GEOMETRIC MONODROMY SEMISIMPLICITY AND MAXIMALITY

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1 GEOMETRIC MONODROMY SEMISIMPLICITY AND MAXIMALITY ANNA CADORET, CHUN-YIN HUI AND AKIO TAMAGAWA Abstract. Let X be a connected scheme, smooth and separated over an agebraicay cosed fied k of characteristic p 0, et f : Y X be a smooth proper morphism and x a geometric point on X. We prove that the tensor invariants of bounded ength d of π 1(X, x) acting on the étae cohomoogy groups H (Y x, F ) are the reduction moduo- of those of π 1(X, x) acting on H (Y x, Z ) for greater than a constant depending ony on f : Y X, d. We appy this resut to show that the geometric variant with F -coefficients of the Grothendieck-Serre semisimpicity conjecture namey that π 1(X, x) acts semisimpy on H (Y x, F ) for 0 is equivaent to the condition that the image of π 1(X, x) acting on H (Y x, Q ) is amost maxima (in a precise sense; what we ca amost hyperspecia ) with respect to the group of Q -points of its Zariski cosure. Utimatey, we prove the geometric variant with F -coefficients of the Grothendieck-Serre semisimpicity conjecture Mathematics Subject Cassification. Primary: 14F20, 20G25; Secondary: 20E Introduction Let X be a connected scheme, smooth and separated over an agebraicay cosed fied k of characteristic p 0 and et f : Y X be a smooth proper morphism. As a the objects are of finite type over the base, we may assume that X = X 0 k0 k for some smooth, separated and geometricay connected scheme X 0 over a finitey generated subfied k 0 k and that f : Y X is the base-change over k of a smooth proper morphism of k 0 -schemes f 0 : Y 0 X 0. In the foowing, we aways use the notation f 0 : Y 0 X 0 /k 0 for such a mode. By the smooth-proper base-change theorem, for every prime p, the higher-direct image sheaves R f 0 Z/ n are ocay constant constructibe hence, for every geometric point x on X, they give rise to continuous actions of the étae fundamenta group π 1 (X 0, x) on (R f 0 Z/ n ) x H (Y x, Z/ n ), (R f 0 Z ) x H (Y x, Z ) and (R f 0 Q ) x H (Y x, Q ). The aim of this paper is to prove the foowing two statements about the restriction of these representations to the geometric étae fundamenta group π 1 (X, x) (note that they are independent of the mode f 0 : Y 0 X 0 /k 0 ). Theorem 1.1. The foowing hods. - (1.1) For 0 (depending on f : Y X), the action of π 1 (X, x) on H (Y x, F ) is semisimpe. The assertion (1.1) is the natura geometric variant with F -coefficients of the Grothendieck-Serre semisimpicity conjecture, stating that the action of π 1 (X 0, x) on H (Y x, Q ) is semisimpe for p (see for instance p. 109 of the version of Tate s Woods Hoe tak in [T65]; see aso Section 11). The geometric variant with Q -coefficients of this conjecture is a ceebrated theorem of Deigne, proved in [D80] (see beow for detais). Theorem 1.2. The assertion (1.1) is equivaent to the foowing. - (1.2) After repacing X by a connected étae cover and for 0 (depending on f : Y X), the image of π 1 (X, x) in the group of Q -points of its Zariski cosure in GL(H (Y x, Q )) is amost hyperspecia. Given a connected semisimpe group G over Q, reca that a compact subgroup Π G(Q ) is caed hyperspecia if there exists a semisimpe group scheme G over Z with generic fiber G and such that Π = G(Z ). When they exist, hyperspecia subgroups are the compact subgroups of maxima voume in G(Q ). We say that a compact subgroup Π G(Q ) is amost hyperspecia if (p sc ) 1 (Π) G sc (Q ) is hyperspecia, where p sc : G sc G denotes the simpy connected cover. We refer to the beginning of Section 8 for further detais. A motivation for the reformuation (1.2) of (1.1) in terms of maximaity is its potentia appications to probems requiring arge monodromy assumptions (see the introduction of [H08] and references therein 1

2 2 ANNA CADORET, CHUN-YIN HUI AND AKIO TAMAGAWA for a survey of appications of arge monodromy resuts). The assertion (1.1) was previousy known in the foowing cases - (1.1.1) when p = 0; - (1.1.2) if one repaces F -coefficients with Q -coefficients; - (1.1.3) when f : Y X is an abeian scheme (or more generay for H 1 (Y x, F )) and p is arbitrary [Z77] or a famiy of K3-surfaces and p 2 [SkZ15] whie (1.2) was previousy known when p = 0 (see for instance [C15, Rem. 2.5]). Let us aso point out that an arithmetic variant of (1.2) hods over a set of primes of density one and for arbitrary systems of compatibe rationa semisimpe -adic representations [La95a]. Let us reca the proofs of (1.1.1), (1.1.2). For (1.1.1), we may assume k C and x X(C). The topoogica fundamenta group Π := π top 1 (Xan C, x) acts on the singuar cohomoogy H := H (Yx an, Z). As H is a finitey generated Z-modue, H F H (Yx an, F ) for 0. Thus, by comparison of singuar/étae cohomoogy and of topoogica/étae fundamenta group, the image of π 1 (X, x) acting on H (Y x, F ) identifies with the image of Π acting on H F for 0. The fact that Π acts semisimpy on H F for 0 then foows formay [CT11, Lem. 2.5] from the Hodge-theoretica fact that Π acts semisimpy on H Q [D71, Cor (a)]. For (1.1.2), by standard speciaization arguments (see for instance Subsection 4.2) we may assume k 0 is a finite fied and k = k 0. Set F := R w f Q and consider the argest maxima semisimpe smooth subsheaf F F. As F x F x is stabe under the action of π 1 (X 0, x), the extension 0 F F F/F 0 corresponds to a cass in H 1 (X, Hom(F/F, F )) F, where F denotes the geometric Frobenius on X. As F is smooth, pure of weight w, Hom(F/F, F ) is smooth, pure of weight 0 hence H 1 (X, Hom(F/F, F )) is mixed of weights 1 and H 1 (X, Hom(F/F, F )) F = 0 (see [D80, (3.4)]). So, when p = 0, the essentia ingredient is comparison between compex and étae topoogy, which provides an underying Z-structure for the action of π 1 (X, x) on H (Y x, Z ) and reduces (1.1.1) to a Hodgetheoretica statement by reduction moduo- for 0. When p > 0, such an underying Z-structure is no onger avaiabe. For Q -coefficients, we can resort to Deigne s theory of weights. But such a theory does not exist for F -coefficients. Our basic strategy is to combine both aspects, namey try and deduce (1.1) from (1.1.2) by a reduction moduo- argument which invoves Deigne s weight theory, in particuar, the foowing two consequences of it: - (Fact 3.1) The H (Y x, Z ) are torsion-free for 0; - (Fact 3.2) The H (Y x, Q ) (: prime p) form a compatibe system of Q-rationa representations. The combination of Fact 3.1 and Fact 3.2 provides a weak repacement for the Z-structure in characteristic 0, aowing us to gue together the various representations with F -coefficients by means of the reduction moduo- of the characteristic poynomias of Frobenii. These ideas have aready been expoited to obtain structura resuts about the image of π 1 (X, x) acting on H (Y x, F ). For instance: - (Fact 3.4) After possiby repacing X by a connected étae cover and for 0, the image of π 1 (X, x) acting on H (Y x, F ) is perfect and generated by its order- eements. This seemingy technica statement aso pays a crucia part in our arguments. However, to achieve the proofs of Theorem 1.1 and Theorem 1.2, more information is required. The way to grasp the missing information is Tannakian: instead of considering ony H (Y x, Z ), we consider a possibe tensor constructions of bounded ength buit from H (Y x, Z ). Behind this is the observation that the image of π 1 (X, x) acting on H (Y x, Q ) is captured by its Zariski cosure whie the image of π 1 (X, x) acting on H (Y x, F ) is captured by its agebraic enveope in the sense of Nori (this is one pace where Fact 3.4 is crucia). Both are agebraic groups hence shoud be reconstructibe from their tensor invariants. Whence the idea to compare the tensor invariants for the action of π 1 (X, x) on H (Y x, Q )

