Can one determine whether a given Eisenstein series ( for GL\_{2}(Q)) is overconvergent, just by looking at the associated Galois representation?



Eisenstein series and overconvergence
In the paper "Lissite de la Courbe de Hecke aux points Eisenstein critiques", Bellaiche and Chenevier completely classify all reducible Galois representations coming from overconvergent eigenforms of tame level 1. They are precisely those coming from either the ordinary family of Eisenstein series, or one of the "critical" Eisenstein series


$E^{crit}\_w(q) = q + \sum a\_nq^n, \quad a\_p=p^{k-1}, \quad a\_\ell=\epsilon(\ell) + \ell^{k-1}$.


Here $w:\mathbb{Z}\_p^\*\rightarrow \overline{\mathbb{Q}\_p^\*}$ is the character $x\mapsto \epsilon(x)x^k$ where $k\geq 2$, $\epsilon$ is finite order, and $(\epsilon,k)\neq(\mathbf{1},2)$.


The classification is Proposition 4 of Section 4.


