## Duke Mathematical Journal

### 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.

#### Article information

Source
Duke Math. J., Volume 167, Number 4 (2018), 743-833.

Dates
Revised: 31 August 2017
First available in Project Euclid: 9 February 2018

https://projecteuclid.org/euclid.dmj/1518166812

Digital Object Identifier
doi:10.1215/00127094-2017-0045

Mathematical Reviews number (MathSciNet)
MR3769677

Zentralblatt MATH identifier
06857029

#### Citation

Liu, Yifeng; Zhang, Shouwu; Zhang, Wei. A $p$ -adic Waldspurger formula. Duke Math. J. 167 (2018), no. 4, 743--833. doi:10.1215/00127094-2017-0045. https://projecteuclid.org/euclid.dmj/1518166812

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