Lissite de la courbe de Hecke de GL(2) aux points Eisenstein critiques

TL;DR

This paper investigates the geometric properties of the Coleman-Mazur eigencurve at Eisenstein points, establishing smoothness and etaleness conditions linked to p-adic zeta values and the reducibility of associated pseudo-characters.

Contribution

It proves smoothness of the eigencurve at Eisenstein points and characterizes etaleness conditions in terms of p-adic zeta values, advancing understanding of eigencurve geometry.

Findings

C is smooth at evil Eisenstein points

Conditions for etaleness relate to p-adic zeta values

Dirichlet L-functions have simple zeros at integers

Abstract

Let p be a prime number and C be the p-adic tame level 1 eigencurve introduced by Coleman-Mazur. We prove that C is smooth at the evil Eisenstein points and we give necessary and sufficient conditions for etaleness of the map to the weight space at these points in terms of p-adic zeta values. A key step is the determination at these points of the schematic reducibility locus of the pseudo-character carried by C restricted to a decomposition group at p. Then, the smoothness appears to be a consequence of the fact that the Dirichlet L-functions only have simple zeros at integers.