The eigencurve at crystalline points with scalar Frobenius and Gross-Stark regulators

TL;DR

This paper investigates the local geometric structure of the $p$-adic eigencurve at certain irregular weight one points, revealing new insights into the associated Galois representations and Hecke algebras.

Contribution

It provides a complete description of the eigencurve's local geometry at $p$-irregular points where traditional methods fail, and shows the non-freeness of related cohomology modules.

Findings

The eigencurve's local structure at irregular points is explicitly characterized.

The associated $p$-adic cohomology group is shown to be non-free over the Hecke algebra.

New phenomena in the Galois representations at irregular weights are identified.

Abstract

A complete description of the local geometry of the $p$-adic eigencurve at $p$-irregular classical weight one cusp forms is given in the cases where the usual $R=T$ methods fall short. As an application, we show that the ordinary $p$-adic \'etale cohomology group attached to the tower of elliptic modular curves $X_1(Np^r)$ is not free over the Hecke algebra, when localized at a $p$-irregular weight one point.