Modulformen und das inverse Galois-Problem

Transcription

1 Modulformen und das inverse Galois-Problem Gabor Wiese Université du Luxembourg Vortrag auf der DMV-Jahrestagung 2012 in Saarbrücken 19. September 2012 Modulformen und das inverse Galois-Problem p.1/19

2 Question of Hilbert: Inverse Galois Problem Given a finite group G. Is there a Galois extension K/Q such that Gal(K/Q) = G? Modulformen und das inverse Galois-Problem p.2/19

3 Question of Hilbert: Inverse Galois Problem Given a finite group G. Is there a Galois extension K/Q such that Gal(K/Q) = G? In this talk focus on two cases: The GL 2 -case: G = PSL 2 (F l d). The GSp 2n -case: G = PSp 2n (F l d). Modulformen und das inverse Galois-Problem p.2/19

4 Introduction: GL 2 -case Consider a cuspidal modular form f = n=1 a nq n (q = e 2πiz ) s.t. a 1 = 1 (normalised), Hecke eigenform, no CM, any weight, on Γ 1 (N), nebentype ψ : (Z/NZ) C. Modulformen und das inverse Galois-Problem p.3/19

5 Introduction: GL 2 -case Consider a cuspidal modular form f = n=1 a nq n (q = e 2πiz ) s.t. a 1 = 1 (normalised), Hecke eigenform, no CM, any weight, on Γ 1 (N), nebentype ψ : (Z/NZ) C. Theorem (Deligne, Shimura, Eichler, Igusa, Serre). For each prime l, Galois representation ρ proj f,l ρ f,l : G Q GL 2 (F l ) nat. proj. PGL 2 (F l ) unramified outside Nl such that for all p Nl Tr(ρ f,l (Frob p )) a p mod l. One speaks of a compatible system. Modulformen und das inverse Galois-Problem p.3/19

6 Introduction: GL 2 -case For f = n=1 a nq n and each prime l, Galois rep. ρ proj f,l ρ f,l : G Q GL 2 (F l ) nat. proj. PGL 2 (F l ). Questions/Tasks: Modulformen und das inverse Galois-Problem p.4/19

7 Introduction: GL 2 -case For f = n=1 a nq n and each prime l, Galois rep. ρ proj f,l ρ f,l : G Q GL 2 (F l ) nat. proj. PGL 2 (F l ). Questions/Tasks: (I) Smallest d such that ρ proj f,l can be defined over F l d? Modulformen und das inverse Galois-Problem p.4/19

8 Introduction: GL 2 -case For f = n=1 a nq n and each prime l, Galois rep. ρ proj f,l ρ f,l : G Q GL 2 (F l ) nat. proj. PGL 2 (F l ). Questions/Tasks: (I) Smallest d such that ρ proj f,l can be defined over F l d? (II) Image of ρ proj f,l? Note: Gal(Q ker(ρproj f,l ) /Q) = ρ proj f,l (G Q). Modulformen und das inverse Galois-Problem p.4/19

9 Introduction: GL 2 -case For f = n=1 a nq n and each prime l, Galois rep. ρ proj f,l ρ f,l : G Q GL 2 (F l ) nat. proj. PGL 2 (F l ). Questions/Tasks: (I) Smallest d such that ρ proj f,l can be defined over F l d? (II) Image of ρ proj f,l? Note: Gal(Q ker(ρproj f,l ) /Q) = ρ proj f,l (G Q). (III) Prove the existence of f such that for fixed l,d: ρ proj f,l (G Q) = PSL 2 (F l d), i.e. realise PSL 2 (F l d) as Galois group over Q. Modulformen und das inverse Galois-Problem p.4/19

10 Introduction: GL 2 -case For f = n=1 a nq n and each prime l, Galois rep. ρ proj f,l ρ f,l : G Q GL 2 (F l ) nat. proj. PGL 2 (F l ). (I) Smallest d such that ρ proj f,l can be defined over F l d? Modulformen und das inverse Galois-Problem p.5/19

11 Introduction: GL 2 -case For f = n=1 a nq n and each prime l, Galois rep. ρ proj f,l ρ f,l : G Q GL 2 (F l ) nat. proj. PGL 2 (F l ). (I) Smallest d such that ρ proj f,l can be defined over F l d? Answer: If ρ f,l is irreducible, then ρ proj f,l can be defined over residue field (above l) of the global field Q( a2 p ψ(p) p N) Modulformen und das inverse Galois-Problem p.5/19

