An explicit trace formula of Jacquet-Zagier type for Hilbert modular forms

TL;DR

This paper derives an explicit trace formula for Hilbert modular forms, generalizing Zagier's formula, and uses it to demonstrate the equidistribution of Satake parameters for forms with fixed weight and increasing levels.

Contribution

It provides a new explicit trace formula for Hilbert modular forms and applies it to establish Satake parameter equidistribution as levels grow.

Findings

Derived an exact average formula for adjoint L-functions of Hilbert cusp forms.

Proved equidistribution of Satake parameters for forms with fixed weight and increasing levels.

Generalized Zagier's formula from elliptic to Hilbert modular forms.

Abstract

We give an exact formula of the average of adjoint $L$-functions of holomorphic Hilbert cusp forms with a fixed weight and a square-free level, which is a generalization of Zagier's formula known for the case of elliptic cusp forms on ${\rm SL}_2(\mathbb{Z})$. As an application, we prove that the Satake parameters of Hilbert cusp forms with a fixed weight and with growing square-free levels are equidistributed in an ensemble constructed by values of the adjoint $L$-functions.