3 GEOMETRIC MONODROMY SEMISIMPLICITY AND MAXIMALITY 3 and H (Y x, F ). This is the core resut of our paper. To state it, we need some notation. For every partition λ of an integer d 0 et c λ Z[S d ] denote the associated Young symmetrizer and write n λ := d! d λ, where d λ is the dimension of the irreducibe representation of the symmetric group S d defined by c λ. Then 1 n λ c λ Z[ 1 n λ ][S d ] is an idempotent. Fix a prime such that d <. Let Λ denote Z or F and et Π be a profinite group acting continuousy on a finitey generated free Λ -modue M. Let S d act on M d on the right; this action commutes with the one of Π thus S λ (M) := 1 n λ c λ (M d ) M d again gives a representation of Π. For λ, λ partitions of integers d, d respectivey, write where ( ) denotes the Λ -dua. S λ,λ (M) := S λ (M) S λ (M ), For a profinite group Π (in practice, Π wi be π 1 (X, x)) acting continuousy on a finitey generated free Z -modue M, consider the foowing equivaent properties (Inv, M) (i) M Π F (M F ) Π ; (ii) H 1 (Π, M)[] = 0; (iii) H 1 (Π, M) is torsion-free. Eventuay, for a group Π 0 acting on a modue M and a morphism Π Π 0, et M Π denote the Π-modue obtained from M by restriction of the action from Π 0 to Π. Theorem 1.3. For a integers d, d 0, partitions λ, λ of d, d respectivey and 0 (depending on f, d, d ) - (1.3.0) the property (Inv, S λ,λ (H (Y x, Z ))) hods. Furthermore (Inv, M π1 (X,x)) hods for every π 1 (X 0, x)-modue M which is of one of the foowing forms - (1.3.1) a torsion-free quotient of S λ,λ (H (Y x, Z )); - (1.3.2) a submodue of S λ,λ (H (Y x, Z )) with torsion-free cokerne. Theorem 1.3 appies in particuar to λ = (d) i.e. c λ = σ S d σ, for which S λ (M) = S d (M) and to λ = (1,, 1) i.e. c λ = σ S d sign(σ)σ, for which S λ (M) = Λ d (M). To prove Theorem 1.3, we reduce to the case where X 0 is a curve and k 0 is finite (this uses Bertini s theorem, de Jong s aterations and speciaization of tame étae fundamenta group). This aows us to use Deigne s theory of weights and the machinery of étae cohomoogy. To obtain Theorem 1.2, we foow the above rough Tannakian strategy. Theorem 1.3 enabes us to show that, for 0, the Nori enveope of the image of π 1 (X, x) acting on H (Y x, F ) identifies with the reduction moduo- of the Zariski cosure G of the image of π 1 (X, x) acting on H (Y x, Z ) (Theorem 7.3). This in turn enabes us to show that there exists a constant C 1 (depending ony on f) such that the image of π 1 (X, x) acting on H (Y x, Z ) has index C in G (Z ) (what we ca weak maximaity see (7.3.2)). Using this, we can give severa equivaent formuations of (1.1), among which are that G is semisimpe (7.5.4) and (1.2) (see Coroary 8.2). The reformuation (7.5.4) raises a genera question: given a connected semisimpe group G over Q together with a faithfu finite-dimensiona Q -representation V and a attice H V, can one expoit tensor invariants data (as in Theorem 1.3) to deduce that the Zariski cosure G of G in GL H is a semisimpe mode of G over Z? This question ed to a first compete proof of Theorem 1.1 (Section 9) which is entirey due to the second author, Chun-Yin Hui. More precisey, the key-resut is that, under mid assumptions, the semisimpicity of G is encoded by a finite expicit ist of tensor invariants dimensions (Theorem 9.1). This criterion is then appied to H := H (Y x, Z ) using Theorem 1.3 and the toos deveoped for the proof of Theorem 1.2. Theorem 9.1 reies on Lie theory and is of independent interest. After this first proof of Theorem 1.1 was obtained, we competed a second proof (Section 10) which is cohomoogica and reminiscent of the argument of Deigne. This second proof requires Theorem 1.3, Fact

4 4 ANNA CADORET, CHUN-YIN HUI AND AKIO TAMAGAWA 3.1, Fact 3.2 and Fact 3.4 but invoves no additiona group-theoretica machinery. We concude by observing that Theorem 1.3 and Theorem 1.1 impy that the positive characteristic variant of the (arithmetic) Grothendieck-Serre-Tate conjectures with F -coefficients foow from the usua Grothendieck-Serre-Tate conjectures (arithmetic, with Q -coefficients) (Coroary 11.1). The paper is divided into three parts. In Part I (Sections 1-6), we review the properties of étae cohomoogy invoved in the proofs of our main resuts and estabish Theorem 1.3; here, Deigne s weight theory is ubiquitous. In Part II (Sections 7-8), we deveop the group-theoretica machinery eading to the proof of Theorem 1.2. Part III (Sections 9-11) is devoted to the proofs of Theorem 1.1 and to the appication to the (arithmetic) Grothendieck-Serre-Tate conjectures with F -coefficients. The second proof of Theorem 1.1 (Section 10) can be read just after Part I. Acknowedgments: Anna Cadoret was party supported by the ANR grant ANR-15-CE Chun-Yin Hui was supported by the Nationa Research Fund, Luxembourg and cofounder under the Marie Curie Actions of the European Commission (FP7-COFUND). Akio Tamagawa was party supported by JSPS KAKENHI Grant Numbers , 15H The authors thank the unknown referees for their comments, which heped improve the exposition of the paper. They aso thank Brian Conrad, Phiippe Gie, Wiberd van der Kaen, Tamas Szamuey and Jiong Tong for their interest and comments. PART I: ÉTALE COHOMOLOGY 2. Notation, conventions 2.1. Given a fied k 0 and an agebraicay cosed fied k containing k 0, write π 1 (k 0, k) (or simpy π 1 (k 0 )) for Aut(k 0 /k 0 ), where k 0 denotes the separabe cosure of k 0 in k. From a scheme-theoretic point of view, writing x for the geometric point Spec(k) Spec(k 0 ), one has π 1 (k 0, k) = π 1 (Spec(k 0 ), x), which justifies the notation. If k 0 is finite, et F k0 π 1 (k 0, k) (or simpy F if k 0 is cear from the context) denote the geometric Frobenius Given a prime and a profinite group Π, et Rep Z (Π) denote the category of finitey generated Z -modues endowed with a continuous action of Π. Let X 0 be a connected scheme and x a geometric point on X 0. Then the fiber functor F F x induces an equivaence 1 from the category S(X 0, Z ) of smooth Z -sheaves on X 0 to Rep Z (π 1 (X 0, x)). Thus, if P is a property of objects in Rep Z (π 1 (X 0, x)) (e.g., torsion-free, torsion, irreducibe, semisimpe etc.) we wi say that F S(X 0, Z ) has P if F x has P. The same considerations appy to the corresponding Q -categories. In the foowing, we wi often impicity identify smooth Z - (resp. Q -) sheaves on X 0 and finitey generated Z - (resp. Q -) modues endowed with a continuous action of π 1 (X 0, x). For every x 0 X 0 and geometric point x over x 0, consider the natura action of π 1 (x 0, x) on F x defined as the composition ρ x0 : π 1 (x 0, x) x 0 π 1 (X 0, x) GL(F x ) := Aut Z (F x ). Assume X 0 is geometricay connected and of finite type over a fied k 0. Let x 0 X 0 and fix a geometric point x : Spec(k) X over x 0 ; write X := X 0 k0 k. Then the sequence π 1 (X, x) π 1 (X 0, x) π 1 (k 0, k) 1 is exact. The functor F F π 1(X,x) x from S(X 0, Z ) to Rep Z (π 1 (k 0, k)) coincides with the goba section functor H 0 (X, ) : S(X 0, Z ) Rep Z (π 1 (k 0, k)) and the action of π 1 (k 0, k) on H 0 (X, F) by transport 1 As X is connected, there aways exists an étae path between two geometric points on X. So this equivaence of categories is independent of x in a canonica way up to fixing an étae path from x to any other geometric point.

