A. We compute the diagonal restriction of the rst derivative with respect to the weight of a p-adic family of Hilbert modular Eisenstein series a ached to a general (odd) character of the narrow class group of a real quadratic eld, and express the Fourier coe cients of its ordinary projection in terms of the values of a distinguished rigid analytic cocycle in the sense of [DV1] at appropriate real quadratic points of Drinfeld's p-adic upper half-plane. is can be viewed as the p-adic counterpart of a seminal calculation of Gross and Zagier [GZ, §7] which arose in their "analytic proof" of the factorisation of di erences 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 eld. Our main identity enriches the dictionary between the classical theory of complex multiplication and its extension to real quadratic elds 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 rst derivatives of certain twisted Rankin triple product p-adic Lfunctions.

We prove that the cuspidal eigencurve Ccusp is etale over the weight space at any classical weight 1 Eisenstein point f and meets transversally each of the two Eisenstein components of the eigencurve C containing f . Further, we prove that the local ring of C at f is Cohen-Macaulay but not Gorenstein and compute the Fourier coefficients of a basis of overconvergent weight 1 modular forms lying in the same generalised eigenspace as f . In addition, we prove an R = T theorem for the local ring of the closed subspace of C given by the union of Ccusp and one Eisenstein component and prove unconditionally, via a geometric construction of the residue map, that the corresponding congruence ideal is generated by the Kubota-Leopoldt p-adic L-function. Finally we obtain a new proof of the Ferrero-Greenberg theorem and Gross' formula for its derivative at the trivial zero.

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A . The values of the so-called Dedekind-Rademacher cocycle at certain real quadratic arguments are shown to be global p-units in the narrow Hilbert class field of the associated real quadratic field, as predicted by the conjectures of [DD06] and [DV21]. The strategy for proving this result combines the approach of [DPV21] with one crucial extra ingredient: the study of infinitesimal deformations of irregular Hilbert Eistenstein series of weight one in the anti-parallel direction, building on the techniques of [BDP].

The values of the Dedekind–Rademacher cocycle at certain real quadratic arguments are shown to be global p-units in the narrow Hilbert class field of the associated real quadratic field, as predicted by the conjectures of Darmon–Dasgupta (2006) and Darmon–Vonk (2021). The strategy for proving this result combines the approach of prior work of the authors (2021) with one crucial extra ingredient: the study of infinitesimal deformations of irregular Hilbert Eisenstein series of weight 1 in the anti-parallel direction.