12 Introduction: GL 2 -case For f = n=1 a nq n and each prime l, Galois rep. ρ proj f,l (II) Image of ρ proj f,l? ρ f,l : G Q GL 2 (F l ) nat. proj. PGL 2 (F l ). Modulformen und das inverse Galois-Problem p.6/19

13 Introduction: GL 2 -case For f = n=1 a nq n and each prime l, Galois rep. ρ proj f,l (II) Image of ρ proj f,l? ρ f,l : G Q GL 2 (F l ) nat. proj. PGL 2 (F l ). Answer: From (I): ρ proj f,l definable over F l d. By Dickson ( 1900): ρ proj f,l (G Q) is PSL 2 (F l d), PGL 2 (F l d) dihedral ( 0 ) A 4,S 4,A 5 Modulformen und das inverse Galois-Problem p.6/19

14 Introduction: GL 2 -case For f = n=1 a nq n and each prime l, Galois rep. ρ proj f,l (II) Image of ρ proj f,l? ρ f,l : G Q GL 2 (F l ) nat. proj. PGL 2 (F l ). Answer: From (I): ρ proj f,l definable over F l d. By Dickson ( 1900): ρ proj f,l (G Q) is PSL 2 (F l d), PGL 2 (F l d) dihedral ( 0 ) A 4,S 4,A 5 huge image induced reducible exceptional Modulformen und das inverse Galois-Problem p.6/19

15 Introduction: GL 2 -case For f = n=1 a nq n and each prime l, Galois rep. ρ proj f,l (II) Image of ρ proj f,l? ρ f,l : G Q GL 2 (F l ) nat. proj. PGL 2 (F l ). Answer: From (I): ρ proj f,l definable over F l d. By Dickson ( 1900): ρ proj f,l (G Q) is PSL 2 (F l d), PGL 2 (F l d) dihedral ( 0 ) A 4,S 4,A 5 Ribet: For almost all l: huge image. huge image induced reducible exceptional Modulformen und das inverse Galois-Problem p.6/19

16 Introduction: GL 2 -case (III) Prove the existence of f such that for fixed l,d: ρ proj f,l (G Q) = PSL 2 (F l d), i.e. realise PSL 2 (F l d) as Galois group over Q. Modulformen und das inverse Galois-Problem p.7/19

17 Introduction: GL 2 -case (III) Prove the existence of f such that for fixed l,d: ρ proj f,l (G Q) = PSL 2 (F l d), i.e. realise PSL 2 (F l d) as Galois group over Q. Partial Answers: Theorem A (W. 2008). Given l, infinitely many d s.t. PSL 2 (F l d) occurs as ρ proj f,l (G Q) (for some f depending on d) with only l and one other prime (dep. on d) ramifying. Modulformen und das inverse Galois-Problem p.7/19

18 Introduction: GL 2 -case (III) Prove the existence of f such that for fixed l,d: ρ proj f,l (G Q) = PSL 2 (F l d), i.e. realise PSL 2 (F l d) as Galois group over Q. Partial Answers: Theorem A (W. 2008). Given l, infinitely many d s.t. PSL 2 (F l d) occurs as ρ proj f,l (G Q) (for some f depending on d) with only l and one other prime (dep. on d) ramifying. Theorem B (Dieulefait, W. 2011). Given d, positive density set of primes L s.t. l L: PSL 2 (F l d) occurs as ρ proj f,l (G Q) with only l and at most three other primes (not dep. on l) ramifying. Modulformen und das inverse Galois-Problem p.7/19

19 Introduction: GL 2 -case (III) Prove the existence of f such that for fixed l,d: ρ proj f,l (G Q) = PSL 2 (F l d), i.e. realise PSL 2 (F l d) as Galois group over Q. Partial Answers: Expected Theorem (W. 2012). Given d. Assume Maeda s conjecture on level 1 modular forms (the Galois closure of the coefficient field of every newform f S k (1) is Sym dim Sk (1)). Then the density of the set of primes l such that PSL 2 (F l d) occurs as ρ proj f,l (G Q) with only l ramifying is 1. (Proof needs to be checked and written up.) Modulformen und das inverse Galois-Problem p.8/19

20 Introduction: GSp 2n -case Generalisation to GSp 2n any n: Theorem A (Khare, Larsen, Savin, 2008). Given l, infinitely many d s.t. PSp 2n (F l d) or PGSp 2n (F l d) occurs as image of the residual Galois representation attached to a suitable automorphic form on GL 2n over Q. Modulformen und das inverse Galois-Problem p.9/19

