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 in the style of Darmon, Pozzi and Vonk. 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.
