STNB2019 (33rd edition)
On the failure of Gorensteinness at weight 1 Eisenstein points of the eigencurve
Presenters
Adel Betina
Abstract
Coleman and Mazur Introduced the p-adic eigencurve, a rigid analytic space parametrizing the system of Hecke eigenvalues of p-adic families of finite slope. We know from the results of Hida and Coleman that the eigencurve is étale over the weight space at classical non critical points of cohomological weight. Moreover, Bellaiche-Dimitrov proved that the eigencurve is smooth at classical p-regular weight one forms and they gave a precise criterion for etalness over the weight space. However, the geometry of the eigencurve is still misterious at classical irregular weight one forms. I will present in this talk a joint work with Dimitrov and Pozzi in which we describe the geometry of the eigencurve at irregular weight one Eisenstein series. Such forms are not cuspidal in a classical sense, but they become cuspidal when viewed as p-adic modular forms. Thus, they give rise to points that belong to the intersection of the Eisenstein locus and the cuspidal locus of the eigencurve. We proved that the cuspidal p-adic eigencurve is etale over the weight space at any irregular classical weight 1 Eisenstein point, and that cuspidal locus meets transversely each of the two Eisenstein components of the eigencurve passing through that point. Moreover, the congruences between cuspidal and Eisenstein families yield a new proof of the Ferrero-Greenberg and Gross-Koblitz theorem on the order of vanishing of the Kubota-Leopoldt p-adic L-function at the trivial zero s = 0; we also obtain the formula for its leading term proved by Gross via a new method. Finally, we prove that the local ring of C at f is Cohen-Macaulay but not Gorenstein and compute the q-expansions of a basis of overconvergent weight 1 modular forms lying in the same generalised eigenspace as f.
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