G-rigid local systems are integral
aa r X i v : . [ m a t h . AG ] S e p G-RIGID LOCAL SYSTEMS ARE INTEGRAL
CHRISTIAN KLEVDAL AND STEFAN PATRIKIS
Abstract.
Let G be a reductive group, and let X be a smooth quasi-projective complexvariety. We prove that any G -irreducible, G -cohomologically rigid local system on X withfinite order abelianization and quasi-unipotent local monodromies is integral. This general-izes work of Esnault and Groechenig when G = GL n , and it answers positively a conjectureof Simpson for G -cohomologically rigid local systems. Along the way we show that the con-nected component of the Zariski-closure of the monodromy group of any such local systemis semisimple. Introduction
A central question of arithmetic geometry is to identify which Galois representations arise,via ´etale cohomology, from algebraic geometry. The Fontaine-Mazur conjecture ([FM95]),asserting that all potentially semistable and almost everywhere unramified semisimple ℓ -adicrepresentations of the absolute Galois group of a number field F do indeed appear in the co-homology of smooth projective varieties over F , is the prototypical and most famous explicitproblem in this area. Carlos Simpson formulated in [Sim92, pg. 9 Conjecture] an analo-gous conjecture that rigid complex representations of the topological fundamental group ofa smooth projective complex variety X necessarily appear in the cohomology of a family ofvarieties over X . In particular ( loc. cit. ), rigid representations should be defined over thering of integers in some number field. Simpson raised these questions for representationsvalued not just in GL n , but in general algebraic groups G .When G = GL n , two striking general results are known. For X an open subvariety of P , Katzhas proven that any GL n -cohomologically rigid (see Definition 1.1) irreducible local systemon X with quasi-unipotent local monodromies is a subquotient of the monodromy repre-sentation of a family of varieties over X ([Kat96, Theorem 8.4]). Esnault and Groechenig([EG18, Theorem 1.1]; see also [EG17]) have proven Simpson’s integrality conjecture, for allsmooth quasi-projective varieties, for GL n -cohomologically rigid irreducible representationswith finite-order determinant and quasi-unipotent local monodromies.In the present paper we generalize the main theorem of [EG18] to rigid representations valuedin general connected reductive groups G . We begin by making precise the basic terms. Definition 1.1.
Let X be a connected smooth quasi-projective variety over C , and let j : X ֒ → X be a good compactification: thus X is smooth projective, and the boundary D = X \ X is a strict normal crossings divisor. Let x ∈ X ( C ) be any base-point, and let π top1 ( X, x ) Date : September 2020.We thank Domingo Toledo for helpful conversations about lattice rigidity theorems, and we thank H´el`eneEsnault and Michael Groechenig for their comments. C.K. was supported by NSF grant DMS-1840190. S.P.was supported by NSF grants DMS-1700759 and DMS-1752313. be the corresponding topological fundamental group. A homomorphism ρ : π top1 ( X, x ) → G ( C ) • is G -irreducible if the image of ρ is not contained in any proper parabolic subgroupof G . • is G -cohomologically rigid if H ( X, j ! ∗ g der ) = 0, where g der is the Lie algebra of thederived group of G , regarded as a local system on X via the composite Ad ◦ ρ . • has quasi-unipotent local monodromy if for all points y in the smooth locus of D andany sufficiently small ball ∆ ⊂ X around y , ρ ( γ ) is quasi-unipotent for a generator γ of π top1 (∆ \ D ∩ ∆) ∼ = Z .Equivalently, ρ is cohomologically rigid if it represents a smooth isolated point on an ap-propriate moduli space of G -local systems; see Proposition 4.7 and Remark 4.8. Note thatif ρ is either G -irreducible or G -cohomologically rigid, there need not exist a faithful finite-dimensional representation r : G → GL n of G such that r ◦ ρ is either GL n -irreducible orGL n -cohomologically rigid; thus the results of [EG18] cannot be used to bootstrap to thecase of general G .We now state the main theorem: Theorem 1.2.
Let X/ C be a connected smooth quasi-projective complex algebraic variety,with a base-point x ∈ X ( C ) . Let G be a split connected reductive group over Z , and let ρ : π top1 ( X, x ) → G ( C ) be a G -irreducible and G -cohomologically rigid local system such that • ρ has quasi-unipotent local monodromy; • the image of the composite homomorphism π top1 ( X, x ) ρ −→ G ( C ) → A ( C ) to the maximal abelian quotient A of G has finite order.Then im( ρ ) , the connected component of the identity of the Zariski closure of the image im( ρ ) of ρ , is semisimple; and there is a number field L with ring of integers O L such that ρ is G ( C ) -conjugate to a homomorphism π top1 ( X, x ) → G ( O L ) .Remark . When G = GL n , and discounting the conclusion that im( ρ ) is semisimple, thisis precisely [EG18, Theorem 1.1]. If ρ comes from geometry, i.e. it arises as the monodromyrepresentation of a sub-local system of R i f ∗ C for a smooth map f : Y → X , then ρ satis-fies the two conditions of the theorem. Indeed, it first follows from the local monodromytheorem that ρ has quasi-unipotent local monodromy. That the abelianized monodromyrepresentation has finite image follows from [Del71, Corollaire 4.2.8.iii(b)]. Remark . Many naturally-occurring local systems are rigid and provably integral by othermeans. For instance, let H be a connected semisimple Lie group with (real) rank at least 2 Replacing G ( C ) in this definition by G ( K ) for some field K , we will tend to abuse notation and say arepresentation is G -irreducible if it is G -absolutely irreducible, i.e. the resulting homomorphism into G ( K )does not factor through a proper parabolic subgroup. -RIGID LOCAL SYSTEMS ARE INTEGRAL 3 and having no compact factors, and let Γ ⊂ H be an irreducible lattice such that Γ · H is isdense in H , with H is the minimal connected normal subgroup such that H/H is is compact.Then Margulis has proven in turn the following remarkable results (see [Mar91, TheoremsIX.6.5, IX.6.15], starting from his lattice superrigidity theorem: • Γ is an arithmetic subgroup. • For any homomorphism ρ : Γ → G ( C ), the Zariski-closure im( ρ ) is semisimple. • For every representation r : Γ → GL n ( C ), H (Γ , r ) is trivial.In particular, with r equal to the adjoint action of G on g der , if the associated locallysymmetric space Γ \ H has the structure of a complex quasi-projective variety, we see that ρ is cohomologically rigid in the sense of Definition 1.1. (See Propositions 4.6 and 4.7;the vanishing condition here is in general stronger than what is needed for cohomologicalrigidity.) Many interesting rigid representations, however, have monodromy groups that arenot lattices in their real Zariski-closures: this is the phenomenon of so-called thin monodromygroups, and famous (hypergeometric) examples have been studied in [DM86], and morerecently [BT14]. In our algebro-geometric setting, the chain of reasoning is reversed: weassume cohomological rigidity, and then deduce the semisimplicity of the monodromy groupand integrality (in place of arithmeticity) of the representation.1.1. Overview of the proof.
