Hecke fields of analytic families of modular forms

Hida, Haruzo

doi:10.1090/S0894-0347-2010-00680-7

Hecke fields of analytic families of modular forms
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by Haruzo Hida

J. Amer. Math. Soc. 24 (2011), 51-80

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

Published electronically: September 8, 2010

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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

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, DOI 10.1515/9781400883943
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, DOI 10.1515/9781400881710
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, DOI 10.24033/asens.1512
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 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 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) Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973, pp. 143–316 (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, DOI 10.5802/aif.2077
Haruzo Hida, Geometric modular forms and elliptic curves, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. MR 1794402, DOI 10.1142/9789812792693
Haruzo Hida, Iwasawa modules attached to congruences of cusp forms, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 2, 231–273. MR 868300, DOI 10.24033/asens.1507
Haruzo Hida, Galois representations into $\textrm {GL}_2(\textbf {Z}_p[[X]])$ attached to ordinary cusp forms, Invent. Math. 85 (1986), no. 3, 545–613. MR 848685, DOI 10.1007/BF01390329
Haruzo Hida, Hecke algebras for $\textrm {GL}_1$ and $\textrm {GL}_2$, 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 $\textrm {GL}_2$ over totally real fields, Ann. of Math. (2) 128 (1988), no. 2, 295–384. MR 960949, DOI 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 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 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, DOI 10.1093/acprof:oso/9780198571025.001.0001
Taira Honda, Isogeny classes of abelian varieties over finite fields, J. Math. Soc. Japan 20 (1968), 83–95. MR 229642, DOI 10.2969/jmsj/02010083
Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Kanô Memorial Lectures, No. 1, Iwanami Shoten Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. MR 0314766
Lawrence C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR 1421575, DOI 10.1007/978-1-4612-1934-7
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) Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973, pp. 361–500. 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 10.4064/aa-26-2-159-174
B. Mazur, Deforming Galois representations, Galois groups over $\textbf {Q}$ (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 16, Springer, New York, 1989, pp. 385–437. MR 1012172, DOI 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, DOI 10.1017/CBO9780511526046
Toshitsune Miyake, Modular forms, Springer-Verlag, Berlin, 1989. Translated from the Japanese by Yoshitaka Maeda. MR 1021004, DOI 10.1007/3-540-29593-3
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, DOI 10.1007/978-3-642-52229-1
Kenneth A. Ribet, On $l$-adic representations attached to modular forms. II, Glasgow Math. J. 27 (1985), 185–194. MR 819838, DOI 10.1017/S0017089500006170
A. J. Scholl, Motives for modular forms, Invent. Math. 100 (1990), no. 2, 419–430. MR 1047142, DOI 10.1007/BF01231194