Algorithmic Treatment of Algebraic Modular Forms Diamant Symposium Fall 2015

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1 Algorithmic Treatment of Algebraic Modular Forms Diamant Symposium Fall 2015 Sebastian Schönnenbeck November 27, 2015 Experimental and constructive algebra Sebastian Schönnenbeck November 27, 2015 Algorithmic Treatment of Algebraic Modular Forms 1 / 20

2 A Classical Problem V = Q n, q : V Q a positive definite quadratic form. L, L V lattices (i.e. Z-submodules in V of rank n) L, L isometric, L = L, iff gl = L for some g O(q); class(l) = {M M = L}. L, L in the same genus, iff for all p prime there is g p O(Q p V, Q p q) with g p(z p L) = Z p L. L, L isometric implies L, L in the same genus but the converse is false. The genus of L decomposes into finitely many isometry classes; genus(l) = class(l 1)... class(l r), r the class number of L. Question: How do we find representatives for the isometry classes in a given genus? Theorem (Eichler, Kneser) Assume L even (i.e. q(l) 2Z) of rank greater or equal 3, det(l) squarefree, and p det(l) prime. Then every class in the genus of L is represented by a lattice M such that Z q L = Z q M q p. Sebastian Schönnenbeck November 27, 2015 Algorithmic Treatment of Algebraic Modular Forms 2 / 20

3 Strong Approximation and the Kneser Method Neighbours L as before. M, N genus(l) are called p-neighbours, M p N if [M : M N] = [N : M N] = p. Theorem (Kneser) For M genus(l) there is an M class(m) and a chain of lattices L = L 0, L 1,..., L k = M such that L 0 p L1 p L2 p... p M. Neighbouring Graph The p-neighbouring graph of genus(l) is the directed weighted graph with vertices the isometry classes in genus(l) and edges class(l i) class(l j) weighted by the number of p-neighbours of L i isometric to L j. Sebastian Schönnenbeck November 27, 2015 Algorithmic Treatment of Algebraic Modular Forms 3 / 20

4 Example Dimension 16 There are two isometry classes in the set of even unimodular lattices of rank 16: class(e 8 E 8) class(d 16). + The 2-neighbouring graph of this genus is: Remark Note that = , number of neighbours does not depend on the class. Note that = Aut(E 8 E 8 ) Aut(D + 16 ). The adjacency matrix of the graph acts as a Hecke operator on the space of modular forms generated by the theta series of the two lattices (Eichler, Andrianov, Yoshida). Sebastian Schönnenbeck November 27, 2015 Algorithmic Treatment of Algebraic Modular Forms 4 / 20

5 Algebraic Modular Forms Notation: G almost simple algebraic group defined over Q, G(R) compact, V a f.d. Q-rational representation of G, A f the finite adeles of Q. Definition [Gross 99] K G(A f ) open and compact. M(V, K) := {f : G(A f ) V f(gxk) = gf(x) for all g G(Q), x G(A f ), k K}, the space of algebraic modular forms of level K and weight V. Remark gkg 1 G(Q) < for all g G(A f ). G(Q)\G(A f )/K <. Let G(A f ) = r i=1 G(Q)γiK and Γi := γikγ 1 i G(Q) then M(V, K) =Q r V Γ i. i=1 Sebastian Schönnenbeck November 27, 2015 Algorithmic Treatment of Algebraic Modular Forms 5 / 20

6 The Hecke Algebra Definition K G(A f ) open and compact. H K := {f : G(A f ) Q f compactly supported and K-bi-invariant} with multiplication by convolution. The Hecke algebra of G with respect to K. Remark H K has the natural basis 1 KγK, KγK G(A f ) /K. Let γ 1, γ 2 G(A f ) with Kγ ik = j γi,jk. The multiplication in HK is 1 Kγ1 K1 Kγ2 K = j,j 1 γ1,j γ 2,j K. If K = p Kp is a product of local factors, the Hecke algebra is the restricted tensor product H K = ph Kp. Sebastian Schönnenbeck November 27, 2015 Algorithmic Treatment of Algebraic Modular Forms 6 / 20

7 The Action of the Hecke Algebra Definition For γ G(A f ) we define the linear operator T (γ) End Q (M(V, K)) via (T (γ)f)(x) = i f(xγ i) where f M(V, K) and KγK = i γik. Remark The additive extension of 1 KγK T (γ) defines an algebra homomorphism H K End Q (M(V, K)). M(V, K) is a semi-simple H K-module. Sebastian Schönnenbeck November 27, 2015 Algorithmic Treatment of Algebraic Modular Forms 7 / 20

8 Integral Forms What would be interesting / computationally well-suited open compact subgroups to consider? Definition Let G GL n be a faithful representation. An integral form G L of G is given by a lattice L Z Q n via G L(O k ) = Stab G(k) (O k L), G L(Z p) = Stab G(Qp)(Z p L) for every finite extension k of Q and every prime p. Remark G(A f ) acts on the integral forms via (g p) pl = L where Z p L = g p(z p L) for all p. Stab G(Af )(L) = p GL(Zp) is an open compact subgroup of G(A f ). G L(Z p) is a hyperspecial maximal compact subgroup of G(Q p) for all but finitely many p. Call L, L (G-)isomorphic if gl = L for some g G(Q). Say L, L in the same genus if γl = L for some γ G(A f ). Sebastian Schönnenbeck November 27, 2015 Algorithmic Treatment of Algebraic Modular Forms 8 / 20

