Gλ semisimple. Both references rely on [BT84, §§ 1.2 and 2.2]. Note also that the reductionof ϕλ proves assertion (a), while (b) follows from having constructed Tλ via Γ-regular elements.
We also remark that [LP95, § 1] assumes purity as part of the definition of a compatible system.
However this hypothesis is not used in the proof of [LP95, Prop. 1.3], nor in the preliminaries
leading up to it, e.g not in [LP92], and it is also not used in the our supplementary references.
We now come to the central result of this subsection:
43

Theorem 5.52. For all `λ  0, the special fiber Gkλ is smooth, and its identity component issaturated in GLn,k .
Proof. By Proposition 5.51 the group scheme Gλ is smooth over Oλ for `λ  0, and hence Gkλis smooth. The theorem now follows from Theorem 5.44, once we verified hypotheses (i)–(iii)
listed at the beginning of Subsection 5.5 for Gkoλ ⊂ GLn,kλ .
For this let R0 be the set of weights of Tkλ acting on V̄λ := kλn and let Φ be the root system of
Gkλ . By Proposition 5.51, the set Φ is also the root system of the reductive group Gλ , and thisshows that conditions (i) and (ii) hold.
For (iii) note, again by Proposition 5.51, that the set R0 is also the set of weights of Tλ for its
action on Eλn . The sets Φ and R0 are the sets of roots of the split motivic group and of the split
motivic representation attached to ρ• over some finite extension F/E, respectively, and hence byCorollary 4.7 these sets are independent of λ. Thus also R = {α − β | α, β ∈ R0 } is independentof λ, and it is the set of weights for the action of Tkλ on End(V̄λ ). It is now clear that condition(iii) must hold for `λ  0.

6Residual saturation in the absolutely irreducible caseIn this section we shall prove two main results on E-rational compatible systems ρ• that are
absolutely irreducible and connected, Theorem 6.11 and Theorem 6.14, and in Theorem 6.8 an
important result on lifting (arbitrary) residually compatible systems. The first theorem showsthat their reductions ρ̄λ are absolutely irreducible for almost all λ, even after restriction to
π1geo (X). For the second, suppose each ρλ takes its image in GLn (Oλ ) and let Gλgeo denote theZariski closure of π1geo (X) therein. Then we show that for almost all λ the Oλ -group scheme
Gλgeo is semisimple connected, and its special fiber is the saturation of ρ̄λ (π1geo (X)). Our proofs
combine L. Lafforgue’s global Langlands conjecture, and a conjecture of de Jong, as proved by
Gaitsgory. Whenever the image is smaller than expected, we construct a congruence between
automorphic forms, to a given automorphic form. Since we manage to bound the candidates
for such a congruence within a finite set, only finitely many congruences are allowed. The firstauthor had learned the usefulness of similar congruence arguments from Ribet, e.g. [Wes04,
Prop. 5.3]. In fact, by our very approach, after some preliminary observations in Subsection 6.1,
we shall in Subsection 6.2 first prove that any (suitably defined) potentially tame residuallycompatible system is the reduction of a compatible system. From this we shall quickly deduce
Theorem 6.11 in Subsection 6.3. Subsection 6.4 contains the proof of Theorem 6.14. A proofof Theorem 6.11 was outlined by us in [Böc16]. However we recently learned that in [Dri15,
Prop. E.10.1] Drinfeld had given such a proof by essentially the same method. Since we give
more details and also want to stress the assertion of Theorem 6.8 we do include proofs.

6.1

On central characters and lifting mod λ representations

This subsection contains some auxiliary results. We begin by defining two natural reductionsof a λ-adic representation ρλ which is a member of an E-rational compatible system ρ• . Then
we provide some results on 1-dimensional compatible systems that will be relevant to the lifting
techniques used in the later subsections, and we also provide two basic results on lifting.
44

Let Λλ be an Oλ -lattice in Eλn that is stable under the action of π1 (X) via ρλ ; the existence ofsuch a lattice can be deduced from π1 (X) being a profinite group. Then the image of ρλ liesin AutOλ (Λλ ), and the latter group has a reduction map to AutFλ (Λλ /$λ Λλ ) ∼= GLn (kλ ). We
Λλdefine ρ̄Λ
:=
ρ̄
as
the
composite
of
ρ
with
the
just
definedreduction
map,
i.e.,
as the actionλλλof π1 (X) on Λλ /$λ Λλ . The semisimplification of ρ̄Λ
wesimplydenotebyρ̄.
λλThe theorem of Brauer and Nesbitt (see [CR62, 30.16]) shows that ρ̄λ is independent of anyΛ∼ Λ
choices. In general ρ̄Λ
λ depends on Λλ . If ρ̄λ is irreducible, then ρ̄λ = ρ̄λ , and so then ρ̄λ is
independent of the choice of Λλ . In this section we mainly deal with ρ̄λ , in Section 7 with ρ̄Λλ.
Remark 6.1. Let E 0 be a finite extension of E and let ρ0• be the compatible system ρ• ⊗E E 0 .
0000
Let λ0 ∈ PE
0 have contraction λ to PE and define Λλ0 := Λλ ⊗Oλ Oλ0 . Then one has a functorial
isomorphisms for the respective reductions0∼ Λ(ρ̄0 )Λ
λ0 = ρ̄λ ⊗kλ kλ0

and ρ̄0λ0 ∼= ρ̄λ ⊗kλ kλ0 ,where kλ → kλ0 is the map of the residue field induced from E → E 0 at λ0 . Both isomorphisms are straightforward. The second uses (again) the theorem of Brauer and Nesbitt and thesemisimplicity of the reductions. It follows that if ρ̄λ is absolutely irreducible, then so is ρλ .
We begin with some preparations for Theorem 6.8 in the following subsection.
Lemma 6.2. Let α1 , . . . , αn be plain of characteristic p, and let x be in |X|.
Q For any partition℘ = (℘1 , . . . , ℘t ) of n and j ∈ {1, . . . , t} define α℘,j as a [κx : κ]-th root of i∈℘j αi and defineρ℘,j,• as the 1-dimensional compatible system ρα℘,j ,• . Let X 0 → X be a finite cover, and let Ξ
be the set of continuous characters τ : π1 (X) → GL1 (E) that are trivial when restricted to π1 (x)and factor via π1 (X 0 )ab,p when restricted to π1 (X 0 ). Then(a) each α℘,j is plain of characteristic p, and hence ρ℘,j,• is well-defined;(b) the set Ξ is finite;(c) each representation
δ̄λ : π1 (X) → GL1 (kλ0 ) such that δ̄λ |π1 (X 0 ) factors via π1ab,p (X 0 ) andQδ̄λ (Frobx ) = i∈℘j (αi (mod λ)) is congruent modulo λ to ρτ,λ ⊗ ρ℘,j,λ for some τ ∈ Ξ.
(d) if all αi are q-Weil numbers of weight w, then all α℘,j are q-Weil numbers of weight w·#℘j .
Proof. For (a) observe that plainness of characteristic p is preserved under the formation of products and roots. Part (b) follows from Theorem 3.16. Regarding (c) note that the 1-dimensional
representation δ̄λ ⊗ (ρ−1℘,j,λ (mod λ)) of π1 (X) is trivial when restricted to π1 (x) and factors via
π1 (X 0 )ab,p , and thus it is of the form ρτ,λ (mod λ) for some τ ∈ Ξ; note that any representation
π1 (X) → GL1 (kλ0 ) that is trivial on π1 (x) and factors via π1 (X 0 )ab,p , has Teichmüller lift in Ξ.
Part (d) is clear from the definition of α℘,j using that x is of degree [κx : κ] over κ.
Lemma 6.3. Suppose that ρ• is n-dimensional. Fix x ∈ |X| and denote by α1 , . . . , αn the rootsof Px (T ) ∈ E[T ] in a fixed algebraic closure E of E. For any partition ℘ = (℘1 , . . . , ℘t ) of n and0j ∈ {1, . . . , t} define α℘,j as in Lemma 6.2. For each λ ∈ PE
and embedding kλ → k λ choose an
isomorphism
t
M
ρ̄λ,j
ρ̄λ ⊗kλ k λ ∼
=
j=1

45

with each ρ̄λ,j irreducible. Then there exists a finite set Ξ of characters τ : π1 (X) → GL1 (E)0such that for each λ ∈ PE
there exists a partition ℘ = (℘1 , . . . , ℘t ) of n and a character tupletτ = (τ1 , . . . , τt ) in Ξ , with (℘, τ ) depending on λ, such that for all j = 1, . . . , t we havedet ρ̄λ,j = ρ̄τj ,λ ⊗ ρ̄℘,j,λ .
0
Proof. Fix a finite cover X 0 → X such that for all λ ∈ PE
the ramification of ρλ |π1 (X 0 ) is pro`λ , cf. Lemma 3.31, and define Ξ as in Lemma 6.2 for this cover, upon noting that α1 , . . . , αnare plain of characteristic p by Remark 2.1. Now the reduction Px (T ) mod λ is the productof the characteristic polynomials of the ρ̄λ,j (FrobxQ), j = 1, . . . , t. Hence there is a partition℘ = (℘1 , . . . , ℘t ) of n such that charpolρ̄λ,j (Frobx ) = i∈℘j (T − αi(mod λ)), and thus[κ :κ]det ρ̄λ,j (Frobx ) = α℘,jx(mod λ) =Y

αi (mod λ).

i∈℘j

The proof of the lemma is now implied by Lemma 6.2(c).
The following result is based on de Jong’s conjecture from [dJ01] and its solution by Gaitsgoryfor ` > 2 in [Gai07], and of de Jong for n ≤ 2. We recall it in the form needed below.
Lemma 6.4 (de Jong and Gaitsgory). Let X be a smooth projective geometrically irreducible
curve over κ and let S ⊂ |X| be a finite subset. Let k be a field of characteristic ` > n withring of Witt vectors W (k) and K = W (k)[1/`]. Let ρ̄ : π1 (X r S) → GLn (k) be a continuous
absolutely irreducible representation, and let η : π1 (X r S) → GL1 (OL ) be a character for somefinite extension L of K such that the reduction of η agrees with det ρ̄.
Then there exists a finite extension L0 of L and a continuous representation ρ : π1 (X) →GLn (OL0 ) with det ρ = η and reduction isomorphic to ρ̄. If moreover ρ̄ is at most tamely
ramified (along S), then the conductor of ρ is bounded by nS.
Proof. Denote by R the universal ring for deformations ρ : π1 (X) → GLn (R) of ρ̄ with det ρ = η,where R ranges over all complete noetherian local OL -algebras with maximal ideal mR and
residue field k = OL /mOL ; see [Maz89] and [dJ01, 3.1-3.3] for the precise definition of thedeformation problem and the existence of a universal deformation ρe: π1 (X) → GLn (R) using
the absolute irreducibility of ρ̄.
Now de Jong’s conjecture [dJ01, Conj. 2.3] for ` > 2 is known by work of Gaitsgory; see [Gai07,
proof of Conj. 1.3]. Therefore by [dJ01, Thm. 3.5], the resulting homomorphism OL → R is a
finite flat complete intersection morphism. In particular R ⊗OL L is a finite non-empty productof finite extensions L0 of L. Fix one L0 . The corresponding map p : R → L0 takes its image inthe ring of integers OL0 of L0 , and defining the representation ρ := p ◦ ρe: π1 (X) → GLn (OL0 )proves the first assertion of the lemma.
Regarding the second assertion, observe that if ρ̄ is at most tamely ramified, the same holdsfor ρ, since the kernel of ρ(π1 (X)) → ρ̄(π1 (X)) is a pro-` group. But the conductor of a tame
representation, at any place of S, of an n-dimensional representation is bounded by n.
For later use, we also state the following variant of Lemma 6.4.

