# An explicit Waldspurger formula for Hilbert modular forms II

## TL;DR

This paper develops an explicit Waldspurger formula for Hilbert modular forms, linking Fourier coefficients of certain preimages to central L-values, enabling computations even when these values vanish.

## Contribution

It extends previous work by providing a construction of preimages via generalized theta series and an explicit formula for central L-values in the Hilbert modular setting.

## Key findings

- Constructed preimages for the Shimura map using generalized theta series.
- Derived an explicit Waldspurger formula relating Fourier coefficients to L-values.
- Enabled computation of central L-values when the main value vanishes.

## Abstract

We describe a construction of preimages for the Shimura map on Hilbert modular forms using generalized theta series, and give an explicit Waldspurger type formula relating their Fourier coefficients to central values of twisted $L$-functions. This formula extends our previous work, allowing to compute these central values when the main central value vanishes.