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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
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Abstract |
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We define a rational invariant associated to singular moduli of discriminants and on the genus-zero Shimura curves of discriminant or . An algorithm is devised to compute this invariant -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
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Mathematical Subject Classification
Primary: 14G35
Secondary: 11G15, 11G18
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Supplementary material |
Milestones
Received: 16 August 2021
Revised: 1 November 2021
Accepted: 4 November 2021
Published: 29 July 2022
Communicated by Amanda Folsom
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Authors |
| Sofia Giampietro | |
| Institute of Mathematics École Polytechnique Fédérale de Lausanne Lausanne Switzerland |
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| Henri Darmon | |
| Department of Mathematics and
Statistics McGill University Montreal, QC Canada |
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