46

Lemma 6.5 ([BHKT16, Thm. 5.14]). Let X be a smooth proper geometrically connected curve
over κ, and S ⊂ |X| be a finite set. Let k be a finite field of characteristic ` 6= p with ` > 2(n−1).
Let G be semisimple connected and split affine algebraic group over W (k) that has an almost
faithful representation i : G ×W (k) k → GLn,k . Let ρ̄ : π1 (X r S) → G(k) be a representation suchthat i ◦ ρ̄ is absolutely irreducible. Then there exists a finite extension E of W (k)[1/`] and acontinuous homomorphism ρ : π1 (X r S) → G(OE ) such that ρ (mod mOE ) = ρ̄.

6.2

Potentially tame mod λ compatible systems lift

0
Definition 6.6. An n-dimensional E-rational mod λ compatible system over L ⊂ PE
is a tuple((ρ̄λ )λ∈L , (Px )x∈|X| ),or for short ρ̄• , that consists of a homomorphism ρ̄λ : π1 (X) → GLn (kλ ) for every λ ∈ L, and amonic degree n polynomial Px ∈ E[T ] for each x ∈ |X|, such that(a) ρ̄λ is semisimple for all λ ∈ L,(b) Px is plain of characteristic p for all x ∈ |X|, and(c) charpolρ̄λ (Frobx ) (T ) = Px (T ) (mod λ) for all (λ, x) ∈ L × |X|
The system is called pure of weight w, if Px is pure of weight w for #κx for all x ∈ |X|.
The notions tame and potentially tame are defined as for compatible systems, using ρ̄λ in placeof ρλ .
For E → E 0 a finite extension, we define the mod λ compatible system ρ̄• ⊗OE OE 0 over L0 , the00subset of PE
0 of places above L, by setting ρ̄µ := ρ̄λ ⊗kλ kµ for µ ∈ L and λ ∈ L its contraction,00and by considering Px ∈ E [T ] via E → E ; the ρ̄µ are again semisimple by [CR62, 69.8].
Remark 6.7. Note the following:(a) If ρ• is an n-dimensional E-rational compatible system, then (ρ̄λ )λ∈PE0 , defined as at the
beginning of Subsection 6.1, is an n-dimensional E-rational mod λ compatible system over0PE
. We then call ρ̄• the reduction of ρ• . If ρ• is pure, tame, potentially tame, then thesame property holds for ρ̄•0(b) Given an E-rational mod λ compatible system over some infinite subset L ⊂ PE
, thepolynomials (Px )x∈|X| are uniquely determined by the representations (ρ̄λ )λ∈L and condition (c). In particular, by Lemma 3.1, there exists up to isomorphism at most one
E-rational compatible system ρ• whose reduction restricted to L agrees with ρ̄• .

Theorem 6.8. Suppose X is a geometrically connected curve over κ and ρ̄• is a potentially tame
E-rational n-dimensional mod λ compatible system over an infinite set L. Then there exists a0
finite extension E 0 of E and an E 0 -rational compatible system ρ• (over PE0 ) whose semisimplified
0
0
0
0
0
reduction is ρ̄• ⊗OE OE over L := {λ ∈ PE 0 | λ lies above some λ ∈ L}.
If ρ̄• is absolutely irreducible for some λ0 , then ρ• is absolutely irreducible. If ρ̄λ ⊗kλ k̄λ isreducible for infinitely many λ ∈ L, then ρ• is reducible.

47

Proof. Let X 0 → X be a finite Galois cover over which all ρ̄• are tame. Fix x ∈ |X| and denoteby α1 , . . . , αn the roots of Px (T ) ∈ E[T ] in a fixed algebraic closure E of E. They are plainof characteristic p. For any partition ℘ = (℘1 , . . . , ℘t ) of n and j ∈ {1, . . . , t} define α℘,j as inLemma 6.2.
0
For each λ ∈ PE
choose a finite extension kλ → kλ0 over which there is an isomorphismρ̄λ ⊗kλ kλ0 ∼
=

tλ
M

ρ̄λ,j

j=1

with each ρ̄λ,j absolutely irreducible. Then following the proof of Lemma 6.3 there exists a0
there exists a partition
finite set Ξ of characters τ : π1 (X) → GL1 (E) such that for each λ ∈ PE
℘λ = (℘λ,1 , . . . , ℘λ,tλ ) of n and a character tuple τ λ = (τλ,1 , . . . , τλ,tλ ) in Ξtλ , such that for all
j = 1, . . . , tλ we havedet ρ̄λ,j = ρ̄τλ,j ,λ ⊗ ρ̄α℘λ ,j .
By Lemma 6.4, we obtain a finite extension L of Eλ and for each j ∈ {1, . . . , tλ } a representation
ρλ,j : π1 (X) → GLnλ,j (OL )that is a lift of ρ̄λ,j with det ρλ,j = ρτλ,j ,λ ⊗ ρα℘λ ,j ,λ . Note also that ρλ,j is absolutely irreducible,
because ρ̄λ,j is so.
Let S be the set of places of κ(X) not in |X|, and let S 0 be the places of κ(X 0 ) above S. Let
Ks be the local field at s, Ks0 0 be the extension corresponding to s0 ∈ S 0 above s. Then any
ρλ,j restricted to ΓKs0 0 is tame. Considering the higher ramification filtration, it follows that
there is a uniform bound M on the wild conductors at any s ∈ S of any ρλ,j , supported on S.
The tame conductor of ρλ,j is bounded by ℘λ,j S. By Theorem 3.30 there is an automorphic
representation Πλ,j of GLn/Aκ(X) with central character τj and conductor bounded by nλ,j S + M
with ρΠλ,j ,λ ∼= ρλ,j ⊗ρ−1℘,j,λ . Define S as the set of tuples (℘, τ , Π1 , . . . , Πt ) where ℘ = (℘1 , . . . , ℘t )is a partition of n, each τ lies in Ξt , each Πj is a cuspidal automorphic representation for GL#℘jover Aκ(X) of conductor bounded by ℘j S + M and central character that agrees with τj . Now
by [BJ79, Cor. 5.3] the set S is finite.
From the finiteness of S it follows that there exists a tuple (℘, τ , Π1 , . . . , Πt ) in S and an infinite
subset L0 of L such that for all λ ∈ L0 we have ρλ,j ∼= ρΠj ,λ ⊗ ρ℘,j,λ . Let E 0 be a finite extension
0tof E over which the compatible system ρ• := ⊕j=1 ρΠj ,• ⊗ ρ℘,j,• is defined. By the choice of L00
we have for all λ ∈ L0 an isomorphism ρ0µ (mod µ) ∼= ρ̄λ ⊗kλ kµ for any µ ∈ PE
0 above λ. This00
implies that the polynomials Px for ρ̄• and for ρ• agree in E [T ] for all x ∈ |X|. But then it
follows thatcharpolρ0µ (Frobx ) = charpolρ̄λ (Frobx ) ⊗kλ kµ
for all x ∈ |X| and µ ∈ L0 . The Čebotarov density theorem now implies that ρ̄• ⊗OE OE 0 is the
reduction of ρ• , and this concludes the proof of the main part.
The first part of the final assertion follows from Remark 6.1 and Lemma 4.1. To see the second
part, let Lred be the set of places in L at which ρ̄λ ⊗kλ k̄λ is reducible, and let S red be the setof tuples (℘, τ , Π1 , . . . , Πt ) in S such that t > 1. Then each ρ̄λ , λ ∈ Lred arises via reduction
from some tuple in S red . Repeating the argument from the previous paragraph, we find that48

(ρ̄λ )λ∈Lred is the reduction of a reducible compatible system ρ0• over some E 0 , where in fact ρ0•0
red
is defined for all λ ∈ PE
, we
0 . By the uniqueness of the compatible system for L and for L
0deduce that ρ• constructed for L is reducible over E .
Remark 6.9. If X is not a curve, the above result can be applied to the restriction of thesystem to any curve C ,→ X. Following [Dri12, § 4] this allows one to extend Theorem 6.8
to arbitrary normal, geometrically irreducible finite type varieties X over κ under a potential
tameness hypothesis.
Question 6.10. We do not know if the hypothesis of potential tameness is needed in Theorem 6.8.
Can one construct counterexamples if L is infinite, or even if L has density one?

6.3

Mod λ absolute irreducibility for almost all λ.

Theorem 6.11 (cf. [Dri15, Prop. E.10.1]). Suppose ρ• is absolutely irreducible and E-rational.
0
the representation ρ̄λ is absolutely irreducible.
Then for almost all λ ∈ PE
Proof. By Lemma 4.1(c) and Lemma 3.1, there is a smooth geometrically irreducible curve
C ⊂ Xreg such that ρ• |π1 (C) is absolutely irreducible. If now ρ̄µ |π1 (C) is absolutely irreducible
for almost all µ, then the same will hold for ρ̄µ ; hence from now on we assume that X is a smoothand geometrically irreducible curve. It follows from Lemma 3.31 that ρ• is potentially tame.
0
at which ρ̄λ ⊗kλ k̄λ is reducible. Assume that L is infinite.
Let L be the set of places of PE
Then by Theorem 6.8, there exists a finite extension E 0 of E and an E 0 -rational compatible000system ρ0• (over PE:= {λ0 ∈ PE0 ) whose semisimplified reduction is ρ̄• ⊗OE OE 0 over L0 |0
λ lies above some λ ∈ L}, and moreover this system is reducible. Now by Remark 6.7(b) wemust have ρ• ⊗E E 0 ∼= ρ0• , which is a contradiction.

Corollary 6.12. Suppose ρ• is absolutely irreducible, connected and E-rational. Then for almost0
the representation ρ̄λ |π1geo (X) is absolutely irreducible.
all λ ∈ PE
0
Proof. Let L be the set of places λ of PE
with `λ > n for which ρ̄λ is absolutely irreducible, butρ̄λ |π1geo (X) is not. Let λ be in L. The short exact sequence 1 → π1geo (X) → π1 (X) → Γκ → 1
induces a short exact sequence

1 → Nλ := ρ̄λ (π1geo (X)) → Hλ := ρ̄λ (π1 (X)) → Cλ → 1,
where Cλ is defined to be the cokernel of the morphism on the left. Clearly Cλ is cyclic of finitenorder. By hypothesis the action of Hλ on Vλ := k λ defined by ρ̄λ is irreducible. By Clifford
theory, cf. [CR62, § 49, 50], the module Vλ is completely reducible for the action of Nλ , andif Wλ is an irreducible Nλ -submodule of Vλ , and Nλ∗ ⊇ Nλ the stabilizer of Wλ in Hλ , thenH
Vλ ∼= IndNλλ∗ Wλ . It follows that the index [Hλ : Nλ∗ ] is at most of size n. Let κ0 be the uniqueextension of κ of degree n! and X 0 → X the base change of X along Spec κ0 → Spec κ. Thenfor all λ ∈ L, the restriction ρ̄λ |π1 (X 0 ) is not absolutely irreducible. Because ρ• is absolutely
irreducible and connected, the same holds for ρ• |π1 (X 0 ) . Thus L is finite by Theorem 6.11.
Corollary 6.13. Suppose ρ• is absolutely irreducible, connected and E-rational. Then for almost
geo
geo
+ sat
0
o
sat o
all λ ∈ PE
we have (ρ̄λ (π1 (X))satkλ ) = (ρ̄λ (π1 (X))kλ ) = (ρ̄λ (π1 (X)) )kλ and this group isconnected semisimple.
49

Proof. Let G = ρ̄λ (π1 (X)) and N = ρ̄λ (π1geo (X)). Then N and N + are normal subgroups of G,
G/N is finite cyclic, and N/N + of order prime to `. By first applying Theorem 3.34 and then0
Corollary 6.12, it follows that N + acts absolutely irreducibly on kλn for almost all λ ∈ PE
. By
sat
sat
+ sat
Proposition 5.18(b) we deduce that Gkλ , Nkλ and (N )kλ have a reductive identity component
o
sat o
+
for almost all λ. Now by Proposition 5.20 we have (Gsatiskλ ) = (Nkλ ) , and because N/N+ sat osat o
+ sat
finite of order prime to `, by Corollary 5.16 we have ((N )kλ ) = (Nkλ ) . Finally ((N )kλ ) isconnected semisimple by Proposition 5.18(c).

