Invent math DOI 10.1007/s00222-013-0469-9 Motives with exceptional Galois groups and the inverse Galois problem Zhiwei Yun Received: 14 December 2011 / Accepted: 10 January 2013 © Springer-Verlag Berlin Heidelberg 2013 Abstract We construct motivic -adic representations of Gal(Q/Q) into ex- ceptional groups of type E7,E8 and G2 whose image is Zariski dense. This answers a question of Serre. The construction is uniform for these groups and is inspired by the Langlands correspondence for function fields. As an appli- cation, we solve new cases of the inverse Galois problem: the finite simple groups E8(F) are Galois groups over Q for large enough primes . Mathematics Subject Classification 14D24 · 12F12 · 20G41 Contents 1 Introduction . ................. 1.1Serre’squestion........................... 1.2Mainresults............................. 1.3 The case of type A1 ......................... 1.4Descriptionofthemotives..................... 1.5Thegeneralconstruction...................... 1.6Conjecturesandgeneralizations.................. 2 Group-theoretic preliminaries . ................. 2.1 The group G ............................ 2.2 Loop groups . ................. 2.3 A class of parahoric subgroups . ................. Z. Yun () Department of Mathematics, Stanford University, 450 Serra Mall, Bldg 380, Stanford, CA 94305, USA e-mail: [email protected] Z. Yun 2.4Resultsoninvolutions....................... 2.5 Minimal symmetric subgroups of G ................ 2.6Aremarkablefinite2-group.................... 3Theautomorphicsheaves........................ 3.1Convention............................. 3.2 Moduli stacks of G-bundles . ................. 3.3 Sheaves on the moduli stack of G-bundles . 3.4 Proof of Theorem 3.2 ........................ 4Constructionofthemotives....................... 4.1GeometricHeckeoperators.................... 4.2Eigenlocalsystems......................... 4.3Descriptionofthemotives..................... 4.4 Proof of Theorem 4.2 ........................ 5 Local and global monodromy . ................. 5.1 Remarks on Gaitsgory’s nearby cycles . 5.2 Local monodromy . ................. 5.3 Global geometric monodromy . ................. 5.4ImageofGaloisrepresentations.................. 5.5 Conjectural properties of the local system . 5.6 Application to the inverse Galois problem . Acknowledgement............................. References................................. 1 Introduction 1.1 Serre’s question About two decades ago, Serre raised the following question which he de- scribed as “plus hasardeuse” (English translation: more risky): Question 1.1 (Serre [36,Sect.8.8]) Is there a motive M (over a number field) such that its motivic Galois group is a simple algebraic group of exceptional type G2 or E8? The purpose of this paper is to give an affirmative answer to a variant of Serre’s question for E7,E8 and G2, and to give applications to the inverse Galois problem. 1.1.1 Motivic Galois groups Let us briefly recall the notion of the motivic Galois group, following [36, Sects. 1 and 2]. Let k and L be number fields. Let Motk(L) be the cat- egory of motives over k with coefficients in L (under numerical equiva- lences). This is an abelian category obtained by formally adjoining direct Motives with exceptional Galois groups summands of smooth projective varieties over k cut out by idempotent cor- respondences with L-coefficients. Assuming the Standard Conjectures, the category Motk(L) becomes a semisimple L-linear Tannakian category (see Jannsen [22, Corollary 2]). Moreover, it admits a tensor structure and a fiber functor ω into VecL, the tensor category of L-vector spaces. For example, one may take ω to be the singular cohomology of the underlying analytic spaces (using a fixed embedding k→ C) with L-coefficients. By Tannakian formal- Mot ism [10], such a structure gives a group scheme Gk over L as the group of tensor automorphisms of ω. This is the absolute motivic Galois group of k. Any motive M ∈ Motk(L) generates a Tannakian subcategory Mot(M) of Mot Motk(L). Tannakian formalism again gives a group scheme GM over L,the group of tensor automorphisms of ω|Mot(M).Thisisthe motivic Galois group of M. Of course Serre’s question could be asked for other exceptional types. Al- though people hoped for an affirmative answer to Serre’s question, the search within “familiar” types of varieties all failed. For example, one cannot find an abelian variety with exceptional motivic Galois groups (see [30, Corol- lary 1.35] for the fact that the Mumford-Tate groups of abelian varieties can- not have exceptional factors, and by [10, Theorem 6.25], the Mumford-Tate group of an abelian variety surjects onto its motivic Galois group), nor does one have Shimura varieties of type E8,F4 or G2. 