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Non-Archimedean Integration and Elliptic Curves over Function Fields

Authors
Journal
Journal of Number Theory
0022-314X
Publisher
Elsevier
Publication Date
Volume
94
Issue
2
Identifiers
DOI: 10.1006/jnth.2001.2735
Keywords
  • Elliptic Curves In Char.P
  • Drinfeld Modular Curves
  • Non-Archimedean Integration
  • P-Adicl-Functions
Disciplines
  • Mathematics

Abstract

Abstract Let F be a global function field of characteristic p and E/ F an elliptic curve with split multiplicative reduction at the place ∞: then E can be obtained as a factor of the Jacobian of some Drinfeld modular curve. This fact is used to associate to E a measure μ E on P 1( F ∞). By choosing an appropriate embedding of a quadratic unramified extension K/ F into the matrix algebra M 2( F), μ E is pushed forward to a measure on a p-adic group G, isomorphic to an anticyclotomic Galois group over the Hilbert class field of K. Integration on G then yields a Heegner point on E when ∞ is inert in K and an analogue of the L -invariant if ∞ is split. In the last section, the same methods are extended to integration on a geometric cyclotomic Galois group.

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