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.

The primary goal of this paper is to study the geometry of the p-adic eigencurve at a point f corresponding to a weight one theta series θ ψ irregular at p. We show that f belongs to exactly three or four irreducible components and study their mutual congruences.In particular, we show that the congruence ideal of one of the CM components has a simple zero at f if, and only if, a certain L -invariant L-(ψ-) does not vanish. Further, using Roy's Strong Six Exponential Theorem we show that at least one amongst L-(ψ-) and L-(ψ −1 -) is non-zero. Combined with a divisibility proved by Hida and Tilouine, we deduce that the anti-cyclotomic Katz p-adic L-function of ψ-has a simple (trivial) zero at s = 0, if L-(ψ-) is non-zero, which can be seen as an anti-cyclotomic analogue of a result of Ferrero and Greenberg. Finally, we propose a formula for the linear term of the two-variable Katz p-adic L-function of ψ-at s = 0 thus extending a conjecture of Gross.

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Abstract. We prove a variant of Oesterlé's conjecture describing p-adic periods of the modular curve X 0 (p), with an additional Γ(2)-structure (and also Γ(3) ∩ Γ 0 (p) if p ≡ 1 (mod 3)). We use de Shalit's techniques and p-adic uniformization of curves with semi-stable reduction.