Explicit mean value theorems for toric periods and automorphic $L$-functions

TL;DR

This paper derives explicit formulas for the average values of toric periods and central automorphic L-values twisted by quadratic characters over number fields, using prehomogeneous zeta functions.

Contribution

It provides the first explicit mean value formulas for toric periods and automorphic L-values in the context of quaternion algebras over number fields.

Findings

Explicit mean value formula for toric periods

Mean value formula for central L-values twisted by quadratic characters

Connection established via prehomogeneous zeta functions

Abstract

Let $F$ be a number field and $D$ a quaternion algebra over $F$. Take a cuspidal automorphic representation $\pi$ of $D_{\mathbb{A}}^\times$ with trivial central character and a cusp form $\phi$ in $\pi$. Using the prehomogeneous zeta function, we find an explicit mean value of the toric periods of $\phi$ with respect to quadratic algebras over $F$. The result can also be written as a mean value formula for the central values of automorphic $L$-functions twisted by quadratic characters.