Inventiones mathematicae

, Volume 84, Issue 1, pp 1–48

Onp-adic analogues of the conjectures of Birch and Swinnerton-Dyer

  • B. Mazur
  • J. Tate
  • J. Teitelbaum
Article

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References

  1. [A] Arnaud, B.: Interpolationp-adique d'un produit de Rankin. University of Orsay, 1984 (Preprint)Google Scholar
  2. [A-V] Amice, Y., Vélu, J.: Distributionsp-adiques associées aux séries de Hecke. Astérisque, No. 24/25. Soc. Math. Fr., Paris, 119–131 (1975)Google Scholar
  3. [A-L] Atkin, O., Li, W.: Twists of newforms and pseudo-eigenvalues ofW-operators. Invent. Math.48, 221–243 (1978)Google Scholar
  4. [D-R] Deligne, P., Rapoport, M.: Schémas de modules de courbes elliptiques. Lect. Notes Math., vol. 349. Berlin-Heidelberg-New York: Springer 1973Google Scholar
  5. [H] Hecke, E.: Vorlesungen über die Theorie der algebraische Zahlen. New York: Chelsea 1948Google Scholar
  6. [I] Iwasawa, K.: Lectures onp-adicL-functions. Ann. Math. Stud., vol. 74. Princeton: Princeton University Press 1972Google Scholar
  7. [K] Katz, N.:p-adic properties of modular schemes and forms. In: Modular Functions of One Variable, III. Lect. Notes Math., vol. 350, pp. 69–191. Berlin-Heidelberg-New York: Springer 1973Google Scholar
  8. [La] Lang, S.: Elliptic Functions, Reading, MA: Addison-Wesley 1973Google Scholar
  9. [L-T] Lang, S., Trotter, H.: Frobenius Distributions inGL 2-Extensions. Lect. Notes Math., vol. 504. Berlin-Heidelberg-New York: Springer 1976Google Scholar
  10. [L] Li, W.: Newforms and functional equations. Math. Ann.212, 285–315 (1975)Google Scholar
  11. [Man] Manin, J.: Parabolic points and zeta functions of modular curves. Izv. Akad. Nauk SSSR (AMS translation)6(1) 19–64 (1972)Google Scholar
  12. [Maz] Mazur, B.: Courbes elliptiques et symboles modulaires. Sémin. Bourbaki exp. No. 414; Lect. Notes Math., vol. 317. Berlin-Heidelberg-New York: Springer 1973Google Scholar
  13. [M-S-D] Mazur, B., Swinnerton-Dyer, P.: Arithmetic of Weil curves. Invent. Math.25, 1–61 (1974)Google Scholar
  14. [M-T1] Mazur, B., Tate, J.: Thep-adic sigma function. (In preparation)Google Scholar
  15. [M-T2] Mazur, B., Tate, J.: Canonical Height Pairings via Biextensions. In: Arithmetic and Geometry, Progr. Math., vol. 35, pp. 195–237. Boston-Basel-Stuttgart: Birkhäuser 1983Google Scholar
  16. [M-W] Mazur, B., Wiles, A.: Class fields of abelian extensions ofQ. Invent. Math.76, 179–330 (1984)Google Scholar
  17. [McC] McCabe, J.:p-adic Theta Functions. Harvard Ph.D. Thesis 1968Google Scholar
  18. [Mo] Morikawa, H.: On theta functions and abelian varieties over valuation fields of rank one, I and II. Nagoya Math. J.20, 1–27 and 231–250 (1962)Google Scholar
  19. [N] Norman, P.:p-adic theta functions. Am. J. Math.107, 617–661 (1985)Google Scholar
  20. [O] Ogg, A.: On the eigenvalues of Hecke operators. Math. Ann.179, 101–108 (1969)Google Scholar
  21. [P] Panciskin, A.: Le prolongementp-adique analytiques des fonctionsL de Rankin. C.R. Acad. Sci. Paris, I295, 51–53 and 227–230 (1982)Google Scholar
  22. [P-R] Perrin-Riou, B.: Descente infinie et hauteurp-adique sur les courbes elliptiques à multiplication complexe. Invent. Math.70, 369–398 (1983)Google Scholar
  23. [Ra] Raynaud, M.: Spécialisation du foncteur de Picard. Publ. Math. I.H.E.S.38, 27–76 (1970)Google Scholar
  24. [R] Roquette, P.: Analytic theory of elliptic functions over local fields. Hamb. Math. Einzelschriften, Neu Folgen, Heft 1, 1970Google Scholar
  25. [Sch] Schneider, P.:p-adic heights. Invent. Math.69, 401–409 (1982)Google Scholar
  26. [S] Serre, J.-P.: Quelques applications de théorème de densité de Chebotarev. I.H.E.S. Publ. Math.54, 323–401 (1981)Google Scholar
  27. [S2] Serre, J.-P.: Une interprétation des congruences relatives à la fonction τ de Ramanujan. Séminaire Delange-Pisot-Poitou: Théorie des nombres Exposé 14 (1967/68)Google Scholar
  28. [S3] Serre, J.-P.: Sur la lacunarité des puissances de ν. Glasgow Math. J.27, 203–221 (1985)Google Scholar
  29. [Sh] Shimura, G.: Introduction to the arithmetic theory of automorphic forms. Publ. Math. Soc. Japan, vol. 11. Iwanami. Shoten and Princeton Univ. Press 1971Google Scholar
  30. [St] Stevens, G.: Arithmetic on Modular Curves Progr. Math., vol. 20. Boston-Basel-Stuttgart: Birkhäuser 1982Google Scholar
  31. [V] Vishik, M.: Nonarchimedean measures connected with Dirichlet series. Math. USSR Sb.28, 216–228 (1976)Google Scholar
  32. [W] Weil, A.: Uber die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. Math. Ann.168, 149–156 (1967); (Oeuvres, vol. III, pp. 165–172)Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • B. Mazur
    • 1
  • J. Tate
    • 1
  • J. Teitelbaum
    • 1
  1. 1.Dept. of MathematicsHarvard UniversityCambridgeUSA

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