Abstract We prove that if F is a nonzero (possibly noncuspidal) vector-valued Siegel modular form of any degree, then it has infinitely many Fourier coefficients which are indexed by half-integral matrices having odd, square-free (and thus fundamental) discriminant. The proof uses an induction argument in the setting forms. Further, as application variant our result and complementing work A. Po...

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 cha...

Fix integers g ≥ 1 and N ≥ 3, and a prime p not dividing N . We show that the systems of Hecke eigenvalues occurring in the spaces of Siegel modular forms (mod p) of dimension g, level N , and varying weight, are the same as the systems occurring in the spaces of Siegel cusp forms with the same parameters and varying weight. In particular, in the case g = 1, this says that the Hecke eigensystem...