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

Abstract

We introduce an alternate set of generators for the Hecke algebra, and give an explicit formula for the action of these operators on Fourier coefficients. Using 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), and show that some of the Hecke operators annihilate 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).