Vol. 15, No. 2, 2022

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A $p$-adic approach to singular moduli on Shimura curves

Sofia Giampietro and Henri Darmon

Vol. 15 (2022), No. 2, 345–365
Abstract

We define a rational invariant 𝒥N(D1,D2) associated to singular moduli of discriminants D1 and D2 on the genus-zero Shimura curves of discriminant N = 6,10 or 22. An algorithm is devised to compute this invariant p-adically using the Cerednik–Drinfeld uniformization of Shimura curves, following the approach described in the thesis of I. Negrini (2017). A formula for the factorization of this invariant is proposed, similar to the formula of Gross and Zagier for differences of classical singular moduli.

Keywords
singular moduli on Shimura curves, $p$-adic uniformization
Mathematical Subject Classification
Primary: 14G35
Secondary: 11G15, 11G18
Supplementary material

Algorithm code

Tables

Milestones
Received: 16 August 2021
Revised: 1 November 2021
Accepted: 4 November 2021
Published: 29 July 2022

Communicated by Amanda Folsom
Authors
Sofia Giampietro
Institute of Mathematics
École Polytechnique Fédérale de Lausanne
Lausanne
Switzerland
Henri Darmon
Department of Mathematics and Statistics
McGill University
Montreal, QC
Canada