5 GEOMETRIC MONODROMY SEMISIMPLICITY AND MAXIMALITY 5 of structure identifies with the abeed arrow ( ) in the commutative diagram: π 1 (x 0, x) π 1 (k 0, k) coker(π 1 (X, x) π 1 (X 0, x)) ρ x0 ( ) GL(F π 1(X,x) x ), which aso shows that the restriction to F π 1(X,x) x of the action of π 1 (x 0, x) on F x factors through the canonica morphism π 1 (x 0, x) π 1 (k 0, k). In particuar, if x 0 X 0 (k 0 ), the induced action of π 1 (x 0, x) on F π 1(X,x) x is independent of x 0. For σ π 1 (x 0, x), write Pσ,x F 0 := det(t Id ρ x0 (σ), F x Q ) Q [T ]. If the residue fied k(x 0 ) at x 0 is a finite fied, we wi simpy write F x0 := F k(x0 ) and Px F 0 := PF F x0,x 0 even simpy P x0 if F is cear from the context). (or Let q be a power of a prime number and w Z. A q-wei number of weight w is an agebraic number α such that ι(α) = q w 2 for every compex embedding ι : Q C. If X 0 is of finite type over Z, foowing [D80, (1.2)], a smooth Z -sheaf F on X 0 is said to be pure of weight w (resp. mixed) if for every cosed points x 0 X 0 [ 1 ] the roots of P x F 0 are k(x 0 ) -Wei numbers of weight w (resp. if F admits a fitration whose successive quotients are pure; the weights of the non-zero quotients are then caed the weights of F). A smooth Z -sheaf F on X 0 is said to be Q-rationa if for every cosed point x 0 X 0 [ 1 ], Px F 0 is in Q[T ]. Given a set L of primes, a system F, L of smooth Z -sheaves on X 0 is said to be Q-rationa compatibe if each of the F is Q-rationa and if for every cosed point x 0 X 0 the poynomias P F x 0 Q[T ] are independent of (for not equa to the residue characteristic of x 0 ) Given a prime and 0 P = n 0 a nt n Q[T ], we define the reduction moduo- P of P to be the reduction moduo- of a(p )P Z[T ], where a(p ) = p p min{vp(an), n 0}. Here, the product is over a rationa primes and v p : Q Z { } is the p-adic vauation. Given an F [T ]-modue M and P Q[T ], we say that M is kied by P if it is kied by P. 3. Some consequences of the Wei conjectures From now on, we retain the notation and conventions of the introduction for f : Y X/k and f 0 : Y 0 X 0 /k 0. By the smooth-proper base-change theorem R f 0 Z is a smooth Z -sheaf on X 0 and (R f 0 Z ) x = H (Y x, Z ) for every geometric point x on X. The foowing facts a rey on the Wei conjectures. Fact 3.1. ([G83] (projective case), [O13, Rem ]) The smooth Z -sheaves R f Z are torsion-free (of finite constant rank) for 0. In particuar, H (Y x, Z ) F H (Y x, F ), for 0. Fact 3.2. ([D80, Cor ]) Assume k 0 is finite (so that X 0 is of finite type over Z). Then R f 0 Z (: prime p) is a Q-rationa compatibe system. Fact 3.3. ([D80, Cor , Cor ], [LaP95, Prop. 1.1], [LaP95, Prop. 2.2]) After possiby repacing X 0 by a connected étae Gaois cover, the Zariski cosure of the image of π 1 (X, x) (resp. π 1 (X 0, x)) acting on H (Y x, Q ) is connected semisimpe (resp. connected) for a p. Fact 3.4. ([CT13, Thm. 1.1], [CT16, Fact 5.1]) After possiby repacing X by a connected étae Gaois cover and for 0 depending ony on f : Y X, the image of π 1 (X, x) acting on H (Y x, F ) is perfect and generated by its order eements.

6 6 ANNA CADORET, CHUN-YIN HUI AND AKIO TAMAGAWA Fact 3.1 is used severa times in the proof of Theorem 1.3: to reduce the proof of Theorem 1.3 to the case where Y X is the base change of a smooth proper morphism Y 0 X 0 with X 0 a curve over a finite fied k 0 (see Subsection 4.1) and then, combined with Fact 3.2, to compare the actions of π 1 (X, x) on H (Y x, Q ) and H (Y x, F ) for 0 (see Subsection 5.1). Fact 3.3 and Fact 3.4 are used in the proof of (1.3.2) (see Subsection 5.3) and they aso pay a crucia part in the proofs of Theorem 1.2 (Part II) and Theorem 1.1 (Part III). Consider the foowing assertions. (Inv, f 0 ) (GSS, f) 4. Preiminary reductions For 0, (Inv, M π1 (X,x)) hods for every π 1 (X 0, x)-modue M which is a torsion-free quotient of H (Y x, Z ) or a submodue of H (Y x, Z ) with torsion-free cokerne. For 0, the action of π 1 (X, x) on H (Y x, F ) is semisimpe. Let (Inv) (resp. (GSS)) denote (Inv, f 0 ) (resp. (GSS, f)) for every smooth proper morphism f 0 : Y 0 X 0. The aim of this section is to prove the foowing. Proposition 4.1. Assume (Inv,f 0 ) (resp. (GSS,f)) hods when k 0 is a finite fied, k = k 0 and X 0 is a smooth, separated and geometricay connected curve over k 0. Then Theorem 1.3 (resp. (1.1)) hods Locaization. We begin with a forma group-theoretic emma. Lemma 4.2. Let Π be a profinite group and U Π a norma open subgroup. Fix a prime not dividing [Π : U]. The foowing hod. - (4.2.1) Let H be a finitey generated free Z -modue endowed with a continuous action of Π. If the sequence 0 H U H U (H F ) U 0 is exact then the sequence 0 H Π H Π (H F ) Π 0 is aso exact. - (4.2.2) Let H be a finitey generated F -modue endowed with a continuous action of Π. If U acts semisimpy on H then Π aso acts semisimpy on H. Proof. The assertion (4.2.1) foows from the fact that H 1 (Π/U, H U )[] = 0 by appying the functor ( ) Π/U to the short exact sequence 0 H U H U (H F ) U 0. The assertion (4.2.2) foows from the fact that for every Π-submodue H H, U-equivariant projector p U : H H and system of representatives π 1,..., π r of Π/U the map p Π : H H defined by p Π (h) = 1 π i p U (πi 1 h) r is a Π-equivariant projector. 1 i r As a resut, to prove Theorem 1.3 (resp. (1.1)) we may base-change f 0 : Y 0 X 0 over X 0 X 0 for any morphism X 0 X 0 inducing a morphism π 1 (X 0, x ) π 1 (X 0, x) with norma open image. This appies for instance to open immersions or connected étae Gaois covers Speciaization. Let Λ denote Z or F. Lemma 4.3. There exist a finite fied k 0, a smooth, separated and geometricay connected curve X 0 over k 0, a smooth proper morphism f 0 : Ỹ0 X 0, and a norma open subgroup U π 1 (X, x), such that the foowing hods. Write k = k 0 and f : Ỹ X for the base-change of f 0 : Ỹ0 X 0 over k. Then, for every geometric point x on X and 0, there is an isomorphism H (Ỹ x, Λ ) H (Y x, Λ ) such that the image of π 1 ( X, x) acting on H (Ỹ x, Λ ) identifies with the image of U acting on H (Y x, Λ ). Furthermore we may assume that the action of π 1 ( X 0, x) on H (Ỹ x, Λ ) factors through the tame étae fundamenta group π 1 ( X 0, x) π t 1 ( X 0, x). Proof. We proceed in two steps. - Reduction to dim(x 0 ) = 1: We may assume X 0 has dimension 1. Using [Jou83, Thm. 6.10] (see [CT13, Ex. 3.1] for detais), we can construct a finitey generated fied extension K 0 of k 0 and a cosed curve C 0 X 0 k0 K 0 smooth, separated, geometricay connected over K 0, such that for every geometric point c on C mapping to x on X the induced morphism π 1 (C, c) π 1 (X, x) is surjective.

