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

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.

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

Algorithm code


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

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