Kneser s p-neighbours and algebraic modular forms

Transcription

1 Kneser s p-neighbours and algebraic modular forms Matthew Greenberg University of Calgary 5 September / 36

2 Agenda 1 introduction 2 adelic automorphic forms 3 algebraic modular forms (after Gross) previous work by others connection with class groups of lattices 4 Kneser s neighbour construction and Hecke operators for orthogonal groups lattice methods isometry class enumeration using p-neighbours (after Kneser) isomorphism testing Plesken & Souvignier 5 unitary group variant 6 to-do list 2 / 36

3 Why compute spaces of automorphic forms? initially: testing the Shimura-Taniyama conjecture, i.e., the modularity of elliptic curves finding interesting number fields via Galois representations associated to modular forms Theorem. (Dembélé, Dembélé-G-Voight, Dieulefait) There exist nonsolvable number fields unramified away from p for p {2, 3, 5, 7}. The proof of the theorem uses explicit computations of Hilbert and Siegel modular forms. gathering evidence for various conjectures that comprise the Langlands program 3 / 36

4 Dembélé s field 4 / 36

5 Roberts polynomial 5 / 36

6 Modular symbols the prototype Theorem. 0 = Div 0 P 1 (Q) is a finitely generated Z[Γ 0 (N)]- module. If 0 = Z[Γ 0 (N)]D Z[Γ 0 (N)]D h, then a modular symbol is determined by its values on the D i. We must enumerate generators D i We need a reduction theory: Given D 0, find w i Z[Γ 0 (N)] such that D = w 1 D 1 + w h D h. All approaches to computation of automorphic forms involve enumeration and reduction steps. 6 / 36

7 Adelic automorphic forms Let F be a totally real number field with ring of integers o and let d = [F : Q] { F = x F v : x v o v for almost all v v }, ô = v o v, F = v F v = R d. 7 / 36

8 Let G be a reductive, algebraic F -group. Set G = G(F ). Let K G(ô) be a compact, open subgroup e.g., K = {( c a d b ) G(ô) : c N ô}, N o. Let K be a maximal compact, connected subgroup of G(F ), e.g., K = SO(n) d. Define the Shimura manifold of level K : ( Y (K ) = G(F )\ G( F )/K G /K Z }{{} H ). 8 / 36

9 ( ) Y (K ) = G(F )\ G( F )/K H. Theorem: h(k ) := G(F )\G( F )/K < Sorting out the diagonal action, where G( F ) = h(k ) i=1 Y (K ) = h(k ) i=1 Γ xi \H G(F )x i K, Γ xi := G(F ) x i K x 1 i. 9 / 36

10 A computational challenge For an algebraic repesentation V of G, decompose H i (Y (K ), V ) into eigenspaces for appropriate Hecke operators. One approach: Hybrid geometric/arithmetic methods (Ash, Gunnells, Yasaki) 10 / 36

11 Algebraic modular forms Setting introduced by Gross (Israel J. Math., 1999) a class of automorphic forms particularly well-suited to calculation G/F connected, reductive algebraic group, G(F ) compact e.g., definite orthogonal groups, definite unitary groups, definite quaternionic unitary groups,... K G( F ) compact open subgroup Since G(F ) is compact, we take K = G(F ). Y (K ) = G(F )\G( F )/K (finite, size h(k )) 0-dimensional Shimura variety 11 / 36

12 Let V be a finite-dimensional, algebraic representation of G. Space of algebraic modular forms, level K, weight V M(V, K ) = {f : G( F )/K V : f (γg) = γf (g), γ G(F )} Suppose G( F ) = h G(F )x i K i=1 f M(V, K ) is determined by {f (x 1 ),..., f (x h )}. If we can represent elements of V, we can represent elements of M(V, K) provided we can find representatives {x i }. We need an enumeration algorithm. 12 / 36

13 Hecke operators f M(V, K ) {f (x 1 ),..., f (x h )} For ϖ G( F ), write K ϖk = i ϖ ik Define T (ϖ) : M(V, K ) M(V, K ) by (T (ϖ)f )(xk ) = i f (xϖ i K ) Knowing {f (x i )}, how do we compute (T (ϖ)f )(x i )? x i ϖ j K = γ i,j x k(i,j) K for some γ i,j G(F ) G(F )-equivariance of f = (T (ϖ)f )(x i ) = j γ i,jf (x k(i,j) ). To compute γ i,j and k(i, j), we need a reduction algorithm. 13 / 36

