Soumya Das

Bounds for the Petersson norms of the pullbacks of Saito–Kurokawa lifts

Abstract Using the amplification technique, we prove that ‘mass’ of the pullback of the Saito–Kurokawa lift of a Hecke eigen form g ∈ S 2 k is bounded by k 1 − 1 210 + e . This improves the previously known bound k for this quantity.

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 H 2 and those on H 1 × C (here 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 any Siegel modular form of degree 2 is determined by its ‘fundamental’ Fourier coefficients.

Sturm-Like Bound for Square-Free Fourier Coefficients

In this short article, we show the existence of an analogue of the classical Sturm’s bound in the context of the square-free Fourier coefficients for cusp forms of square-free levels. This number is a cut-off to determine a cusp form from its initial few square-free Fourier coefficients. We also mention some questions in this regard.