9 Some (very informal) remarks on buildings Associated to G(Q p) there is a so-called building B, a simplicial complex with a G(Q p)-action and the following properties: B is a union of apartments A which are in bijection with the maximal Q p-split tori of G. Each apartment is an affine Euclidean space. Each apartment A corresponding to T comes with an extended affine Weyl group, W = X (T ) W 0, where W 0 is the usual (finite) Weyl group (w.r.t. T ). The extended affine Weyl group is an extension of an infinite Coxeter group by an abelian group. Every compact subgroup fixes a point in B. The stabilizers of vertices are always maximal compact subgroups. There is a Bruhat decomposition G(Q p) = w W IwI where I is the pointwise stabilizer of a maximal simplex (chamber). New concept of p-neighbourship (of certain lattices/ integral forms/ open compact subgroups) via closeness in the affine building. Sebastian Schönnenbeck November 27, 2015 Algorithmic Treatment of Algebraic Modular Forms 9 / 20

10 Example: The affine building of type G 2 The group G 2 can be realized as the automorphism group of the (8-dim.) octonion algebra O (Dickson-double of the Hamilton quaternions). The affine building for G 2(Q p) can be realized as follows: There are 3 labels for the vertices, V = V 1 V 2 V 3, corresponding to 3 (G 2(Q p)-)classes of orders in O p. Hyperspecial vertices correspond to maximal orders. Adjacency is given by inclusion (V 1 V 2 V 3). An apartment in this building looks as follows: Sebastian Schönnenbeck November 27, 2015 Algorithmic Treatment of Algebraic Modular Forms 10 / 20

11 Algorithmic Questions Aim: Compute the action of T (γ) on M(V, K) (where K comes from an integral form G L). Approach Decompose G(A f ) = i G(Q)µiK, compute Γi = µikµ 1 i G(Q) and V Γ i. Decompose KγK = j γjk. For i, j write γ jµ i as g µ i k for some i. Aspects to consider What do we have to know in order to make this work? Decide for two integral forms if they are G-isomorphic / compute the stabilizer of a lattice in G(Q). Be able to compute a system of representatives for genus(g L). Decompose double cosets into left cosets. Sebastian Schönnenbeck November 27, 2015 Algorithmic Treatment of Algebraic Modular Forms 11 / 20

12 Stabilizers and Isometries G GL n, L, L < Q n lattices. G(Q) GL n(q) compact G(Q) fixes a definite inner product on Q n. G(Q) O n(q) Stab G(Q) (L) Stab On(Q)(L). Stab On(Q)(L) computable (Plesken-Souvignier-algorithm) and finite Finding Stab G(Q) (L) reduced to a finite problem. Same idea for isometry testing: Find O n-isometry g : L L All isometries are given by g Stab On(Q)(L) Finite problem. Example: G 2 The group G 2 can be realized as the automorphism group of the (8-dim.) octonion algebra O (Dickson-double of the Hamilton quaternions). G 2 fixes the inner product (x, y) xȳ. L < O lattice then Stab G2 (L) is the stabilizer of the multiplication (which can be thought of as an element of V V V ) in Stab O8 (L). E.g. L a maximal order then Stab O8 (L) = , Stab G2 (L) = Sebastian Schönnenbeck November 27, 2015 Algorithmic Treatment of Algebraic Modular Forms 12 / 20

13 Genus Enumeration Question: How do you compute representatives of genus(l) starting at L? Almost Strong Approximation If G is simply connected and of certain type then there is a finite set Ω of primes such that for all p / Ω the lattice L p is hyperspecial and we can find representatives of genus(l) as p-neighbours of L. Question: How do you know when to stop? Mass Formula Let genus(l) = class(l 1)... class(l r) and set mass(genus(l)) := r i=1 1 Stab G (L i). Then we can compute mass(genus(l)) from information only on the local structure of L. Sebastian Schönnenbeck November 27, 2015 Algorithmic Treatment of Algebraic Modular Forms 13 / 20

14 Examples Some Genera for G 2 Genus Class Number Mass Decomposition max. order 1 type 2 at 3, max. else 2 type 2 at 5, max. else 3 type 3 at 7, max. else Some Genera for Sp 4 Compact forms of Sp can be found as unitary groups over (definite) quaternion algebras D, integral forms via O D-lattices (where O D is a maximal order). Genus Representative Class Number Mass Decomposition O 2 D, D = O 2 D, D = O 2 D, D = O 2 D, D = ( 1, 1 Q ( 2, 5 Q ( 2, 13 Q ( 1, 23 Q ) ) ) ) Sebastian Schönnenbeck November 27, 2015 Algorithmic Treatment of Algebraic Modular Forms 14 / 20

