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Relations d’orthogonalité sur les groupes de Mordell-Weil. (Relations of orthogonality on the Mordell-Weil groups). (French) Zbl 0607.14014
Théorie des nombres, Sémin. Paris 1984-85, Prog. Math. 63, 33-39 (1986).
[For the entire collection see Zbl 0593.00007.]
Let E be an elliptic curve with complex multiplication by a quadratic imaginary field K, and assume that E is defined over K. Let \({\mathfrak p}\) be a prime of K of \(norm\quad p\) that is a prime of good reduction for E. For a number field F, galois over K with Galois group G, one defines the finite dimensional vector space V to be E(F)\(\otimes {\mathbb{Q}}\), where E(F) is the Mordell-Weil group of E. The vector space V is naturally a K[G]- module equipped with a \({\mathbb{Q}}_ p\)-valued quadratic form \(h_{{\mathfrak p}}\), the p-adic height function. Using the quadratic form \(h_{{\mathfrak p}}\) one can define a \({\mathbb{Q}}\)-bilinear symmetric form \((\quad,\quad)_{{\mathfrak p}}\) on \(V\times V\) by \((P,Q)_{{\mathfrak p}}=(h_{{\mathfrak p}}(P+Q)-h_{{\mathfrak p}}(P)-h_{{\mathfrak p}}(Q))\). The main result of this paper gives a necessary and sufficient condition for two elements P and Q of V to be orthogonal with respect to \((\quad,\quad)_{{\mathfrak p}}.\) The relevant condition is the vanishing of a certain bilinear form on a submodule of K[G]\(\times K[G]\) that is associated to the pair (P,Q).
Reviewer: S.Kamienny

MSC:
14G25 Global ground fields in algebraic geometry
14K22 Complex multiplication and abelian varieties
14H52 Elliptic curves
14H45 Special algebraic curves and curves of low genus