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?)

  • [ABV] D. Mumford, Abelian Varieties, Tata Institute of Fundamental Research Studies in Mathematics, Oxford University Press, New York, 1994. MR 2514037 (2010e:14040)
  • [ACM] G. Shimura, Abelian Varieties with Complex Multiplication and Modular Functions, Princeton University Press, Princeton, NJ, 1998. MR 1492449 (99e:11076)
  • [AME] N. M. Katz and B. Mazur, Arithmetic Moduli of Elliptic Curves, Annals of Math. Studies 108, Princeton University Press, Princeton, NJ, 1985. MR 772569 (86i:11024)
  • [BAL] N. Bourbaki, Livre II: Algèbre, Chapitre 5, Hermann, Paris, 1959. MR 0174550 (30:4751);
  • [Ca] H. Carayol, Sur les représentations $ l$-adiques associées aux formes modulaires de Hilbert, Ann. Sci. École Norm. Sup. (4) 19 (1986), 409-468. MR 870690 (89c:11083)
  • [Ch] C.-L. Chai, Every ordinary symplectic isogeny class in positive characteristic is dense in the moduli, Inventiones Math. 121 (1995), 439-479. MR 1353306 (96f:11082)
  • [Ch1] C.-L. Chai, A rigidity result for $ p$-divisible formal groups, Asian J. Math. 12 (2008), 193-202. MR 2439259 (2009f:14089)
  • [Ch2] 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).
  • [DR] P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, in ``Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972)'', pp. 143-316. Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973. MR 0337993 (49:2762)
  • [GV] E. Ghate and V. Vatsal, On the local behaviour of ordinary $ \Lambda$-adic representations, Ann. Inst. Fourier (Grenoble) 54 (2004), 2143-2162. MR 2139691 (2006b:11050)
  • [GME] H. Hida, Geometric Modular Forms and Elliptic Curves, World Scientific, Singapore, 2000. MR 1794402 (2001j:11022)
  • [H86a] H. Hida, Iwasawa modules attached to congruences of cusp forms, Ann. Sci. Ecole Norm. Sup. 4th series 19 (1986), 231-273. MR 868300 (88i:11023)
  • [H86b] H. Hida, Galois representations into $ GL_2(\mathbb{Z}_p[[X]])$ attached to ordinary cusp forms, Inventiones Math. 85 (1986), 545-613. MR 848685 (87k:11049)
  • [H86c] H. Hida, Hecke algebras for $ GL_1$ and $ GL_2$, Sém. de Théorie des Nombres, Paris 1984-85, Progress in Math. 63 (1986), 131-163. MR 897346 (88i:11078)
  • [H88] H. Hida, On $ p$-adic Hecke algebras for $ GL_2$ over totally real fields, Ann. of Math. (2) 128 (1988), 295-384. MR 960949 (89m:11046)
  • [H00] H. Hida, Adjoint Selmer groups as Iwasawa modules, Israel Journal of Math. 120 (2000), 361-427. MR 1809628 (2001k:11094)
  • [H10] H. Hida, The Iwasawa $ \mu$-invariant of $ p$-adic Hecke $ L$-functions, Ann. of Math. (2) 172 (2010), 41-137.
  • [HM] H. Hida and Y. Maeda, Non-abelian base change for totally real fields, Olga Taussky Todd memorial issue, Pacific Journal of Math. (1997), 189-217. MR 1610859 (99f:11068)
  • [HMI] H. Hida, Hilbert modular forms and Iwasawa theory, Oxford University Press, 2006. MR 2243770 (2007h:11055)
  • [Ho] T. Honda, Isogeny classes of abelian varieties over finite fields, J. Math. Soc. Japan 20 (1968), 83-95. MR 0229642 (37:5216)
  • [IAT] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press, Princeton, NJ, and Iwanami Shoten, Tokyo, 1971. MR 0314766 (47:3318)
  • [ICF] L. C. Washington, Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, 83, Springer, New York, 1982. MR 1421575 (97h:11130)
  • [K] N. M. Katz, Serre-Tate local moduli, in ``Surfaces Algébriques'', Lecture Notes in Math. 868 (1978), 138-202. MR 638600 (83k:14039b)
  • [La] R. P. Langlands, Modular forms and $ \ell $-adic representations, in ``Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972)'', pp. 361-500. Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973. MR 0354617 (50:7095)
  • [Lo] J. H. Loxton, On two problems of R. M. Robinson about sum of roots of unity, Acta Arithmetica 26 (1974), 159-174. MR 0371852 (51:8069)
  • [M] B. Mazur, Deforming Galois representations, in ``Galois group over $ \mathbb{Q}$'', MSRI publications 16 (1989), 385-437. MR 1012172 (90k:11057)
  • [MFG] H. Hida, Modular Forms and Galois Cohomology, Cambridge Studies in Advanced Mathematics 69, Cambridge University Press, Cambridge, England, 2000. MR 1779182 (2002b:11071)
  • [MFM] T. Miyake, Modular Forms, Springer, New York-Tokyo, 1989. MR 1021004 (90m:11062)
  • [MW] B. Mazur and A. Wiles, On $ p$-adic analytic families of Galois representations, Compositio Math. 59 (1986), 231-264. MR 860140 (88e:11048)
  • [NAA] S. Bosch, U. Güntzer and R. Remmert, Non-Archimedean analysis. A systematic approach to rigid analytic geometry. Grundlehren der Mathematischen Wissenschaften, Vol. 261, Springer, Berlin, 1984. MR 746961 (86b:32031)
  • [R] K. Ribet, On $ l$-adic representations attached to modular forms II, Glasgow Math. J. 27 (1985), 185-194. MR 819838 (88a:11041)
  • [S] A. J. Scholl, Motives for modular forms, Inventiones Math. 100 (1990), 419-430. MR 1047142 (91e:11054)

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
Email: hida@math.ucla.edu

DOI: https://doi.org/10.1090/S0894-0347-2010-00680-7
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

American Mathematical Society