On classical weight one forms in Hida families

Mladen Dimitrov[1]; Eknath Ghate[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

Journal de Théorie des Nombres de Bordeaux (2012)

  • Volume: 24, Issue: 3, page 669-690
  • ISSN: 1246-7405

Abstract

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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.

How to cite

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Dimitrov, Mladen, and Ghate, Eknath. "On classical weight one forms in Hida families." Journal de Théorie des Nombres de Bordeaux 24.3 (2012): 669-690. <http://eudml.org/doc/251134>.

@article{Dimitrov2012,
abstract = {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.},
affiliation = {Laboratoire Paul Painlevé Université Lille 1, U.F.R. de Mathématiques 59655 Villeneuve d’Ascq cedex, France; School of Mathematics Tata Institute of Fundamental Research Homi Bhabha Road, Mumbai 400005, India},
author = {Dimitrov, Mladen, Ghate, Eknath},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Hida family; -adic cusp form; Hecke algebra; Galois representation},
language = {eng},
month = {11},
number = {3},
pages = {669-690},
publisher = {Société Arithmétique de Bordeaux},
title = {On classical weight one forms in Hida families},
url = {http://eudml.org/doc/251134},
volume = {24},
year = {2012},
}

TY - JOUR
AU - Dimitrov, Mladen
AU - Ghate, Eknath
TI - On classical weight one forms in Hida families
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2012/11//
PB - Société Arithmétique de Bordeaux
VL - 24
IS - 3
SP - 669
EP - 690
AB - 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.
LA - eng
KW - Hida family; -adic cusp form; Hecke algebra; Galois representation
UR - http://eudml.org/doc/251134
ER -

References

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  1. J. Bellaïche and M. Dimitrov, On the Eigencurve at classical weight one points. Preprint (2012). Zbl06556667
  2. K. Buzzard and R. Taylor, Companion forms and weight 1 forms. Ann. of Math., 149 (1999), 905–919. Zbl0965.11019MR1709306
  3. K. Buzzard, Analytic continuation of overconvergent eigenforms. J. Amer. Math. Soc., 16 (2003), 29–55. Zbl1076.11029MR1937198
  4. S. Cho and V. Vatsal, Deformations of induced Galois representations. J. Reine Angew. Math., 556 (2003), 79–98. Zbl1041.11039MR1971139
  5. M. Emerton, R. Pollack and T. Weston, Variation of Iwasawa invariants in Hida families. Invent. Math., 163 (2006), 523–580. Zbl1093.11065MR2207234
  6. A. Fischman, On the image of Λ -adic Galois representations. Ann. Inst. Fourier, Grenoble, 52 (2002), no. 2, 351–378. Zbl1020.11037MR1906479
  7. E. Ghate and N. Kumar, Control theorems for ordinary 2 -adic families of modular forms. In preparation. Zbl1307.11068
  8. E. Ghate and V. Vatsal, On the local behaviour of ordinary Λ -adic representations. Ann. Inst. Fourier, Grenoble, 54 (2004), no. 7, 2143–2162. Zbl1131.11341MR2139691
  9. H. Hida, Galois representations into GL 2 ( Z p [ [ X ] ] ) attached to ordinary cusp forms. Invent. Math., 85 (1986), 545–613. Zbl0612.10021MR848685
  10. Iwasawa modules attached to congruences of cusp forms. Ann. Sci. Ecole Norm. Sup. (4), 19 (1986), 231–273. Zbl0607.10022MR868300
  11. J-P. Serre, Modular forms of weight one and Galois representations. Proc. Sympos. Univ. Durham, Durham (1975), Academic Press, London, 1977, 193–268. Zbl0366.10022MR450201
  12. A. Wiles, On ordinary λ -adic representations associated to modular forms. Invent. Math., 94 (1988), 529–573. Zbl0664.10013MR969243

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