21 Introduction: GSp 2n -case Generalisation to GSp 2n any n: Joint work with Sara Arias-de-Reyna and Luis Dieulefait: (I) Determine projective field of definition of compatible system of symplectic Galois representations. (DONE. Explain now.) Modulformen und das inverse Galois-Problem p.10/19

22 Introduction: GSp 2n -case Generalisation to GSp 2n any n: Joint work with Sara Arias-de-Reyna and Luis Dieulefait: (I) Determine projective field of definition of compatible system of symplectic Galois representations. (DONE. Explain now.) (II) Classify images of symplectic representations under some constraint. (DONE. Show result now.) Modulformen und das inverse Galois-Problem p.10/19

23 Introduction: GSp 2n -case Generalisation to GSp 2n any n: Joint work with Sara Arias-de-Reyna and Luis Dieulefait: (I) Determine projective field of definition of compatible system of symplectic Galois representations. (DONE. Explain now.) (II) Classify images of symplectic representations under some constraint. (DONE. Show result now.) (III) Generalise Theorem B. (ALMOST DONE, subject to a promised theorem by others). Modulformen und das inverse Galois-Problem p.10/19

24 Inner twists Let K be a field, K separable closure. Consider: ρ proj : G Q ρ GSp2n (K) nat. proj. PGSp 2n (K). Modulformen und das inverse Galois-Problem p.11/19

25 Inner twists Let K be a field, K separable closure. Consider: ρ proj : G Q ρ GSp2n (K) nat. proj. PGSp 2n (K). Def.: ρ proj 1 ρ proj 2 if M GSp 2n (K) s.t. ρ proj 1 = (Mρ 2 M 1 ) proj. Modulformen und das inverse Galois-Problem p.11/19

26 Inner twists Let K be a field, K separable closure. Consider: ρ proj : G Q ρ GSp2n (K) nat. proj. PGSp 2n (K). Def.: ρ proj 1 ρ proj 2 if M GSp 2n (K) s.t. ρ proj 1 = (Mρ 2 M 1 ) proj. Qu.: Smallest L K s.t. ρ proj (G Q PGSp 2n (L))? Modulformen und das inverse Galois-Problem p.11/19

27 Inner twists Let K be a field, K separable closure. Consider: ρ proj : G Q ρ GSp2n (K) nat. proj. PGSp 2n (K). Def.: ρ proj 1 ρ proj 2 if M GSp 2n (K) s.t. ρ proj 1 = (Mρ 2 M 1 ) proj. Qu.: Smallest L K s.t. ρ proj (G Q PGSp 2n (L))? Simple observations: Let ǫ : G Q K char. (ρ ǫ) proj = ρ proj. Modulformen und das inverse Galois-Problem p.11/19

28 Inner twists Let K be a field, K separable closure. Consider: ρ proj : G Q ρ GSp2n (K) nat. proj. PGSp 2n (K). Def.: ρ proj 1 ρ proj 2 if M GSp 2n (K) s.t. ρ proj 1 = (Mρ 2 M 1 ) proj. Qu.: Smallest L K s.t. ρ proj (G Q PGSp 2n (L))? Simple observations: Let ǫ : G Q K char. (ρ ǫ) proj = ρ proj. Suppose ρ proj 1 ρ proj 2. Put ǫ(g) := M 1 ρ 1 (g)mρ 2 (g) 1 K. ρ 1 ρ 2 ǫ. Modulformen und das inverse Galois-Problem p.11/19

29 Inner twists Qu.: Smallest L K s.t. ρ proj (G Q PGSp 2n (L))? Galois action on coefficients: for σ G K consider σ ρ : G Q ρ GSp2n (K) σ GSp 2n (K). Modulformen und das inverse Galois-Problem p.12/19

30 Inner twists Qu.: Smallest L K s.t. ρ proj (G Q PGSp 2n (L))? Galois action on coefficients: for σ G K consider σ ρ : G Q ρ GSp2n (K) σ GSp 2n (K). Def.: A pair (σ,ǫ) with σ G K and ǫ : G Q K character is called an inner twist if σ ρ ρ ǫ ( ( σ ρ) proj ρ proj ). Modulformen und das inverse Galois-Problem p.12/19

