Ramification of the eigencurve at classical RM points

TL;DR

This paper investigates the local structure of the eigencurve at classical RM points, establishing isomorphisms with universal deformation rings and analyzing ramification, with implications for Hecke algebras of Hilbert modular forms.

Contribution

It proves an isomorphism between local eigencurve rings and universal deformation rings at specific points, and provides criteria for ramification index, also establishing smoothness of related Hecke algebras.

Findings

Isomorphism between eigencurve local rings and deformation rings at RM points.

Criterion for ramification index being exactly 2.

Smoothness of Hecke algebras at certain Eisenstein points.

Abstract

J.Bella\"iche and M.Dimitrov have shown that the $p$-adic eigencurve is smooth but not etale over the weight space at $p$-regular theta series attached to a character of a real quadratic field $F$ in which $p$ splits. We proof in this paper the existence of an isomorphism between the subring of the completed local ring of the eigencurve at these points fixed by the Atkin-Lehner involution and an universal ring representing a pseudo-deformation problem, and one gives also a precise criterion for which the ramification index is exactly $2$. We finish this paper by proving the smoothness of the nearly ordinary and ordinary Hecke algebras for Hilbert modular forms over $F$ at the cuspidal-overconvergent Eisenstein points which are the base change lift for $\mathrm{GL}(2)_{/F}$ of these theta series.