Geometry of the eigencurve at critical Eisenstein series of weight 2
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 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 .

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 .

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

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

[1] J. Bellaïche, Critical p-adic L-functions. Invent. Math.189, (2012), 1–60. | MR

[2] J. Bellaïche, Introduction to the Conjecture of Bloch and Kato. Available at http://people.brandeis.edu/~jbellaic/BKHawaii5.pdf.

[3] J. Bellaïche, G. Chenevier, Lissité de la Courbe de Hecke de GL_2 aux Points Eisenstein Critiques. J. Inst. Math. Jussieu.5, (2006), 333–349. | MR | Zbl

[4] J. Bellaïche, G. Chenevier, Families of Galois representations and Selmer groups. Astérisque.324, (2009), xii+314. | MR | Zbl

[5] J. Bellaïche, , G. Chenevier, Sous-groupes de GL 2 et arbres, J. Algebra.410, (2014),501–525. | MR | Zbl

[6] J.Bellaïche, M. Dimitrov, On the Eigencurve at classical weight one points. ArXiv e-prints (2013).

[7] K.Buzzard, Eigenvarieties. L-functions and Galois representations, London Math. Soc. Lecture Note Ser.320 (2007),59–120. | MR | Zbl

[8] R.Coleman, B.Mazur, The eigencurve. Galois representations in arithmetic algebraic geometry(Durham, 1996), London Math. Soc. Lecture Note Ser.254 (1998),1–113. | MR | Zbl

[9] R. Coleman, F. Q. Gouvêa, N. Jochnowitz, E 2 , Θ, and overconvergence. Int. Math. Res. Not. (1995), 23–41. | MR | Zbl

[10] M. Dimitrov, E. Ghate, On classical weight one forms in Hida families. J. Théor. Nombres Bordeaux. 24 (2012), 669–690. | Numdam | MR | Zbl

[11] H. Hida, Galois representations into GL 2 (Z p [[X]]) attached to ordinary cusp forms. Invent. Math. 85 (1986), 545–613. | MR | Zbl

[12] M. Kisin, Overconvergent modular forms and the Fontaine-Mazur conjecture. Invent. Math. 153 (2003), 373–454. | MR | Zbl

[13] N. Koblitz, 2-adic and 3-adic ordinals of (1/j)-expansion coefficients for the weight 2 Eisenstein series. Bull. London Math. Soc. 9 (1977), 188–192. | MR | Zbl

[14] T. Miyake, Modular Forms. Springer-Verlag, Berlin. (1989), x+335, translated from the Japanese by Yoshitaka Maeda. | MR | Zbl

[15] J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of number fields. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. 323 (2008), xvi+825. | MR | Zbl

[16] A. G. M. Paulin, Local to global compatibility on the eigencurve. Proc. Lond. Math. Soc. (3) 103 (2011), 405–440. | MR | Zbl

[17] A. G. M. Paulin, Geometric level raising and lowering on the eigencurve. Manuscripta Math. 137 (2012), 129–157. | MR | Zbl

[18] C. Soulé, On higher p-adic regulators. Algebraic K-theory, Evanston 1980 (Proc. Conf., Northwestern Univ., Evanston, Ill., (1980), Lecture Notes in Math. 854 (1981), 372–401. | MR | Zbl

[19] W. A. Stein, Modular Forms, A Computational Approach. Book available at http://modular.math.washington.edu/books/modform/modform/index.html. | MR

[20] R. Taylor, Galois representations associated to Siegel modular forms of low weight. Duke Math. J. 63 (1991), 281–332. | MR | Zbl

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