6.4Mod λ saturatedness for almost all λ.

0Let ρ• be an E-rational compatible system. For each λ ∈ PE
, let Λλ ⊂ Eλn be a π1 (X)-stablegeo
Oλ lattice. Define Gλ (resp. Gλ ) as the Zariski closure of ρλ (π1 (X)) (resp. ρλ (π1geo (X))) ingeo
geo
AutOλ (Λλ ) ∼= GLn (Oλ ). Define Gkλ := Gλ ⊗Oλ kλ and Gkλ := Gλ ⊗Oλ kλ .

Theorem 6.14. Suppose that ρ• is absolutely irreducible, connected and that ρ• |π1geo (X) is con0
the following hold:nected.14 Then for almost all λ ∈ PE(a) Gkgeo
is connected, semisimple and saturated; thus Gλgeo is connected semisimple over Oλ .
λgeo(b) We have ρ̄λ (π1geo (X))satkλ = Gkλ .
geo
geo
sat
sat
(c) If Ggeo
λ = Gλ , then ρ̄λ (π1 (X))kλ = Gkλ = Gkλ = ρ̄λ (π1 (X))kλ .

(d) If Ggeoλ ( Gλ , denoting by Zλ the central torus of GLn/Oλ , then we have(i) Gλ = Gλgeo Zλ , the group Gλ is connected reductive over Oλ , ando= Gkgeo.
(ii) ρ̄λ (π1 (X))satkλ
λ0: By Lemma 4.1(c),Proof. We first assert that Gλgeo is smooth over Oλ for almost all λ ∈ PE
there exists a 1-dimensional compatible system δ• such that Gρ• |πgeo (X) ,λ = Gρ• ⊗δ• |πgeo (X) ,λ , and1
1
at the same time Gρ• ⊗δ• ,λ has finite order determinant. This implies Gρ• |πgeo (X) ,λ = Goρ• ⊗δ• ,λ by
1

Proposition 4.5. Clearly there is a finite cover X 0 → X such that ρ• ⊗δ• |π1 (X 0 ) and ρ• ⊗δ• |π1geo (X 0 )are connected. Because ρ• |π1geo (X) is connected, we have Gρ• |πgeo (X) ,λ = Gρ• ⊗δ• |π (X 0 ) ,λ . Now our11
assertion follows from Proposition 5.51.
0For (a) let L be the set of places λ ∈ PE
such that ρ̄• |π1geo (X) is absolutely irreducible and Gλgeo
0is smooth over Oλ . By Corollary 6.12 and the above assertion, the set PE
r L is finite. For
geo
geo
Λλ
λ ∈ L we have ρ̄λ = ρ̄λ and it follows that ρ̄λ (π1 (X)) ⊂ Gkλ (kλ ) by the definition of Gλgeo .
Hence the action of Gkgeo
on kλn via its embedding into GLn,kλ is irreducible. By the proof ofλ[Win02, Thme. 1] the identity component of Gkgeo
is reductive – the argument deduces fromλthe semisimplicity of the action that the unipotent radical of (Gkgeo)o , which is a characteristicλsubgroup, must vanish. It follows from [Con11, Prop. 3.1.3] that (Gλgeo )o is a reductive groupover Oλ that is open and closed in Gλgeo . Since Gλgeo was defined as the closure of the connectedgeo
are connected reductive; theygroup Ggeo
= (Gλgeo )o and hence Gλgeo and Gkgeo
λ , we deduce Gλλgeo
are semisimple because Gλ is semisimple. From Theorem 5.52 (and the reduction in the firstis also saturated.
paragraph), it follows that Gkgeo
λ14 See Corollary 7.6 for a formulation that does not require any connectedness hypothesis.

50

geo
To prove (b) note first that by (a) we have ρ̄λ (π1geo (X))satkλ ⊆ Gkλ for λ ∈ L, since the group on0
the right is saturated. Let X → X be a finite cover such that ρ̄λ (π1geo (X 0 )) is `λ -generated foralmost all λ; it exists by Remark 3.35. Note that this property is preserved under any further
cover X 00 → X 0 since the index [ρ̄λ (π1geo (X 0 )) : ρ̄λ (π1geo (X 00 ))] is smaller than `λ for almost all
geo
geo
sat
λ. Set Hkλ := ρ̄λ (π1 (X))satkλ and Hkλ := ρ̄λ (π1 (X))kλ , and observe by Corollary 6.13 thatgeo
Hkλ is semisimple and connected for almost all λ. Enlarging the finite base field κ to its uniqueextension of degree n!, and using Proposition 5.20, we may assume from now on that Hkgeo= Hkλλfor almost all λ. After another cover from Lemma 3.31, and then by applying Corollary 3.33,
we may further assume that each ρλ |π1 (X 0 ) is at most `λ -ramified, and that there is a smooth
curve C on X 0 such that ρλ (π1 (X 0 )) = ρ0λ (π1 (C)) for ρ0λ := ρλ |π1 (C) for all λ; in particular ρ0• is
geotame. To complete the proof of (b) it will suffice to prove that ρ̄λ (π1geo (C))satkλ = Gkλ for almost
geo
0
all λ ∈ L. Note also that Gλ and Gλ are unchanged if we replace X by X by the connectivity
hypotheses of the theorem, and that Hkλ = ρ̄λ (π1geo (C))satkλ , for almost all λ.

Let L0 ⊆ L be the set of places λ where ρ̄0λ (π1geo (C)) is `λ -generated, Hkgeo= Hkλ , and Gλ , Gλgeo
λ0
are smooth over Oλ , and `λ > 2(n − 1), and ρ̄λ is absolutely irreducible. By the results quoted0
r L0 is finite. By the reduction argument from the first paragraph, to prove (b) we
so far PE
00
0
may also assume that Gλ = Ggeo
λ . Let L ⊂ L be the set of places λ at which Hkλ ( Gkλ , and000fix λ ∈ L . Let kλ → kλ be a finite extension such that Hkλ0 := Hkλ ⊗kλ kλ0 is split, connected
and semisimple. Denote by Vkλ0 the representation of Hkλ0 via its embedding into SLn . Thenfrom Lemma 5.37 and Proposition 5.40. we obtain a semisimple connected group scheme Hλover W (kλ0 ) and a representation Vλ of Hλ defined over W (kλ0 ) whose reductions to kλ0 agreewith Hkλ0 and Vkλ0 . Since W (kλ0 ) is regular local, we may identify Vλ with W (kλ0 )n . Because Hλis semisimple connected, its image under the representation ιλ : Hλ → AutW (k) (Vλ ) must thenlie in SLn,W (kλ0 ) .
At this point, Lemma 6.5 provides a representation ρ00λ : π1 (C) → Hλ (Oλ0 ) where Oλ0 is thevaluation ring of some finite extension Eλ0 of W (kλ0 )[1/`λ ]. The composition ιλ ◦ ρ00λ : π1 (C) →SLn (Oλ0 ) is then a lift of ρ̄0λ . Since ρ̄0λ is absolutely irreducible, so is the lift, if regarded as arepresentation into SLn (Eλ0 ). Because det ρ̄0λ is trivial and `λ > n, det ιλ ◦ ρ00λ is trivial. Notealso that the monodromy group of ιλ ◦ ρ00λ has smaller dimension than Gλ , because Hkλ ( Gkλand both are connected. Since ρ̄0λ is `λ -tame, this also holds for ρ00λ , and hence the conductor ofιλ ◦ ρ00λ is at most nS where S is the set of places of κ(C) not in |C|, and so this conductor isindependent of λ. Let S be the set of cuspidal automorphic representations Π for GLn over Aκ(C)of conductor bounded by nS and with trivial central character. It follows from Theorem 3.30
that there is an automorphic representation Πλ in S with ρΠλ ,λ ∼= ιλ ◦ ρ00λ . Arguing as at the end
00of the proof of Theorem 6.11, it follows that first S and then L is finite, proving (b). Part (c) is
geo
sat
00
immediate from (a), (b) and Gkgeo= ρ̄λ (π1geo (X))satkλ ⊆ ρ̄λ (π1 (X))kλ ⊆ Gkλ = Gkλ for λ ∈ LrL .
λ0
We now turn to (d), and thus assume Ggeoλ ( Gλ . To prove (i), we let X → X and δ• be as ingeo
the first paragraph. Then ρ• ⊗ δ• |π1 (X 0 ) has trivial determinant and Gρ• ,λ = Gρ• ⊗δ• ,λ . It follows
that Gλgeo ⊆ Gλ ⊆ Zλ Gλgeo . As det ρ• cannot have finite order in (d), det must surject Gλ onto
Gm , mapping the semisimple group Gλgeo to 1. Since Zλ is 1-dimensional connected, it follows
that Gλ ⊆ Zλ Gλgeo must be an equality. Also, because Gλgeo is connected semisimple over Oλand Zλ is a torus, the group Gλ is connected reductive over Oλ , and completes the proof of (i).
Finally (ii) follows from (a) and (i), by which Gkλ is saturated and Gλ (kλ )/Gλgeo (kλ ) is finite oforder prime to `λ .

51

Remark 6.15. Theorem 6.14 for Q-rational compatible systems but only for a set of primes ofdensity one and not all but finitely many primes is due to Larsen; cf. [Lar95]. The result for
Q-rational compatible systems that are cohomological in the sense of the introduction is provedin [CHT17].
Remark 6.16. Given [CHT17, Thm. 1.2], the following argument, whose details remain to be completed, gives a proof of Theorem 6.14 for arbitrary Q-rational absolutely irreducible connectedcompatible systems ρ• such that ρ• and ρ• |π1geo (X) are connected: Lafforgue’s proof [Laf02] of
the Langlands correspondence for global function fields, partially quoted in Theorem 3.30, showsthat ρ• occurs in the étale cohomology of a (not necessarily smooth) scheme over the function
field. Presumably by an argument like [BGP, Thm. 7.3] there is a finite number of smooth projective varieties over the function field such that ρ• occurs in the direct sum of their étale `-adic
cohomologies (though for each ` it is not clear in which summand). Now [CHT17, Thm. 1.2]
gives the semisimplicity and saturatedness of the reduction for the direct sum for all `  0.
This in turn should allow one to deduce the same for the factor corresponding to ρ` , and hence
Theorem 6.14 for ρ• .

