Duke Mathematical Journal

A p-adic Waldspurger formula

Yifeng Liu, Shouwu Zhang, and Wei Zhang

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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
Received: 19 October 2014
Revised: 31 August 2017
First available in Project Euclid: 9 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1518166812

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

Mathematical Reviews number (MathSciNet)
MR3769677

Zentralblatt MATH identifier
06857029

Subjects
Primary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10]
Secondary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35] 11J95: Results involving abelian varieties

Keywords
p-adic L-function p-adic Waldspurger formula

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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