# Eisenstein series and overconvergence

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

• 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? – Ramsey Aug 22 '11 at 15:53
• 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. – jkl Aug 23 '11 at 6:57

$E^{crit}_w(q) = q + \sum a_nq^n, \quad a_p=p^{k-1}, \quad a_\ell=\epsilon(\ell) + \ell^{k-1}$.
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 $(\epsilon,k)\neq(\mathbf{1},2)$.