Lifting Puzzles for Siegel Modular Forms

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1 Department of Mathematics University of California, Los Angeles April, 007

2 What are Siegel modular forms Multivariate modular forms Arithmetic group Upper half-plane Automorphy factor Growth condition

3 Ingredients Computing Modular Forms Arithmetic group Γ n = Sp n (Z) = {M GL n (Z) : M t JM = J}, J = ( 0 In I n 0 Upper half-plane: H n = {Z M n (R) : Z t = Z, Im(Z ) > 0. ).

4 Siegel Modular Forms Let M k (Γ n ) = Mk n be the space of Siegel modular forms of weight k and degree n. I.e., F Mk n iff F : h n C is holomorphic, F (AZ + B)(CZ + D) 1 «= det(cz + D) k A B F(Z ) for all Γ C D n F(Z ) = P T 0 a(t )eπitr(tz ) where T runs over all positive semi-definite even integral n n matrices. If the expansion of F is only supported on positive definite forms, it s a cusp form, i.e. F S n k.

5 An example in S 1 sage: M = SiegelModularForms(1) sage: S = M.cuspidal_subspace() sage: S.basis_as_list_of_forms() [0*q^(0, 0, 0) + 10*q^(1, 0, 1) + -13*q^(1, 0, ) *q^(1, 0, 4) + 1*q^(1, 1, 1) + -88*q^(1, 1, 1, 3) *q^(1, 1, 4) *q^(, 0, ) *q^(, 0, 4) *q^(, 1, ) *q^( 38950*q^(, 1, 4) + 784*q^(,, ) *q^(, *q^(,, 4) *q^(3, 0, 3) * *q^(3, 1, 3) *q^(3, 1, 4) *q *q^(3,, 4) *q^(3, 3, 3) *q^( *q^(4, 0, 4) *q^(4, 1, 4) *q^(4, 3, 4) *q^(4, 4, 4)]

6 M k (Γ 1 (N)) is computable Theorem (Lots of People) The space of modular forms of weight k invariant under Γ(N) is computable, i.e., there is an algorithm that takes as input k, N, B and outputs a basis of q-expansions for M k (Γ 1 (N)) to precision O(q B ). Based on Eichler-Shimura. Mostly due to Manin. Algorithm is implemented in SAGE.

7 Computing S n k n = 1: Modular Symbols (w/ and w/o level) n = : Generators known explicitly (w/o level) Poor-Yuen give dimension formulas for prime level (joint w/ D. Yuen) computed level 1 forms up to weight 5, Hecke data for p =, 3 n = 3: Generators known, dimension formulae known (w/o level) n = 4: Generators unknown, dimension formulae unknown Poor-Yuen lets you compute spaces for low weight (currently only up to weight 16)

8 L-functions in degree 1 Definition (L-function) Let f = a n q n Sk 1 be a simultaneous Hecke eigenform with Hecke eigenvalue λ p with respect to T p. The L-function associated to f is given by L(f, s) = n 1 a n n s = p 1 a p p s + p k 1 p s

9 Satake Parameters Theorem Let λ Hom Q (H n,p, C) be nontrivial. There exists α = (α 0,p, α 1,p,..., α n,p ) C n+1 such that the following diagram commutes: H n,p Ω λ Q[x ±1 0, x ±1 1,..., x n ±1 ] Wn > x i := α i > C

10 L-functions in higher degree Definition (Standard L-function) Let F be a simultaneous eigenform with Satake parameters α. The standard L-function associated to F is given by L(F, s, st) = p L p (F, s, st) where L p (F, s, st) = (1 p s ) n (1 α i,p p s )(1 α 1 i=1 i,p p s ).

11 Ikeda Lift Computing Modular Forms Theorem (Ikeda, 001) For even k, n and (n + k)/, each eigenform f S1 k corresponds to an eigenform I n (f ) S (n+k)/ n such that the standard L-function factors L(I n (f ), s, st) = ζ(s) n i=1 L(f, s + n + k i).

12 Miyawaki Lift Computing Modular Forms Theorem (Ikeda, 006) Let k, n + r and ˆk = (n + r + k)/ be even integers with r n. Each pair of eigenforms f S k 4 1 and g Sˆk r corresponds to an element M n (f, g) Sˆk n via M n (f, g)(z n ) = I n+r (f )(Z n Z r ), g c (Z r ), where, is the Petersson inner product. If M n (f, g) is nontrivial then it is an eigenform and L(M n (f, g), s, st) = L(g, s, st) n i=r+1 L(f, s + ˆk i).

13 Computing in Degree 4 Theorem (Poor-Yuen, 007) Using their restriction method and vanishing theorems they have computed bases of eigenforms for n = 4 and k 16. (joint w/ Poor-Yuen) for weight 14 all forms are either Miyawaki or Ikeda lifts

14 S 4 16 Computing Modular Forms (joint w/ Poor-Yuen) h 46 ( ( β)x 819x )( ( β)x 4096x ) 1 ( ( β)x 048x )( ( β)x 104x ) h 46 (104 + ( β)x + 819x )(048 + ( β)x x ) ( ( β)x + 048x )(819 + ( β)x + 104x ) h 34 ( ( β)x 4096x )( ( β)x 048 ) 3 ( x x + 36x x 4 ) h 34 (048 + ( β)x x )( ( β)x ) 4 ( x x + 36x x 4 ) h 36 ( 51 + ( γ)x 104x )( ( γ)x 51x ) 5 (51 + 3x + 56x )(56 + 3x + 51x ) h 36 (51 + ( γ)x + 104x )(104 + ( γ)x + 51x ) 6 (51 + 3x + 56x )(56 + 3x + 51x ) h 7 8 ( x 0755x 640x x 4 ) ( x 0755x 580x x 4 )

15 Status Computing Modular Forms What we know: h 5 and h 6 are Miyawaki lifts h 1 and h are Ikeda lifts No space of cusp forms of degree 1, any level, has the same irrationality as h 3, h 4, and h 7 No space of cusp forms of degree and level 1 has the same irrationality as h 3, h 4, and h 7 What we believe: they are lifts (non-unimodular Satake parameters) they are previously unidentified lifts or