On the eigencurve at classical weight 1 points. (English) Zbl 1404.11047

Bellaïche, Joël; Dimitrov, Mladen

Duke Math. J. 165, No. 2, 245-266 (2016).

Summary: We show that the \(p\)-adic eigencurve is smooth at classical weight \(1\) points which are regular at \(p\) and give a precise criterion for étaleness over the weight space at those points. Our approach uses deformations of Galois representations.

MSC:

11F33	Congruences for modular and \(p\)-adic modular forms
11F11	Holomorphic modular forms of integral weight
11F80	Galois representations

Keywords:

weight one modular forms; weight \(1\) modular forms; deformations of Galois representations; eigencurve

References:

[1]	J. Bellaïche, Critical \(p\)-adic \(L\)-functions , Invent. Math. 189 (2012), 1-60. · Zbl 1318.11067 · doi:10.1007/s00222-011-0358-z
[2]	J. Bellaïche, Eigenvarieties and p-adic L-functions , book in preparation. · Zbl 0942.65053
[3]	J. Bellaïche and G. Chenevier, Families of Galois representations and Selmer groups , Astérisque 324 , Soc. Math. France, Paris, 2009. · Zbl 1192.11035
[4]	A. Betina, Ramification of the eigencurve at classical RM points , in preparation. · Zbl 0679.32009
[5]	A. Brumer, On the units of algebraic number fields , Mathematika 14 (1967), 121-124. · Zbl 0171.01105 · doi:10.1112/S0025579300003703
[6]	K. Buzzard, “Eigenvarieties” in \(L\)-functions and Galois Representations (Durham, England, 2004) , London Math. Soc. Lecture Note Ser. 320 , Cambridge Univ. Press, Cambridge, 2007, 59-120. · Zbl 1230.11054 · doi:10.1017/CBO9780511721267.004
[7]	G. Chenevier, Familles \(p\)-adiques de formes automorphes pour \(\mathrm{GL}_{n}\) , J. Reine Angew. Math. 570 (2004), 143-217.
[8]	G. Chenevier, Une correspondance de Jacquet-Langlands \(p\)-adique , Duke Math. J. 126 (2005), 161-194. · Zbl 1070.11016 · doi:10.1215/S0012-7094-04-12615-6
[9]	S. Cho and V. Vatsal, Deformations of induced Galois representations , J. Reine Angew. Math. 556 (2003), 79-98. · Zbl 1041.11039 · doi:10.1515/crll.2003.025
[10]	R. Coleman, Classical and overconvergent modular forms , Invent. Math. 124 (1996), 215-241. · Zbl 0851.11030 · doi:10.1007/s002220050051
[11]	R. Coleman and B. Mazur, “The eigencurve” in Galois Representations in Arithmetic Algebraic Geometry (Durham, England, 1996) , London Math. Soc. Lecture Note Ser. 254 , Cambridge Univ. Press, Cambridge, 1998, 1-113. · Zbl 0932.11030 · doi:10.1017/CBO9780511662010.003
[12]	P. Deligne and J.-P. Serre, Formes modulaires de poids \(1\) , Ann. Sci. Éc. Norm. Supér. (4) 7 (1974), 507-530. · Zbl 0321.10026 · doi:10.24033/asens.1277
[13]	M. Dimitrov, “On the local structure of ordinary Hecke algebras at classical weight one points” in Automorphic Forms and Galois Representations, vol. 2 (Durham, England, 2011) , London Math. Soc. Lecture Note Ser. 415 , Cambridge Univ. Press, Cambridge, 2014, 1-16.
[14]	M. Dimitrov and E. Ghate, On classical weight one forms in Hida families , J. Théor. Nombres Bordeaux 24 (2012), 639-660. · Zbl 1271.11060 · doi:10.5802/jtnb.816
[15]	M. Emsalem, H. Kisilevsky, and D. Wales, Indépendance linéaire sur \(\bar{\mathbb {Q}}\) de logarithmes \(p\)-adiques de nombres algébriques et rang \(p\)-adique du groupe des unités d’un corps de nombres , J. Number Theory 19 (1984), 384-391. · Zbl 0547.12003 · doi:10.1016/0022-314X(84)90079-9
[16]	H. Hida, Galois representations into \(\mathrm{GL}_{2}(\mathbf{Z}_{p}[[X]])\) attached to ordinary cusp forms , Invent. Math. 85 (1986), 545-613. · Zbl 0612.10021 · doi:10.1007/BF01390329
[17]	K. Kitagawa, “On standard \(p\)-adic \(L\)-functions of families of elliptic cusp forms” in \(p\)-adic Monodromy and the Birch and Swinnerton-Dyer Conjecture (Boston, 1991) , Contemp. Math. 165 , Amer. Math. Soc., Providence, 1994, 81-110. · Zbl 0841.11028 · doi:10.1090/conm/165/01611
[18]	B. Mazur, J. Tate, and J. Teitelbaum, On \(p\)-adic analogues of the conjectures of Birch and Swinnerton-Dyer , Invent. Math. 84 (1986), 1-48. · Zbl 0699.14028 · doi:10.1007/BF01388731
[19]	L. Nyssen, Pseudo-représentations , Math. Ann. 306 (1996), 257-283. · Zbl 0863.16012 · doi:10.1007/BF01445251
[20]	R. Rouquier, Caractérisation des caractères et pseudo-caractères , J. Algebra 180 (1996), 571-586. · Zbl 0857.16013 · doi:10.1006/jabr.1996.0083
[21]	A. Wiles, On ordinary \(\lambda\)-adic representations associated to modular forms , Invent. Math. 94 (1988), 529-573. · Zbl 0664.10013 · doi:10.1007/BF01394275