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

Michael A. Daas

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arXiv · 2309.17251

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

Michael A. Daas

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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.

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Michael A. Daas. 2023-09-29. CM-values of $p$-adic $\Theta$-functions. https://arxiv.org/abs/2309.17251

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