7

Independence of lattices and main result on saturation

Suppose ρ• is E-rational compatible system which is a direct sum of absolutely irreducible compatible systems and has semisimple monodromy. In the first subsection we shall explain that thesame monodromy groups, even integrally, arise from an absolutely irreducible compatible system,
up to a finite kernel. In the second subsection, this will allow us in Corollary 7.6 to generalize
Theorem 6.14 to any system ρ• , i.e., the integral monodromy groups are smooth semisimple and0. This generalizes some of
their special fibers are the saturation of ρ̄• , for almost all λ ∈ PE
the main results from [CHT17] from Q- to E-rational compatible systems. Our Corollary 7.7
gives an alternative proof of [CHT17, Thm. 1.2]. In Remark 7.8 we indicate on argument bywhich their main result implies Corollary 7.6 for Q-rational systems. Finally, in Corollary 7.90, as conjectured by Larsen
we deduce that the groups Gρ• ,λ are unramified for almost all λ ∈ PE
and Pink (cf. [LP95, Conj. 5.4]).

7.1

Saturation under changing representations

In this subsection we prove a criterion for when two representations whose generic monodromy
groups differ by a central isogeny have saturations of their reductions that also differ by such an
isogeny. Throughout this subsection, we let ` be in PQ0 with ` > 2 and we let K/Q` be a finite
extension with the ring of integers O, residue field k and maximal ideal m.
Lemma 7.1. Let M be a semisimple connected group over K. Let α0 : M → GLn0 be a faithful,and α : M → GLn be an almost faithful representation. Let ϕ : π1 (X) → M (K) be a homomorphism with Zariski dense image. Define ρ = α ◦ ϕ, ρ0 = α0 ◦ ϕ and ρ00 = ρ ⊕ ρ0 , let Λ ⊂ K n
0
000
and Λ0 ⊂ K n be π1 (X)-stable O-lattices, and set Λ00 := Λ ⊕ Λ0 . Denote by ρ̄Λ , ρ̄0,Λ and ρ̄00,Λ
the corresponding reductions modulo m, cf. the beginning of Subsection 6.1. Denote by MK ,M0K and M00K the image of M under α, α0 and α ⊕ α0 , respectively, and by M, M0 , M00 theO-group schemes that are the Zariski closures of MK in AutO (Λ), of M0K in AutO (Λ0 ), and of
M00K in AutO (Λ)×AutO (Λ0 ) ⊂ AutO (Λ00 ), respectively. Suppose that52

(i) the group schemes M, M0 , M00 are smooth over O;(ii) the identity components of their special fibers Mk , M0k and M00k are saturated;(iii) the kernel of α is of order strictly less than `.
(iv) the saturation ρ̄Λ (π1geo (X))satk is semisimple connected and agrees with Mk .
Denote by pr0 : AutO (Λ) × AutO (Λ0 ) → AutO (Λ0 ) and pr : AutO (Λ) × AutO (Λ0 ) → AutO (Λ) thenatural projection homomorphisms. We add the index O for their restrictions prO : M00 → M
and pr0O : M00 → M0 . Then the following hold:(a) M0 over O and M0k are connected semisimple, and the analogous assertion holds for M00 .
(b) Mk , M0k and M00k are saturated in GLn,k , GLn0 ,k and GLn+n0 ,k , respectively.
(c) pr0O is an isomorphism; prO is a central étale isogeny, and an isomorphism if α is faithful.
00

0

00
0,Λ
0
(d) The inclusions ρ̄00,Λ (π1geo (X))sat(π1geo (X))satk ⊆ Mk and ρ̄k ⊆ Mk from (b) are equalities.

Proof. Denote by prk : M00k → Mk and pr0k : M00k → M0k the morphisms induced from prO and
pr0O on special fibers, and consider the diagrams

3M
O

ρ



prO
ρ00

π1 (X)/ M00  

pr0O
ρ0


+ M0  / AutO (Λ)
O

Λ geo
sat 
4 ρ̄ (π1 O (X))k
sat

ρ̄Λ

pr

00

/ AutO (Λ) × AutO (Λ0 )π1 (X)

ρ̄00,Λ

/ AutO (Λ0 )

ρ̄0,Λ

0

ι

*

(pr0 )sat
k

0

ι00



/ M00
k

k


ρ̄0,Λ (π1geo (X))sat
k

/ Mk
O

prk

prk

/ ρ̄00,Λ00 (π geo (X))sat  
1

pr0



pr0k
ι0


/ M0 ,kwhere on the right we display the relevant situation on special fibers, and on the left over O.
The closed immersion ι exists by hypotheses, the existence of ι0 , ι00 still needs a proof.
Since α is almost faithful and α0 is faithful, the maps M → M00K and M → M0K are isomorphisms
and M → MK is a central isogeny with finite kernel. In particular dim M = dim MK =dim M0K = dim M00K , and by (i) the groups M, M0 and M00 all have the same relative dimension
over O. Since the restriction of prO and pr0O to the generic fibers are clearly surjective, the maps
pr0O and prO are surjective homomorphisms by the definition of M, M0 and M00 . Hence prk andpr0k are surjective, as well. For dimension reasons, their kernels must be finite. The unipotent
radical of (M00k )o maps to a normal unipotent subgroup of Mk , and the induced map has finite
kernel. Since Mk is connected semisimple, the unipotent radical of (M00k )o must be trivial andhence (M00k )o is reductive. An analogous argument applies to centers and it follows that (M00k )o is
semisimple. From the surjectivity of pr0k it follows similarly that (M0k )o is semisimple. As in the
proof of Theorem 6.14, by [Con11, Prop. 3.1.3] it follows that (M0 )o and (M00 )o are semisimple,
and open and closed in M0 a M00 , respectively. Since by assumption their generic fibers areconnected, this proves (a). Part (b) follows immediately from (a) and (ii). Note that (b) also
gives the existence of ι0 and ι00 as well as the inclusions in (e).
Concerning (c) note first that by [Con11, Prop. 6.1.10] the maps prO and pr0O in (c) and (d)are isogenies. Hence their kernels are finite flat over O by [Con11, Def. 3.3.9]. By flatness and53

considering generic fibers, it follows that the kernel of pr0O is trivial so that pr0O is an isomorphism
and that the kernel of prO is flat over O of order strictly less than `. In particular Ker prO is
finite étale over O, so that Ker prk is separable and hence central. Now by [Con11, Prop. 3.3.10]
pr is central. If moreover α is faithful, then prO is an isomorphism by the proof for pr0O . This
proves (c).
We finally prove (d). We first note that by Proposition 5.19, the maps prsatand (pr0 )sat
are
k
k
geo
sat
00,Λ00well-defined and surjective. By (iv) it follows that dim ρ̄(π1 (X))k ≥ dim Mk and fromthe existence of ι00 and from (c) the reverse equality follows. Since M00k is semsimple connected,
every proper closed subscheme has smaller dimension, and we find that ι00 is an isomorphism.
0
We already observed that (pr0 )sat
k is surjective, and that prk is an isomorphism. Hence we deduce
0
geo
00
0
dim ρ̄0,Λ (π1 (X))satk ≥ dim Mk = dim Mk , where the equality is due to (c). Arguing as above,
the inclusion ι0 is the identity as well. This completes (d).
Corollary 7.2. Let ρ : π1 (X) → GLn (K) be a continuous homomorphism, and let Λ and Λ0 be
O-lattices in K n that are stable under the action of π1 (X) via ρ. Let M and M0 be the O-groupschemes that are the Zariski closures of ρ(π1 (X)) in AutO (Λ) and AutO (Λ0 ), respectively, andlet M00 be the Zariski closure of (ρ ⊕ ρ)(π1 (X)) ⊆ AutO (Λ) × AutO (Λ0 ). Suppose that(i) the generic fiber M := M ⊗O K(∼= M0 ⊗O K) is semisimple and connected;(ii) the O-group schemes M, M0 and M00 are smooth;(iii) the identity components of their special fibers Mk , M0k and M00k are saturated;(iv) the saturation ρ̄Λ (π1geo (X))satk is semisimple and agrees with Mk .
00
∼ 0Then ρ̄Λ (π1geo (X))satk is isomorphic to Mk , M = M and both are semisimple connected over O.

Proof. Let g ∈ GLn (K) be such that Λ0 = gΛ, so that AutO (Λ0 ) = g AutO (Λ0 )g −1 . The result
is now a direct consequence of Lemma 7.1 applied to ρ and ρ0 = gρg −1 .

7.2

Geometric mod λ semisimplicity for almost all λ.

0, let Λλ ⊂ Eλn be a π1 (X)-stable OλLet ρ• be an E-rational compatible system. For each λ ∈ PE
geo∼
lattice. Define Gλ (resp. Gλ ) as the Zariski closure of Gλ (resp. Ggeoλ ) in AutOλ (Λλ ) = GLn (Oλ ).
geo
geo
Define Gkλ := Gλ ⊗Oλ kλ and Gkλ := Gλ ⊗Oλ kλ . The following is the main result of this section.
It generalizes the part of Theorem 6.14 concerning geometric monodromy groups to compatible
systems that are not necessarily absolutely irreducible.
0
Theorem 7.3. Suppose that ρ• |π1geo (X) is connected.15 Then for almost all λ ∈ PE
geo
geo
sat
(a) the group Gλgeo is semisimple and connected over Oλ , and ρ̄Λλ (π1 (X))kλ = Gkλ , and(b) ρ̄Λλ is a semisimple representation, and thus isomorphic to ρ̄λ .
15 Because of Corollary 3.37, here we do not need to assume that ρ•

54

is semisimple.