31 Inner twists Qu.: Smallest L K s.t. ρ proj (G Q PGSp 2n (L))? Galois action on coefficients: for σ G K consider σ ρ : G Q ρ GSp2n (K) σ GSp 2n (K). Def.: A pair (σ,ǫ) with σ G K and ǫ : G Q K character is called an inner twist if σ ρ ρ ǫ ( ( σ ρ) proj ρ proj ). ρ has complex multiplication (CM) if σ = id, ǫ 1. Modulformen und das inverse Galois-Problem p.12/19

32 Inner twists Qu.: Smallest L K s.t. ρ proj (G Q PGSp 2n (L))? Galois action on coefficients: for σ G K consider σ ρ : G Q ρ GSp2n (K) σ GSp 2n (K). Def.: A pair (σ,ǫ) with σ G K and ǫ : G Q K character is called an inner twist if σ ρ ρ ǫ ( ( σ ρ) proj ρ proj ). ρ has complex multiplication (CM) if σ = id, ǫ 1. Suppose ρ is irreducible and has no CM. Then: σ ρ ρ ǫ σ(tr(ρ(frob p ))) = Tr(ρ(Frob p ))ǫ(frob p ) unramified p. Modulformen und das inverse Galois-Problem p.12/19

33 Inner twists Def.: H ρ := ǫ ker(ǫ) G Q for ǫ occuring in an inner twists. Γ ρ := {σ G K σ occurs in an inner twist}. K ρ := K Γ ρ, called projective field of definition of ρ. Modulformen und das inverse Galois-Problem p.13/19

34 Inner twists Def.: H ρ := ǫ ker(ǫ) G Q for ǫ occuring in an inner twists. Γ ρ := {σ G K σ occurs in an inner twist}. K ρ := K Γ ρ, called projective field of definition of ρ. Theorem (Arias-de-Reyna, Dieulefait, W., 2012). Suppose ρ Hρ is irreducible. Then: (1) ρ such that ρ proj ρ proj and ρ proj factors through K ρ. (2) K ρ is the smallest subfield of K with this property. Morale: The inner twists determine the smallest field over which ρ proj can be defined. Modulformen und das inverse Galois-Problem p.13/19

35 Compatible systems Let n N, L/Q Galois number field, N,k N, ψ : G Q L, for all p N: P p (X) = X 2n a p X 2n 1 + L[X]. A compatible system ρ is: Modulformen und das inverse Galois-Problem p.14/19

36 Compatible systems Let n N, L/Q Galois number field, N,k N, ψ : G Q L, for all p N: P p (X) = X 2n a p X 2n 1 + L[X]. A compatible system ρ is: for each λ place of L a Galois representation ρ λ : G Q GSp 2n (L λ ) such that abs. irred., unramified outside Nl (for Λ l), p Nl : charpoly(ρ λ (Frob p )) = P p, similitude factor of ρ λ is ψχ k l (for χ l cyclotomic char.). Modulformen und das inverse Galois-Problem p.14/19

37 Compatible systems Let n N, L/Q Galois number field, N,k N, ψ : G Q L, for all p N: P p (X) = X 2n a p X 2n 1 + L[X]. A compatible system ρ is: for each λ place of L a Galois representation ρ λ : G Q GSp 2n (L λ ) such that abs. irred., unramified outside Nl (for Λ l), p Nl : charpoly(ρ λ (Frob p )) = P p, similitude factor of ρ λ is ψχ k l (for χ l cyclotomic char.). Sources: algebraic, essentially conjugate self-dual cuspidal automorphic representations for GL 2n over Q. Modulformen und das inverse Galois-Problem p.14/19

38 Compatible systems Let n N, L/Q Galois number field, N,k N, ψ : G Q L, for all p N: P p (X) = X 2n a p X 2n 1 + L[X]. A compatible system ρ is: for each λ place of L a Galois representation ρ λ : G Q GSp 2n (L λ ) such that abs. irred., unramified outside Nl (for Λ l), p Nl : charpoly(ρ λ (Frob p )) = P p, similitude factor of ρ λ is ψχ k l (for χ l cyclotomic char.). Sources: algebraic, essentially conjugate self-dual cuspidal automorphic representations for GL 2n over Q. We consider: ρ λ (residual representation), ρ proj λ, and ρ proj λ. Modulformen und das inverse Galois-Problem p.14/19

39 Compatible systems Let ρ be a compatible system. Def.: (σ,ǫ) (with σ Gal(L/K) and ǫ : G Q L ) inner twist of ρ if σ(a p ) = a p ǫ(frob p ) for all p N. Modulformen und das inverse Galois-Problem p.15/19

