A B\"ocherer-Type Conjecture for Paramodular Forms

TL;DR

This paper extends B"ocherer's conjecture to paramodular forms, relating central L-values of quadratic twists to Fourier coefficients, proving it for Gritsenko lifts and providing numerical evidence otherwise.

Contribution

It formulates a new conjecture for paramodular forms, proves it for Gritsenko lifts, and offers numerical evidence for non-lift cases.

Findings

Proved the conjecture for Gritsenko lifts.

Numerical evidence supports the conjecture for non-lift forms.

Extended B"ocherer's conjecture to a broader class of forms.

Abstract

In the 1980s B\"ocherer formulated a conjecture relating the central value of the quadratic twists of the spinor L-function attached to a Siegel modular form F to the coefficients of F . He proved the conjecture when F is a Saito-Kurokawa lift. Later Kohnen and Kuss gave numerical evidence for the conjecture in the case when F is a rational eigenform that is not a Saito-Kurokawa lift. In this paper we develop a conjecture relating the central value of the quadratic twists of the spinor L-function attached to a paramodular form and the coefficients of the form. We prove the conjecture in the case when the form is a Gritsenko lift and provide numerical evidence when it is not a lift.