Proof. By Convention 2.2 ρ• is semisimple, and using the connectivity hypothesis, it is clear that
we may pass from the base field κ to a finite extension, so that we can assume that ρ• is connected
as well. We also observe that to prove the theorem, we may pass from E to a finite extension E 0 : If
λ is unramified for E 0 /Q, then the formation of the Zariski closure commutes with the unramified
base change Oλ → Oλ0 0 , and being semisimple connected descends from the base change to
geosat
the original situation. Moreover we are given closed immersions ρ̄Λλ (π1 (X))kλ ,→ GLn,k andgeo
Gkλ ,→ GLn,k , and if these immersions are the same under a base change k → k 0 , then the given
immersions are the same: if Ik0 , Jk0 are the kernels in the coordinate ring k 0 [GLn ] describing
these immersions, then Ik0 = Jk0 , and their invariants under Gal(k 0 /k) are the original kernels
and hence the same.
Thus
L from now on we also assume that ρ• is a direct sum of absolutely irreducible representationsi ρi,• . Since ρ• is connected, as in the proof of Proposition 4.5 one has surjections Gρ• ,λ →Gρi,• ,λ , and so the ρi,• are connected. Hence by Lemma 4.1 we may twist each ρi,• by anunramified compatible system δi,• , such that each ρi,• ⊗ δi,• has finite order determinant, andhence trivial determinant by Lemma 4.4. Since the twist does not change ρ• |π1geo (X) , we assume
from now on that each summand ρi,• has trivial determinant.
If necessary, we now enlarge E again, so that a split motivic triple is defined over E; denote it by(E, M, α). Then by Corollary 4.14(d) we have an almost faithful representation β of M , and a0
we have a homomorphism ϕλ : π1 (X) → M (Eλ )compatible system ρ0• such that for every λ ∈ PE
and a commuting diagramρλπ1 (X)ϕλ/ M (Eλ )ρ0λ0 GLn (Eλ )5α⊗E Eλβ⊗E Eλ). GL 0 (E ).
nλ0Let Λ0λ ⊂ Eλn be an Oλ -lattice stable under the action of π1 (X) under β ⊗E Eλ , so that0
Λ00λ := Λλ ⊕ Λ0λ ⊂ Eλn+n is stable for π1 (X) acting via (α ⊕ β) ⊗E Eλ . Define for ? ∈ {λ, kλ }and ∗ ∈ {0, 00} the group schemes G?∗ and G?∗,geo starting from Λ∗λ in the same way as G? and G?geo
was constructed. By Proposition 5.51, the Oλ -group schemes Gλ , Gλ0 , Gλ00 are smooth for almost0
all λ ∈ PE
. By Theorem 5.52, the identity components of their special fibers Gkλ , Gk0 λ , Gk00λ are00saturated for almost all λ ∈ PE
. By Theorem 6.14, for almost all λ ∈ PE
one has00,geo
ρ̄λ0,Λ (π1geo (X))satkλ = Gkλ ,and both groups are semisimple connected over kλ . Note also that the size of kernel of β ⊗E Eλis clearly independent of λ. Thus by Lemma 7.1 (with the roles of ρ0 and ρ reversed), we have for
geo
geo
sat
0
almost all λ ∈ PE
that ρ̄Λλ (π1 (X))k is semisimple connected and equal to Gkλ and that thegeogroup scheme Gλ is semisimple connected over Oλ . This proves part (a). Part (b) is immediatefrom (a) and Proposition 5.18.
The following result summarizes the Q-rational case.
Corollary 7.4. Let ρ• be Q-rational connected and with semisimple split motivic group. Thenfor almost all ` ∈ PQ0 the following hold:55

(a) The Z` -group scheme G` is smooth semisimple connected.
sat
+
geo
Λ geo
(b) One has ρ̄Λ= G` (F` )+ .
` (π1 (X)) F = GF` , and hence ρ̄` (π1 (X))`(c) For almost all `, one has G` (Z` )+ ⊆ ρ` (π1geo (X)) ⊆ G` (Z` ).
(d) Assertions (b) and (c) hold with π1 (X) in place of π1geo (X).
Q(e) There exists a finite subset L ofQPQ0 , a compact open subgroup
HLgeo of `∈L G` (Q` ) and aQ
finite cover X 0 → X such that `∈P 0 rL G` (Z` )+ × HLgeo = `∈P 0 ρ` (π1geo (X 0 )).
Q

Q

Proof. By our hypotheses G` = Go` = Ggeo
` , and now part (a) is immediate from Theorem 7.3,
as is the first half of part (b). To see the second assertion in (b), we may pass to a cover of X,
geo
so that by Theorem 3.34, we can assume that ρ̄Λ` (π1 (X)) is `-generated for almost all `. In
geo
sat
Λ geo
that case, by Remark 5.15 the group ρ̄` (π1 (X))F` is the Nori envelope of ρ̄Λ` (π1 (X)). The
equality now follows from [Nor87, Thm. B], and this concludes the proof of (b).
In (c), only the inclusion on the left is at stake. By (b), we know the left inclusion after reduction
modulo `. Let π` : Ge` → G` be the universal cover of G` . Then by [Ste68, Thms. 12.4 and 12.6],
the image of Ge` (F` ) in G` (F` ) is precisely G` (F` )+ . Let Γ ⊂ Ge` (Z` ) be the inverse image ofgeoρΛ` (π1 (X)) under π` . Then by (b), the result just quoted and [Lar95, Prop. 2.6], we have forall `  0 that Γ = Ge` (Z` ), and this implies (c). Assertion (d) is clear since the claims in (b) and(c) are weaker if one replaces π1geo (X) by π1 (X).
For (e) let X 0 → X be a finite cover as in Theorem 3.34 so that ρ• |π1 (X 0 ) is independent andalmost all ρ` (π1geo (X 0 )) are `-generated. Going through the above arguments we see that theregeo
0+
is a finite set L of primes such that for all ` ∈/ L we now
have ρ̄Λ` (π1 (X )) = G` (F` ) in (b) and
Q
geo
geo
hence ρ` (π1 (X 0 )) = G` (Z` )+ in (c). Define HL := `∈L ρ` (π1 (X)). Then (e) follows using
the independence of ρ• .
Corollary 7.5. Suppose that ρ• is E-rational with semisimple split motivic group. Then
Y
Y




(
ρλ π1geo (X) ⊂ (ρλ π1 (X)
0
λ∈PE

0
λ∈PE

is an open subgroup.
Proof. It suffices to consider ρ0• := ResE/Q ρ• . By Lemma 3.27 its split motivic group is semisimple, and hence we shall assume E = Q from now on and work with ρ• . For the proof of thecorollary, we may pass to a finite cover, and hence by Theorem 3.13 we may assume that ρ• isalso connected. By Theorem 3.34 we may further assume that ρ• is independent. Now from
Corollary 7.4(d) and (e), Q
it follows that there exists a finite subset L of PQ0 and compact open
subgroups HLgeo ⊂ HL ⊂ `∈L G` (Q` ) such thatY`∈PQ0 rLG` (Z` )+ × HLgeo =

Y 

(∗)

(π1geo (X)) ⊆

`∈PQ0

Y 
`∈PQ0

(π1 (X)) ⊆Y

G` (Z` ) × HL .

`∈PQ0 rL

Because G` is semisimple for ` ∈ PQ0 r L and contained in GLn,Q` , the quotients G` (Z` )/G` (Z` )+are finite abelian of cardinality bounded independent of `, e.g. [BGP18, Lem. 3.8]. Since HLgeo ⊂b and so there is a finite extension κ0 /κ
HL is open its index is finite. Now π1 (X)/π1geo (X) ∼= Z,
after which the inclusion labelled (∗) becomes an equality for X 0 = X ⊗κ κ0 in place of X.
56

Corollary 7.6. Suppose that ρ• is an E-rational compatible system of representations of π1 (X),0and that the lattices Λλ , λ ∈ PE
are stable under π1geo (X) (but not necessarily π1 (X)). Then for
geo,o
geo
sat,o
0
almost all λ ∈ PE the group Gλis semisimple over Oλ , and ρ̄Λ= Gkgeo,o.
λ (π1 (X))kλλ
Note that because of Corollary 3.37 the corollary also holds if ρ• is not semisimple.
Proof. Either by the above remark, or by Convention 2.2, we shall assume that ρ• is semisimple.
Clearly it suffices to prove the result after passing to a finite coefficient extension E 0 /E, since
E 0 /E is unramified for almost all λ: the formation of the reduction is clearly compatible withthis (cf. Remark 6.1); but also the formation of Gλgeo as a Zariski closure, since taking the
Zariski closure commutes with base change under unramified extensions Oλ0 0 /Oλ ; cf. the proofof [LP95, Prop. 1.3]; and finally the formation of saturation is compatible with base change (anddescent) by Corollary 5.7. Thus we assume from now on that ρ• is a direct sum of absolutely
irreducible compatible systems ρi,• . We may also twist the ρi,• according to Lemma 4.1(c) bycharacters of Γκ so that the ρi,• have finite order determinant, since the assertion we prove only
concerns π1geo (X).
Since the assertion of the corollary only concerns identity components, we may also freely replace
X by a finite cover. Thus by Theorem 3.13 we may assume that ρ• and the ρi,• are connected,
and have trivial determinant. This might destroy the absolute irreducibility of the ρi,• . But by
passing through the above steps finitely many times (the degree of ρ• is finite), we may finallyassume that the ρi,• are absolutely irreducible, and then by Proposition 4.5 that ρ• is connected
and has semisimpleQsplit motivic
group.
7.5,

 Now

 and a further finite cover we
Q by Corollary
may assume that ( λ∈P 0 ρλ π1geo (X) = ( λ∈P 0 ρλ π1 (X) . In particular, the lattices ΛλEEare stable under (the new) π1 (X). Now the corollary follows from Theorem 7.3.
Theorem 7.3 also yields a proof of a main result of [CHT17] by independent methods, as is shownby the following result and its proof.
Corollary 7.7 ([CHT17, Thm. 1.2]). Let k be an algebraically closed field of characteristic p,
let X be a smooth k-scheme and let f : Y → X be a smooth proper morphism. Then the actionof π1 (X) on Ri f∗ F` is semisimple for almost all ` ∈ PQ0 .
Proof. By a standard reduction argument, cf. [CHT17, Prop. 4.1], it suffices to prove the resultin the case when k = κ for some finite extension κ of Fp , and when f is the base change underκ → k of a smooth morphism f0 : Y0 → X0 between smooth geometrically connected κ-schemes.
In the latter case, by [Del80, Cor. 3.3.9], for each i, the family ρ• of Galois representations
attached to the family of lisse sheaves (Ri f0∗ Q` )`∈PQ0 on X is a Q-rational compatible system,
pure of weight i. It carries a natural integral structure given by the family of Z` -lattices Λ`
defined by (the generic fiber of) Ri f0∗ Z` , ` ∈ PQ0 . Let ρ̄` be the induced representation of π1 (X)on Λ` /`Λ` .
To prove the corollary, we may clearly pass to a finite extension of X, and hence by Theorem 3.34
we may assume that the images ρ` (π1geo (X)) are pro-` generated for almost all `. From Theogeo
geo
geo
sat
is therem 7.3 we deduce that ρ̄Λ` (π1 (X))F` = GF` is semisimple for almost all `, where G`
geo
geo
i
Zariski closure of ρ` (π1 (X)) in AutZ` (Λ` ). Now the action of π1 (X) on R f∗ F` is via ρ̄Λ`,and since its saturation is semisimple, it follows from the last assertion of Proposition 5.18 that
π1geo (X) acts semisimply via ρ̄Λ` , as was to be shown.
57

Remark 7.8. Combining Remark 6.16 with the reduction argument given in Theorem 7.3 should
prove Theorem 7.3 for arbitrary Q-rational compatible systems ρ• with semisimple split motivic
group from the results in [CHT17, Thm. 1.2] by Cadoret, Hui and Tamagawa without invoking
any results from Section 6.
The following result answers in the affirmative Conjecture 5.4 of [LP95] for representations of
π1 (X) with X a normal variety over Fp .
0
Corollary 7.9. Suppose that ρ• is an E-rational compatible system. Then for almost all λ ∈ PE
,
geo,o
o
the groups Gρ• ,λ and Gρ• ,λ are unramified, i.e., they are quasi-split and become split over anunramified extension.

Proof. The assertions are unaffected by replacing X by a finite cover, and so by Theorem 3.13
and Theorem 3.34 we may assume that ρ• and ρ• |π1geo (X) are connected, and that ρλ (π1geo (X))
0
. By Theorem 7.3, for almost all λ the group Ggeo
is `λ -generated for almost all λ ∈ PEρ• ,λ is thegeneric fiber over a connected semisimple group scheme over Oλ . It follows that group scheme
as well as its generic fiber are quasi-split and become split over an unramified extension; see[Con11, 5.2.14]. This proves the corollary for Ggeo,oρ• ,λ .
derNext observe that by Lemma 3.9, we have Ggeo,oρ• ,λ = Gρ• ,λ , since in our situation both aregeoconnected. But now if B is a Borel subgroup in Gρ• ,λ , then its image B ad in the adjoint group(Goρ• ,λ )ad is a Borel subgroup, and hence the preimage of B ad under Goρ• ,λ → (Goρ• ,λ )ad is a Boreloosubgroup in Goρ• ,λ . Thus whenever Ggeo,oρ• ,λ is quasi-split, then so is Gρ• ,λ . That Gρ• ,λ becomes
split over an unramified extension of Eλ for almost all λ is a consequence of the existence of a
split motivic triple over a finite extension F of E, since F/E is unramified for almost all λ.

