Maass relations for Saito-Kurokawa lifts of higher levels

Abstract

It is known that among Siegel modular forms of degree $2$ and level $1$ the only functions that violate the Ramanujan conjecture are Saito-Kurokawa lifts of modular forms of level $1$. These are precisely the functions whose Fourier coefficients satisfy Maass relations. More generally, the Ramanujan conjecture for $\mathrm{GSp}_4$ is predicted to fail only in case of CAP representations. It is not known though whether the associated Siegel modular forms (of various levels) still satisfy a version of Maass relations. We show that this is indeed the case for the ones related to P-CAP representations. Our method generalizes an approach of Pitale, Saha and Schmidt who employed representation-theoretic techniques to (re)prove this statement in case of level $1$. In particular, we compute and express certain values of a global Bessel period in terms of Fourier coefficients of the associated Siegel modular form. Moreover, we derive a local-global relation satisfied by Bessel periods, which allows us to combine those computations with a characterization of local components of CAP representations.