The proof follows the arguments of [EG18], and indeed ourdebt to that paper will be evident throughout. The essential idea is, having shown the rigidrepresentation is defined over the ring of Σ-integers O K, Σ ⊂ K for some number field K andfinite set of places Σ, to check integrality at each λ ∈ Σ by specializing ρ to characteristic p and using results of Drinfeld and Lafforgue on the existence of compatible systems of λ -adic representations. Such arguments are considerably subtler for general G than for GL n ,since in general the semisimple conjugacy classes associated to Frobenius elements do notuniquely characterize G -irreducible λ -adic representations. In particular, our argument mustkeep track of monodromy groups in a way that [EG18] does not, and we rely on Drinfeld’swork [Dri18] for the existence of the requisite compatible systems.Here is a more detailed section-by-section outline of the proof, restricting for notationalsimplicity to the case where X is projective. In §
3, we prove the local integrality conditionneeded for the main theorem: granted that ρ factors as π top1 ( X, x ) ρ −→ G ( O K, Σ ), this reducesus to checking that for each λ ∈ Σ, the composite ρ λ : π top1 ( X, x ) ρ −→ G ( O K, Σ ) → G ( K λ )can be conjugated into G ( O K λ ).The initial factorization of ρ through some G ( O K, Σ ) is obtained, following [Sim92] and[EG18], by studying a suitable moduli space of G -local systems on X . In §
4, we recallthe construction and basic properties of these moduli spaces. Our proof begins in earnest in §
5. We consider a set S of (isomorphism classes of) G -local systems satisfying the hypothesesof the theorem. Using that the moduli space of G -local systems is finite type, it is deduced(here is the key input from cohomological rigidity) that S is finite, and that there exists anumber field K and a finite set of places Σ such that each element of S is conjugate to arepresentation ρ : π top1 ( X, x ) → G ( O K, Σ ). For each ρ ′ ∈ S and each place λ of K , we obtain, C. KLEVDAL AND S. PATRIKIS via extending scalars, a representation ρ ′ λ : π top1 ( X, x ) → G ( K λ ), and we denote the collec-tion of these homomorphisms by S λ . The final step of the proof (and the most interesting)is to deduce the integrality of our original ρ λ , for each λ ∈ Σ, from the integrality of themembers of S λ ′ for a fixed λ ′ Σ (note that in § § λ and λ ′ are reversed).We indicate this last step in more detail. The crucial inputs are results of L. Lafforgue([Laf02]) on the Langlands correspondence over function fields, and results of Drinfeld([Dri18], building on [Dri12]) that promote Lafforgue’s work to construct λ -adic compan-ions for λ ′ -adic representations—and even for suitable G ( K λ ′ )-representations—of the fun-damental group of a smooth variety of any dimension over a finite field. To exploit theseresults, following [EG18], we spread the complex variety X out and and take a fiber X s overa finite field. In the remainder of § S λ ′ can be specialized anddescended to (´etale) G -local systems { ρ ′ λ ′ ,s } ρ ′ ∈S on X s . This step is subtler than in the caseof GL n ; it requires attending to the monodromy groups of the specialized representations(Proposition 5.6), establishing along the way their (connected components’) semisimplicity(Corollary 5.7).In §
6, we use the work of Lafforgue and Drinfeld mentioned above to produce a collection { ρ ′ λ,s } ρ ′ ∈S of λ -adic companions (which are necessarily integral) of these ρ ′ λ ′ ,s . The semisim-plicity of the monodromy groups is essential here too, in order to make use of [Dri18].From the collection of ρ ′ λ,s on X s , we construct via tame specialization G -local systems ρ ′ λ : π top1 ( X, x ) → G ( O K λ ) on our original X (over C ). It is then shown that the S localsystems ρ ′ λ constructed in this way are pairwise distinct and satisfy all of the defining prop-erties of elements of S . By counting, our original ρ λ must belong to this set, each memberof which is integral at λ (by virtue of arising from ´etale local systems). We conclude thatfor all λ ∈ Σ, ρ λ can be conjugated into G ( O K λ ), and then we are done by the results of § Notation
For a connected reductive group G , we let G der denote the derived group, G ad denote theadjoint group, G sc denote the simply-connected cover of G der (equivalently, of G ad ), and Z G denote the center. For a detailed treatment of how these constructions carry over to the caseof the base scheme Spec( Z ), and indeed much more generally, we refer the reader to [Con14](or to the original constructions in [Gro64]), especially [Con14, Theorem 3.3.4, Example5.1.7, Theorem 5.3.1 and following].If X is a complex variety and x ∈ X ( C ), we write π top1 ( X, x ) to denote the fundamental groupbased at x of the topological space X ( C ) with its analytic topology. If X is a scheme, and x : Spec(Ω) → X is a separably-closed base-point, we let π ´et1 ( X, x ) denote the corresponding´etale fundamental group. If j : X ֒ → X is a good compactification, so that X is regularand D = X \ X is a strict normal crossings divisor, we let π ´et ,t ( X, x ) be the quotient of the´etale fundamental group corresponding to the fully faithful embedding of finite ´etale covers Y → X tamely ramified along D (i.e. such that the valuation v D i on k ( X ) of any irreduciblecomponent D i of D is tamely ramified in k ( Y )) into the category of all finite ´etale covers of X . For a reference that this is a Galois category, we refer the reader to [GM06, Theorem -RIGID LOCAL SYSTEMS ARE INTEGRAL 5 K is a number field or a local field, we denote its ring of integers by O K . In the numberfield case, if Σ is a set of finite places of K , we let O K, Σ denote the localization away fromΣ of O K . We write K ∞ for the product of the completions of K at its infinite places, andwe write A ∞ K for the finite ad`eles of K .3. Local integrality condition
The next proposition gives the criterion we will apply in § Proposition 3.1.
Let Γ be a finitely generated group, and let G be a connected reductivegroup over Z . Let ρ : Γ → G ( O K, Σ ) be a homomorphism, where Σ is a finite set of finiteplaces of K . Assume that for each λ ∈ Σ , the representation ρ λ : Γ → G ( K λ ) is G ( K λ ) -conjugate to a representation Γ → G ( O K λ ) . Then there exists a finite extension L/K suchthat ρ is G ( L ) -conjugate to a representation Γ → G ( O L ) .Proof. Each representation ρ λ : Γ → G ( K λ ) is conjugate to a G ( O K λ )-valued representation.That is, for each λ ∈ Σ, there is some g ′ λ ∈ G ad ( K λ ) such that g ′ λ ρg ′ λ − : Γ → G ( O K λ ) (alsowriting g ′ λ for any lift to G ( K λ )). The map G sc ( K λ ) → G ad ( K λ ) is surjective, so we canfind g λ ∈ G sc ( K λ ) lifting g ′ λ . It follows that the image of g λ in G der ( K λ ) conjugates ρ λ intoa G ( O K λ )-valued representation. We denote this representation ρ g λ λ .Each g λ lies in G sc ( L ( λ )) for some finite extension L ( λ ) of K λ , which we may assume to beGalois. In particular, ρ g λ λ has image contained in G ( L ( λ )) ∩ G ( O K λ ) = G ( O L ( λ ) ). Since Σ isfinite, we can by class field theory find a Galois (in fact solvable) extension of number fields L over K such that for all λ ∈ Σ, and all places ν of L above λ , L ν is isomorphic (over K λ )to L ( λ ), and moreover L has no real embeddings.Consider the element g A ∞ L = ( g ν ) ∈ G sc ( A ∞ L ) whose ν -th component is, for ν above λ ∈ Σ,the image of g λ defined above under any K λ -isomorphism L ( λ ) ∼ −→ L ν , and the identityotherwise. Then (the image in G der of) g A ∞ L conjugates the representation ρ A ∞ L : Γ ρ −→ G ( O K, Σ ) → G ( K ) → G ( L ) → G ( A ∞ L )into a homomorphism Γ → G ( Q O L ν ). Now suppose that γ , . . . , γ n ∈ G ( A ∞ L ) are the imagesunder ρ A ∞ L of a set of generators of Γ. Let π : G sc → G der be the quotient map and considerthe map conj : G sc ( A ∞ L ) → n Y i =1 G ( A ∞ L ) , g ( π ( g ) γ π ( g ) − , . . . , π ( g ) γ n π ( g ) − ) . Then U = conj − ( G ( Q O L ν ) n ) is open and non-empty since it contains g A ∞ L . Since G sc issimply-connected, and G sc ( L ∞ ) is non-compact, the strong approximation theorem [Kne66]implies there is an element g ′ ∈ G sc ( L ) ∩ U . Then the element π ( g ′ ) is the desired elementof G ( L ). (cid:3) C. KLEVDAL AND S. PATRIKIS Moduli of representations
In this section G is a split connected reductive group over a field K of characteristic zero,with center Z G and adjoint group G ad = G/Z G . Let Γ be a finitely-generated group withpresentation Γ = h r , . . . , r k | { s α } α ∈ B i , where the set B indexes the relations s α . Let b : G → A be the maximal abelian quotient of G , and fix a homomorphism θ : Γ → A ( K ). Let Rep G (Γ , θ ) be the affine K -variety of repre-sentations of Γ with abelianization equal to θ , that is for a K -scheme T , Rep G (Γ , θ )( T ) is theinverse image of Γ θ −→ A ( K ) → A ( T ) under Hom(Γ , G ( T )) → Hom(Γ , A ( T )). More explicitly,the map Rep G (Γ , θ )( T ) → G k ( T ) given by ρ ( ρ ( r ) , . . . , ρ ( r k )) identifies Rep G (Γ , θ ) withthe closed subscheme of G k given by given by the conditions s α ( g , . . . , g k ) = 1 for α ∈ B and b ( g i ) = θ ( g i ) for i = 1 , . . . , k . (Note that since G k is Noetherian, finitely many of therelations s α suffice to describe this subscheme.) There is an action of G ad on Rep G (Γ , θ ) byconjugating homomorphisms. Let Loc G (Γ , θ ) be the resulting stack quotient,Loc G (Γ , θ ) = [Rep G (Γ , θ ) /G ad ] . If T is a K -scheme then Loc G (Γ , θ )( T ) is the groupoid with objects ( E , f ) and isomorphisms( E ′ , f ′ ) ∼ −→ ( E , f ) given respectively as commutative diagrams E Rep G (Γ , θ ) T f E ′ E Rep G (Γ , θ ) T f ′ ≃ f where E → T is a (left) G ad -torsor, and f : E →
Rep G (Γ , θ ) is a G ad -equivariant morphism(likewise for ( E ′ , f ′ )). The isomorphism E ′ → E is required to be G ad -equivariant. Definition 4.1. If T is a K -scheme, and ρ a representation ρ : Γ → G ( T ) whose abelianiza-tion is θ , we denote by [ ρ ] in Loc G (Γ , θ )( T ) the object ( G ad T , f ρ ), where f ρ : G ad T → Rep G (Γ , θ )is given by f ρ ( g ) = gρg − . Lemma 4.2. If Ω is an algebraically closed field (containing K ), then Loc G (Γ , θ )(Ω) can beidentified with the groupoid of conjugacy classes of representations Γ → G (Ω) with abelian-ization equal to θ .Proof. The groupoid C of conjugacy classes of morphisms Γ → G (Ω) has objects beingrepresentations Γ → G (Ω), whose abelianization is θ . A morphism ρ → ρ is given by innerautomorphisms Ad( g ) : G (Ω) → G (Ω) for g ∈ G ad (Ω) such that ρ = Ad( g ) ◦ ρ , so there isa morphism only when ρ = gρ g − for some g ∈ G ad (Ω).Given a representation ρ : Γ → G (Ω) with abelianization θ , we have the associated G -localsystem [ ρ ] = ( G adΩ , f ρ ) in Loc G (Γ , θ )(Ω). The isomorphism ρ → gρg − for g ∈ G ad (Ω) givesa morphism of G adΩ -torsors R g − : ( G adΩ , f ρ ) → ( G adΩ , f gρg − ) by R g − ( h ) = hg − on Ω-points.This gives an isomorphism [ ρ ] → [ gρg − ] in Loc G (Γ , θ ) since f ρ ( h ) = hρh − = hg − gρg − gh − = f gρg − ( hg − ) . -RIGID LOCAL SYSTEMS ARE INTEGRAL 7 Thus, ρ [ ρ ] is a functor C →
Loc G (Γ , θ )(Ω).To see that it is fully faithful, notice that both Hom( ρ , ρ ) and Hom([ ρ ] , [ ρ ]) can naturallybe identified the the set of h ∈ G ad (Ω) such that ρ = hρ h − .We now show that the functor is fully faithful. By definition, an object of Loc G (Γ , θ )(Ω)is a G ad -torsor E →
Spec(Ω) and a G ad -equivariant map ϕ : E →
Rep G (Γ , θ ). Since Ω isalgebraically closed, there is a point s ∈ E (Ω), which gives a G ad -equivariant isomorphism π : G adΩ ≃ −→ E which is g gs on Ω points. Let ρ = f ( s ). Since ϕ ( gs ) = gϕ ( s ) g − , π gives anisomorphism [ ρ ] = ( G ad , f ρ ) ≃ −→ ( E , ϕ ). (cid:3) We define a substack IrrLoc G (Γ , θ ) of Loc G (Γ , θ ) consisting of the objects of Loc G (Γ , θ )( S )whose base change along any geometric point Spec(Ω) → S yields via the identification ofLemma 4.2 a conjugacy class of G -irreducible representations. Recall that a representationΓ → G (Ω) is G -irreducible if the image is not contained in any proper parabolic subgroup. Proposition 4.3.
