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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