7 GEOMETRIC MONODROMY SEMISIMPLICITY AND MAXIMALITY 7 (Note that, if x is fixed, we cannot ensure that there exists a geometric point c on C mapping to x but this is not a probem since, to prove Lemma 4.3, we may repace x by any other geometric point - see Footnote 1). Here, we write K := K 0 k and C := C 0 K0 K. So the concusion foows from the fact that the resuting representation π 1 (C 0, c) π 1 (X 0, x) GL(H (Y x, Λ )) identifies (moduo the isomorphism H ((Y X C) c, Λ ) H (Y x, Λ ) given by the smooth-proper base-change theorem) with the representation π 1 (C 0, c) GL(H ((Y X C) c, Λ )) associated to the base-change Y 0 X0 C 0 C 0. - Reduction to dim(x 0 ) = 1, k 0 < + and k = k 0 : From the above, we may assume that X 0 is a smooth, separated, geometricay connected curve over k 0. After enarging k 0, we may assume it has a smooth compactification X cpt 0 over k 0. From de Jong s ateration theorem [Be96, Prop ], for every u X cpt 0 \ X 0 there exists an open subgroup U u of the inertia group I u π 1 (X 0, x) at u such that the image of U u in GL(H (Y x, Q )) is unipotent for a p. The image of U u in GL(H (Y x, Z )) is aso a pro- group for 0 (Fact 3.1). Hence, after repacing X 0 by a connected étae Gaois cover (Subsection 4.1), we may assume that π 1 (X 0, x) GL(H (Y x, Z )) factors through the tame étae fundamenta group π 1 (X 0, x) π1 t(x 0, x) and even that π 1 (X 0, x) GL(H (Y x, F )) does (Fact 3.1). Then by the standard speciaization arguments (speciaization of tame étae fundamenta group, smooth-proper base-change for étae cohomoogy), we may assume that k 0 is finite and k = k End of proof of Proposition 4.1. Lemma 4.2 and Lemma 4.3 show that if (Inv, f 0 ) (resp. (GSS, f)) hods when X 0 is a smooth, separated, geometricay connected curve over a finite fied k 0 and k = k 0 then (Inv) (resp. (GSS)) hods. We now expain why (Inv) impies Theorem 1.3. Consider the d-fod fiber product f [d] 0 : Y [d] 0 := Y 0 X0 X0 Y 0 X 0. By construction (Y [d] ) x = (Y x ) [d] and, as H (Y x, Z ) is torsion-free for 0, the Künneth formua (for both Z - and F -coefficients) shows that the horizonta arrows in the canonica commutative square of graded π 1 (X 0, x)-modues H ((Y [d] ) x, Z ) H ((Y [d] ) x, F ) H (Y x, Z ) d H (Y x, F ) d are isomorphisms. This induces a commutative square, whose horizonta arrows are sti isomorphisms H ((Y [d] ) x, Z ) π1(x,x) F (H (Y x, Z ) d ) π1(x,x) F H ((Y [d] ) x, F ) π 1(X,x) (H (Y x, F ) d ) π 1(X,x) On the other hand, as f [d] 0 : Y [d] 0 X 0 is a smooth proper morphism, (Inv, f [d] 0 ) impies that for 0 (depending on f : Y X, d) the eft vertica arrow is an isomorphism as we. We now concude the proof of Theorem 1.3. For 0 (depending on f : Y X, d, d ), H (Y x, Z ) is torsion-free so Lemma 4.4 beow reduces the assertion of Theorem 1.3 to the statement that (Inv, M π1 (X,x)) hods for every π 1 (X 0, x)-modue M which is a torsion-free quotient of H (Y x, Z ) d (H (Y x, Z ) ) d or a submodue of H (Y x, Z ) d (H (Y x, Z ) ) d with torsion-free cokerne. But by Poincaré duaity and the Künneth formua, H (Y x, Z ) d (H (Y x, Z ) ) d is isomorphic to H (Y [d+d ] x, Z ) (after suitabe Tate twists) as π 1 (X 0, x)-modue. So the concusion foows from (Inv, f [d+d ] 0 ). Lemma 4.4. Let Π be a profinite group acting continuousy on a finitey generated torsion-free Z -modue M and et 0 d, d < be integers. Then, for every pair of partitions λ, λ of d, d respectivey, S λ,λ (M) is a direct factor of M d M d as a Π-modue. Proof. Write S λ 1 (M) := (1 n λ c λ )(M d ) M d ; this is again a Π-submodue and we have a direct sum decomposition M d = S λ (M) S λ (M).

