Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

Hecke fields of analytic families of modular forms


Author: Haruzo Hida
Journal: J. Amer. Math. Soc. 24 (2011), 51-80
MSC (2010): Primary 11E16, 11F11, 11F25, 11F27, 11F30, 11F33, 11F80
DOI: https://doi.org/10.1090/S0894-0347-2010-00680-7
Published electronically: September 8, 2010
MathSciNet review: 2726599
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We make finiteness conjectures on the composite of Hecke fields of classical members of a $p$-adic analytic family of slope 0 elliptic modular forms in the vertical case (with fixed level varying weight). In the horizontal case (fixed weight varying $p$-power level), we prove the corresponding statements.


References [Enhancements On Off] (What's this?)

References
  • David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay; by Hindustan Book Agency, New Delhi, 2008. With appendices by C. P. Ramanujam and Yuri Manin; Corrected reprint of the second (1974) edition. MR 2514037
  • Goro Shimura, Abelian varieties with complex multiplication and modular functions, Princeton Mathematical Series, vol. 46, Princeton University Press, Princeton, NJ, 1998. MR 1492449
  • Nicholas M. Katz and Barry Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, vol. 108, Princeton University Press, Princeton, NJ, 1985. MR 772569
  • N. Bourbaki, Éléments de mathématique. I: Les structures fondamentales de l’analyse. Fascicule XI. Livre II: Algèbre. Chapitre 4: Polynomes et fractions rationnelles. Chapitre 5: Corps commutatifs, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1102, Hermann, Paris, 1959 (French). Deuxième édition. MR 0174550
  • Henri Carayol, Sur les représentations $l$-adiques associées aux formes modulaires de Hilbert, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 3, 409–468 (French). MR 870690
  • Ching-Li Chai, Every ordinary symplectic isogeny class in positive characteristic is dense in the moduli, Invent. Math. 121 (1995), no. 3, 439–479. MR 1353306, DOI https://doi.org/10.1007/BF01884309
  • Ching-Li Chai, A rigidity result for $p$-divisible formal groups, Asian J. Math. 12 (2008), no. 2, 193–202. MR 2439259, DOI https://doi.org/10.4310/AJM.2008.v12.n2.a3
  • C.-L. Chai, Families of ordinary abelian varieties: canonical coordinates, $p$-adic monodromy, Tate-linear subvarieties and Hecke orbits, preprint 2003 (available at: www.math.upenn.edu/˜chai).
  • P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Springer, Berlin, 1973, pp. 143–316. Lecture Notes in Math., Vol. 349 (French). MR 0337993
  • Eknath Ghate and Vinayak Vatsal, On the local behaviour of ordinary $\Lambda $-adic representations, Ann. Inst. Fourier (Grenoble) 54 (2004), no. 7, 2143–2162 (2005) (English, with English and French summaries). MR 2139691
  • Haruzo Hida, Geometric modular forms and elliptic curves, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. MR 1794402
  • Haruzo Hida, Iwasawa modules attached to congruences of cusp forms, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 2, 231–273. MR 868300
  • Haruzo Hida, Galois representations into ${\rm GL}_2({\bf Z}_p[[X]])$ attached to ordinary cusp forms, Invent. Math. 85 (1986), no. 3, 545–613. MR 848685, DOI https://doi.org/10.1007/BF01390329
  • Haruzo Hida, Hecke algebras for ${\rm GL}_1$ and ${\rm GL}_2$, Séminaire de théorie des nombres, Paris 1984–85, Progr. Math., vol. 63, Birkhäuser Boston, Boston, MA, 1986, pp. 131–163. MR 897346
  • Haruzo Hida, On $p$-adic Hecke algebras for ${\rm GL}_2$ over totally real fields, Ann. of Math. (2) 128 (1988), no. 2, 295–384. MR 960949, DOI https://doi.org/10.2307/1971444
