2023/09/29 by Michael A. Daas, Daas, Michael A.
Mathematics · #11F33 #11F80 (Primary) 11G10 #11G15 (Secondary) #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2309.17251
openalex publication_date 2023/09/29 · openalex created_date 2023/10/03 · openalex updated_date 2026/07/28
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 Θ-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.