8Adelic openness for some E-rational compatible systems

Let ρ• be an absolutely irreducible connected E-rational with simple adjoint split motivic group
M , cf. Definition 8.6, and suppose that the split motivic representation α is the adjoint representation of M . We shall deduce adelic openness in the sense of Hui-Larsen for the algebraic andgeometric monodromy groups attached to such a ρ• . We shall make crucial use of [Pin98] by
Pink. In Remark 8.16 we shall also indicate with a sketch of proof how far one can push adelic
openness results by the methods developed here and in [Pin98]. From the results in this section,
starting from Corollary 7.4, only, we will also deduce a second proof of Theorem 6.14, and thus
by the results of Section 7 of Theorem 7.3. Hence by Remark 7.8 this proof builds, up to someunchecked compatibilities, directly on [CHT17] and avoids the use of any results from Section 6.

8.1

Summary of parts of [Pin98]

We recall some context from [Pin98] specialized toQthe case of local fields of characteristic zero.
In the following by L we denote a finite product i Li of local non-archimedean fields of characteristic zero. Different Li may or may not have the same residue characteristic. By a local
non-archimedean field of characteristic zero we mean a finite extension of some field Q` . We
regard L as a commutative semisimple ring. By K we denote a (topologically) closed subring of58

L. It may have fewer components. By G we denote a semisimple` group over L which is fiberwise
absolutely simple adjoint; i.e., G is a disjoint unionofgroupsi Gi where each Gi is absolutelyQ
simple adjoint over Li . Note also that G(L) = i G(Li ). Moreover by Γ we denote a compact
subgroup of G(L) that is fiberwise Zariski dense.
Definition 8.1. A quasi-model of (L, G, Γ) is a triple (K, H, ϕ) consisting of(i) a semisimple closed subring K of L over which L is of finite type,(ii) a fiberwise absolutely simple adjoint group H over K,(iii) an isomorphism ϕ : H ⊗K L → G,such that Γ is contained in ϕ(H(K)).16
The triple (L, G, Γ) is minimal if for every quasi-model (K, H, ϕ) one has L = K.
A quasi-model (K, H, ϕ) is called minimal, if the triple (K, H, ϕ−1 (Γ)) is minimal.
Fix a triple (L, G, Γ) as above. A main abstract result on minimal quasi-models is the following:
Theorem 8.2 ([Pin98, Thm. 3.6]).

(a) A minimal quasi-model (K, H, ϕ) exists for (L, G, Γ).

(b) The subring K ⊂ L in (a) is unique, and (H, ϕ) are determined up to unique isomorphism.
In particular, if (K 0 , H 0 , ϕ0 ) is a quasi-model for (L, G, Γ) and if (K, H, ϕ) is a minimal quasimodel for (K 0 , H 0 , (ϕ0 )−1 (Γ)), then (K, H, ϕ0 ◦ ϕ) is a minimal quasi-model for (L, G, Γ).
However [Pin98] gives also a fairly concrete description of a minimal quasi-model. In the following
result, we only quote the part on the minimal field of definition. The proof follows immediatelyfrom [Pin98, Thms. 2.3 and 2.14] and [Pin98, Prop. 3.14]; we omit details.
Theorem 8.3 (cf. [Pin98, Prop. 0.6]). Let AdG : G → Aut(Lie G) be the fiberwise adjoint representation of G. Let KAd,G/L ⊂ L be the closure in L of the Q-algebra generated by the set{Tr(AdG (γ)) | γ ∈ Γ}. Then there exists a minimal quasi-model of the form (KAd,G/L , H, ϕ).
Also relevant to us is the following result:
Proposition 8.4 ([Pin98, Cor. 3.8]). Let Γ0 ⊂ Γ be a closed normal subgroup. Suppose that Γ0
is also fiberwise Zariski dense in G. If (L, G, Γ) is minimal, then so is (L, G, Γ0 ).
The following is the main result of [Pin98] on open images; note that in characteristic zeroisogenies are separable and ϕ is an isomorphism for any quasi-model (K, H, ϕ).
Theorem 8.5 ([Pin98, Thm. 0.2]). If (K, H, ϕ) is a minimal quasi-model for (L, G, Γ), then the
inclusion Γ ,→ G(L) factors via H(K) and Γ is open in H(K). Conversely, if Γ ⊂ G(L) is open
then (L, G, Γ) is minimal.
The converse part is rather simple to prove by considering the Q` dimension of Lie G.
16 In [Pin98, § 3], ϕ can be a totally inseparable isogeny and dϕ can vanish at some factors. Both are notpossible in characteristic zero.

59

8.2

Simple adjoint compatible systems

In this section, we shall deduce some adelic openness results for certain E-rational compatible
systems which we call (simple) adjoint. Using the just quoted results from Pink, for such systems
we can apply Weil restriction to get into the setting of Q-rational systems. To the latter we apply
Corollaries 7.4 and 7.6. We then also indicate how, starting from Corollary 7.4, one can prove
Theorem 6.14 for general E-rational systems from its analog for Q-rational systems.
Definition 8.6. We call a compatible system simple adjoint if it is connected semisimple, if its
split motivic group M is simple adjoint and if its split motivic representation
is AdM . We callQ
it adjoint, if its split motivic group M isNsemisimple adjoint, say M = i Mi with Mi simple,
and if its split motivic representation is i AdMi .
Example 8.7. Let ρ• be an E-rational compatible system. Then after possibly enlarging E,(a) there exists an E-rational compatible system ρe• whose monodromy groups Gρe• ,λ are semi0
there exists a surjective homosimple and of adjoint type, and such that for all λ ∈ PE
morphism Gρ• ,λ → Gρe• ,λ with central kernel, and(b) there exists an adjoint compatible system whose split motivic group is the adjoint groupof the split motivic group of ρ• .
To see (a), by Proposition 4.3 we may assume that ρ• is absolutely completely reducible after
enlarging E. Let ρ• = ⊕i∈I ρi,• be the corresponding decomposition
into absolutely irreducibleLcompatible systems ρi,• of dimension ni . Then ρe• :=Adρi,• is an E-rational compatible
system. Since the center of each ρi,• is the center of the ambient GLni intersected with Gρi,• ,λ ,the groups Gρe• ,λ are semisimple and of adjoint type, and (a) follows readily.
For (b) let ρeMbe the M -compatible system from Corollary 4.13, where again we possibly have•and aQsplit motivic triple (E, M, α) for ρe• exist over E. Since N
M is split
to enlarge E so that ρeM•simple adjoint, we have M = i Mi with Mi absolutely simple, split adjoint. Let β := i AdMi .
is the searched-for E-rational adjoint compatible system for (b).
Then β ◦ ρeM•
Definition 8.8. If ρ• is an E-rational compatible system, we call E minimal for ρ• , if E/Q isgenerated by the coefficients of the polynomials Px , x ∈ |X|.
We deduce several immediate consequences:
Corollary 8.9. Suppose ρ• is E-rational simple adjoint. Let E 0 ⊂ E be the field of definitionof the polynomials Px , x ∈ |X|, for ρ• . Then the following hold:(a) There exists an E 0 -rational compatible system ρ0• , such that ρ• = ρ0• ⊗E 0 E, i.e., E 0 is
minimal for ρ0• .
L 0Q(b) For any ` ∈ PQ0 , the images of π1 (X) and π1geo (X) underρλ0 are open inGρ0• ,λ0 (Eλ0 0 ).
λ0 |`

λ0 |`

0
0
(c) The Q-rational compatible system ρQ• := ResE /Q ρ• is semisimple, and its monodromy
groups areYGρQ• ,` =ResE 0 /Q` Gρ0• ,λ0 .
λ0 |`

60

Proof. Let M be the split motivic group of ρ• and let n = dim M , so that ρ• is n-dimensional.
Let E0 be the subfield of E generated by the coefficients of T n−1 of the polynomials Px ∈ E[T ],x ∈ |X|. In the`course of the proof we shall see that E0 = E 0 . For ` ∈ PQ0 consider L = ⊕λ|` Eλ =E ⊗Q Q` , G = λ0 |` Gρ• ,λ and ρL = ⊕λ|` ρλ = AdG (by hypothesis). Let K be the closure of E0
in L. Then K = E0 ⊗Q Q` and it is embedded into L via the completion at ` of E0 ,→ E. Thenfor all x ∈ |X| one has Tr(ρL (Frobx )) ∈ K = E0 ⊗Q Q` , and from the definition of E0 and using
the Čebotarov density theorem and the continuity of Tr ◦ρL it is clear that K = KAd,G/L .
Let (K, HK , ϕ) be a minimal quasi-model from Theorem 8.3 for Γ = ρL (π1 (X)). DefineρK : π1 (X) → Aut(Lie HK ) as the composite of π1 (K) → Γ with Γ ⊂ HK (K) and then with
AdH . Because AdK ⊗K L = AdG , and so ρK ⊗K L = ρL we havecharpolρK (Frobx ) (T ) = charpolρL (Frobx ) (T ) = Px (T )for all x ∈ |X|. It follows that Px (T ) ∈ E0 ⊗Q Q` [T ], and this for all ` ∈ PQ0 , and thus
Px (T ) ∈ E0 [T ]. Since this holds for all x ∈ |X|, we find E0 = E 0 . Now define ρ0λ0 := ρK ⊗K Eλ0 00Then it is clear that (ρ0λ0 )λ0 ∈PE0 0 forms a compatible system definedfor λ0 ∈ PE
0 above `.
0over E , and (a) is immediate. Part (b) follows from Theorem 8.5 and Proposition 8.4 (we haveGρ0• ,λ0 (Eλ0 0 ) = HK (Eλ0 0 )).
Q
The image of π1 (X) under (ResE 0 /Q ρ0• )` lies in the Q` -points of λ0 |` ResE 0 /Q` Gρ0• ,λ0 which issemismiple and a closed subgroup of ResE 0 /Q GLn ⊂ GLn[E 0 :Q] . By (b) the image is open in thissemisimple subgroup, and this implies both parts of (c).
Definition 8.10. For an E-rational compatible system and σ ∈ Gal(Q/Q), we define the σconjugate σρ• as the σ(E)-rational compatible system ρ• ⊗σE σ(E).
Alternatively, if ρ• = (ρ• , (Px )x∈|X| ) is a E-rational compatible and σ ∈ Gal(Q/Q), then σρ• =0be the place(σρ• , (Pxσ )x∈|X| ) can be described as follows, where for λ ∈ Pσ(E) we let λ|σ ∈ PEσunder λ via σ : E → σ(E): one has (σρ• )λ = ρλ|σ ⊗Eλ|σ Eλ and Px = σ(Px ).
Remark 8.11. Suppose E is minimal for the E-compatible system ρ• . Then σρ• is isomorphic toρ• if and only if σ ∈ Gal(Q/Q) fixes E. The ‘if’-direction is obvious. For the ‘only if’-directionnote that if σρ• is isomorphic to ρ• , then Pxσ = Px ∈ Q[T ] for all x ∈ |X|. But by the minimalityof E this implies that σ fixes E.
Lemma 8.12. Let E be a number field, let I be a finite set, for each i ∈ I, let Ei be a subfield of
E and let ρi,• be an Ei -rational simple adjoint compatible system such that each Ei is minimalfor ρi,• . Suppose thatN(i) the split motivic group of ρ• := i∈I (ρi,• ⊗Ei E) is the product of the split motivic groupsof the ρi,• (over a suitable extension F of E),(ii) no ρi,• is σ-conjugate to some ρj,• for i 6= j for some σ ∈ Gal(Q/Q).
Then for any ` ∈ PQ0 , the subgroup
O M
i∈I

ρi,λi




 YY
π1geo (X) ⊂Gρi,• ,λi (Ei,λi )
i∈I

λi |`

61

λi |`

(13)

0
is open, where the λi run through allNplaces of PE
above `, and the same holds with π1 (X)igeoin place Qof π1 (X). Moreover ρ• := i∈I (ResEi /Q ρi,• )` is semisimple with monodromy groupGρ• ,` = i∈I GResEi /Q ρi,• ,` .

