CM-values of $p$-adic $\Theta$-functions

TL;DR

This paper establishes a $p$-adic analogue of Gross and Zagier's work on singular moduli, proving conjectures related to CM-values of $p$-adic $ heta$-functions through algebraic and analytic methods.

Contribution

It introduces a novel $p$-adic framework for CM-values, providing two proofs and connecting classical CM-theory with modern $p$-adic techniques.

Findings

Proved conjectures on factorization of rational invariants of CM-points.

Developed algebraic and analytic proofs involving $p$-adic deformations.

Computed Frobenius traces and established modularity via an $R = T$ theorem.

Abstract

We prove a $p$-adic version of the work by Gross and Zagier on the differences between singular moduli by proving a set of conjectures by Giampietro and Darmon, who investigated the factorisation of a rational invariant associated to a pair of CM-points on a genus zero Shimura curve, obtained as the ratio of the CM-values of $p$-adic $\Theta$-functions. As did Gross and Zagier, we give two proofs; an algebraic proof using CM-theory, and more interestingly, also an analytic proof using $p$-adic infinitesimal deformations of Hilbert Eisenstein series. Since there are no explicit formulae for its cuspidal $p$-adic deformations, we instead compute the Frobenius traces of the appropriate Galois deformation, and show their modularity via an $R = T$ theorem. This approach aims to bridge the gap between classical CM-theory and the more recent $p$-adic advances in the theory of real multiplication.