On the Fourier Coefficients of Siegel modular Forms

Abstract

We prove that if F is a non-zero (possibly non-cuspidal) vector-valued Siegel modular form of any degree, then it has infinitely many non-zero 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 of vector-valued modular forms. In an Appendix, as an application of a variant of our result and building upon the work of A. Pollack, we show how to obtain an unconditional proof of the functional equation of the spinor L-function of a holomorphic cuspidal Siegel eigenform of degree 3.

On mod p singular modular forms

“…here Z * denotes the r × r-submatrix in the upper left corner of Z and C n,r (Z) is an appropriate (Klingen) parabolic subgroup of Sp n (Z). One can use the explicit formulas for the Fourier coefficients of such modular forms from [3,4,18] to deduce a nice non-divisibility result from (7.1) about special values of certain L-functions: To get a smooth formulation, we use the primitive Fourier coefficients instead of the ordinary Fourier coefficients, see [6]: Then the primitive Fourier coefficients of E n,r (f ) are given (up to some factor not depending on T and up to a critical value of the standard L-function D(f, s) by a special value of the Rankin convolution R(f, θ r T ) of f and the degree r theta series attached to T , defined by…”

“…In Böcherer-Raghavan [8] (see page 82 and 83), the notion of "primitive Fourier coefficient" was introduced; we denote it by a * (S) for S positive definite. Namely, a * (S) is defined by the formula…”

Cuspidality and the growth of Fourier coefficients of modular forms : A survey

“…Whereas this statement is not very difficult to prove for elliptic modular forms (cf. [Koh10], [BD14], [BK19], or Proposition 3.1) it becomes more interesting -and non-trivial for higher rank groups. Here we would be working with Siegel's modular group acting on Siegel's upper half-space, but most of the paper generalizes without much issue to other kinds of tube domains.…”

“…This allows one to ignore the convergence properties of D F (s). The cases covered (essentially) were M k (Γ), k > 2n, c < k − n; -(vi) In [BK19], a nice approach adapted from [Miy89], which transfers the problem to the behaviour of F (Z) as det(Y ) → 0, was successfully carried out. The approach works smoothly when the Fourier expansion of F is supported on positive definite matrices (at a cusp), but otherwise requires some additional tricks to reduce to this situation.…”

“…Now we talk about the quantity k 0 in Question 1.1. Whereas the large values of k 0 in [BD14] occurred from the poles of D F (s), in [BK19] it was partly due to the use of Lipschitz formula. Since our argument will avoid all of these, we do not face any such issue.…”