Geometry of the eigencurve at critical Eisenstein series of weight 2

Majumdar, Dipramit

Journal de théorie des nombres de Bordeaux, Tome 27 (2015) no. 1, pp. 183-197.

Dans cet article, nous montrons que la série d’Eisenstein critique de poids 2, $E_2^{cri{t}_p}$, définit un point lisse dans la courbe de Hecke $ℂ\left(l\right)$, où $l$ est un nombre premier différent de $p$. Nous montrons également que $E_2^{cri{t}_p,or{d}_l}$ définit un point lisse dans la courbe de Hecke pleine ${ℂ}^{full}\left(l\right)$ et que le point défini par $E_2^{cri{t}_p,or{d}_{l_1},or{d}_{l_2}}$ est non lisse dans la courbe de Hecke pleine ${ℂ}^{full}\left(l_1{l}_2\right)$. En outre, nous montrons que $ℂ\left(l\right)$ est étale sur l’espace des poids au point défini par $E_2^{cri{t}_p}$. En conséquence, nous montrons que la conjecture d’abaissement du niveau de Paulin n’est pas valide pour $E_2^{cri{t}_p,or{d}_l}$.

In this paper we show that the critical Eisenstein series of weight 2, $E_2^{cri{t}_p}$, defines a smooth point in the eigencurve $ℂ\left(l\right)$, where $l$ is a prime different from $p$. We also show that $E_2^{cri{t}_p,or{d}_l}$ defines a smooth point in the full eigencurve ${ℂ}^{full}\left(l\right)$ and $E_2^{cri{t}_p,or{d}_{l_1},or{d}_{l_2}}$ defines a non-smooth point in the full eigencurve ${ℂ}^{full}\left(l_1{l}_2\right)$. Further, we show that $ℂ\left(l\right)$ is étale over the weight space at the point defined by $E_2^{cri{t}_p}$. As a consequence, we show that level lowering conjecture of Paulin fails to hold at $E_2^{cri{t}_p,or{d}_l}$.

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