Geometry of the Hilbert cuspidal eigenvariety at weight one Eisenstein points

Abstract: In this talk, we will report on a joint work with Adel Betina and Mladen Dimitrov about the geometry of the Hilbert cuspidal eigenvarity at a point $f$ coming from a weight one Eisenstein series irregular at a single prime $P$ of the totally real field $F$ above $p$.

Assuming Leopoldt's conjecture for $F$ at $p$, we show that the nearly ordinary cuspidal eigenvariety is étale at f over the weight space when $[F_P:Q_p]\geq[F:Q]−1$, and hence, the ordinary eigencurve is étale over the weight space as well. When $F_P=Q_p$ we show that the eigenvariety is smooth at $f$, while in all the remaining cases, we prove that the eigenvariety is never smooth at $f$.

If time permits, we will also discuss some applications in Iwasawa Theory and a new proof of the rank 1 Gross-Stark conjecture.

number theory

Audience: researchers in the topic

UCLA Number Theory Seminar