14 My goal Develop a unified approach and systematic algorithms for computing with algebraic modular forms based on lattice algorithms. Implement them. (Help!) For the rest of the talk, I ll describe some progress with definite unitary and orthogonal groups. 14 / 36

15 Previous work Lansky & Pollack G = G 2 over Q key fact: G 2 (Q)G 2 (Ẑ) = G 2( F ) Dembélé, Dembélé & Donnelly F /Q totally real, B/F totally definite quaternion algebra, G = B principal ideal testing/ideal principalization Cunningham, Dembélé B = H Q( 5), G = GU 2 (B) Loeffler U(3) relative to Q( 7)/Q some clever ad-hoc methods 15 / 36

16 Orthogonal groups Let Q M n (o) be such that Q = Q t, Q 0. be a symmetric matrix such that Q v M n (o v ) is positive-definite for all v. For an o-algebra R, define G(R) = O(Q)(R) = {A GL n (R) : A t QA = Q}. Since Q 0, G(F ) is compact. Thus, we may consider algebraic modular forms for G. 16 / 36

17 Lattices Let L and M be o-lattices in F n. If there is an x G(F ) (resp. x G( F )) such that M = xl (resp. M = x L), then we say that L and M are equivalent (locally equivalent). Write gen(l) for the set of lattices in F n locally equivalent to L. K := stab G( F ) L is a maximal compact subgroup of G( F ). Thus, G( F )/K gen(l), Y (K ) = G(F )\G( F )/K G(F )\ gen(l). The genus of L is the union of finitely many equivalence classes. Thus, elements of M(V, K ) can be viewed as V -valued functions of lattices. 17 / 36

18 Lattice enumeration Kneser s method Enumeration of quadratic forms in n variables Scharlau & Hemkemeier, Math. Comp. (1998) implementation of Kneser s method as an algorithm, large scale computations; Schulze-Pillot approach in ranks 3 and 4 using almost-canonical forms for Gram matrices. 18 / 36

19 Let p o be a prime ideal. Definition: Lattices L and M in F n are p-neighbours if pl L M, pm L M, and dim k(p) L/L M = dim k(p) M/L M = 1. The p-neighbours of L are easy to construct! They all have for form L(v) := p 1 v + {x L x t Qy p} for v such that v t Qv p. In nice situations, the p-neighbours of L are locally equivalent to L and we can find representatives for G(F )\ gen(l) by constructing more and more p-neighbours. 19 / 36

20 We study p-neighbours more closely in the following situation: ( ) 0 In Q Fp. I n 0 Equivalently, there is a basis e 1,..., e n, f 1,..., f n of F 2n such that ei t Qe i = fi t Qf i = 0, ei t Qf i = 1 for all i. The quadratic space (Fe i + Ff i, Q) is called a hyperbolic plane. Changing to this basis, we have {( ) } A B G(F p ) GL C D 2n (F p ) : At B and C t D skew symmetric, A t D + B t, C = I n {( ) } A 0 T (F p ) 0 A 1 G(F p ) : A diagonal. 20 / 36

21 Theorem: (Elementary divisors) Let L and M be two unimodular lattices in Fp 2n. Then there is a hyperbolic basis of L, and integers such that is a hyperbolic basis of M. e 1,..., e n, f 1,..., f n a 1 a 2 a n 0, π a 1 e 1,..., π an e n, π a 1 f 1,..., π an f n 21 / 36

22 Corollary: G(F p ) acts transitively on the set of unimodular lattices L F 2n p. Corollary: The p-neighbours of L are locally equivalent to L and are unimodular if L is. Let K p = GL 2n (o p ) G(F p ), + = {diag(π a 1,..., π an, π a 1,..., π an ) : a 1 a n 0} T (F p ). Corollary: (p-adic Cartan decomposition) Let x G(F p ). Then the double coset K p xk p contains a unique element of / 36

23 p-neighbours and Hecke operators Let ϖ = diag(π, 1,..., 1, π 1, 1,..., 1) +. The following sets are in canonical bijection: 1 The p-neighbours of the lattice o 2n. 2 K p ϖk p /K p (Hecke operators!), 3 the set of lattices in F 2n with elementary divisors π, 1,..., 1, π 1, 1,..., 1 with respect to o 2n. 4 the set of isotropic lines in o 2n /po 2n, 5 the set of k(p)-rational points of the hypersurface in P 2n 1 /k(p) cut out by the quadratic form. 23 / 36

