Hostname: page-component-848d4c4894-jbqgn Total loading time: 0 Render date: 2024-06-17T05:50:56.388Z Has data issue: false hasContentIssue false
  • English
  • Français

Galois action on some ideal section points of the abelian variety associated with a modular form and its application

Published online by Cambridge University Press:  22 January 2016

Fumiyuki Momose
Fumiyuki Momose*
Affiliation:
Department of Mathematics, Faculty of Science, University of Tokyo, Hongo, Tokyo 113, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For an integer N let X1(N) be the modular curve defined over Q which corresponds to the modular group Γ1(N) To each primitive cusp form f ═ Σ amqm, a1═1, (Γ normalized new form in the sense of [1]) on Γ1(N) of weight 2, there corresponds a factor Jf of the jacobian variety of X1(N) (cf. Shimura [19]).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 91 , September 1983 , pp. 19 - 36
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1983

References

[1] Atkin, A. O. L. and Lehner, J., Hecke operators on Γ0(ℳ), Math. Ann., 185(1970), 134–160.Google Scholar
[2] Deligne, P., Formes modulaires et représentations Z-adiques, sém. Bourbaki, 1968/1969, exposé n° 355, Lecture Notes in Math., 179, 139–172. Berlin-Heidelberg-New York: Springer 1971.Google Scholar
[3] Deligne, P. and Rapoport, P., Schémas de modules des courbes elliptiques, vol. II of the Proceedings of the International Summer School on Modular Functions, Antwerp (1972). Lecture Notes in Math., 349, Berlin-Heidelberg-New York: Springer 1973.Google Scholar
[4] Doi, K. and Yamauchi, M., On the Hecke operators for Γ0(N)and class fields over quadratic number fields, J. Math. Soc. Japan, 25 (1973), 629–643.Google Scholar
[5] Ishii, H., Congruences between cusp forms and the fundamental units of real quadratic number fields, to appear.Google Scholar
[6] Ishikawa, H., Explicit formula of the traces of Hecke operators for Γ0(N), J. Fac. Sci. Univ. Tokyo, 21 (1974), 357–376.Google Scholar
[7] Katz, N., p-adic properties of modular schemes and modular forms, vol. III of Proceedings of the International Summer School on Modular Functions, Antwerp (1972), Lecture Notes in Math., 350, Berlin-Heidelberg-New York, 69–190 (1973).Google Scholar
[8] Koike, M., On certain abelian varieties obtained from new forms of weight 2 on Γ0(34) and Γ0(35), Nagoya Math. J., 62 (1976), 29–39.CrossRefGoogle Scholar
[9] Katz, N., Congruences between cusp forms and linear representations of Galois group, Nagoya Math. J., 64 (1976), 63–85.Google Scholar
[10] Momose, F., On the ιadic representations attached to modular forms, J. Fac. Sci. Univ. Tokyo, 28 (1981), 89–109.Google Scholar
[11] Ohta, M., The representation of Galois group attached to certain finite group schemes, and its application to Shimura′s theory, Algebraic Number Theory, Papers contributed for the International Symposium, Kyoto 1976, Japan Societyz for the Promotion for Science.Google Scholar
[12] Raynaud, M., Schémas en groupes de type (p, …, p), Bull. Soc. Math. France, 102(1974), 241–280.Google Scholar
[13] Ribet, K. A., Endomorphisms of semi-stable abelian varieties over number fields, Ann. of Math., 101(1975), 555–562.Google Scholar
[14] Ribet, K. A., On ιadic representations attached to modular forms, Invent. Math., 28(1975), 245–275.Google Scholar
[15] Ribet, K. A., Twists of modular forms and endomorphisms of abelian varieties, Math. Ann., (1980), 239–244.Google Scholar
[16] Ribet, K. A., Endomorphism algebras of abelian varieties attached to newforms of weight 2, Séminaire D.P.P., 1979-80.Google Scholar
[17] Saito, H., On a decomposition of spaces of cusp forms and trace formula of Hecke operators, Nagoya Math. J., 80(1980), 129–165.Google Scholar
[18] Shimura, G., On elliptic curves with complex multiplication as factors of the jacobians of modular function fields, Nagoya Math. J., 43 (1971), 199–208.Google Scholar
[19] Shimura, G., On the factors of jacobian variety of a modular function field, J. Math. Soc. Japan, 25 (1973), 523–544.Google Scholar
[20] Shimura, G., Class fields over real quadratic fields and Hecke operators, Ann. of Math., 95 (1972), 130–190.CrossRefGoogle Scholar
[21] Shimura, G., Introduction to the Arithmetic Theory of Automorphic Functions, Pub. Math. Soc. Japan, No. 11, Tokyo-Princeton, 1971.Google Scholar