## 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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