8 8 ANNA CADORET, CHUN-YIN HUI AND AKIO TAMAGAWA Simiary we have M d = S λ (M ) S λ (M ). This impies that S λ,λ (M) is a direct factor of M d M d Proposition 4.1 thus reduces the proof of Theorem 1.3 to the foowing specia case. Assume X 0 is a smooth, separated and geometricay connected curve over a finite fied k 0 and that k = k 0. Then Theorem 4.5. For 0 (depending on f : Y X), - (4.5.0) (Inv, H (Y x, Z )) hods. Furthermore (Inv, M π1 (X,x)) hods for every π 1 (X 0, x)-modue M which is of one of the foowing forms - (4.5.1) a torsion-free quotient of H (Y x, Z ); - (4.5.2) a submodue of H (Y x, Z ) with torsion-free cokerne. 5. Proof of Theorem Proof of (4.5.0). We may assume that x is a geometric point on X over x 0 X 0 (k 0 ) and that R f Z is torsion-free (Fact 3.1). In particuar, we have the short exact sequence 0 R f Z R f Z R f F 0 and the assertion of Theorem 4.5 is equivaent to the fact that the canonica (injective) morphism H 0 (X, R f Z ) F H 0 (X, R f F ) is an isomorphism. This, in turn, amounts to showing that H 1 (X, R f Z )[] = 0. To show this, we compute - in two ways - the characteristic poynomia of F (:= F k0 ) acting on H 1 (X, R w f Z )[]. On the one hand, we have H 1 (X, R w f Z )[] H 0 (X, R w f F ) H w (Y x, F ) π 1(X,x) H w (Y x, F ) H w (Y x, Z ) H w (Y x, Q ), which shows that F acting on H 1 (X, R w f Z )[] is kied by P w := P Rw f 0 Q x 0 = det(t Id F, H w (Y x, Q )) Q[T ], which is independent of ( p) and whose roots are k 0 -Wei numbers of weight w [D80, Cor ]. Here we use that x 0 X 0 (k 0 ) hence (Subsection 2.2), that the action of F on H 0 (X, R w f F ) identifies with the restriction of the action of F x0 on H w (Y x, F ). On the other hand, from Lemma 5.1 beow, the characteristic poynomia of F acting on H 1 (X, R w f Z )[] divides the characteristic poynomia of F acting on H 1 (X, R w f Z ) F. As we aso have a canonica F -equivariant embedding H 1 (X, R w f Z ) F H 1 (X, R w f F ), Lemma 5.3 beow shows that there exists P w+1 Q[T ], which is independent of ( p), whose roots are k 0 -Wei numbers of weights w + 1 and such that F acting on H 1 (X, R w f Z )[] is kied by P w+1 for 0. The concusion thus foows from the fact that P w, P w+1 Q[T ] are coprime hence that P w, P w+1 F [T ] are coprime as we for 0. Lemma 5.1. Let H be a finitey generated Z -modue equipped with a Z -inear automorphism F. Then the characteristic poynomia of F acting on H[] aways divides the characteristic poynomia of F acting on H F. Proof. As Z is a P.I.D., the short exact sequence of Z -modues 0 H tors H H/H tors 0 aways spits. In particuar H tors F H F. So it is enough to show that the characteristic poynomia of F acting on H[] aways divides the characteristic poynomia of F acting on H tors F. Hence we may assume H is finite. Then the exact sequence of Z [F ]-modues 0 H[] H H H F 0 shows that, in the Grothendieck group of Z [F ]-modues of finite ength we have [H[]] = [H F ]. In particuar, H[] and H F have the same F -semisimpification. Lemma 5.2. Let Λ denote Q (resp. F ). Let Y 0 be a scheme, separated and of finite type over a finite fied k 0 and et Y denote the base-change of Y 0 over k := k 0. Then for every integer w 0 there exists P w,y0 Q[T ], which is independent of ( p) and whose roots are k 0 -Wei numbers of weights w and such that F acting on H w c (Y, Λ ) is kied by P w,y0 Q[T ] (resp. for 0 (depending on Y )).

9 GEOMETRIC MONODROMY SEMISIMPLICITY AND MAXIMALITY 9 Proof. (See [Ge00, Lemma 4.1]). The assertion hods for smooth proper Y ( [D80, Cor ] for Λ = Q pus Fact 3.1 for Λ = F ). Aso, if the assertion hods over a finite extension k 0 k 0 then it hods over k 0. Indeed, if d := [k 0 : k 0] and F k 0 = Fk d 0 acting on H w c (Y, Λ ) is kied by P w,y0 k0 k 0 Q[T ] whose roots are k 0 -Wei numbers of weights w then F k 0 is kied by P w,y0 := P w,y0 k0 k 0 (T d ) Q[T ] whose roots are k 0 -Wei numbers of weights w. Thus, in the foowing, we wi impicity aow finite fied extensions of the base fied (for instance the divisor D, ateration Y etc. introduced beow may ony be defined over a finite extension of k 0, but this does not affect the argument). Eventuay, by topoogica invariance of étae cohomoogy, we may assume that Y is reduced. We proceed by induction on the dimension of Y. The 0-dimensiona case is straightforward. Assume the assertion of the emma hods for r-dimensiona reduced schemes, separated and of finite type over k 0. Fix an (r + 1)-dimensiona scheme Y 0, separated and of finite type over k 0. Write D for the union of the intersections of the pairs of distinct irreducibe components of Y. Then the ocaization exact sequence for cohomoogy with compact support H w c (Y \ D, Λ ) H w c (Y, Λ ) H w c (D, Λ ) and the induction hypothesis for D show that, without oss of generaity, Y may be assumed to be integra. From de Jong s aterations theorem [dj96, Thm. 4.1], there exists a genericay étae ateration φ : Y Y and an open embedding Y Y cpt into a scheme Y cpt smooth and projective over k such that Y cpt \ Y is a strict norma crossing divisor. Fix a non-empty open subscheme U Y such that U := Y Y U U is finite étae and write D := Y \ U. Again, the ocaization exact sequence for cohomoogy with compact support and the induction hypothesis for D show that it is enough to prove the caim for U. As U U is a finite étae morphism of degree say δ, one has the trace morphism which induces the mutipication-by-δ morphism δ : H w c (U, Λ ) res H w c (U, Λ ) tr H w c (U, Λ ). So, as soon as > δ, H w c (U, Λ ) is a direct factor of H w (U, Λ ) as an F -modue. Hence it is enough to prove the caim for U. Write D := Y cpt \ U. Then the ocaization exact sequence for cohomoogy with compact support H w 1 c (D, Λ ) H w c (U, Λ ) H w c (Y cpt, Λ ), the induction hypothesis for D and the fact that the assertion of the emma hods for the smooth projective scheme Y cpt yied the concusion. Lemma 5.3. Let Λ denote Q (resp. F ). With the notation of Theorem 4.5 there exists P w+1 Q[T ] whose roots are k 0 -Wei numbers of weights w + 1 and such that F acting on H 1 (X, R w f Λ ) is kied by P w+1 (resp. for 0). Proof. One may assume that Y 0 is connected, hence irreducibe. Then Y is equidimensiona, say, of dimension d Y. From the Leray spectra sequence E v,w 2 = R v g R w f Λ R v+w f 0 g 0 (gf) Λ for Y 0 X0 Spec(k 0 ), one sees that H 1 (X, R w f Λ ) = E 1,w 2 = E 1,w (reca that X is a curve) is a subquotient of R 1+w (gf) Λ = H 1+w (Y, Λ ) H 2d Y w 1 c (Y, Λ )(d Y ) (the second isomorphism is Poincaré duaity). Now, take P 2dY w 1,Y 0 Q[T ] as in Lemma 5.2 and et δ denote its degree. Then, for Λ = Q (resp. F ), F acting on H 1 (X, R w f Λ ) is kied by T δ P 2dY w 1,Y 0 ( k 0 d Y T 1 ) Q[T ], whose roots are k 0 -Wei numbers of weights w + 1, as desired Proof of (4.5.1). We may assume X 0 is affine. From (4.5.0) we have an F -equivariant injective morphism H 1 (π 1 (X, x), H w (Y x, Z )) H 1 (π 1 (X, x), H w (Y x, Z )) Q = H 1 (X, R w f Q ). From the Λ = Q case of Lemme 5.3, F acting on H 1 (X, R w f Q ) is kied by P w+1 Q[T ] independent of. So the same is true for H 1 (π 1 (X, x), H w (Y x, Z )). On the other hand, as X has cohomoogica dimension 1, the canonica F -equivariant morphism is surjective. This shows that H 1 (π 1 (X, x), H w (Y x, Z )) H 1 (π 1 (X, x), M)

