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 𝜋 of 𝐷_𝔸^× with trivial central character and a cusp form 𝜙 in 𝜋. Using the prehomogeneous zeta function, we find an explicit mean value of the toric periods of 𝜙 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.