Determining Hilbert Modular Forms by Central Values of Rankin-Selberg Convolutions: The Weight Aspect

TL;DR

This paper proves that primitive Hilbert cusp forms are uniquely identified by the central values of their Rankin-Selberg convolutions with all forms of varying weights over totally real fields, extending previous results.

Contribution

It generalizes prior work to totally real fields and establishes weight aspect uniqueness of Hilbert cusp forms via central L-values.

Findings

Primitive Hilbert cusp forms are determined by central L-values.

Extension of previous results to totally real number fields.

Weight aspect analogue of recent work by the authors.

Abstract

The purpose of this paper is to prove that a primitive Hilbert cusp form $\mathbf{g}$ is uniquely determined by the central values of the Rankin-Selberg $L$-functions $L(\mathbf{f}\otimes\mathbf{g}, \frac{1}{2})$, where $\mathbf{f}$ runs through all primitive Hilbert cusp forms of weight $k$ for infinitely many weight vectors $k$. This work is a generalization of a result of Ganguly, Hoffstein, and Sengupta to the setting of totally real number fields, and it is a weight aspect analogue of the authors recent work.