The substack
IrrLoc G (Γ , θ ) ⊂ Loc G (Γ , θ ) is open.Proof. For a K -scheme S and ( E , f ) ∈ Loc G (Γ , θ )( S ) we say that ( E , f ) is G -reducible at s ∈ S if for some (equivalently any) a geometric point s : Spec(Ω) → S lying over s , theconjugacy class of representations Γ → G (Ω) corresponding to ( E s , f s ) is G -reducible. Wewill show that the locus of s ∈ S for which ( E , f ) is G -reducible is closed.Fix a maximal torus and Borel subgroup T ⊂ B ⊂ G , with corresponding positive roots Φ + and simple roots ∆. For any finite set S ⊂ ∆, let P S be the associated standard parabolic,and consider the bundle π : Z = Z S ( E , f ) := G ad \ ( E × ( G/P S )) → S, where G ad acts diagonally (using that the left multiplication action of G on G/P S factorsthrough G ad ). We get a Γ action on Z via the map E →
Rep G (Γ , θ ): if x ∈ E ( U ) maps via f to ρ x : Γ → G ( U ) in Rep G (Γ , θ )( U ) then for ( x, y ) ∈ ( E ×
G/P S )( U ) define γ · ( x, y ) := ( x, ρ x ( γ ) y ) γ ∈ Γ . This descends to an action on Z ( U ) since if we change representatives to ( gx, gy ) then γ · ( gx, gy ) = ( gx, ρ gx ( γ ) gy ) = ( gx, gρ x ( γ ) g − gy ) = ( gx, gρ x ( γ ) y ) . We claim that the locus π ( Z S ( E , f ) Γ ) ⊂ S consists of all of the points s ∈ S such thatfor any (equivalently every) geometric point s : Spec(Ω) → S lying over s , the conjugacyclass im( f s : E s (Ω) → Rep G (Γ , θ )(Ω)) contains a representation ρ : Γ → G (Ω) whose imageis contained in P S (Ω). Indeed, given s ∈ S and s a geometric point above s , we can write s = π ([ x, y ]) where [ x, y ] is the class of ( x, y ) ∈ ( E s × G/P S )(Ω) in Z s (Ω). By multiplying( x, y ) by a suitable element of G (Ω), we can and do assume that y = e , the identity coset in G/P S (Ω). Thus s ∈ Z S ( E , f ) Γ (Ω) ⇐⇒ ( x, e ) = γ ( x, e ) = ( x, ρ x ( γ ) e ) for all γ ∈ Γ , ⇐⇒ ρ x ( γ ) ∈ Stab G (Ω) ( e ) = P S (Ω) for all γ ∈ Γ , ⇐⇒ im( f s ( x ) = ρ x ) ⊂ P S (Ω) C. KLEVDAL AND S. PATRIKIS
It follows from this description that the locus of s ∈ S where ( E , f ) is G -reducible is the set [ S ( ∆ π ( Z S ( E , f ) Γ )For each S the fixed point locus Z S ( E , f ) Γ is closed, so π ( Z S ( E , f ) Γ ) is closed (since Z → S is proper), and hence the finite union of such sets is closed. (cid:3) Now suppose we are given γ , . . . , γ N ∈ Γ, and conjugacy classes K i ⊂ G defined over K for i = 1 , . . . , N . Note that these are locally closed subschemes of G . We consider the locallyclosed substack M = M G (Γ , θ, { ( γ i , K i ) } ) ⊂ IrrLoc G (Γ , θ ) , (1)where an object ( E , f ) ∈ IrrLoc G (Γ , θ )( T ) is in M ( T ) if for every geometric point t ∈ T (Ω),and any representation ρ : Γ → G ad (Ω) with [ ρ ] = ( G adΩ , f ρ ) ≃ ( E t , f t ), we have that ρ ( γ i ) ∈K i (Ω) for i = 1 , . . . , N . Note that this is independent of the choice of ρ , since any other ρ ′ satisfying ( G adΩ , f ρ ′ ) ≃ ( E t , f t ) will be G ad (Ω) conjugate to ρ by lemma 4.2.To see that the stack M is locally closed in IrrLoc G (Γ , θ ), argue as follows: Let ( E , f ) be anobject of IrrLoc G (Γ , θ )( T ), and let T ⊂ T be set of points t ∈ T such that for any geometricpoint t ∈ T (Ω) over t , ( E t , f t ) ∈ M (Ω). Let E = E × T T . This is G ad stable, so it suffices toshow that E is locally closed. Let ρ : Γ → G ( E ) be the representation associated to f . Each ρ ( γ i ) is a map E → G and E = N \ i =1 ρ ( γ i ) − ( K i ) . Since each K i is locally closed in G , the intersection is locally closed. In summary, we havethe following proposition. Proposition 4.4.
Let Γ be a finitely-generated group, K a field, G a split connected reductivegroup over K with maximal abelian quotient A . Fix • θ : Γ → A ( K ) a representation; • γ , . . . , γ N ∈ Γ ; • conjugacy classes K i ⊂ G defined over K (for i = 1 , . . . , N ).Remark . The case of interest for this paper is when K is a number field. Then the stack M of conjugacy classes of G -irreducible representations ρ : Γ → G withabelianization θ , such that ρ ( γ i ) ∈ K i for all i = 1 , . . . , N is an algebraic stack of finite typeover K . Now suppose ρ : Γ → G ( K ) is a G -irreducible representation such that ρ ( γ i ) ∈ K i ( K ). The G -local system [ ρ ] = ( G ad , f ρ ) associated to ρ is then an object of M ( K ) (for θ = ρ ab0 ). Proposition 4.6.
Let g der be the Lie algebra of G der . Then the Zariski tangent space T [ ρ ] M is the kernel of H (Γ , g der ( K )) res −→ n M i =1 H ( γ Z i , g der ( K )) . -RIGID LOCAL SYSTEMS ARE INTEGRAL 9 Proof.
