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

TL;DR

This paper proves that primitive Hilbert cusp forms can be uniquely identified by the central values of their Rankin-Selberg convolutions with other forms, extending Luo's theorem to totally real fields.

Contribution

It generalizes Luo's theorem to totally real number fields, showing forms are determined by central L-values across infinitely many levels.

Findings

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

The result extends Luo's theorem from rational to totally real fields.

Infinitely many levels are used to establish the uniqueness.

Abstract

In this paper, we 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 level $\mathfrak{q}$ for infinitely many prime ideals $\mathfrak{q}$. This result is a generalization of a theorem of Luo to the setting of totally real number fields.