In this paper we show that the critical Eisenstein series of weight 2, , defines a smooth point in the eigencurve , where is a prime different from . We also show that defines a smooth point in the full eigencurve and defines a non-smooth point in the full eigencurve . Further, we show that is étale over the weight space at the point defined by . As a consequence, we show that level lowering conjecture of Paulin fails to hold at .
Dans cet article, nous montrons que la série d’Eisenstein critique de poids 2, , définit un point lisse dans la courbe de Hecke , où est un nombre premier différent de . Nous montrons également que définit un point lisse dans la courbe de Hecke pleine et que le point défini par est non lisse dans la courbe de Hecke pleine . En outre, nous montrons que est étale sur l’espace des poids au point défini par . En conséquence, nous montrons que la conjecture d’abaissement du niveau de Paulin n’est pas valide pour .
@article{JTNB_2015__27_1_183_0, author = {Dipramit Majumdar}, title = {Geometry of the eigencurve at critical {Eisenstein} series of weight 2}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {183--197}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {27}, number = {1}, year = {2015}, doi = {10.5802/jtnb.898}, zbl = {06554402}, mrnumber = {3346969}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.898/} }
TY - JOUR AU - Dipramit Majumdar TI - Geometry of the eigencurve at critical Eisenstein series of weight 2 JO - Journal de théorie des nombres de Bordeaux PY - 2015 SP - 183 EP - 197 VL - 27 IS - 1 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.898/ DO - 10.5802/jtnb.898 LA - en ID - JTNB_2015__27_1_183_0 ER -
%0 Journal Article %A Dipramit Majumdar %T Geometry of the eigencurve at critical Eisenstein series of weight 2 %J Journal de théorie des nombres de Bordeaux %D 2015 %P 183-197 %V 27 %N 1 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.898/ %R 10.5802/jtnb.898 %G en %F JTNB_2015__27_1_183_0
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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