Diagonal restrictions of p-adic Eisenstein families

Abstract

We compute the diagonal restriction of the first derivative with respect to the weight of a p-adic family of Hilbert modular Eisenstein series attached to a general (odd) character of the narrow class group of a real quadratic field, and express the Fourier coefficients of its ordinary projection in terms of the values of a distinguished rigid analytic cocycle in the sense of Darmon and Vonk (Duke Math J, to appear, 2020) at appropriate real quadratic points of Drinfeld’s p-adic upper half-plane. This can be viewed as the p-adic counterpart of a seminal calculation of Gross and Zagier (J Reine Angew Math 355:191–220, 1985, §7) which arose in their “analytic proof” of the factorisation of differences of singular moduli, and whose inspiration can be traced to Siegel’s proof of the rationality of the values at negative integers of the Dedekind zeta function of a totally real field. Our main identity enriches the dictionary between the classical theory of complex multiplication and its extension to real quadratic fields based on RM values of rigid meromorphic cocycles, and leads to an expression for the p-adic logarithms of Gross–Stark units and Stark–Heegner points in terms of the first derivatives of certain twisted Rankin triple product p-adic L-functions.