Canonical Height Pairings via Biextensions

  • Barry Mazur
  • John Tate
Part of the Progress in Mathematics book series (PM, volume 35)


The object of this paper is to present the foundations of a theory of p-adic-valued height pairings
$$A\left( K \right) \times A'\left( K \right) \to {Q_p}$$
, where Λ is a abelian variety over a global field K, and A′ is its dual. We say “pairings” in the plural because, in contrast to the classical theory of ℝ-valued) canonical height, there may be many canonical p-adic valued pairings: as we explain in § 4, up to nontrivial scalar multiple, they are in one-to-one correspondence with ℤ p -extensions L/K whose ramified primes are finite in number and are primes of ordinary reduction (1.1) for A.


Elliptic Curf Abelian Variety Group Scheme Finite Index Algebraic Space 
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  1. [1]
    M. Actin, Algebraization of formal moduli: 1, Global Analysis. Papers in honor of K. Kodaira. (1970) 21–71. Princeton Univ. Press.Google Scholar
  2. [2]
    I. Barsotti, Considerations on theta-functions, Symposia Mathematica (1970) p. 247.Google Scholar
  3. [3]
    S. Bloch, A note on height pairings, Tamagawa numbers and the Birch and Swinnerton-llyer conjecture, Inv. Math. 58 (1980), 65–76.Google Scholar
  4. [4]
    D. Bernardi, Hauteur p-adique sur les courbes elliptiques. Séminaire de Théorie des Nombres, Paris 1979–80, Séminaire Delange-Pisot-Poitou. Progress in Math. series vol. 12. Birkhäuser, Boston-Basel-Stuttgart. (1981) 1–14.Google Scholar
  5. [5]
    L. Breen, Fonctions thêta et théorème du cube, preprint of the Laboratoire associé au C.N.R.S. N° 305, Université de Rennes I, 1982.Google Scholar
  6. [6]
    V. Cristante, Theta functions and Barsotti- Tate groups,Annali della Scuola Normale Superiore di Pisa, Serie IV, 7 (1980) 181215.Google Scholar
  7. [7]
    M. Harris, Systematic growth of Mordell-Weil groups of abelian varieties in towers of number fields,Inv. Math. 51 (1979) 123141.Google Scholar
  8. [8]
    D. Knutson, Algebraic Spaces, Lecture Notes in Math. 203, Springer Berlin-Heidelberg-New York, 1971.Google Scholar
  9. [9]
    P. F. Kurchanov, Elliptic curves of infinite rank over F-extensions, Math. USSR Sbornik, 19, 320–324 (1973).CrossRefGoogle Scholar
  10. [10]
    J. Manin, The Tate height of points on an abelian variety. Its variants and applications. Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 1363–1390. (AMS Translations 59 (1966) 82–110 ).Google Scholar
  11. [11]
    B. Mazur, Rational points of abelian varieties with values in towers of number fields, Inv. Math. 18 (1972) 183–266.MathSciNetzbMATHGoogle Scholar
  12. [12]
    D. Mumford, Biextensions of formal groups, in the Proceedings of the Bombay Colloquium on Algebraic Geometry, Tata Institute of Fundamental Research Studies in Mathematics 4, London, Oxford University Press 1968.Google Scholar
  13. [13]
    D. Mumford, A remark on Mordell’s conjecture, Amer. J. Math. 87 (1965) 1007–1016.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    J. P. Murre, On contravariant functors from the category of pre-schemes over a field into the category of abelian groups, Pub. Math. I. H. E. S. no 23, 1964.Google Scholar
  15. [15]
    A. Néron, Quasi-fonctions et hauteurs sur les variétés abéliennes, Annals of Math. 82, n° 2, (1965) 249–331.Google Scholar
  16. [16]
    A. Néron, Hauteurs et fonctions thêta. Rend. Sci. Mat. Milano 46 (1976) 111–135.zbMATHCrossRefGoogle Scholar
  17. [17]
    A. Néron, Fonctions thêta p-adiques, Symposia Mathematica, 24 (1981) 315–345.Google Scholar
  18. [18]
    A. Néron, Fonctions thêta p-adiques et hauteurs p-adiques, pp. 149–174, in Séminaire de theorie des nombres (Séminaire DelangePisot-Poitou), Paris 1980–81, Progress in Math., Vol 22, Birkhäuser Boston-Basel-Stuttgart (1982).Google Scholar
  19. [19]
    P. Norman, Theta Functions. Handwritten manuscript.Google Scholar
  20. [20]
    B. Perrin-Riou, Descente infinie et hauteur p-adique sur les courbes elliptiques à multiplication complexe. To appear in Inv. Math.Google Scholar
  21. [21]
    M. Rosenblum, in collaboration with S. Abramov, The BirchSwinnerton-Dyer conjecture mod p. Handwritten manuscript.Google Scholar
  22. [22]
    P. Schneider, p-adic height pairings I,to appear in Inv. Math.Google Scholar
  23. [23]
    J. Tate, Variation of the canonical height of a point depending on a parameter. To appear in Amer. J. Math.Google Scholar
  24. [24]
    Ju. G. Zarhin, Néron Pairing and Quasicharacters, Izv. Akad. Nauk. SSSR Ser. Mat. Tom 36 (1972) No. 3, 497–509. (Math. USSR Izvestija, Vol. 6 (1972) No. 3, 491–503.)Google Scholar
  25. [EGA I]
    A. Grothendieck, Éléments de géométrie algébrigue. Publications Mathématiques, I. H. E. S./Paris (1961).Google Scholar
  26. [SGA 2]
    A. Grothendieck, Cohomologie locale des faisceaux cohérents et Théorèmes de Lefschetz locaux et globaux, North-Holland Publishing Company - Amsterdam (1968).Google Scholar
  27. [SGA 3]
    M. Demazure and A. Grothendieck et al, Séminaire de Géométrie Algébrique du Bois Marie 1962/64. Schémas en Groupes II. Lecture Notes in Mathematics. 152, Springer, Berlin-Heidelberg-New York (1970).Google Scholar
  28. [SGA 7 I]
    A. Grothendieck et al, Séminaire de Géometrie Algébrique du Bois Marie. 1967/69. Groupes de Monodromie en Géométrie Algébrique. Lecture Notes in Mathematics. 288, Springer, BerlinHeidelberg-New York (1972).Google Scholar

Copyright information

© Springer Science+Business Media New York 1983

Authors and Affiliations

  • Barry Mazur
    • 1
  • John Tate
    • 1
  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA

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