DRAFT - - DRAFT - - DRAFT - - DRAFT - - DRAFT - - On The Spezialschar Of Maass

Let M (n) k be the space of Siegel modular forms of degree n and even weight k. In this paper firstly a certain subspace Spez(M (2n) k ) the Spezialschar of M (2n) k is introduced. In the setting of the Siegel three-fold it is proven that this Spezialschar is the Maass Spezialschar. Secondly an embedding of M (2) k into a direct sum ⊕ ⌊ k 10 ⌋ ν=0 Sym 2 Mk+2ν is given. This leads to a basic characterization of the Spezialschar property. The results of this paper are directly related to the non-vanishing of certain special values of L-functions related to the Gross-Prasad conjecture. This is illustrated by a significant example in the paper. Introduction Hans Maass introduced and applied in a series of papers [Ma79I],[Ma79II] and [Ma79III] the concept of a Spezialschar to prove the Saito-Kurokawa conjecture [Za80]. Let M (2) k be the space of Siegel modular forms of degree 2 and weight k. Let A be the set of positive semidefinite half-integral matrices of degree 2. Hence T ∈ A can be identified with the quadratic form T = [n, r,m]. A modular form F ∈ M (2) k is in the Spezialschar if the Fourier coefficients A(T ) of F satisfy the relation A([n, r,m]) = ∑ d|(n,r,m) dA([ nm d2 , r d , 1]) (1) for all ∈ A. The space of such special forms is nowadays called the Maass Spezialschar M k . The purpose of this paper is twofold. First we introduce the concept of the Spezialschar Spez(M (2n) k ) for Siegel modular forms of even degree 2n. This is done in terms of the Hecke algebra H attached to Siegel modular forms of degree n. Let us fix the embedding Spn × Spm −→ Spn+m ( a b c d ) × ( ã b̃ c̃ d̃ ) 7→ ( a 0 0 ã b 0 0 b̃ c 0 0 c̃ d 0 0 d̃ ) . (2)