  • Haruzo Hida, Adjoint Selmer groups as Iwasawa modules. part B, Proceedings of the Conference on $p$-adic Aspects of the Theory of Automorphic Representations (Jerusalem, 1998), 2000, pp. 361–427. MR 1809628, DOI https://doi.org/10.1007/BF02834845
  • H. Hida, The Iwasawa $\mu$–invariant of $p$–adic Hecke $L$–functions, Ann. of Math. (2) 172 (2010), 41–137.
  • Haruzo Hida and Yoshitaka Maeda, Non-abelian base change for totally real fields, Pacific J. Math. Special Issue (1997), 189–217. Olga Taussky-Todd: in memoriam. MR 1610859, DOI https://doi.org/10.2140/pjm.1997.181.189
  • Haruzo Hida, Hilbert modular forms and Iwasawa theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2006. MR 2243770
  • Taira Honda, Isogeny classes of abelian varieties over finite fields, J. Math. Soc. Japan 20 (1968), 83–95. MR 229642, DOI https://doi.org/10.2969/jmsj/02010083
  • Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Kanô Memorial Lectures, No. 1. MR 0314766
  • Lawrence C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR 1421575
  • N. Katz, Serre-Tate local moduli, Algebraic surfaces (Orsay, 1976–78) Lecture Notes in Math., vol. 868, Springer, Berlin-New York, 1981, pp. 138–202. MR 638600
  • R. P. Langlands, Modular forms and $\ell $-adic representations, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Springer, Berlin, 1973, pp. 361–500. Lecture Notes in Math., Vol. 349. MR 0354617
  • J. H. Loxton, On two problems of R. W. Robinson about sums of roots of unity, Acta Arith. 26 (1974/75), 159–174. MR 371852, DOI https://doi.org/10.4064/aa-26-2-159-174
  • B. Mazur, Deforming Galois representations, Galois groups over ${\bf Q}$ (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 16, Springer, New York, 1989, pp. 385–437. MR 1012172, DOI https://doi.org/10.1007/978-1-4613-9649-9_7
  • Haruzo Hida, Modular forms and Galois cohomology, Cambridge Studies in Advanced Mathematics, vol. 69, Cambridge University Press, Cambridge, 2000. MR 1779182
  • Toshitsune Miyake, Modular forms, Springer-Verlag, Berlin, 1989. Translated from the Japanese by Yoshitaka Maeda. MR 1021004
  • B. Mazur and A. Wiles, On $p$-adic analytic families of Galois representations, Compositio Math. 59 (1986), no. 2, 231–264. MR 860140
  • S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 261, Springer-Verlag, Berlin, 1984. A systematic approach to rigid analytic geometry. MR 746961
  • Kenneth A. Ribet, On $l$-adic representations attached to modular forms. II, Glasgow Math. J. 27 (1985), 185–194. MR 819838, DOI https://doi.org/10.1017/S0017089500006170
  • A. J. Scholl, Motives for modular forms, Invent. Math. 100 (1990), no. 2, 419–430. MR 1047142, DOI https://doi.org/10.1007/BF01231194

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 11E16, 11F11, 11F25, 11F27, 11F30, 11F33, 11F80

Retrieve articles in all journals with MSC (2010): 11E16, 11F11, 11F25, 11F27, 11F30, 11F33, 11F80


Additional Information

Haruzo Hida
Affiliation: Department of Mathematics, University of California, Los Angeles, Los Angeles, California 90095-1555
MR Author ID: 213427
Email: hida@math.ucla.edu

Received by editor(s): June 19, 2009
Received by editor(s) in revised form: February 8, 2010, and April 29, 2010
Published electronically: September 8, 2010
Additional Notes: The author is partially supported by the NSF grant: DMS 0753991 and DMS 0854949, and part of this work was done during the author’s stay in January to March 2010 at the Institut Henri Poincaré - Centre Emile Borel. The author thanks this institution for its hospitality and support.
Article copyright: © Copyright 2010 American Mathematical Society