10 10 ANNA CADORET, CHUN-YIN HUI AND AKIO TAMAGAWA (5.2.1) F acting on H 1 (π 1 (X, x), M) is kied by a poynomia P w+1 Q[T ] independent of and M and whose roots are k 0 -Wei numbers of weights w + 1. The assertion (5.2.1) impies that H 1 (π 1 (X, x), M)[] is aso kied by P w+1. To concude, consider the diagram 0 M π1(x,x) F (M F ) π 1(X,x) H w (Y x, Z ) M M F, H 1 (π 1 (X, x), M)[] 0 which shows that F acting on (M F ) π 1(X,x) is kied by a poynomia P w Q[T ] independent of and M and whose roots are k 0 -Wei numbers of weight w Proof of (4.5.2). Let Π denote the image of π 1 (X, x) acting on H w (Y x, Z ). From Fact 3.3, Π ab is finite (see e.g. [CT12, Thm. 5.7]). In particuar, Π acts on det(m) through a finite quotient, which has to be of order dividing 1. But, on the other hand Π is generated by its -Syow subgroups (Fact 3.4) so Π acts triviay on det(m). This shows that the canonica isomorphism M Hom(Λ m 1 M, det(m)) = (Λ m 1 M) det(m), v v (where m denotes the Z -rank of M) induces a π 1 (X, x)-equivariant isomorphism M (Λ m 1 M). As M H w (Y x, Z ) has torsion-free cokerne, M is a torsion-free π 1 (X 0, x)-quotient of H w (Y x, Z ) hence Λ m 1 M is a torsion-free π 1 (X 0, x)-quotient of Λ m 1 H w (Y x, Z ), which is itsef a π 1 (X 0, x)-quotient of H (m 1)(2d f w) (Y x [m 1], Z )((m 1)d f ) for 0. So we are reduced to (4.5.1). 6. Summary Our goa in the remaining parts of this paper is to prove Theorem 1.2 and Theorem 1.1. We fix the notation and conventions which wi be used from now on and review the information we have coected so far. Let k 0 be a fied finitey generated over F p and contained in an agebraicay cosed fied k and et f 0 : Y 0 X 0 be a smooth proper morphism of k 0 -schemes, with X 0 smooth separated, geometricay connected over k 0. Let f : Y X denote the base-change of f 0 : Y 0 X 0 over k. Assume 0 so that R f Z is torsion-free of constant rank r (Fact 3.1). Let Π (resp. Π 0 ) denote the image of π 1 (X, x) (resp. π 1 (X 0, x)) acting on H (Y x, Z ) =: H and et Π denote the image of π 1 (X, x) acting on H (Y x, F ) =: H. Write V := H Z Q. Let G GL H denote the Zariski cosure of Π (endowed with the reduced subscheme structure), G := G,Q GL V its generic fiber, G := G,F GL H its specia fiber. The scheme G coincides with the Zariski cosure of Π in GL V. Let aso G 0 denote the Zariski cosure of Π 0 in GL V. By construction - (6.1) G GL H is fat over Z. In particuar, dim(g ) = dim(g ). From Lemma 4.2, we may assume (Fact 3.3, Fact 3.4) - (6.2.1) Π is perfect and generated by its order eements for 0 ; - (6.2.2) G is connected semisimpe and G 0 is connected for every p. Eventuay, Theorem 1.3 reads - (6.3) For every integers d, d 0, partitions λ, λ of d, d and 0 (depending on d, d ) the property (Inv, M) hods for (6.3.1) Every Π 0 -modue quotient S λ,λ (H ) M which is torsion-free;

11 GEOMETRIC MONODROMY SEMISIMPLICITY AND MAXIMALITY 11 (6.3.2) Every Π 0 -submodue M S λ,λ (H ) with torsion-free cokerne. PART II: SEMISIMPLICITY VERSUS MAXIMALITY 7. Structure of G ; first reformuations of (1.1) 7.1. Group-theoretica preiminaries. Let Λ denote Z or F. Given a cosed subgroup Π of GL r (Λ ), write Π + Π for the (norma cosed) subgroup of Π generated by its -Syow subgroups. Given a finitey generated Λ -modue H, write H := s,t 0 H s (H ) t. For an integer d 1, set T d (H) := s,t d H s (H ) t H. Let H be an r-dimensiona F -vector space. Given a subgroup Π GL(H ), et Π GL H denote its agebraic enveope, in the sense of Nori [N87] that is the agebraic subgroup generated by the oneparameter groups φ g : A 1 F GL H t exp(t og(g)) for g Π of order. Here exp(n) := 0 i 1 ni i! for a nipotent n End(H ) and og(u) := (1 u) i 1 i 1 i for a unipotent u GL(H ). By construction Π is a smooth agebraic subgroup of GL H, connected and generated by its unipotent subgroups. Furthermore, the foowing hod. Lemma (7.1.1) For 0 depending ony on r, we have Π (F ) + = Π +. In particuar, Π+ -submodues and Π -submodues of H coincide. - (7.1.2) There exists d 1 depending ony on r such that Π is the stabiizer of (T d (H )) Π in GL H, for 0 depending ony on r. - (7.1.3) For every d 1 and 0 depending ony on d, r we have (T d (H )) Π = (T d (H )) Π+. Proof. The first part of (7.1.1) is [N87, Thm. B] and the second part foows from the first part by construction of Π. The assertion (7.1.2) foows from [CT16, Lem. 4.1]. For (7.1.3), the incusion (T d (H )) Π (T d (H )) Π+ hods as soon as r (reca that for r the ony eements in GL r (F ) of order a power of are those of order ). For the opposite incusion, fix an isomorphism H F r. Then for every v (T d (H )) Π+ and g Π of order each component of the vector equation exp(t og(g))v v = 0 is a poynomia in t with degree 2d(r 1) and has at east distinct roots. So for > 2d(r 1) the image of φ g is contained in the stabiizer of v in GL H. This shows (T d (H )) Π (T d (H )) Π+. A semisimpe group scheme over Z is a smooth affine group scheme whose geometric fibers are connected semisimpe agebraic groups. Then a the geometric fibers have the same root data [SGA3, XXII, Prop. 2.8]. We say that a semisimpe group scheme over Z is simpy connected if its fibers are. Furthermore, we have Lemma 7.2. Let G be a simpy connected semisimpe group scheme over Z. Then G(Z ) = G(Z ) +. Proof. This foows from the fact that the kerne of the reduction-moduo- morphism G(Z ) G(F ) is a pro- group and that, as G F is simpy connected, G(F ) = G(F ) + [St68, 12].

