A p-adic Waldspurger formula

Abstract

In this article, we study $p$-adic torus periods for certain $p$-adic-valued functions on Shimura curves of classical origin. We prove a $p$-adic Waldspurger formula for these periods as a generalization of recent work of Bertolini, Darmon, and Prasanna. In pursuing such a formula, we construct a new anti-cyclotomic $p$-adic $L$-function of Rankin–Selberg type. At a character of positive weight, the $p$-adic $L$-function interpolates the central critical value of the complex Rankin–Selberg $L$-function. Its value at a finite-order character, which is outside the range of interpolation, essentially computes the corresponding $p$-adic torus period.