On classical weight one forms in Hida families
Journal de théorie des nombres de Bordeaux, Tome 24 (2012) no. 3, pp. 669-690.

Nous effectuons une estimation précise du nombre de spécialisations classiques en poids un d’une famille non-CM de formes modulaires propres ordinaires cuspidales. Nous donnons aussi des exemples où plusieurs familles se spécialisent sur la même forme de poids un.

We give precise estimates for the number of classical weight one specializations of a non-CM family of ordinary cuspidal eigenforms. We also provide examples to show how uniqueness fails with respect to membership of weight one forms in families.

DOI : 10.5802/jtnb.816
Classification : 11F80, 11F33, 11R23
Dimitrov, Mladen 1 ; Ghate, Eknath 2

1 Laboratoire Paul Painlevé Université Lille 1, U.F.R. de Mathématiques 59655 Villeneuve d’Ascq cedex, France
2 School of Mathematics Tata Institute of Fundamental Research Homi Bhabha Road, Mumbai 400005, India
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Dimitrov, Mladen; Ghate, Eknath. On classical weight one forms in Hida families. Journal de théorie des nombres de Bordeaux, Tome 24 (2012) no. 3, pp. 669-690. doi : 10.5802/jtnb.816. http://archive.numdam.org/articles/10.5802/jtnb.816/

[BD12] J. Bellaïche and M. Dimitrov, On the Eigencurve at classical weight one points. Preprint (2012).

[BT99] K. Buzzard and R. Taylor, Companion forms and weight 1 forms. Ann. of Math., 149 (1999), 905–919. | MR | Zbl

[B03] K. Buzzard, Analytic continuation of overconvergent eigenforms. J. Amer. Math. Soc., 16 (2003), 29–55. | MR | Zbl

[CV03] S. Cho and V. Vatsal, Deformations of induced Galois representations. J. Reine Angew. Math., 556 (2003), 79–98. | MR | Zbl

[EPW06] M. Emerton, R. Pollack and T. Weston, Variation of Iwasawa invariants in Hida families. Invent. Math., 163 (2006), 523–580. | MR | Zbl

[F02] A. Fischman, On the image of Λ-adic Galois representations. Ann. Inst. Fourier, Grenoble, 52 (2002), no. 2, 351–378. | Numdam | MR | Zbl

[GK12] E. Ghate and N. Kumar, Control theorems for ordinary 2-adic families of modular forms. In preparation.

[GV04] E. Ghate and V. Vatsal, On the local behaviour of ordinary Λ-adic representations. Ann. Inst. Fourier, Grenoble, 54 (2004), no. 7, 2143–2162. | Numdam | MR | Zbl

[H85] H. Hida, Galois representations into GL 2 (Z p [[X]]) attached to ordinary cusp forms. Invent. Math., 85 (1986), 545–613. | MR | Zbl

[H86] Iwasawa modules attached to congruences of cusp forms. Ann. Sci. Ecole Norm. Sup. (4), 19 (1986), 231–273. | MR

[S77] J-P. Serre, Modular forms of weight one and Galois representations. Proc. Sympos. Univ. Durham, Durham (1975), Academic Press, London, 1977, 193–268. | MR | Zbl

[W88] A. Wiles, On ordinary λ-adic representations associated to modular forms. Invent. Math., 94 (1988), 529–573. | MR | Zbl

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