Geometry of the eigencurve at critical Eisenstein series of weight 2
Journal de théorie des nombres de Bordeaux, Volume 27 (2015) no. 1, pp. 183-197.

In this paper we show that the critical Eisenstein series of weight 2, E 2 crit p , defines a smooth point in the eigencurve (l), where l is a prime different from p. We also show that E 2 crit p ,ord l defines a smooth point in the full eigencurve full (l) and E 2 crit p ,ord l 1 ,ord l 2 defines a non-smooth point in the full eigencurve full (l 1 l 2 ). Further, we show that (l) is étale over the weight space at the point defined by E 2 crit p . As a consequence, we show that level lowering conjecture of Paulin fails to hold at E 2 crit p ,ord l .

Dans cet article, nous montrons que la série d’Eisenstein critique de poids 2, E 2 crit p , définit un point lisse dans la courbe de Hecke (l), où l est un nombre premier différent de p. Nous montrons également que E 2 crit p ,ord l définit un point lisse dans la courbe de Hecke pleine full (l) et que le point défini par E 2 crit p ,ord l 1 ,ord l 2 est non lisse dans la courbe de Hecke pleine full (l 1 l 2 ). En outre, nous montrons que (l) est étale sur l’espace des poids au point défini par E 2 crit p . En conséquence, nous montrons que la conjecture d’abaissement du niveau de Paulin n’est pas valide pour E 2 crit p ,ord l .

DOI: 10.5802/jtnb.898
Classification: 11F85,  11F11,  11F80
Dipramit Majumdar 1

1 Indian Statistical Institute, Bangalore Centre 8th Mile, Mysore Road, RVCE Post, Bangalore, India 560059
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Dipramit Majumdar. Geometry of the eigencurve at critical Eisenstein series of weight 2. Journal de théorie des nombres de Bordeaux, Volume 27 (2015) no. 1, pp. 183-197. doi : 10.5802/jtnb.898. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.898/

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