A p-adic approach to singular moduli on Shimura curves

Giampietro, Sofia; Darmon, Henri

doi:10.2140/involve.2022.15.345

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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

DOI: 10.2140/involve.2022.15.345

Abstract

We define a rational invariant $𝒥_{N} (D_{1}, D_{2})$ associated to singular moduli of discriminants $D_{1}$ and $D_{2}$ 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.

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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

Vol. 15, No. 2, 2022