  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.
