A formula for the action of Hecke operators on half-integral weight Siegel modular forms and applications

TL;DR

This paper derives an explicit formula for Hecke operators on half-integral weight Siegel modular forms, computes eigenvalues, and demonstrates stability of certain subspaces under these operators, with applications to Fourier coefficients and modular form isomorphisms.

Contribution

It introduces new generators for the Hecke algebra, provides explicit formulas for their action, and proves stability of the Kitaoka subspace under all Hecke operators.

Findings

Eigenvalues of Hecke operators on average Siegel theta series are computed.

The eigenvalues are bounded in terms of Fourier coefficient bounds.

The Kitaoka subspace is shown to be stable under all Hecke operators.

Abstract

We introduce an alternate set of generators for the Hecka algebra, and give an explicit formula for the action of these operators on Fourier coefficients. With this, we compute the eigenvalues of Hecke operators acting on average Siegel theta series with half-integral weight (provided the prime associated to the operators does not divide the level of the theta series). Next, we bound the eigenvalues of these operators in terms of bounds on Fourier coefficients. Then we show that the half-integral weight Kitaoka subspace is stable under all Hecke operators. Finally, we observe that an obvious isomorphism between Siegel modular forms of weight $k+1/2$ and "even" Jacobi modular forms of weight $k+1$ is Hecke-invariant (here the level and character are arbitrary).