12 12 ANNA CADORET, CHUN-YIN HUI AND AKIO TAMAGAWA 7.2. Structure of G and weak maximaity. We are now abe to prove the main resut of this section. Theorem 7.3. For 0, we have Π = G. In particuar, - (7.3.1) For 0, G (F ) + is perfect and G = G (F ); - (7.3.2) (Weak maximaity) There exists an integer C r 1 depending ony on r such that, for 0, [G (Z ) : Π ] C r (hence Π = G (Z ) + ). Proof. Assume 0 so that (6.2.1), (6.2.2) and the concusions of Lemma 7.1 hod. By construction, (H ) Π = (H ) G and G is contained in the stabiizer of (H ) Π. As stabiizers commute with arbitrary base-changes, G is contained in the stabiizer of (H ) Π F hence, a fortiori in the stabiizer of T d (H ) Π F for every integer d 1. On the other hand, from (7.1.2), there exists an integer d 1 depending ony on r such that Π is the stabiizer of (T d (H )) Π in GL H. Then, up to increasing, we have T d (H ) Π F = T d (H ) Π (Theorem 1.3). By (6.2.1) and (7.1.3) this shows that G Π. For the opposite incusion, as Π is integra (being smooth and connected), it is enough to show that dim(g ) dim( Π ). This foows from dim(g ) dim( Π ) [La10, Thm. 7] and (6.1). Then (7.3.1) foows from (6.2.1) and (7.1.1) whie the first part of (7.3.2) foows from [La10, Thm. 7 (3)]. The assertion Π = G (Z ) + then foows from the fact that Π + G (Z ) + Π for > C r and (6.2.1). Coroary 7.4. The subgroup scheme G GL H is connected, smooth over Z for 0. Proof. The key point is that Π = G for 0 (Theorem 7.3). Then, the assertion about smoothness foows from the fact that G is fat (6.1) and of finite type over Z and from the smoothness of Π (as observed in the paragraph before Lemma 7.1) whie the assertion about connectedness foows from (6.2.2) and the connectedness of Π First reformuations of (1.1). Using Coroary 7.4, we obtain the foowing reformuations of (1.1). Coroary 7.5. The foowing assertions are equivaent: - (7.5.1) the action of Π on H is semisimpe for 0; - (7.5.2) the action of Π (equivaenty G ) on H is semisimpe for 0; - (7.5.3) Π (equivaenty G ) is semisimpe for 0; - (7.5.4) G is a semisimpe group scheme over Z for 0. Proof. Assume 0 so that (6.2.1), (6.2.2), (6.3) and the concusions of Lemma 7.1 and Coroary 7.4 hod. The equivaence (7.5.1) (7.5.2) foows from (6.2.1) and (7.1.1). The fact that (7.5.2) impies that Π is reductive is standard. Π is then automaticay semisimpe since it it generated by its unipotent subgroups. This shows (7.5.2) (7.5.3). The equivaence (7.5.3) (7.5.4) is by definition (since (6.2.2), Coroary 7.4 hods). The impication (7.5.3) (7.5.2) foows for instance from [J97, Prop. 3.2] (see aso [La95b, Thm. 3.5]). 8. Semisimpicity versus maximaity Let G be a connected semisimpe group over Q. Write p sc : G sc G and p ad : G G ad for the simpy connected cover and adjoint quotient of G respectivey. Reca that the Bruhat-Tits buiding [Ti79] B := B(G sc, Q ) is equipped with a natura action of G ad (Q ) and that G(Q ) acts on B through its image in G ad (Q ). There is a bijective correspondence between - semisimpe modes G of G over Z ; - hyperspecia points b B, given by G B G(Z) [Ti79, 3.8.1]. Aso given an isogeny φ : G G and if G b, G b are respectivey the semisimpe modes over Z of G and G corresponding to a hyperspecia point b B then φ : G G extends uniquey to a morphism φ b : G b G b of group schemes over Z. A compact subgroup Π G(Q ) of the form Π = G(Z ) for some semisimpe mode G of G over Z (or, equivaenty, such that Π G(Q ) is the stabiizer of a hyperspecia point in B) is caed hyperspecia. Hyperspecia subgroups, when they exist, are the compact subgroups of G(Q ) of maxima voume [Ti79, 3.8.2]. In particuar, a G(Q )-conjugate of a hyperspecia subgroup is again hyperspecia.

13 GEOMETRIC MONODROMY SEMISIMPLICITY AND MAXIMALITY 13 A compact subgroup Π G(Q ) is caed amost hyperspecia if (p sc ) 1 (Π) G sc (Q ) is hyperspecia. Lemma 8.1. Let G be a connected semisimpe group over Q. Let G (resp. G sc ) be a smooth, connected group scheme over Z with generic fiber G (resp. G sc ). Assume - (8.1.1) G sc is semisimpe over Z ; - (8.1.2) (p sc ) 1 (G(Z ) + ) is a norma subgroup of G sc (Z ) such that G sc (Z )/(p sc ) 1 (G(Z ) + ) is abeian. Then for 0 depending ony on the dimension of G, G is semisimpe over Z. Proof. For a profinite group Π which is an extension of a finite group by a pro- group et N(Π) denote the product of the orders of the groups (counted with mutipicities) appearing in the non-abeian part of the composition series of Π. Note that if Π Π are two such groups then N(Π ) N(Π). Let H be a connected, smooth, affine group scheme over Z ; write H := H F foowing hod. for its specia fiber. The (1) The non-abeian parts of the composition series of H(Z ) and H(F ) (resp. H(Z ) + and H(F ) + ) coincide. This is because the reduction moduo- map is surjective with pro- kerne. H(Z ) H(F ) (2) Write H ss := H /R(H ), where R(H ) is the sovabe radica of H. Then the non-abeian parts of the composition series of H(F ), H ss(f ), H ss(f ) + and H(F ) + coincide. Indeed, first, as H ss is a semisimpe group, H ss(f )/H ss(f ) + is abeian for 0. More precisey, et µ denote the kerne of the simpy connected cover of H ss. Since H1 (F, µ (F )) is of order dividing µ (F ) (e.g. [S68, XIII, 1, Prop. 1]), H ss(f )/H ss(f ) + embeds into H 1 (F, µ (F )) for prime to the order of µ. The assertion then foows from the fact that the rank of H ss (hence the order of µ ) is bounded as varies. As a resut, the non-abeian parts of the composition series of H ss(f ) and H ss(f ) + coincide. Next, Lang s theorem [L58] gives a short exact sequence 1 R(H )(F ) H(F ) H ss (F ) 1. Hence the non-abeian parts of the composition series of H(F ) and H ss (F ) coincide. Furthermore, the above short exact sequence induces a short exact sequence 1 R(H )(F ) H(F ) + H(F ) + H ss (F ) + 1. Hence the non-abeian parts of the composition series of H(F ) + and H ss (F ) + coincide. (3) Let H i, i I denote the amost simpe factors of H ss. Then [Ti64, Main Thm.] the non-abeian part of the composition series of H ss(f ) is precisey the famiy of the H i (F ) + /(Z(H i (F )) H i (F ) + ), i I for 0. As the kerne and the cokerne of i I H i(f ) H ss(f ), the Z(H i (F )) and H i (F )/H i (F ) +, i I a have order bounded from above by a constant depending ony on the rank of H ss, there exists a constant c > 0 depending ony on the rank of H ss such that N(H ss (F )) H ss (F ) cn(h ss (F )). If d denotes the dimension of H ss, this aso impies [N87, Lemma 3.5] ( 1) d c N(H ss (F )) ( + 1) d. As G sc (Z )/(p sc ) 1 (G(Z ) + ) is abeian, the non-abeian parts of the composition series of G sc (Z ) and (p sc ) 1 (G(Z ) + ) coincide and as the kerne of p sc : (p sc ) 1 (G(Z ) + ) G(Z ) + is abeian, we have N(G sc (F )) = N(G sc (Z )) = N((p sc ) 1 (G(Z ) + )) N(G(Z ) + ) = N(G(F ) + ) = N(G(F )) = N(G ss F (F )). Let d denote the common dimension of G F have ( 1) d Gsc (F ) c c and G sc F and et d ss denote the dimension of G ss F. Then we N(G sc (F )) N(G ss F (F )) ( + 1) dss.

