Petersson norms of not necessarily cuspidal Jacobi modular forms and applications

Abstract

We extend the usual notion of Petersson inner product on the space of cuspidal Jacobi forms to include non-cuspidal forms as well. This is done by examining carefully the relation between certain "growth-killing" invariant differential operators on ${\mathbf{H}}_2$ and those on ${\mathbf{H}}_1\times \mathbf{C}$ (here ${\mathbf{H}}_n$ denotes the Siegel upper half space of degree $n$). As applications, we can understand better the growth of Petersson norms of Fourier Jacobi coefficients of Klingen Eisenstein series, which in turn has applications to finer issues about representation numbers of quadratic forms, and as a by-product we also show that \textit{any} Siegel modular form of degree $2$ is determined by its `fundamental' Fourier coefficients.

41pp

References in corpus (3)

• Distinguishing Hermitian cusp forms of degree 2 by a certain subset of all Fourier coefficients
• Lifting from pairs of two elliptic modular forms to Siegel modular forms of half-integral weight of even degree
• A decomposition of the Fourier-Jacobi coefficients of Klingen Eisenstein series

Cited by in corpus (1)

• Cones of special cycles of codimension 2 on orthogonal Shimura varieties