For any point ( E , f ) ∈ M ( K [ ε ]), let ( E , f ) = i ∗ ( E , f ) with i the closed immersionSpec( K ) → Spec( K [ ε ]). The tangent space T [ ρ ] M consists of the points v = ( E , f ) in M ( K [ ε ]) such that ( E , f ) ≃ −→ ( G ad , f ρ ) over K (the isomorphism not being part of thedata). Fix one such isomorphism ϕ . By the formal criterion of smoothness, the sectionSpec( K ) −→ G ad ϕ − −−→ E extends to give a section of E →
Spec( K [ ε ]), hence E is trivial.By choosing a trivialization ϕ : E ∼ −→ G adSpec( K [ ε ]) (extending ϕ on the closed fiber) we get arepresentation ρ = f ◦ ϕ − (1) : Γ → G ( K [ ε ]) whose composition with G ( K [ ε ]) → G ( K ) is ρ , and which also maps γ i to an element of K i ( K [ ε ]).The map α ( γ ) = ρ ( γ ) ρ ( γ ) − gives an element α ∈ Z (Γ , g der ( K )) (the “fixed determinant”condition ensures that α is valued in g der ( K ) ⊂ g ( K )). The manipulation ρ ( γβ ) ρ ( γβ ) − = ρ ( γ ) ρ ( β ) ρ ( β ) − ρ ( γ ) − , = ρ ( γ ) ρ ( γ ) − · Ad( ρ )( γ ) (cid:0) ρ ( β ) ρ ( β ) − (cid:1) . verifies the cocycle condition, and it is similarly easy to see that changing the trivialization ϕ modifies α by a coboundary.Therefore there is a well defined map T [ ρ ] M → H (Γ , g der ( K )). It is injective because onecan reconstruct the point v from the class [ α ] by taking ρ = αρ , and then v = [ ρ ]. On theother hand, given a cocycle α the point [ αρ ] is in T [ ρ ] M if and only if for each i = 1 , . . . , n we have αρ ( γ i ) ∈ K i ( K [ ε ]). The restriction res([ α ]) is trivial if and only if for each i thereis some ξ i ∈ g der ( K ) such that α ( γ i ) = ξ i ρ ( γ i ) ξ − i ρ − ( γ i ), hence αρ ( γ i ) = ξ i ρ ( γ ) ξ − i , and thus αρ ( γ i ) belongs to K i ( K [ ε ]). We conclude that the image of T [ ρ ] M in H (Γ , g der ( K ))is the kernel of res : H (Γ , g der ( K )) → ⊕ ni =1 H ( γ Z i , g der ( K )). (cid:3) Back to geometry.
We now specialize to the case of interest. Let X be a smoothconnected quasi-projective complex variety, fix a base-point x ∈ X ( C ), and let j : X ֒ → X be a good compactification with D = X \ X the strict normal crossings divisor written asthe union D = ∪ Ni =1 D i of its irreducible components D i . Let D sing be the singular locus of D and U = X \ D sing so there is a factorization j : X a ֒ −→ U b ֒ −→ X . For i = 1 , . . . , N , fix y i ∈ D i ∩ U , ∆ i ⊂ U a small open ball around y i , and x i ∈ ∆ × i = ∆ i \ ( D i ∩ ∆ i ). Choose T i ∈ π top1 ( X, x ) that generates the image of Z ∼ = π top1 (∆ × i , x i ) → π top1 ( X, x ) (with the mapdepending on a fixed choice of path from x ending in ∆ × i .Let K be a characteristic zero field. Suppose K i is a conjugacy class defined over K for each i = 1 , . . . , N , fix a character θ : π top1 ( X, x ) → A ( K ), and define M := M G ( π top1 ( X, x ) , θ, { ( T i , K i ) } ) ⊂ Loc G ( π top1 ( X, x ) , θ ) , the moduli stack of G -irreducible G -local systems on X with abelianization θ and monodromy K i around D i , as constructed above. Let ρ : π top1 ( X, x ) → G ( K ) be a homomorphismthat is G -irreducible, has abelianization θ , and has monodromy ρ ( T i ) ∈ K i ( K ). Then theassociated G -local system [ ρ ] gives an element of M ( K ). Let g der be the locally constantsheaf on X corresponding to the representation π top1 ( X, x ) ρ −→ G ( K ) Ad −→ GL( g der ( K )) . Proposition 4.7.
The tangent space of M at the point [ ρ ] is the finite-dimensional K -vectorspace H ( U, a ∗ g der ) .Proof. Giving the Lie algebra g der ( K ) the structure of a π ( X, x )-module vial Ad ◦ ρ , thereis a commutative diagram0 T [ ρ ] M H ( π top1 ( X ) , g der ( K )) ⊕ Ni =1 H ( π (∆ × i ) , g der ( K )) H ( X, g der ) ⊕ Ni =1 H (∆ × i , g der | ∆ × i ) res ≃ ≃ res and Proposition 4.6 shows that the top row is exact. The result follows from the identificationof the kernel of the map on the bottom row with H ( U, a ∗ g der ) (which can be seen using theLeray spectral sequence for a ). (cid:3) Remark . As was noted in [EG18, Remark 2.4], we have H ( U, a ∗ g der ) = H ( X, j ! ∗ g der ),where j ! ∗ g der is the intermediate extension.5. Arithmetic descents
Reduction to the adjoint case.
We begin the proof of Theorem 1.2 by reducing tothe case in which G is a simple (split, as we always assume) adjoint group: Lemma 5.1.
It suffices to prove Theorem 1.2 in the case where G is a simple adjoint group.Proof. Consider ρ as in Theorem 1.2. The projectivization P ( ρ ) : π top1 ( X, x ) ρ −→ G ( C ) → G ad ( C ) is G ad -irreducible, G ad -cohomologically rigid, and has quasi-unipotent local mon-odromy (as is evident from the definitions). Note that P ( ρ ) having these properties is equiv-alent to its projections to each simple factor of G ad having these properties. Thus, under theassumption of the Lemma, we may assume that after replacing ρ by a G ( C )-conjugate, P ( ρ )factors through G ad ( O K ) for some number field K . We claim that ρ itself factors through G ( O L ) for some finite extension L/K . Indeed, ρ of course factors through the preimageof im( P ( ρ )) in G ( C ). By our assumption that ρ has finite image in A ( C ), im( ρ ) is furthercontained in G der ( C ) · Z G ( Z ), since any point of finite order in Z G ( C ) is contained in Z G ( Z ).Thus it suffices from the long-exact sequence in cohomology to check that the image of eachof the generators of the finitely-generated group π top1 ( X, x ) in G ad ( O K ) is itself in the imageof G sc ( O L ) → G ad ( O L ) for some finite extension L/K . Since ker( G sc → G ad ) is a product(over the simple factors) of group schemes µ r , it suffices to find O L trivializing any givenclass in H (Spec( O K ) , µ r ). The (fppf) Kummer sequence yields a short-exact sequence O × K / ( O × K ) r → H (Spec( O K ) , µ r ) → Pic( O K )[ r ] . The desired L exists because every element of the class group of O K becomes trivial afterpassage to some finite extension O L , and we can adjoin an r th root to trivialize an elementof O × K / ( O × K ) r . Finally, semisimplicity of the connected monodromy of P ( ρ ) clearly impliesthat of ρ itself, as above using that ρ has finite image in A ( C ). (cid:3) -RIGID LOCAL SYSTEMS ARE INTEGRAL 11 In this section and §
6, we will therefore prove:
Theorem 5.2.
Theorem 1.2 is true if G is simple and adjoint. Algebraicity of the local system.
Let X be a smooth quasi-projective variety over C and ( X, U, D, . . . ) a compactification as in the first paragraph of 4.1. Let G be a splitsimple adjoint group over Z , and let ρ : π top1 ( X, x ) → G ( C ) be a representation defining a G -local system on X such that • ρ is G -cohomologically rigid, i.e. H ( U, a ∗ g der ) = 0. • ρ is G -irreducible. • The local monodromies (i.e. ρ ( T ) , . . . , ρ ( T N ) with T i a small enough loop around D i as in 4.1) are quasi-unipotent.Richardson showed ([Ric88, Theorem 3.6, Theorem 4.1], valid over fields of characteristiczero) that G -irreducibility of a closed subgroup H ⊂ G is equivalent to H being reductivewith centralizer C G ( H ) that is finite modulo the center Z G of G (in our case, simply finite).In particular, im( ρ ) enjoys these two properties.Our strategy, following [EG18], to prove Theorem 5.2 is to define a finite set S of G -localsystems with [ ρ ] ∈ S , and to prove that all elements of S are integral. For this, choose aninteger h ≥ ρ ( T i ) is quasi-unipotent of index dividing h for all i = 1 , . . . , N .Let S = S ( G, h ) be the set of conjugacy classes of representations ρ : π top1 ( X, x ) → G ( C )that are G -irreducible, G -cohomologically rigid, and such that each ρ ( T i ) is quasi-unipotentof index dividing h . Lemma 5.3.