15 Coset Decomposition Aim: Decompose a double coset KγK into left cosets. First observation: Since H K = ph Kp we only need to do this locally. For simplicity assume: G split at p, K p nice. Structure of p-adic Groups (Bruhat-Tits) I K p G(Q p) Iwahori subgroup, W (= X W 0) the extended affine Weyl group. G(Q p) = w W IwI, Kp = w W Kp IwI for some W Kp W. H I is an algebra with basis T w, w W and multiplication T wt w = T ww if l(ww ) = l(w) + l(w ), T 2 s = (p 1)T s + p for the simple reflections s. e := [K p : I] 1 w W Kp T w H I is an idempotent and H Kp = ehie. Coset Decomposition Bruhat-Tits ( 65): Explicit formula to decompose IwI, w W into I-left cosets. Lansky-Pollack (2001): Explicit formula to decompose K pwk p, w W into K p-left cosets. Sebastian Schönnenbeck November 27, 2015 Algorithmic Treatment of Algebraic Modular Forms 15 / 20

16 Example G 2 G of type G 2 (simply connected and adjoint) with extended Dynkin diagram (at split prime p): G 2 : If K open compact, with K p hyperspecial maximal compact (W Kp = s 1, s 2 ), then the local Hecke algebra H Kp is a polynomial ring in two variables generated by the characteristic functions on the double cosets T 1 := K ps 0K p and T 2 := K ps 0s 1s 2s 1s 0K p. T 1 decomposes into q(q 5 + q 4 + q 3 + q 2 + q + 1) left cosets. T 2 decomposes into q 5 (q 5 + q 4 + q 3 + q 2 + q + 1) left cosets. Sp 4 In the analogous situation for Sp 4 (simply connected but not adjoint) there are also two generators which decompose into q(q 3 + q 2 + q + 1) and q 3 (q 3 + q 2 + q + 1) left cosets, respectively. Sebastian Schönnenbeck November 27, 2015 Algorithmic Treatment of Algebraic Modular Forms 16 / 20

17 Venkov s Method Aim: Give an alternative method to compute neighbouring operators. Idea: Use a generalization of the following method attributed to Venkov: L, L two lattices then mass(genus(l)) [Stab G(L) : Stab G (L, L )] [Stab G (L ) : Stab G (L, L )] = mass(genus(l )). Now: p prime such that G is split at p and L p is hyperspecial, genus(l) = [L 1]... [L r]. Neighbouring Operator Set N p Z r r the matrix with (N p) i,j = {L L p-neighbour of L i, L = Lj}. (1) The neighbouring operator of genus(l) at p (or the Hecke operator T (s 0) where s 0 is the fundamental reflection at p not fixing L) or just the adjacency matrix of the neighbouring graph. Fix a lattice M L such that L, L are p-neighbours iff L L genus(m) (very often possible). Sebastian Schönnenbeck November 27, 2015 Algorithmic Treatment of Algebraic Modular Forms 17 / 20

18 The Venkov Transfer Observation We can write genus(m) = [M 1]... [M s] where each M i is a sublattice of L j for some j. The Transfer Matrix Set T L M the matrix given by (T L M ) i,j = {M M L i, M = Mj} and define T M L the other way around. Remark There are often far fewer M genus(m), M L than there are p-neighbours of L. (T L M ) i,j Stab Mj = (T M L ) j,i Stab Li. Sebastian Schönnenbeck November 27, 2015 Algorithmic Treatment of Algebraic Modular Forms 18 / 20

19 The Neighbouring Operator Theorem T M L T L M = a I r + N p where a is the number of lattices in the genus of M contained in L. Generalization We can generalize this method to include other operators and modular forms of arbitrary weight: Let K 1, K 2 be two open compact subgroups and let K 2 = k i(k 1 K 2). Then we get an operator T 1 2 : M(V, K 1) M(V, K 2), f f where f (x) = i f(xk i). This is well-defined and generalizes the T L M from above. Sebastian Schönnenbeck November 27, 2015 Algorithmic Treatment of Algebraic Modular Forms 19 / 20

20 Some Remarks Properties of the operators T 1 2 T L M is the matrix of T 1 2 where K 1 is the stabilizer of L, K 2 is the stabilizer of M and V is the trivial representation. T 1 2 and T 2 1 are adjoint to each other with respect to certain scalar products. In particular we can compute T 2 1 if we know T 1 2 (and some other easily obtained information). T 2 1 T 1 2 : M(V, K 1) M(V, K 1) acts as an element of the Hecke algebra (w.r.t. K 1). Under certain conditions we have explicit knowledge of the cosets appearing in T 2 1 T 1 2 (e.g. if K 1 and K 2 differ only at one (split) prime and G is simply connected). The local Hecke algebra for simply connected groups of type C n at hyperspecial primes is generated by n operators of the form T 2 1 T 1 2. In general these operators do not generate the whole Hecke algebra. Sebastian Schönnenbeck November 27, 2015 Algorithmic Treatment of Algebraic Modular Forms 20 / 20