40 Compatible systems Let ρ be a compatible system. Def.: (σ,ǫ) (with σ Gal(L/K) and ǫ : G Q L ) inner twist of ρ if σ(a p ) = a p ǫ(frob p ) for all p N. Def.: Γ ρ := {σ Gal(L/K) σ occurs in an inner twist of ρ }. K ρ := L Γ ρ, called projective field of definition of ρ. Modulformen und das inverse Galois-Problem p.15/19

41 Compatible systems Let ρ be a compatible system. Def.: (σ,ǫ) (with σ Gal(L/K) and ǫ : G Q L ) inner twist of ρ if σ(a p ) = a p ǫ(frob p ) for all p N. Def.: Γ ρ := {σ Gal(L/K) σ occurs in an inner twist of ρ }. K ρ := L Γ ρ, called projective field of definition of ρ. Theorem 1 (Arias-de-Reyna, Dieulefait, W., 2012). Assume moreover: ρ is strictly compatible with regular Hodge-Tate weights and ρ λ is absolutely irreducible for almost all λ. Then for almost all places λ of L, the residue field at λ of K ρ is K ρλ. Modulformen und das inverse Galois-Problem p.15/19

42 Compatible systems Theorem 1 (Arias-de-Reyna, Dieulefait, W., 2012). Assume moreover: ρ is strictly compatible with regular Hodge-Tate weights and ρ λ is absolutely irreducible for almost all λ. Then for almost all places λ of L, the residue field at λ of K ρ is K ρλ. Morale: The global field K ρ (depending only on the inner twists) determines the projective field of definition of ρ proj λ. This field is the GSp 2n -replacement of Q( a2 p ψ(p) p N). Modulformen und das inverse Galois-Problem p.16/19

43 Classification result Theorem 2 (Arias-de-Reyna, Dieulefait, W., 2012). Let l 5 and ρ : G GSp 2n (F l ) be irreducible. Assume: ρ(g) contains a non-trivial transvection. Then either ρ(g) PSp 2n (F l ) (huge image) or ρ is induced from a lower dimensional representation. Modulformen und das inverse Galois-Problem p.17/19

44 Classification result Theorem 2 (Arias-de-Reyna, Dieulefait, W., 2012). Let l 5 and ρ : G GSp 2n (F l ) be irreducible. Assume: ρ(g) contains a non-trivial transvection. Then either ρ(g) PSp 2n (F l ) (huge image) or ρ is induced from a lower dimensional representation. Morale: Our replacement of Dickson s theorem for GL 2 : Recall: ρ proj f,l (G Q) is PSL 2 (F l d), PGL 2 (F l d) dihedral ( 0 ) A 4,S 4,A 5 huge image induced reducible exceptional Modulformen und das inverse Galois-Problem p.17/19

45 Inverse Galois Problem Theorem 3 (Arias-de-Reyna, Dieulefait, W., 2012). Let ρ be as in Theorem 1. Assume moreover: ρ λ (G Q ) contains a transvection for almost all λ. Good dihedral prime (Khare, Wintenberger, Larsen, Savin): prime q, suitable character δ : G Qq 2n L of order 2t (t prime), 2n (t 1) such that ρ λ GQ q IndG Qq G Qq 2n (δ). Then for all d t 1 2n, the set of places λ of L such that PSp 2n (F l d) or PGSp 2n (F l d) equals ρ proj λ (G Q ) has a positive density. Modulformen und das inverse Galois-Problem p.18/19

46 Inverse Galois Problem Theorem 3 (Arias-de-Reyna, Dieulefait, W., 2012). Let ρ be as in Theorem 1. Assume moreover: ρ λ (G Q ) contains a transvection for almost all λ. Good dihedral prime (Khare, Wintenberger, Larsen, Savin): prime q, suitable character δ : G Qq 2n L of order 2t (t prime), 2n (t 1) such that ρ λ GQ q IndG Qq G Qq 2n (δ). Then for all d t 1 2n, the set of places λ of L such that PSp 2n (F l d) or PGSp 2n (F l d) equals ρ proj λ (G Q ) has a positive density. Morale: If such a ρ exists, then we obtain the desired application to the inverse Galois problem. Modulformen und das inverse Galois-Problem p.18/19

47 Vielen Dank für Ihre Aufmerksamkeit. Modulformen und das inverse Galois-Problem p.19/19