Siegel cusp forms of degree 2 are determined by their fundamental Fourier coefficients

Abstract

Abstract. We prove that a Siegel cusp form of degree 2 for the full modular group is determined by its set of Fourier coefficients a(S) with 4 det(S) ranging over odd squarefree integers. As a key step to our result, we also prove that a classical cusp form of half-integral weight and level 4N , with N odd and squarefree, is determined by its set of Fourier coefficients a(d) with d ranging over odd squarefree integers, a result that was previously known only for Hecke eigenforms.

Petersson norms of not necessarily cuspidal Jacobi modular forms and applications

“…(ii) Secondly, we answer a question raised in the paper by A. Saha [25] affirmatively by showing that if F is a non-zero Siegel modular form of degree 2, then it has infinitely many non-zero 'fundamental' Fourier coefficients, i.e., a(F, T ) = 0 with − det(2T ) a fundamental discriminant. This follows from the finer asymptotics in Theorem 6.8 along with a certain 'Ω'-result for Fourier coefficients of elliptic cusp forms.…”

“…This follows from the finer asymptotics in Theorem 6.8 along with a certain 'Ω'-result for Fourier coefficients of elliptic cusp forms. Previously this was only known from [25] in the case of cusp forms. See Proposition 7.7.…”

“…7.2. Determination by Fundamental Fourier coefficients.In[25], a remarkable result was proved: all Siegel cusp forms F of degree 2 are determined by a F (T ) such that − det(2T ) is a fundamental discriminant. In the same paper, it was asked (cf.…”

On the Fourier Coefficients of Siegel modular Forms

“…When the set S as above consists of all fundamental discriminants, A. Saha [33] proved an affirmative result on S 2 k ; this result has applications to the representation theory of automorphic forms, see the discussion in [33,Introduction]. It is of course desirable to generalise the results of [33] to higher degree Siegel cusp forms (including vector-valued modular forms), and also to include the space of Eisenstein series. In fact, these aspects were mentioned as 'difficult open' problems in [33, remark 2.6, 2.7].…”

“…More importantly, these primitive theta components are crucial for us since we are led to deal with levels which are squares, and these levels does not satisfy the conditions of [1,Thm. 2] or [33,Thm. 2].…”