Can one determine whether a given Eisenstein series ( for GL_{2}(Q)) is overconvergent, just by looking at the associated Galois representation?
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2$\begingroup$ Can you say a bit more about what you mean by "Eisenstein series" in this context? Something you get by $p$-adically interpolating classical Eisenstein series as the weight varies? Something more highfalutin? $\endgroup$– RamseyCommented Aug 22, 2011 at 15:53
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$\begingroup$ Sorry, about being imprecise. I meant the one obtained by p-adic interpolation. Basically, given a p-adic modular forms, how can we characterise whether it is overconvergent by looking at the Galois representation? Recent theorems of Emerton, Kisin answer the question in certain situations. $\endgroup$– jklCommented Aug 23, 2011 at 6:57
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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.
answered Aug 24, 2011 at 17:37