24 M(K, V ) = {f : gen(o 2n ) V : f (γl) = γf (L) γ G(F )} ϖ = diag(π, 1,..., 1, π 1, 1,..., 1) +, G(F )\ gen(o 2n ) = {[L 1 ],..., [L h ]}, (T (ϖ)f )(L i ) = j f (M i,j ) {M i,j } p-ngbrs. of L i = j = j f (γ i,j L k(i,j) ) M i,j = γ i,j L k(i,j), γ i,j G(F ) γ i,j f (L k(i,j) ). 24 / 36

25 Reduction We must be able to test lattices for isomorphism. algorithm due to Plesken and Souvignier matches up short vectors, uses other tricks to try to rule out isomorphism as early as possible also used to compute automorphism group of a lattice really nice Magma implementation over Q Allows you to compute isometries that respect a sequence of auxilliary quadratic forms. This allows for enumeration of lattices over totally real fields. Contributors: Souvignier, Kohel, Steel, Nebe, Unger 25 / 36

26 Unitary groups associated to CM fields E/F CM extension, [F : Q] = d o and O rings of integers of F and E, respectively For an o-algebra R, define Since E/F is CM, G(R) = {x GL n (O R) : x t x = 1}. G(F ) = {x GL n (C) : x t x = 1} = U(n) d G(F ) = U(n) d is compact p split in E/F = G(F p ) = {(x, y) GL n (E F p ) = GL n (F p ) 2 : (y t, x t )(x, y) = 1} = GL n (F p ), (x, y) x 26 / 36

27 Hermitian lattices Let L be a Hermitian lattice in E n. As in the orthogonal case, K := stab G( F ) L is a maximal compact subgroup of G( F ). G( F )/K is in bijection with gen(l), the set of Hermitian lattices in E n locally isomorphic to L. G(F )\G( F )/K is in bijection with the isomorphism classes of lattices in gen(l). Again, algebraic modular forms can be viewed as functions on lattices. 27 / 36

28 Enumeration Hoffman, Manuscripta Math. (1991) variant of Kneser s method for unitary groups, calculations by hand (?) Schiemann, J. Symbolic Comput. (1998) computer implementation of unitary variant of Kneser s method, large scale computations Class numbers of Hermitian lattices 28 / 36

29 Neighbours Suppose p splits in E/F, p = P P. Definition: M is a P-neighbour of L if PM L M, P L M, and dim k(p) M/L M = dim k( P) L/L M = 1. Constructing P-neighbours Let {x i } PL be representatives for P( PL/pL) P n 1 (k(p)). Set L(x i ) = P 1 x i + {y L : x i, y P} The L(x i ) are well defined and distinct. They are the P-neighbours of L. 29 / 36

30 Enumeration algorithm keep generating P-neighbours, testing for equivalence along the way. Siegel-type mass formula tells you when to stop Since the Magma implementation of the Plesken-Souvignier Q-lattice isometry testing algorithm allows you to require that an isometry respect a list of additional binary forms, it can be used to test for isometry of Hermitian lattices for CM fields: If 1 = b 1,..., b 2d is a basis of E/Q, then a matrix γ GL n (E) = GL 2dn (Q) is unitary if and only the forms, i : E n E n Q, x, y = Tr E/Q b i x, y are γ-invariant. Note that, 1 is symmetric and positive-definite. 30 / 36

31 Neighbours and Hecke operators Suppose X pl PL is such that X is a (k 1)-plane in P( PL/pL), 1 k n 1. We can define a (P, k)-neighbour L(X ) of L such that {(P, k)-neighbours} K p ϖ k K p /K p, where We have: ϖ k = diag(π,... π, 1,..., 1). }{{}}{{} k n k (T (ϖ k )f )(L) = X f (L(X )). 31 / 36

32 Class numbers: U(3) relative to Q( d)/q d h time (s) d h time (s) / 36

33 Hecke operators: U(3) for Q( 7)/Q p time (s) p time (s) p time (s) p time (s) / 36

34 Hecke operators: U(3) for Q( 23)/Q p time (s) p time (s) / 36

35 Hecke operators: U(3) for Q(ζ 7 )/Q(ζ 7 ) + h = 2; representatives computed in 1.65 seconds p time (s) p time (s) / 36

36 Other variants/to-do list Write more code! Improve my code! Compile? nonsplit primes level structure Adapt to to other groups where there is a sufficiently explicit Bruhat-Tits theory, i.e., for which the Bruhat-Tits buildings can be described in terms of lattice chains. Quaternionic unitary groups? Exceptional lie groups? Any improvements to the Plesken-Souvignier algorithm would yield speed-ups. 36 / 36