14 14 ANNA CADORET, CHUN-YIN HUI AND AKIO TAMAGAWA When +, this forces d ss = d, as desired. Coroary 8.2. The assertions of Coroary 7.5 are aso equivaent to - (8.2) Π G (Q ) is an amost hyperspecia subgroup for 0. Proof. Assume 0 so that (6.2.1), (6.2.2), (6.3), (7.3.2) and the concusion of Coroary 7.4 hod. Assume (7.5.4) hods. Then G corresponds to a hyperspecia point b B, which aso gives rise to a simpy connected semisimpe mode G sc over Z of the simpy connected cover p sc : G sc G with the property that G sc (Z ) is the stabiizer of b in G sc (Q ) and the isogeny p sc : G sc G extends uniquey to a morphism p sc : G sc G of group schemes over Z. Furthermore, p sc : G sc G induces a morphism (Fact 7.2) G sc (Z ) G (Z ) + with the foowing properties. - (i) The diagram G sc (Q ) p sc G (Q ) G sc (Z ) G (Z ) + is cartesian. Indeed, since p sc : G sc G is open (even a oca isomorphism), G (Z ) + p sc (G sc (Q )) G (Z ) is open hence cosed, hence compact. So (p sc ) 1 (G (Z ) + ) is compact in G sc (Q ) as an extension of the compact group G (Z ) + p sc (G sc (Q )) by a finite group. Aso, (p sc ) 1 (G (Z ) + ) contains G sc (Z ). Thus, by maximaity of G sc (Z ), we have (p sc ) 1 (G (Z ) + ) = G sc (Z ). - (ii) The homomorphism G sc (Z ) G (Z ) + is surjective. Indeed, et g G (Z ) +. Since the image of p sc : G sc (Q ) G (Q ) is norma and its cokerne is of exponent bounded by a constant depending ony on the rank of G, g ies in the image of p sc : G sc (Q ) G (Q ) for 0 (compared with the rank of G ), that is, by (i), in the image of (p sc ) 1 (G (Z ) + ) = G sc (Z ). (8.2) thus foows from (7.3.2). The impication (8.2) (7.5.4) foows from Lemma 8.1 and (7.3.2). This concudes the proof of Theorem 1.2. PART III: SEMISIMPLICITY 9. Lie-theoretic proof of Theorem 1.1 The resuts expained here are entirey due to the second author, Chun-Yin Hui. They ed to the first compete proof of Theorem Semisimpe modes and good attices. Let G be a connected semisimpe group over Q of dimension δ and rank s. Let V be a faithfu, r-dimensiona Q -representation of G. Fix a attice H V ; this defines a mode GL H of GL V over Z. Let G denote the Zariski cosure of G inside GL H (endowed with the reduced subscheme structure). Then G is a fat mode of G over Z. Under mid assumptions, we give a criterion in terms of tensor-invariants data to ensure that G is a semisimpe group scheme over Z. Write G := G,F and G for its identity component. Let Rep f Z (G ) denote the category of finitey generated free Z -modues M together with a morphism of Z -group schemes G GL M. Define H : Rep f Z (G ) Z 0 M dim F (M G F ) dim Q (M G Q ). Let T G be a maxima torus. We wi say that T admits a nice mode with respect to H if T spits over a finite extension E of Q and if the cosed embedding T,E G s m,e GL V,E extends to a cosed embedding G s m,o GL H,O, where O denotes the ring of integers of E. Theorem 9.1. Assume G contains a maxima torus which admits a nice mode with respect to H. Then,

15 GEOMETRIC MONODROMY SEMISIMPLICITY AND MAXIMALITY 15 - (9.1.1) G is smooth over Z ; - (9.1.2) The quotient G rd of G by its unipotent radica is a reductive group of rank s (and the root system of G rd is a subsystem of the root system of G,F,Q ); - (9.1.3) For 0 (depending ony on r) the foowing hods. Let g H H denote the Lie agebra of G. Then G is semisimpe over Z if and ony if H (M) 0 for M = Λ n g, n = 1,..., δ. Proof. As Spec(O ) Spec(Z ) is fat, G,O GL H,O coincides with the Zariski cosure of G,E (endowed with its reduced subscheme structure) and G,E = G,E [BrT84, 1.2.6]. So to perform the proof, we may base-change to O, E. For simpicity, we assume O = Z and E = Q beow. Fix a Bore subgroup T B G ; write Φ := Φ(G, T ) for the root system and et Φ + Φ denote the set of positive roots defined by B. For α Φ, et g,α g,q = Lie(G ) g(v ) and U,α G denote the corresponding root space and group respectivey. Let T G s m,z, U,α G denote the Zariski cosure of T, U,α; by construction T, U,α are fat over Z. For α Φ, g,α g(h ) is a free Z -modue of rank 1. Let N,α g,α g(h ) be a Z -basis (in particuar N,α F 0). Then for r, the cosed embedding x,α : G a,z G, t exp(tn,α) induces an isomorphism of Z -group schemes onto a cosed subgroup scheme of G which coincides with U,α G. By [BrT84, (iii)] the T -equivariant morphism induced by mutipication U,α T U,α G α Φ + α Φ + induces an isomorphism onto a dense open Z -subscheme of G. In particuar G is smooth over Z, G G is an open subgroup scheme, smooth, affine over Z [BrT84, 2.2.5] and G contains a spit torus of rank s. As the reductive rank is ower semicontinous (e.g. [Gi13, Thm ]), G has reductive rank s. Furthermore, reduction moduo- induces a canonica isomorphism of G s m-modues g F Lie(G ). The atter impies that the root system of G rd is a subsystem of the root system of,f G,Q (since Lie(G rd) is a quotient of Lie(G )). We now turn to the proof of (9.1.3). Let G u denote the unipotent radica of G and write g := Lie(G ), g u := Lie(Gu and duaizing, ), grd := Lie(G rd ). This gives rise to a decomposition of the adjoint representation 0 g u g g rd 0, 0 (g rd ) g (gu ) 0. As G u acts triviay on grd (observe that for g G u the conjugation automorphism c g on G descends to the identity map on G rd = G /Gu ), the action of G on grd factors through the adjoint representation of G rd. Suppose now the condition on H is satisfied. Caim 1: For 0 (depending ony on r), G rd is semisimpe. Proof of Caim 1. Compute the dimension of the Lie agebra z of the center of G rd dim F (z ) dim F ((g rd )Grd ) (1) = dim F ((g rd ) Grd ) = dim F ((g rd ) G ) (2) dim F ((g )G ) (3) dim Q ((g,q ) G ) (1) = dim Q (g G,Q ) (4) = 0, where (1) is by the semisimpicity [J97, Prop. 3.2] (see aso [Sp68, Cor. 4.3]) and sef-duaity of the adjoint representation of the reductive group G rd (resp. G ) for 0 compared with the rank (resp. for a ), (2) is because g rd is a submodue of g, (3) is the assumption H (g ) 0 and (4) is because G is semisimpe. Caim 2: For every integer n and for 0 (depending ony on n) the foowing hods. Consider a pair of rank n connected semisimpe groups G over F and G over Q. Assume dim(g) < dim(g ). Then there