#### 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 ${\mathsc{𝒥}}_{N}\left({D}_{1},{D}_{2}\right)$ 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.

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

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