1.1.2 Motivic Galois representations Let be a prime number. Fix an embedding L→ Q. For a motive M ∈ Motk(L),wehavethe-adic realization H(M, Q) which is a continuous Gal(k/k)-module. Let V be a finite dimensional Q-vector space. We call a continuous rep- resentation ρ : Gal(k/k) → GL(V ) motivic if there exists a motive M ∈ Motk(L) (for some number field L) such that V ⊗ Q is isomorphic to H(M, Q ) as Gal(k/k)-modules. Let G be a reductive algebraic group over Q. A continuous representa- tion ρ : Gal(Q/Q) → G(Q) is called motivic if for some faithful algebraic ρ representation V of G, the composition Gal(Q/Q) −→ G(Q) → GL(V ) is motivic. Fix an embedding L→ Q,wehaveanexactfunctorH(−, Q) : Motk(L) → Rep(Gal(k/k),Q) by taking étale cohomology with Q-coeffi- cients, on which the Galois group Gal(k/k) acts continuously. We call this the -adic realization functor. For a motive M ∈ Motk(L),wedefinethe- adic motivic Galois group of M to be the Zariski closure of the image of the representation ρM, : Gal(k/k) → GL(H(M, Q)). We denote the -adic motivic Galois group of M by GM,, which is an algebraic group over Q.It =∼ Mot ⊗ Q is expected that GM, GM L (see [36, Sect. 3.2]). Z. Yun 1.2 Main results We will answer Serre’s question for -adic motivic Galois groups instead of the actual motivic Galois groups, because their existence depends on the Stan- dard Conjectures. Main Theorem 1.2 Let G be a split simple adjoint group of type A1,E7,E8 or G2. Let be a prime number. Then there exists an integer N ≥ 1 and a continuous representation : P1 −{ ∞} → Q ρ π1 Z[1/2N] 0, 1, G( ) such that (1) For each geometric point Spec k → Spec Z[1/2N], the restriction of ρ P1 −{ ∞} to the geometric fiber k 0, 1, : : P1 −{ ∞} → Q ρk π1 k 0, 1, G( ) has Zariski dense image. (2) The restriction of ρ to a rational point x : Spec Q → P1 −{0, 1, ∞}: ρx : Gal(Q/Q) → G(Q) is either motivic (if G is of type E8 or G2) or becomes motivic when restricted to Gal(Q/Q(i)) (if G is of type A1 or E7). 1 (3) There exist infinitely many rational points {x1,x2,...} of PQ −{0, 1, ∞} such that ρxi are mutually non-isomorphic and all have Zariski dense image. Corollary 1.3 For G a simple adjoint group of type A1,E7,E8 or G2, there exist infinitely many non-isomorphic motives over Q (if G is of type E8 or G2) or Q(i) (if G is of type E7) whose -adic motivic Galois groups are isomorphic to G. In particular, Serre’s question for -adic motivic Galois groups has an affirmative answer for A1,E7,E8 and G2. 1.2.1 A known case In [11], Dettweiler and Reiter constructed a rank seven rigid local system on 1 PQ −{0, 1, ∞} whose geometric monodromy is dense in G2. The restriction of this local system to a general rational point gives a motivic Galois repre- sentation whose image is dense in G2. We believe that our construction in the G2 case gives the same local system as theirs (see Remark 5.12). On the other hand, Gross and Savin [19] gave a candidate G2-motive in the cohomology of a Siegel modular variety. Motives with exceptional Galois groups 1.2.2 Application to the inverse Galois problem The inverse Galois problem for Q asks whether every finite group can be realized as a Galois group of a finite Galois extension of Q. A lot of finite simple groups are proved to be Galois groups over Q,see[29]and[37], yet the problem is still open for many finite simple groups of Lie type. We will be concerned with finite simple groups G2(F) and E8(F),where is a prime. By Thompson [40] and Feit and Fong [14], G2(F) is known to be a Galois group over Q for all primes ≥ 5. However, according to [29, Chap. II, Sect. 10], E8(F) is known to be a Galois group over Q only for ≡±3, ±7, ±9, ±10, ±11, ±12, ±13, ±14 (mod 31). As an application of our main construction, we solve new instances of the inverse Galois problem. Theorem 1.4 For sufficiently large prime , the finite simple group E8(F) can be realized as Galois groups over any number field. 1.3 The case of type A1 To illustrate the construction in the Main Theorem, we give here an analog of the Main Theorem for G = PGL2, in which case the motives involved are more familiar. Let k = Fq be a finite field of characteristic not 2. The con- struction starts with an automorphic form of G = SL2.LetT ⊂ B ⊂ G be the diagonal torus and the upper triangular matrices. Let F = k(t) be the func- P1 O tion field of k. For each place v of F ,let v,Fv and kv be the correspond- ing completed local ring, local field and residue field. For each v,wehave the Iwahori subgroup Iv ⊂ G(Ov) which is the preimage of B(kv) ⊂ G(kv) under the reduction map G(Ov) → G(kv). = We will consider irreducible automorphic representations π v πv of G(AF ).