Formulas for central critical values of twisted L-functions attached to paramodular forms

TL;DR

This paper explores formulas for central critical values of twisted L-functions associated with paramodular forms, extending B"ocherer's conjecture and providing proofs for lifts and numerical evidence for nonlifts.

Contribution

It generalizes B"ocherer's conjecture to paramodular forms of prime level, even weight, and real character twists, with more explicit formulations and proofs for lifts.

Findings

Proved the conjecture for lifts of paramodular forms.

Provided numerical evidence supporting the conjecture for nonlifts.

Extended the conjecture to include twists by real characters.

Abstract

In the 1980s B\"ocherer formulated a conjecture relating the central values of the imaginary quadratic twists of the spin L-function attached to a Siegel modular form $F$ to the Fourier coefficients of $F$. This conjecture has been proved when $F$ is a lift. More recently, we formulated an analogous conjecture for paramodular forms $F$ of prime level, even weight and in the plus-space. In this paper, we examine generalizations of this conjecture. In particular, our new formulations relax the conditions on $F$ and allow for twists by real characters. Moreover, these formulation are more explicit than the earlier version. We prove the conjecture in the case of lifts and provide numerical evidence in the case of nonlifts.