Proof. Observe first that we can express (13) also by the use of Weil restrictions asYρ` (π1geo (X)) ⊂GResEi /Q ρi,• ,` (Q` ).
i∈Iwhere we use Corollary 8.9(c) on the right.
Q Also, by Corollary 8.9(c), the groups GResEi /Q ρi,• ,` aresemisimple connected, and hence so is i∈I GResEi /Q ρi,• ,` . We need to show that this product isequal to G` = G⊗i∈I (ResEi /Q ρi,• )` ,` , since then the openness follows from Corollary 3.10. Turning
things around, it suffices to establish openness for a single ` ∈ PQ0 .
Choose ` such that ` is split in E/Q, so that for all i and λi ∈ Pi above ` and λ ∈ PE above `we have Ei,λi = Q` and Eλ = Q` . To match
our settingQwith Theorem 8.3 and Theorem 8.5, letQ
L = ⊕i Li with Li = ⊕λi |` Ei,λi , G = Gi with Gi = λi |` Gρi,• ,λi , and let Γ be the image of
π1geo (X) in G(L) under ⊗i∈I ⊕γi |` ρi,λi , and Γi the image of Γ under projection to Gi (Li ). By
our hypothesis on Ei and ρi,• , and by Corollary 8.9(b), the triple (Li , Gi , Γi ) is minimal.
Let now (K, H, ϕ) be a minimal quasi-model of (L, G, Γ). We assume that K is strictly smaller
than L, which is a finite sum of copies of Q` . Then there exists a simple factor of K, whichmust be a copy of Q` , which embeds diagonally into at least two factors of L. Since (Li , Gi , Γi )is minimal, no two factors can lie in the same i. Hence to derive a contradiction, it suffices toassume from now on that I = {1, 2}. By the definition of K, and considering the factor justsingled out, we have
0(i’) a diagonal embedding ι : Q` → E1,λ1 × E2,λ2 for suitable places λi of PE
above `,
i'(ii’) a semisimple Q` -group H0 and an isomorphism H0 ⊗ιQ` (E1,λ1×E2,λ2 ) −→ Gρ1,• ,λ1×Gρ2,• ,λ2 ,(iii’) a homomorphism ρ0 : π1 (X) → H0 (Q` ),such that, if we denote by ϕ0 the diagonal embedding H0 → Gρ1,• ,λ1 ×Gρ2,• ,λ2 induced from (ii’),
we have ϕ0 ◦ ρ0 = ρ1,λ1 × ρ2,λ2 . Since ι is continuous, it is the canonical isomorphism Q` ∼= Ei,λion the factors for i = 1, 2. Interpreted now as an isomorphism ι` : E1,λ1 → E2,λ2 the map ιidentifies ρ1,λ1 with ρ2,λ2 . Let ιλ : Ei → Ei,λ denote the completion homomorphism. Then for
all x ∈ |X| we have ι` ◦ ιλ1 (P1,x ) = ιλ2 (P2,x ). This defines an isomorphism E1 → E2 . By σ wedenote an extension to Gal(Q/Q). Then (σρ1,• )λ2 = ρ2,λ2 , and from Lemma 3.1 it follows thatσρ1,• = ρ2,• . This was excluded by hypothesis (ii) and thus yields a contradiction.
The following result shows that Galois conjugacy of compatible systems as tensor factors is anobstruction to an open image theorem, i.e, that hypothesis (ii) in Lemma 8.12 cannot be omitted.
It should be compared with [Loe16, Thm. 3.2.2] where, by different methods, Galois conjugacy
of compatible systems was identified as an obstruction to big image in pairs.
Proposition 8.13. Suppose ρ• is E-rational simple adjoint and E is minimal for ρ• . Let F/E
be finite Galois over Q, let I be a non-empty subset of HomQ (E, F ), and fix σ0 ∈ I. Then62

(a) the split motivic group of ρ0• :=groups of the σρ• ,Nσ∈I (σρ• ⊗σ(E) F ) is the product of the split motivic(b) for any ` ∈ PQ0 , the subgroup
O M
σ∈I

(σρ• )λσ




 YY
Gσρ• ,λσ (σ(E)λσ )π1geo (X) ⊂
σ∈I

λσ |`

is open in the diagonal embedding of
run over all places of σ(E) above `.

λσ |`

Q

λ|` Gσ0 ρ• ,λ (Eλ ) under−1σ∈I σσ0 , where the λσQ

σ0 = id. That the inclusion in (b)Proof. It suffices to prove the lemma forQI = Emb(E, Q), and Q
factors via the diagonal embedding of λ|` Gρ• ,λ (Eλ ) under σ∈I σ is clear. The open image
statement follows from Corollary 8.9(b) applied to the single factor for σ0 . This proves (b).
N
Regarding (a), we deduce from Lemma 3.5 that ResE/Q ρ• ⊗Q F = σ∈I (σρ• ⊗σ(E) F ), and thus
the split motivic
Qgroup in question is that of ResE/Q ρ• . Now it follows from Corollary 8.9(c) that
GResE/Q ρ• ,` = λ|` (ResEλ /Q` Gρ• ,λ ). Moreover for µ0 ∈ PF0 above ` we have, by Lemma 3.3,Q
Q
that λ|` (ResEλ /Q` Gρ• ,λ ⊗Q` Fµ0 ) = σ∈I Gσρ• ,µ0 . This completes the proof of (a).
We now combine Lemma 8.12 with Corollary 7.4 to deduce some further results.
Theorem 8.14. Let E, I, Ei , ρi,• be as in Lemma 8.12, and assume that all hypothesis of that0nlemma are satisfied. Let furthermore for each i and each λi ∈ PE
denote by Λi,λi ⊂ Ei,λani
i
Oi,λi -lattice stable under the action by π1 (X) via ρi,λi , and attach to this an Oi,λi -group scheme
Gi,λi as above Theorem 6.14.17 Then for almost all ` ∈ PQ0 the following hold:Q Q(a) The Z` -group scheme G` := i∈I λi |` ResOi,λi /Z` Gi,λi is smooth and semisimple.
(b) One has
 YYiReski,λi /F` ρ̄Λ
i,λiπ1geo (X)sati∈I λi |`and soQ Q
i∈I λi |`

geo
i
ρ̄Λ
i,λi (π1 (X))

=

F`

YY

geo
sat
i
Reski,λi /F` (ρ̄Λi,λi (π1 (X))ki,λ ) = GF` ,ii∈I λi |`+ Q Q
+ Q Q
geo
i
=
ρ̄Λ
=
Gi,λi (ki,λi )+ = G` (F` )+ .
i,λi π1 (X)i∈I λi |`i∈I λi |`(c) For almost all `, one has
YY

YY
 YY
Gi,λi (Oi,λi )+ ⊆
ρλi π1geo (X) ⊆Gi,λi (Oi,λi ) = G` (Z` ).
i∈I λi |`i∈I λi |`i∈I λi |`(d) Assertions (b) and (c) hold with π1 (X) in place of π1geo (X).
(e) There is a finite subset L of PQ0 and a compact open subgroup HL ⊂Q Q Q

Gρi,• ,λi (Ei,λi )`∈L i∈I λi |`such that Y YY`∈PQ0 i∈I λi |`
ρi,λi (π1geo (X)) ⊇YYY`∈PQ0 rLi∈I λi |`geo
17 Note that under our hypothesis we have Gi,λi = Gi,λi .

63

Gi,λi (Oi,λi )+ × HL .

Proof. Let us first consider the case #I = 1 and in this case omit the index i from the notation.
0
Suppose `  0. Then ` is unramified in E/Q, so that for all λ ∈ PE
above ` the ring Oλ isunramified over Z` and hence étale. Moreover by Proposition 5.51 the schemes Gλ are smoothwith semisimple generic fiber. Then by [BLR90, §7.5, Prop. 5] the Weil restriction
ResOλ /Z` Gλ
Q
is smooth connected with semisimple generic fiber, and hence so is G`0 := λ|` ResOλ /Z` Gλ .
Define ρQ• := ResE/Q ρ• . Then the lattice Λ` := ⊕λ|` Λλ is preserved under the action of π1 (X)Q
under ρ` , and we define H` as the Zariski closure in Autπ1 (X) (Λ` ) where π1 (X) acts via ρQ• .
Because `  0 the group scheme H` is smooth over Z` by Proposition 5.51. By its definition,H` ⊂ G`0 is a closed immersion, and by Corollary 8.9 the inclusion is an isomorphism on genericfibers. Since both are smooth connected group schemes and of the same relative dimension overZ` , they must be isomorphic. We deduce from Corollary 7.4 that both are also semisimple andsatgeo
that their special fiber satisfies ρ̄Λ` (π1 (X)) F = HF` .
`It follows that ResOλ /Z` Gλ is semisimple for each λ above ` and hence similar to Lemma 3.5 thateach Gλ is smooth semisimple connected above Oλ . From Proposition 5.35(e) we furthermore
deduce thatsat Y
sat
geo
geo
ρ̄Λ
Reskλ /F` ρ̄Λ` (π1 (X)) F ⊂λ (π1 (X)) k .
`

λ

λ|`

By the previous paragraph, the group HF` is equal to the left hand side, butQat the same timeclearly contains the right hand side. By reducing H` = G`0 , we deduce HF` = λ|` Reskλ /F` Gkλ .
satgeo
Hence we must have ρ̄Λ
λ (π1 (X)) k = Gkλ , giving the first assertion of (b) for #I = 1.
λRegarding the second part of (b), from our definitions and from what we proved so far, we haveYY++ Y Λ geo
geo
geo
ρ̄Λ
=
ρ̄Λ
⊆
ρ̄λ (π1 (X))+ ⊆Gλ (kλ )+ = G` (F` )+ .
` (π1 (X))λ (π1 (X))
λ|`

