CM-values of p-adic Theta-functions

Daas, M.A. (2024)

Doctoral Thesis

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.

• CM-theory
• Shimura curves
• Quaternion algebras
• Galois representations
• Deformation theory
• p-adic modular forms
• Hilbert modular forms
• Eisenstein series

• All authors: Daas, M.A.
• Supervisor: Vonk, J.B.; Stevenhagen, P.
• Committee: Derks, G.L.A.; Holmes, D.S.T.; Charollois, P.; Darmon, H.; Pozzi, A.
• Qualification: Doctor (dr.)
• Awarding Institution: Mathematical Institute (MI), Faculty of Science, Leiden University
• Date: 2024-10-30