The set S is finite, and there exist a number field K ⊂ C and a finite set Σ of finite places of K such that each element of S is represented by some representation ρ : π top1 ( X, x ) → G ( O K, Σ ) .Proof. There is a finite collection K , . . . , K M ⊂ G ( C ) of conjugacy classes of quasi-unipotentelements of index dividing h . Indeed, one can choose a finite number of semisimple elementsthat represent any semisimple conjugacy class of elements of order dividing h (the h -torsionin T ( C ), for T a maximal torus of G will do). The conjugacy classes with a fixed semisimplepart s correspond to unipotent conjugacy classes in C G ( s ). As C G ( s ) is connected reductive(since s is semisimple), it has finitely many unipotent conjugacy classes, and a fortiori thesame then holds for C G ( s ) (all unipotent elements of C G ( s ) belong to C G ( s ) ).The K i are the complex points of a locally closed algebraic subvariety of G K for some commonnumber field K . By abuse of notation, we write K i ⊂ G K for this variety. For each N -tuple J = ( j , . . . , j N ) with 1 ≤ j , . . . , j N ≤ M , let M J = M G ( π top1 ( X, x ) , { ( T i , K j i ) } ) be themoduli space defined in (1) of §
4, that is the collection of G -irreducible G -local systems on X with monodromy around D i in K j i . Note that since G is adjoint, the condition fixing theprojection G → A = { } is vacuous. Let N = ∪ J M J as J runs over the finite set of N -tuplesas above. By Proposition 4.7 the set S ( G, h ) is contained in the isolated points of the stack N , and since N is finite type by Proposition 4.4, there are finitely many isolated points. The index of a quasi-unipotent element g ∈ G ( C ) is the order of g s is the Jordan decomposition g = g s g u . More precisely, suppose ρ : π top1 ( X, x ) → G ( C ) has ρ ( T i ) ∈ K j i for i = 1 , . . . , N . If[ ρ ] ∈ S ( G, h ) we claim that the image of the map f = f ρ : G C → Rep G ( π top1 ( X, x ) , { T i , K j i } ),given by g gρg − , is a connected component. If not, im( f ) is contained in some closedconnected subspace Z ) im( f ). Let v ∈ T f (1) Z \ T f (1) im( f ρ ). Then v corresponds to acocycle α v ∈ Z ( π top1 ( X, x ) , g der ( C )), with action Ad ◦ ρ on g der ( C ). The cocycles from T f (1) im( f ) are precisely the coboundaries, so our choice of v , along with Proposition 4.7shows that the cohomology class [ α v ] is a non zero element of H ( U, a ∗ g der ( C )). This contra-dicts the fact that ρ is G -cohomologically rigid. Thus im( ρ ) is a connected component. AsRep G ( π top1 ( X, x ) , { T i , K j i } ) C is finite type over C , there are finitely many connected compo-nents. We conclude that S ( G, h ) is finite as it injects into the set of connected componentsof finitely many representation varieties of the above type.The group Aut( C / Q ) acts on the isomorphism classes of objects of N ( C ), and permutesthe finite set S ( G, h ). We claim that if ρ : π top1 ( X, x ) → G ( C ) represents a G -local system[ ρ ] ∈ S ( G, h )( C ), then ρ is defined over Q (and hence over a number field K since π top1 ( X, x )is finitely-generated). This can be seen using the pseudo-character tr ρ = (Θ n ) n ≥ of ρ (for background on pseudo-characters, including the notation we use, see [BHKT19, § n is the map Θ n : Z [ G n ] G → Map( π top1 ( X, x ) n , C ) given by Θ n ( f )( γ , . . . , γ n ) = f ( ρ ( γ ) , . . . , ρ ( γ n )). The pseudo-characterof σ ρ is then ( σ Θ n ( f ))( γ ) = σ (Θ n ( f )( γ )) for γ = ( γ , . . . , γ n ) ∈ π top1 ( X, x ) n . If ρ cannot bedefined over Q , then by [Laf18, Proposition 11.7] (see also [BHKT19, Theorem 4.5]) there issome f ∈ Z [ G n ] G and some γ ∈ π top1 ( X, x ) n such that Θ n ( f )( γ ) is transcendental over Q . Itfollows that there is an infinite set H ⊂ Aut( C / Q ) such that ( σ Θ n ( f ))( γ ) = ( τ Θ n ( f ))( γ ) forany σ = τ ∈ H , and hence by loc. cit. , σ ρ is not conjugate to τ ρ . Thus the set of conjugacyclasses of { σ ρ } σ ∈ H is infinite and contained in S ( G, h ) which contradicts finiteness of S ( G, h ).Choose representatives ρ i , i = 1 , . . . , N of the conjugacy classes in S ( G, h ). By the abovediscussion each ρ i is defined over a number field K . Enlarging K , we can assume each ρ i isdefined over the same K . Since π top1 ( X, x ) is finitely generated, each ρ i is defined over O K, Σ for Σ a finite set of finite places. Enlarging Σ, we may assume that ρ , . . . , ρ N are all definedover O K, Σ . (cid:3) By the previous lemma we choose once and for all representations ρ i : π top1 ( X, x ) → G ( O K, Σ ) , i = 1 , . . . , N , which form a complete set of representatives for the conjugacy classes in S ( G, h ).Fix λ ∈ Spec( O K, Σ ) a finite place. Consider the representation ρ ◦ , top i,λ : π top1 ( X, x ) → G ( O K, Σ ) → G ( O K λ ) . Composing with G ( O K λ ) → G ( K λ ) we get a representation ρ top i,λ (and similarly we get repre-sentations ρ top i,λ ′ for λ ′ ∈ Σ). Since G ( O K λ ) is a profinite group, the representations ρ ◦ , top i,λ , ρ top i,λ of π top1 ( X, x ) factor through π ´et1 ( X, x ). We denote the factored representations by ρ ◦ i,λ and ρ i,λ .The next step is to spread X out over a scheme S of finite type over Spec( Z ) and then tospecialize to characteristic p as in [EG18]. More precisely, there exists S a connected regular -RIGID LOCAL SYSTEMS ARE INTEGRAL 13 scheme of finite type over Spec( Z ) with a complex generic point η : Spec( C ) → S such that( j : X ֒ → X, x, D, D i ) is the fiber over η of ( j S : X S ֒ → X S , x S , D S , D i,S ), where X S is smoothand projective over S , D S = ∪ i D i,S ⊂ X S is a relative strict normal crossings divisor whosecomplement is X S , and x S ∈ X S ( S ). If s ∈ S ( k ) let ( j s : X s → X s , x s , D s , D i,s ) be the dataover k obtained as the base change of the corresponding data over S via s . Remark . Strictly speaking, for the proof of Proposition 6.2 we will need a little more:Fix a curve C ⊂ X that is smooth and projective, does not meet D sing and meets D \ D sing transversely (for the existence of C , see [Jou79, Theorem 6.3]). We take x ∈ C , and choosepoints y i ∈ D i ∩ C . We may (and do) choose S such that ( j : X ֒ → X, x, D, D i , C, y i )spreads out over S to ( j S : X S ֒ → X S , x S , D S , D i,S , C S , y i,S ) satisfying the above constraintson j S , X S , D S , D i,S and such that C S is smooth and projective over S and C S intersects D S transversely. Write C s and y i,s for the fiber of C S and y i,S over s . The reader may safelyignore the curve C until the proof of Proposition 6.2.We now choose a finite field k and a closed point s ∈ S ( k ) such that p = char k is coprimeto • the prime ℓ under λ ; • all places under Σ; • for each i , the cardinality of the image of the reduction¯ ρ ◦ i,λ : π ´et1 ( X, x ) → G ( k ( λ ))of ρ ◦ i,λ to the residue field k ( λ ) of O K λ ; and • the index of quasi-unipotence of the ρ i .Given these choices, the p -part of π ´et1 ( X, x ) is in the kernel of ρ ◦ i,λ , hence ρ ◦ i,λ factors throughthe prime-to- p fundamental group π ´et1 ( X, x ) → π ´et ,p ′ ( X, x ). If s is a geometric point over s , there is a discrete valuation ring R and a map Spec( R ) → S mapping the generic pointof Spec( R ) to η and the special point to s . By pulling back X S to Spec( R ), there is aprime-to- p specialization map on prime-to- p fundamental groups giving an isomorphismSp : π ´et ,p ′ ( X, x ) ∼ −→ π ´et ,p ′ ( X s , x s ) (see [LO10, Corollary A.12], or originally [sga71]). Thus weget representations ρ ◦ i,λ,s : π ´et1 ( X s , x s ) ։ π ´et ,p ′ ( X s , x s ) Sp − −−−→ π ´et ,p ′ ( X, x ) ρ ◦ i,λ −−→ G ( O K λ )and again denote by ρ i,λ,s : π ´et ,p ′ ( X s , x s ) → G ( O K λ ) → G ( K λ )the composites into G ( K λ ). Remark . We say that ρ i,λ,s has quasi-unipotent local monodromy with index dividing h along D j ( j = 1 , . . . , N ) if it maps any pro-generator of the tame inertia group at D j (i.e.,the absolute Galois group of the fraction field of the strict henselization of the DVR O X ¯ s ,D j, ¯ s )to a quasi-unipotent element whose h th power is unipotent. This property is inherited fromthe corresponding property of ρ i . Indeed, the monodromy of ρ i at D j is the same as the There is also a surjective tame specialization map π ´et1 ( X, x ) = π ´et ,t ( X, x ) → π ´et ,t ( X s , x s ) (see [LO10,Corollary A.12]) which we will use later (particularly in the proof of Proposition 6.2). monodromy of ρ i | C at y j for our chosen curve C above. Similarly, the the monodromy of ρ i,λ,s at D j,s is the same as the monodromy of it’s restriction ρ i,λ,s | C s to C s at y i,s by Abhyankar’slemma [sga71, XIII Proposition 5.2] (noting that these local systems are tamely ramified).Thus we reduce to showing that the monodromy of ρ i,λ,s | C s at y i,s is quasi-unipotent of indexdividing h , which follows from [EG18, Lemma 3.2] (or [DK73, XIV 1.1.10]).5.3. Arithmetic descent.
