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

Abstract

Let $F$ be a number field and $D$ a quaternion algebra over $F$. Take a cuspidal automorphic representation $\ddot{I}€$ of $D_{\mathbb{A}}^{\times }$ with trivial central character and a cusp form $\ddot{I}•$ in $\ddot{I}€$. Using the prehomogeneous zeta function, we find an explicit mean value of the toric periods of $\ddot{I}•$ 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.

Appendix by authors and Shun'ichi Yokoyama