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Arithmetic and Geometry pp 195–237Cite as

Birkhäuser

Canonical Height Pairings via Biextensions

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Part of the book series: Progress in Mathematics ((PM,volume 35))

Abstract

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.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Harvard University, Cambridge, Massachusetts, 02138, USA

    Professor Barry Mazur & Professor John Tate

Authors
  1. Professor Barry Mazur
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  2. Professor John Tate
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Editor information

Editors and Affiliations

  1. Mathematics Department, Massachusetts Institute of Technology, 02139, Cambridge, MA, USA

    Michael Artin

  2. Mathematics Department, Harvard University, 02138, Cambridge, MA, USA

    John Tate

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© 1983 Springer Science+Business Media New York

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Mazur, B., Tate, J. (1983). Canonical Height Pairings via Biextensions. In: Artin, M., Tate, J. (eds) Arithmetic and Geometry. Progress in Mathematics, vol 35. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4757-9284-3_9

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  • DOI: https://doi.org/10.1007/978-1-4757-9284-3_9

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-0-8176-3132-1

  • Online ISBN: 978-1-4757-9284-3

  • eBook Packages: Springer Book Archive

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