Geometry of the eigencurve at critical Eisenstein series of weight 2

TL;DR

This paper investigates the geometric properties of the eigencurve at certain Eisenstein series of weight 2, revealing smoothness, non-smoothness, and étaleness properties, and implications for the level lowering conjecture.

Contribution

It provides new insights into the local geometry of the eigencurve at critical Eisenstein series of weight 2, including smoothness and étaleness results, and shows the failure of a conjecture.

Findings

E_{2}^{crit_{p}} is smooth in the eigencurve C(l).

E_{2}^{crit_{p},ord_{l}} is smooth in the full eigencurve C^{full}(l).

E_{2}^{crit_{p},ord_{l_{1}},ord_{l_{2}}} is non-smooth in C^{full}(l_{1}l_{2}).

Abstract

In this paper we show that the critical Eisenstein series of weight 2, E_{2}^{crit_{p}}, is smooth in the eigencurve C(l), where l is a prime. We also show that E_{2}^{crit_{p},ord_{l}} is smooth in the full eigencurve C^{full}(l) and E_{2}^{crit_{p},ord_{l_{1}},ord_{l_{2}}} is non-smooth in the full eigencurve C^{full}(l_{1}l_{2}). Further, we show that, E_{2}^{crit_{p}}, is \'etale over the weight space in the eigencurve C(l). As a consequence, we show that level lowering conjecture of Paulin fails to hold at E_{2}^{crit_{p},ord_{l}}.