We would like to say that the G -local system [ ρ i,λ,s ] on X s isthe pullback of a local system on X s , or equivalently that there exists a representation ρ i,λ,s making the diagram commute π ´et1 ( X s , x s ) π ´et1 ( X s , x s ) G ( K λ ) ρ i,λ,s ρ i,λ,s The next proposition (a variant on [EG18, Proposition 3.1], [Sim92, Theorem 4]) says thatthis indeed can be arranged, after possibly replacing k by a finite extension. Proposition 5.6.
After replacing k by a finite extension there is a representation ρ i,λ,s : π ´et ,p ′ ( X s , x s ) → G ( Q λ ) whose restriction to π ´et ( X s , x s ) is ρ i,λ,s and whose image has the same Zariski-closure as ρ i,λ, ¯ s .Proof. The point x s ∈ X s is rational, so it splits the “prime-to- p homotopy sequence”1 → π ´et ,p ′ ( X s , x s ) → π ´et ,p ′ ( X s , x s ) → π ´et1 ( s, s ) → , which is by definition the pushout of the homotopy exact sequence for π ´et1 ( X s , x s ) along π ´et1 ( X ¯ s , x ¯ s ) → π ´et ,p ′ ( X ¯ s , x ¯ s ). For σ ∈ π ´et1 ( s, s ) we also denote by σ its image in π ´et ,p ′ ( X s , x s )under the splitting from x s . Let ρ σi,λ : π ´et1 ( X, x ) → G ( K λ ) , γ ρ i,λ,s ( σ · Sp( γ ) · σ − ) , where Sp : π ´et1 ( X, x ) → π ´et ,p ′ ( X s , x s ) is the specialization map. We claim that the G -localsystem [ ρ σ, top i,λ ] represented by ρ σ, top i,λ , the pullback of ρ σi,λ to π top1 ( X, x ), has quasi-unipotentlocal monodromies of index dividing h (i.e. [ ρ σ, top i,λ ] ∈ N ( K λ )). Indeed, the proof of [EG18,Lemma 3.2] (which proves the claim when G = GL n ) goes through verbatim for G , since theonly thing that is used about GL n is that the union of conjugacy classes of quasi-unipotentelements of index dividing h is closed; and this is true for G also.The map π ´et1 ( s, s ) → N ( K λ ) given by σ ρ σ, top i,λ is continuous for the profinite topology on π ´et1 ( s, s ) and the λ -adic topology on N ( K λ ). The G -local system [ ρ top i,λ ] represented by ρ top i,λ isisolated, so there is an open subgroup U with [ ρ σ, top i,λ ] = [ ρ top i,λ ] for all σ ∈ U . Thus replacing k by a finite extension, we may assume that for all σ ∈ π ´et1 ( s, ¯ s ), ρ σ, top i,λ is G ( Q λ )-conjugateto ρ top i,λ (we fix an extension of λ to Q ). By specializing, we see that ρ σi,λ,s = T ( σ ) ρ i,λ,s T ( σ ) − for some T ( σ ) ∈ G ( Q λ ), where ρ σi,λ,s ( γ ) = ρ i,λ,s ( σγσ − ). -RIGID LOCAL SYSTEMS ARE INTEGRAL 15 The element T ( σ ) is determined up to multiplication by an element of the centralizer C G ( ρ i,λ,s ). Note that G -irreducibility of ρ i,λ, ¯ s implies that the geometric monodromy group G i,λ := im( ρ i,λ, ¯ s ) is reductive, and that its centralizer C G ( G i,λ ) is finite (by [Ric88, Theorem4.1], using also that G is semisimple, so has finite center). T ( σ ) normalizes G i,λ , and itfollows formally that σ T ( σ ) defines a homomorphism π ´et1 ( s, ¯ s ) → N G ( G i,λ ) /C G ( G i,λ ) . We claim that, after replacing k by a finite extension, we may assume T ( σ ) is an innerautomorphism of G i,λ . We use part of the argument of [Del80, Corollaire 1.3.9]. Indeed,any automorphism of G i,λ induces, arguing successively, automorphisms of G i,λ , Z G i,λ , and(to abbreviate) Z = Z G i,λ . Let r : G → GL V be a faithful finite-dimensional represen-tation, and consider the finite set of characters X of Z in V . Although T ( σ ) is onlydefined up to C G ( G i,λ ), conjugating by any representative T ∈ N G ( G i,λ ) of T ( σ ) showsthat as Z -representations, r | Z and r T | Z are isomorphic (as before, the notation means r T ( z ) = r ( T zT − ); this is independent of the choice of representative T ). Thus T ( σ ) actsby permutations on the finite set X , and it follows that a finite power T ( σ ) n fixes X point-wise. Faithfulness of the representation r implies that these characters generate the characterlattice X • ( Z ), and therefore T ( σ ) n = T ( σ n ) acts trivially on Z . The set of outer automor-phism of a (not necessarily connected) reductive group that are trivial on the maximal centraltorus is finite, so enlarging n we have that T ( σ n ) acts on G i,λ as an inner automorphism.Replacing k by its degree n extension, we then conclude that σ T ( σ ) factors π ´ et ( s, ¯ s ) → G i,λ · C G ( G i,λ ) /C G ( G i,λ ) ∼ −→ G i,λ /Z G i,λ . We then obtain a “projective” representation P ( ρ i,λ,s ) : π ´et ,p ′ ( X s , x ¯ s ) → G i,λ /Z G i,λ ( Q λ )whose restriction to π ´ et ( X ¯ s , ¯ s ) is P ( ρ i,λ, ¯ s ) as follows: the rational point x s induces a semi-direct product decomposition π ´ et ( X s , s ) ∼ = π ´ et ( X ¯ s , ¯ s ) ⋊ π ´ et ( s, ¯ s ), and then we set P ( ρ i,λ,s )( g, σ ) = P ( ρ i,λ, ¯ s )( g ) · T ( σ ) . The obstruction to lifting P ( ρ i,λ,s ) to a G i,λ -valued representation lies in H ( π ´et ,p ′ ( X s , x ¯ s ) , Z G i,λ ),which we analyze using the Hochschild-Serre spectral sequence. The key observation is thatthe existence of ρ i,λ, ¯ s to begin with implies that the obstruction vanishes after restriction to π ´et ,p ′ ( X ¯ s , x ¯ s ). Since k has cohomological dimension 1, vanishing of the full obstruction followsafter possibly enlarging k once more to kill a class in H ( π ´et1 ( s, ¯ s ) , H ( π ´et ,p ′ ( X ¯ s , x ¯ s ) , Z G i,λ )).Finally, by once more enlarging k we may assume that our lift ρ i,λ,s : π ´et ,p ′ ( X s , x ¯ s ) → G i,λ ( Q λ )restricts to the given ρ i,λ, ¯ s : the two differ by an element of the finite group H ( π ´et ,p ′ ( X ¯ s , x ¯ s ) , Z G i,λ ),and after enlarging k any such homomorphism extends to π ´et ,p ′ ( X s , x ¯ s ) (the outer-action of π ´et1 ( s, ¯ s ) on this H trivializes after a finite restriction, and then we apply the inf-res se-quence). By construction, this descended representation has image with Zariski closureequal to G i,λ . (cid:3) Having established the arithmetic descent, we deduce the following important consequence:
Corollary 5.7.
The identity component G i,λ is semisimple.Proof. The descent ρ i,λ,s has (not necessarily connected) reductive monodromy group G i,λ .It follows from [Del80, Corollaire 1.3.9] that G i,λ is semisimple, since it is also the identitycomponent of the geometric monodromy group. (cid:3) Constructing Companions
Application of results of Drinfeld.
