In this article, we study -adic torus periods for certain -adic-valued functions on Shimura curves of classical origin. We prove a -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 -adic -function of Rankin–Selberg type. At a character of positive weight, the -adic -function interpolates the central critical value of the complex Rankin–Selberg -function. Its value at a finite-order character, which is outside the range of interpolation, essentially computes the corresponding -adic torus period.
"A -adic Waldspurger formula." Duke Math. J. 167 (4) 743 - 833, 15 March 2018. https://doi.org/10.1215/00127094-2017-0045