A p-adic approach to singular moduli on Shimura curves

Abstract

We define a rational invariant ${\mathcal{𝒥}}_N(D_1,D_2)$ 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.