Waldspurger formula over function fields

TL;DR

This paper establishes a function field analogue of the Waldspurger formula, linking central L-values to toric periods, thereby advancing understanding of non-vanishing criteria and supporting the Gross-Prasad conjecture over function fields.

Contribution

It derives a new formula expressing central L-values as ratios of global and local toric periods in the function field setting, providing a key tool for non-vanishing results.

Findings

Provides a necessary and sufficient criterion for non-vanishing of central L-values.

Supports the Gross-Prasad conjecture for SO(3) over function fields.

Establishes a new relation between L-values and toric periods in function fields.

Abstract

In this paper, we derive a function field version of the Waldspurger formula for the central critical values of the Rankin-Selberg L-functions. This formula states that the central critical L-values in question can be expressed as the "ratio" of the global toric period integral to the product of the local toric period integrals. Consequently, this result provides a necessary and sufficient criterion for the non-vanishing of these central critical L-values, and supports the Gross-Prasad conjecture for $SO(3)$ over function fields.