λ|`

λ|`

The outer terms are equal by Corollary 7.4(b), and this completes the proof of (b). Moreoverpart (c) is immediate from our notation and Corollary 7.4(c). Assertion (d) is now clear since
the claims in (b) and (c) are weaker if one replaces π1geo (X) by π1 (X). Part (e) follows from
Corollary 7.4(e) applied to ρQ• . Finally, if #I > 1 then one has to also invoke Lemma 8.12. Weleave the details to the reader.
e → G be its
We now follow [HL15]. Let G be a semisimple group over a field M , and let πG : G
universal cover. Using that commutators with central elements are trivial, the commutator map
e×G
e→Ge is seen to factor via a map [·, ·]∼ : G × G → G.
e For i ≥ 1 and Γ a subgroup of[·, ·] : G∼ie
G(M ), denote by [Γ, Γ] the subset of G(M ) of i-fold products of the set {[g, h]∼ | g, h ∈ Γ}.
If for any ` ∈ PQ0 one has a semisimple group G` over Q` , and if Γ is a closed subgroup ofQQ∼ie ` ).
as a subset of `∈P 0 G(Q`∈P 0 G(Q` ), then we also define in the same way [Γ, Γ]Q

Q

In [HL15, § 2], Hui and Larsen introduce the notion of a special adelic group. We alter their definition by having as an indexing set PQ0 and not the set set of all rational primes. An exampleQQ
of such a group is `∈P 0 rL Ge` (Z` ) × HL with HL open compact in `∈L G` (Q` ), where G` is asQe p ) := Q0 0 Ge` (Q` ), and similarly without
in Theorem 8.14 or Corollary 7.4. We abbreviate G(AQ`∈PQ

the tilde, where the restriction is with respect to Ge` (Z` ) ⊂ Ge` (Q` ) for ` ∈ PQ0 r L.
64

Corollary 8.15 (cf. [HL15, Conj. 1.3]). Suppose either of the following hypotheses:(i) Let ρ• be a Q-rational
connected compatible system with semisimple split motivic group,Qand define Γ := `∈P 0 ρ` (π1geo (X)) ⊂ G(ApQ ).
Q(ii) Let the notation and hypotheses be as in Theorem 8.14 and define Y YYΓ :=ρi,λi (π1geo (X)) ⊂ G(ApQ ).
`∈PQ0 i∈I λi |`e p ), which is equal to [Γ, Γ]∼t for some t.
Then [Γ, Γ]∼ generates a special adelic subgroup of G(AQe p ).
Moreover [Γ, Γ]∼2 contains a special adelic subgroup of G(AQ
Proof. Using [HL15, Thm. 3.8], under (i) the assertion follows from Corollaries 7.4 and 7.6, andunder (ii) it is a consequence of the explicit description of Γ given in Theorem 8.14(e).
Remark 8.16. We have singled out in this section compatible systems ρ• with simple adjoint
split motivic groups and their adjoint representation as split motivic representations to prove anadelic openness theorem for E-rational systems with E ) Q using the results from [Pin98]. The
results in op.cit. also allow to show the following:
Suppose ρ• is an E-rational compatible system whose split motivic group is semisimple andsimple. Let E0 be the subring of E generated over Q by {Tr(ρλ (Frobx )) | x ∈ |X|}. Thenby [Pin98, Thm. 2.14] for all `  0 one can define a simple semisimple group G` over E0,` :=(E0 ) ⊗Q Q` such that ⊕λ|` ρλ can be obtain by base change E0,` → E ⊗Q Q` from a representation
ρ0` : π1 (X) → GLn,E0,` with Zariski closure G` .
For finitely many ` it may happen in loc.cit. that ρ` takes its image in the units of a non-split
central simple algebra D` of dimension n2 over E0 . Even then one can define a group G` that is
a closed subgroup of the unit group of D` . The exceptional primes ` are not controlled by op.cit.
In some cases, for instance if the split motivic representation is the adjoint representation, theset of such exceptional places is empty.
Let further F ⊃ E be a number field over which a split motivic triple (F, M, α) exists for ρ• andfrom Corollary 4.13 exists, such that ρ ⊗E F ∼over which the M -compatible system ρM= α ◦ ρM
• .
•
M
Ad
Ad
Define ρ• := AdM ◦ρ• and let Ead be the subring of F generated over Q by {Tr(ρλ (Frobx )) |x ∈ |X|}. If E0 = Ead , then by [Pin98, Prop. 0.6(c)] for every `  0 in PQ0 the representation ρ`
has open image in G` (E0,` ).
A careful reworking of the results of this section will thenQgive an adelic openness result of the0
image of π1geo (X) in a suitably defined restricted product `∈P 0 G(E` ).
Q

Using only results from Section 8 given above and results up now to and including Section 5,
we shall prove the following corollary which by Remark 6.16 allows one to give another proof of
Theorem 6.14 based on [CHT17]. It basically says that Theorem 6.14 for Q-rational compatible
systems implies Theorem 6.14 for E-rational compatible systems.
Corollary 8.17. Consider the following statement (∗)E : For all E-rational absolutely irreducible
compatible systems ρ• such that ρ• and ρ• |π1geo (X) are connected and all choices of π1 (X)-stable65

0lattices Λλ of Oλn under the action of ρλ , it holds that for all but finitely many λ ∈ PE
one hasequality of semisimple groupssat,oρ̄λ (π1geo (X))sat= Gkgeo,kλ = ρ̄λ (π1 (X))kλλ

where Gλgeo is defined as the Zariski closure of ρλ (π1geo (X)) in AutOλ (Λ).
Then (∗)Q implies (∗)E for all number fields E.
Proof. To prove the corollary, we may apply the following reductions: It follows from Corolsat,ogeo
satlary 5.16 that ρ̄λ (π1geo (X))satkλ = ρ̄λ (π1 (X))kλ , and so it suffices to prove that ρ̄λ (π1 (X))kλ =geo
Gkλ , and that these groups are semisimple. By Lemma 4.1 and passing to a finite extension
X 0 → X, we may assume that the split motivic group of ρ• is semisimple. By Lemma 7.1,
it suffices
Q to prove the corollary in the case where the split motivic group is an adjoint group
M = i∈I Mi with Mi simple adjoint for some finite set I. By Lemma 7.1 and Corollary 4.14
we may further assume that ρ• = ⊗i∈I ρi,• , where the split motivic representation of ρi,• is adMi
on Mi and the trivial representation on the factors Mi0 , i0 6= i (cf. Remark 4.15(a)). By Corollary 8.9, we can assume that
S each ρi,• is Ei -rational with Ei minimal for ρi,• . We partition I
into a disjoint union I = j∈J Ij , where i and i0 belong to the same class if and only if ρi,• and
ρi0 ,• are σ-conjugate for some σ ∈ Gal(Q/Q). Because of Proposition 5.9, we may assume forthe proof that for each i ∈ I and σ ∈ Gal(Q/Q), the system σρi,• is isomorphic to some ρi0 .•with i0 ∈ I.
For each j ∈ J we choose one ij ∈ Ij , and we define ρQ• := ⊗j∈J ρij ,• . By Theorem 8.14, which
in fact only uses (∗)Q and not the full strength of Corollary 7.4, we have for `  0
 YYsat Y Y
Λij 
geoReskij ,λj /F` ρ̄ij ,λπ(X)=
Reskij ,λj /F` Gigeo,
1jj ,ki ,λjF`j∈J λj |`jj∈J λj |`where the notation is as in Theorem 8.14. Let F/Q be a finite Galois extension containing all
Ei (and E), and for each ` denote by k` the residue field of F above `. Note also, that by (∗)Q
the groups on both sides are semisimple for all `λ  0.
Set Λ0 := σΛi for i ∈ I with ρi,• ∼= σρi ,• for some σ ∈ Gal(Q/Q), and observe that the lattices
i

j

j

0,geo
Λ0i are independent of any choices by Remark 8.11 since Ei is minimal for ρi,• . Define Gi,λ
as
i
geo
0
the Zariski closure of ρi,λi (π1 (X)) in AutOλi (Λi ). Then for `  0 base change of the above tok` gives
 YY 0sat Y Y 0,geoΛ=
Gi,ki,λ ⊗kλi k` .
ρ̄i,λi i ⊗ki,λi k` π1geo (X)k`i∈I λi |`ii∈I λi |`Now for `  0 the field F/Q is unramified
above `. Using the notation ρF• := ⊗i∈I ρi,• ⊗Ei F ,
Q
0,geo
F
0
F
Λµ := ⊗i∈I Λi ⊗Ei,λi Fµ , and Gµ = i∈I, Gi,λ
⊗
F
where
µ
restricts
to λi on Ei , we
Ei,λi
µ
i
regroup the previous formula to

Y
Y
sat
F
geo
ρ̄F,Λ
π
(X)
=
GkFµ .
1kµk`

0µ|`,µ∈PF
F

This implies ρ̄F,Λ
kµ

0µ|`,µ∈PFsat
π1geo (X) kµ = GkFµ for all µ ∈ PF0 with `µ  0, and we still have that thegroups GkFµ are semisimple over kµ . It follows that the smooth group schemes GµF are smooth
66

sat
sat
F
geo
and split semisimple over Oµ for `µ  0, and that ρ̄F,Λ
π1geo (X) kµ = ρ̄Fµ π1 (X) kµ . Bykµ
our reduction to the case where I is closed under Galois conjugation, we have in fact thatρF• = ρ• ⊗E F . Hence the previous equality and compatibility of saturation with descent bysatCorollary 5.7 implies that ρ̄λ π1geo (X) k = Gk0,geo
.
λ
λ
sat
geo
geo
From Theorem 5.52 we have the inclusion ρ̄Λ
λ π1 (X) kλ ⊂ Gkλ . An explicit computation
invoking expn from Subsection 5.1 shows that saturation commutes with semisimplification,
sat
sat
geo
geo
i.e., that one has a surjection prλ : ρ̄Λ
λ π1 (X) kλ → ρ̄λ π1 (X) kλ . Invoking the previous
paragraph, we have
sat
sat
geo
dim ρ̄λ π1geo (X) k = dim Gk0,geo= dim Gλ = dim Gkgeo≥ dim ρ̄Λ
λ π1 (X) k .
λ
λ
λ

λ

Since all groups involved are smooth, prλ must be an isomorphism. Therefore the above inclusion
is an equality and Gkgeo
is semisimple.
λRemark 8.18. By the following argument the semisimplicity assertion in Corollary 8.17 implies
that the ρ̄λ are absolutely irreducible for almost all λ:Let Tλ be a split torus of Gλ over Oλ ; cf. Proposition 5.51. By Proposition 5.51 and Proposition 4.6, the weights for the action of Tλ on Fλn and on kλn via ρλ and ρ̄λ are the same andindependent of λ for `λ  0. The representations of generic fiber of the split group Gλ on Fλn
are irreducible and hence the representation of Gkλ on kλn is irreduciblefor `λ  0, cf. Proposition 5.40. Now apply Proposition 5.18 to deduce that ρ̄kλ π1geo (X) acts semisimply on kλn , andhence absolutely irreducible since this is not changed under saturation.

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Gebhard Böckle
Computational Arithmetic Geometry
IWR (Interdisciplinary Center for Scientific Computing)
University of Heidelberg
Im Neuenheimer Feld 368
69120 Heidelberg, Germany
E-mail address: gebhard.boeckle@iwr.uni-heidelberg.de
Wojciech Gajda
Faculty of Mathematics and Computer Science
Adam Mickiewicz University
Umultowska 87
61614 Poznań, Poland
E-mail adress: gajda@amu.edu.pl
Sebastian Petersen
Universität Kassel
Fachbereich 10
Wilhelmshöher Allee 73
34121 Kassel, Germany
E-mail address: petersen@mathematik.uni-kassel.de

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