We have for each i the representations ρ i,λ,s : π ´et1 ( X s , x ¯ s ) → G ( K λ ) ⊂ G ( Q λ ) , whose images have Zariski-closures equal to the semisimple (not necessarily connected) sub-groups G i,λ ⊂ G over Q λ ; recall that by construction im( ρ i,λ,s ) has the same Zariski-closureas im( ρ i,λ, ¯ s ). Also, since i ranges over a finite set, we may assume that the same s works forall i . For any place λ ′ of Q not above p , we will construct a λ ′ -companion ρ i,λ ′ ,s with “thesame” algebraic monodromy group as ρ i,λ,s . Let ( G i , φ λ ) be a pair consisting of a reductivegroup G i over Q and an isomorphism G i ⊗ Q Q λ φ λ −→ G i,λ . We also fix an embedding of groupsover Q , α : G i → G Q , such that ( α ⊗ Q λ ) ◦ φ − λ is G ( Q λ )-conjugate to the given G i,λ ⊂ G .We recall the main theorem of [Dri18]. For a characteristic zero field E , let Pro-ss( E )denote the groupoid whose objects are pro-semisimple (not necessarily connected) groupschemes over E , and whose morphisms are group scheme isomorphisms modulo conjugationby the connected component of the identity. There is an extension of scalars equivalenceof categories Pro-ss( Q ) ∼ −→ Pro-ss( Q λ ) when λ is any place of Q (see [Dri18, Proposition2.2.5] for details). To the smooth variety X over the finite field k , Drinfeld associates foreach λ an object of Pro-ss( Q ) as follows: consider the inverse limit ( ˆΠ λ , r λ ) over all pairs( H, r ) consisting of a semisimple group H over Q λ and a continuous homomorphism withZariski-dense image r : π ´et1 ( X s , x ¯ s ) → H ( Q λ ). Via the above equivalence, we descend ˆΠ λ to an object ˆΠ ( λ ) of Pro-ss( Q ). If we let Π Fr denote the set of all Frobenius elements in π ´et1 ( X s , x ¯ s ) as in [Dri18, § r λ induces a diagram of setsΠ Fr → ( ˆΠ ( λ ) (cid:12) ˆΠ λ ) )( Q ) → π ´et1 ( X s , x ¯ s ) . (2)(Note that the GIT quotient, which as in [Dri18] we will also denote by [ ˆΠ ( λ ) ], is actually well-defined, even though ˆΠ ( λ ) is only defined up to conjugation by the connected component.)The main theorem of [Dri18] is then: Theorem 6.1 ([Dri18]) . For any two non-archimedean places λ and λ ′ of Q not above p ,there is a unique isomorphism ˆΠ ( λ ) ∼ −→ ˆΠ ( λ ′ ) in Pro-ss( Q ) carrying diagram (2) for λ to theanalogous diagram for λ ′ . We now return to our representations ρ i,λ,s . By construction and [Dri18, Proposition 2.3.3], ρ i,λ,s induces a surjection of (pro-)semisimple groups over Q , ˆΠ ( λ ) → G i , well-defined up to G i -conjugation. -RIGID LOCAL SYSTEMS ARE INTEGRAL 17 Composing with the inverse of the isomorphism in Theorem 6.1 and extending scalars Q → Q λ ′ , we obtain a surjection ˆΠ λ ′ → G i,λ ′ , where we abbreviate G i,λ ′ = G i ⊗ Q Q λ ′ ; this map isalso defined up to G i, Q λ ′ -conjugation. This furnishes us with the desired companion, namelythe composite π ´et1 ( X s , x ¯ s ) ˆΠ λ ′ ( Q λ ′ ) G i,λ ′ ( Q λ ′ ) . r λ ′ ρ i,λ ′ ,s It is evident from the construction that the Zariski-closure im( ρ i,λ ′ ,s ) is equal to G i,λ ′ , andthat ρ i,λ,s and ρ i,λ ′ ,s are everywhere locally compatible (in the sense made precise by diagram(2)). Proposition 6.2. As i = 1 , . . . , N varies, the representations ρ i,λ ′ , ¯ s : π ´et ( X ¯ s , x ¯ s ) → G i,λ ′ ( Q λ ′ ) (1) are tamely ramified;(2) are G -cohomologically rigid;(3) are pair-wise distinct as G ( Q λ ′ ) -conjugacy classes; and(4) have quasi-unipotent local monodromies with index dividing h (in the sense of Remark5.5).Proof. (1): By [KS10, Proposition 4.2], it is enough to check all restrictions of ρ i,λ ′ ,s toregular curves mapping to X s , and with image not contained in D , are tamely ramified. Let Z → X s be one such curve and fix a geometric base point z ∈ Z and path from (the imageof) z to x ¯ s in X s . Choose a faithful representation R : G i → GL n over Q and write R i,λ,s forthe composition π ´et1 ( Z, z ) → π ´et1 ( X s , x ¯ s ) ρ i,λ,s −−−→ G ( Q λ ) R −→ GL n ( Q λ ) , and similarly for R i,λ ′ ,s . These are, by the compatibility of ρ i,λ,s and ρ i,λ ′ ,s , compatiblerepresentations. Considering them as representations of the Galois group of the function field k ( Z ) of Z , [Del73, Th´eor`eme 9.8] shows that the local representations of R i,λ,s and R i,λ ′ ,s ateach place of k ( Z ) have isomorphic semisimplified Weil-Deligne representations; regardinga Weil-Deligne representation as a pair consisting of a Weil group representation r (smoothon the inertia subgroup) and a nilpotent matrix N , this semisimplification remembers therestriction of r to the inertia subgroup, and in particular it detects whether or not therepresentation is tamely ramified. As ρ i,λ,s factors through the prime-to- p quotient, it istame, so [KS10, Proposition 4.2] (which is a corollary of Abhyankar’s lemma [sga71, XIIIProposition 5.2]) shows that R i,λ,s is tamely ramified. We conclude from [Del73] that R i,λ ′ ,s is also tamely ramified. Since this holds for all such Z → X s , we conclude again by [KS10,Proposition 4.2] that ρ i,λ ′ ,s (equivalently, R ◦ ρ i,λ ′ ,s ) is tamely ramified.(4): As explained in Remark 5.4, we have a smooth curve C ⊂ X that is a completeintersection with smooth complement; C does not meet the singular locus of D and intersectseach D j transversely. We also have points x ∈ X \ D and y j ∈ D j ∩ C . Recall that S ischosen as in Remark 5.4 so that ( C, y j ) (along with these good properties) spreads out to( C S , y j,S ) over S , and ( C s , y j,s ) is the fiber over s . As in (1), choose a faithful representation R : G i → GL n over Q . Write R i,λ,s for the λ -adic local system on the curve C s given byrestricting ρ i,λ,s to C s and then composing with R (and similarly for R i,λ ′ ,s ). The index ofquasi-unipotence around D j of a representation of π ´et ,t ( X s , x ¯ s ) is equal to the index around y j of its restriction to π ´et ,t ( C s , x ¯ s ): indeed, locally at y j the divisor D j is the vanishing locusof a function f j that restricts (via O X ¯ s ,y j → O C ¯ s,yj ) to a uniformizer of C ¯ s at y j (since theintersection is transverse). The claim then follows by Abhyankar’s Lemma ([sga71, XIIIProposition 5.2]) upon noting that a Kummer cover ramified along D j of degree n pulls backto a Kummer cover ramified along y j of degree n .As noted in 5.5, the index of quasi-unipotence of ρ i,λ,s around D j divides h , so the sameholds for R i,λ,s around y j . We again invoke Deligne’s theorem [Del73, Th´eor`eme 9.8] andfind that R i,λ ′ ,s has the same index of quasi-unipotence around y j , dividing h . Finally, thesame observation just used implies that ρ i,λ ′ ,s has index of quasi-unipotence along D j alsodividing h .(2): Let A i,λ,s be the local system on X s corresponding to the representation π ´et1 ( X s , x s ) ρ i,λ,s −−−→ G ( Q λ ) → GL( g der ( Q λ ))and likewise for A i,λ ′ ,s . The determinant of each irreducible component of A i,λ,s is finiteorder since G i,λ is semisimple, and hence A i,λ,s is pure of weight zero by Lafforgue [Laf02].The sheaves A i,λ,s and A i,λ ′ ,s are compatible by the remark above.We argue as in the proof of Theorem 1.1 of [EG18]. Compatibility gives an equality of L -functions L ( X s , A i,λ,s ) = L ( X s , A i,λ ′ ,s )and since A i,λ,s is tame and weight 0, [EG18, Lemma 3.4] shows that h ( X s , j ! ∗ A i,λ ′ ,s ) = h ( X s , j ! ∗ A i,λ,s ) . The latter is 0, as a consequence of local acyclicity and the fact that ρ top i,λ is cohomologicallyrigid. Thus ρ i,λ ′ ,s is cohomologically rigid as required.(3): If ρ i,λ ′ ,s were G ( Q λ ′ )-conjugate to ρ j,λ ′ ,s , then again invoking [Dri18, Proposition 2.3.3] wefind that these maps are induced by homomorphisms ˆΠ ( λ ′ ) → G Q that are G ( Q )-conjugate,and hence that ρ i,λ,s and ρ j,λ,s are G ( Q λ )-conjugate (note that G is connected). Moreover, thesame argument works if we replace k ( s ) by any finite extension. We claim the same for therepresentations ρ i,λ ′ , ¯ s . We will first check that ρ i,λ ′ ,s and ρ i,λ ′ , ¯ s have the same Zariski-closure(namely G i,λ ′ ), by the corresponding property for the λ -companions. Write G i,λ ′ , ¯ s for theZariski-closure of the image of ρ i,λ ′ , ¯ s (for any λ ′ , including λ ). Since the commutator subgroupof G i,λ ′ is contained in G i,λ ′ , ¯ s , and G i,λ ′ is semisimple, G , der i,λ ′ = G i,λ ′ is equal to G i,λ ′ , ¯ s .Thus G i,λ ′ /G i,λ ′ , ¯ s is isomorphic to the quotient of component groups π ( G i,λ ′ ) /π ( G i,λ ′ , ¯ s ).If this were non-trivial, it would be generated by the image of the geometric Frobenius F s ∈ π ´et1 ( s, ¯ s ) x s −→ π ´et1 ( X s , x ¯ s ), and, for some integer n , F ns would have image in π ( G i,λ ′ , ¯ s ).Thus after replacing k ( s ) by a finite extension, we may assume G i,λ ′ , ¯ s = G i,λ ′ , as desired.Now assume that ρ i,λ ′ , ¯ s and ρ j,λ ′ , ¯ s are conjugate by an element of G ( Q λ ′ ). Replacing ρ i,λ ′ , ¯ s by such a conjugate, we may assume the two homomorphisms ρ i,λ ′ ,s and ρ j,λ ′ ,s are literallyequal on π ´et1 ( X ¯ s , x ¯ s ). Let F s ∈ π ´et1 ( X s , x ¯ s ) be the image of the generating geometric Frobeniuselement of π ´et1 ( s, ¯ s ). Then for all γ ∈ π ´et1 ( X ¯ s , x ¯ s ), ρ i,λ ′ , ¯ s ( F s γF − s ) = ρ j,λ ′ , ¯ s ( F s γF − s ), hence -RIGID LOCAL SYSTEMS ARE INTEGRAL 19 ρ j,λ ′ ,s ( F s ) − ρ i,λ ′ ,s ( F s ) belongs to the center Z G j,λ ′ (we have used the above assertion aboutequality of arithmetic and geometric monodromy groups). As this center is finite, replacing k ( s ) by a finite extension we have ρ i,λ ′ ,s = ρ j,λ ′ ,s , a contradiction: indeed, then their λ -companions, and consequently ρ i,λ, ¯ s and ρ j,λ, ¯ s , would then be equivalent. (cid:3) Deduction of the main theorem.
