An explicit Waldspurger formula for Hilbert modular forms

TL;DR

This paper constructs explicit preimages for the Shimura map on Hilbert modular forms and derives a Waldspurger-type formula linking Fourier coefficients to central L-values, applicable to various levels and fields.

Contribution

It provides a new explicit construction of preimages for the Shimura map and a Waldspurger formula for Hilbert modular forms over arbitrary fields.

Findings

Explicit preimages for the Shimura map constructed.

Waldspurger-type formula relating Fourier coefficients to L-values derived.

Applicable to nontrivial levels and arbitrary base fields under certain conditions.

Abstract

We describe a construction of preimages for the Shimura map on Hilbert modular forms, and give an explicit Waldspurger type formula relating their Fourier coefficients to central values of twisted L-functions. Our construction is inspired by that of Gross and applies to any nontrivial level and arbitrary base field, subject to certain conditions on the Atkin-Lehner eigenvalues and on the weight.