On the Eigencurve at classical weight one points

TL;DR

This paper proves the smoothness of the p-adic Eigencurve at certain classical weight one points and provides criteria for its etaleness over the weight space, using Galois representation deformations.

Contribution

It establishes smoothness and etaleness criteria of the Eigencurve at classical weight one points, advancing understanding of its local geometry.

Findings

The Eigencurve is smooth at regular classical weight one points.

A precise criterion for etaleness over the weight space is provided.

Deformations of Galois representations are used in the analysis.

Abstract

We show that the p-adic Eigencurve is smooth at classical weight one points which are regular at p and give a precise criterion for etaleness over the weight space at those points. Our approach uses deformations of Galois representations.