Proof of Theorem 1.2.
Now, since the companions ρ i,λ ′ , ¯ s are tamely ramified, we can pullthem back along the tame specialization map ([LO10, Corollary A.12]) π ´et ,t ( X, x ) ։ π ´et ,t ( X ¯ s , x ¯ s )to obtain complex local systems ρ top i,λ ′ : π top1 ( X, x ) → G i,λ ′ ( Q λ ′ ) ⊂ G ( Q λ ′ )with image in fact contained in G ( Z λ ′ ) (because they are constructed from ´etale local sys-tems). The Zariski closure of im( ρ top i,λ ′ ) is G i,λ ′ since this is true for im( ρ i,λ ′ ,s ) and the tamespecilization map is surjective. Thus ρ top i,λ ′ has (not necessarily connected) semisimple mon-odromy. By the Betti-´etale comparison isomorphism, the ρ top i,λ ′ are still cohomologically rigid(now as local systems on X ), and they are still inequivalent and G -irreducible with quasi-unipotent local monodromy of index dividing h . If we apply a field isomorphism ι : C ∼ −→ / Q Q λ ′ to the elements of the finite set S ( G, h ), we obtain a set of N distinct G ( Q λ ′ )-local systemson X that are G -irreducible, G -cohomologically rigid, and with monodromy at infinity quasi-unipotent of index dividing h , and these are by construction a complete set of such localsystems. Our ρ top i,λ ′ , for i = 1 , . . . , N , are thus up to G ( Q λ ′ )-isomorphism a full collection ofrepresentatives of this set ι S ( G, h ). Returning to our original ρ : π top1 ( X, x ) → G ( O K, Σ ), wesee that for any λ ′ above a place in Σ, the composite π top1 ( X, x ) ρ −→ G ( O K, Σ ) ⊂ G ( Q λ ′ )is also a member of ι S ( G, h ). We conclude that ρ can be conjugated by an element of G ( Q λ ′ )into G ( Z λ ′ ), since this holds for each ρ top i,λ ′ . Combining this argument for all places in Σ withour integrality criterion, Lemma 3.1, we deduce that for some finite extension L/K , ρ is G ( L )-conjugate to a homomorphism π top1 ( X, x ) → G ( O L ), concluding the proof. (cid:3) References [BHKT19] Gebhard B¨ockle, Michael Harris, Chandrashekhar Khare, and Jack A. Thorne,ˆ G -local systems on smooth projective curves are potentially automorphic , ActaMath. (2019), no. 1, 1–111. MR 4018263[BT14] Christopher Brav and Hugh Thomas, Thin monodromy in Sp(4) , Compos. Math. (2014), no. 3, 333–343. MR 3187621[Con14] Brian Conrad,
Reductive group schemes , Autour des sch´emas en groupes, Panora-mas et Synth`eses [Panoramas and Syntheses], vol. 42-43, Soci´et´e Math´ematiquede France, Paris, 2014, p. 458.[Del71] Pierre Deligne,
Th´eorie de Hodge. II , Inst. Hautes ´Etudes Sci. Publ. Math. (1971),no. 40, 5–57. MR 0498551 (58 [Del73] ,
Les constantes des ´equations fonctionnelles des fonctions l , Modularfunctions of one variable II, Springer, 1973, pp. 501–597.[Del80] ,
La conjecture de Weil. II , Inst. Hautes ´Etudes Sci. Publ. Math. (1980),no. 52, 137–252. MR 601520 (83c:14017)[DK73] P. Deligne and N. Katz,
Groupes de monodromie en g´eom´etrie alg´ebrique. II ,Springer-Verlag, Berlin, 1973, S´eminaire de G´eom´etrie Alg´ebrique du Bois-Marie1967–1969 (SGA 7 II), Dirig´e par P. Deligne et N. Katz, Lecture Notes in Math-ematics, Vol. 340. MR 50
Monodromy of hypergeometric functions and non-lattice integral monodromy , Inst. Hautes ´Etudes Sci. Publ. Math. (1986), no. 63,5–89. MR 849651[Dri12] Vladimir Drinfeld,
On a conjecture of Deligne , Mosc. Math. J. (2012), no. 3,515–542, 668. MR 3024821[Dri18] , On the pro-semisimple completion of the fundamental group of a smoothvariety over a finite field , Adv. Math. (2018), 708–788. MR 3762002[EG17] H´el`ene Esnault and Michael Groechenig,
Rigid connections and F -isocrystals , toappear in Acta Mathematica (2017).[EG18] H´el`ene Esnault and Michael Groechenig, Cohomologically rigid local systems andintegrality , Selecta Math. (N.S.) (2018), no. 5, 4279–4292. MR 3874695[FM95] Jean-Marc Fontaine and Barry Mazur, Geometric Galois representations , Ellipticcurves, modular forms, & Fermat’s last theorem (Hong Kong, 1993), Ser. NumberTheory, I, Int. Press, Cambridge, MA, 1995, pp. 41–78. MR 1363495 (96h:11049)[GM06] Alexandre Grothendieck and Jacob P Murre,
The tame fundamental group of aformal neighbourhood of a divisor with normal crossings on a scheme , vol. 208,Springer, 2006.[Gro64] A. Grothendieck,
Sch´emas en groupes. III: Structure des sch´emas en groupesr´eductifs , Springer-Verlag, Berlin, 1962/1964. MR 43
Th´eor`emes de Bertini et applications , Universit´e LouisPasteur, D´epartement de Math´ematique, Institut de Recherche Math´ematiqueAvanc´ee, Strasbourg, 1979. MR 658132 (83k:13002)[Kat96] Nicholas M. Katz,
Rigid local systems , Annals of Mathematics Studies, vol. 139,Princeton University Press, Princeton, NJ, 1996. MR 1366651 (97e:14027)[Kne66] Martin Kneser,
Strong approximation , Algebraic Groups and Discontinuous Sub-groups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), vol. 187, 1966, p. 196.[KS10] Moritz Kerz and Alexander Schmidt,
On different notions of tameness in arith-metic geometry , Mathematische Annalen (2010), no. 3, 641.[Laf02] Laurent Lafforgue,
Chtoucas de Drinfeld et correspondance de Langlands , Invent.Math. (2002), no. 1, 1–241. MR 1875184 -RIGID LOCAL SYSTEMS ARE INTEGRAL 21 [Laf18] Vincent Lafforgue,
Chtoucas pour les groupes r´eductifs et param´etrisation deLanglands globale , J. Amer. Math. Soc. (2018), no. 3, 719–891. MR 3787407[LO10] Max Lieblich and Martin Olsson, Generators and relations for the ´etale funda-mental group , Pure Appl. Math. Q. (2010), no. 1, Special Issue: In honor ofJohn Tate. Part 2, 209–243. MR 2591190[Mar91] Gregori A Margulis, Discrete subgroups of semisimple lie groups , vol. 17, SpringerScience & Business Media, 1991.[Ric88] R. W. Richardson,
Conjugacy classes of n -tuples in Lie algebras and algebraicgroups , Duke Math. J. (1988), no. 1, 1–35. MR 952224[sga71] Revˆetements ´etales et groupe fondamental , Lecture Notes in Mathematics, Vol.224, Springer-Verlag, Berlin-New York, 1971, S´eminaire de G´eom´etrie Alg´ebriquedu Bois Marie 1960–1961 (SGA 1), Dirig´e par Alexandre Grothendieck. Augment´ede deux expos´es de M. Raynaud. MR 0354651[Sim92] Carlos T. Simpson,
Higgs bundles and local systems , Inst. Hautes ´Etudes Sci.Publ. Math. (1992), no. 75, 5–95. MR 1179076 (94d:32027)
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