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On a Galois extension with restricted ramification related to the Selmer group of an elliptic curve with complex multiplication. (English) Zbl 0738.11059

The paper exploits the analogy between algebraic number fields and algebraic function fields. In particular by the algebraic description of the algebraic \(p\)-adic height pairing of an elliptic curve \(\mathcal E\) defined over a finite extension \(F\) of \(\mathbb{Q}\), resp. \(\mathbb{F}_ q(t)\). According to Schneider the height pairing of the Selmer group of \(\mathcal E\) is induced by a duality of certain flat cohomology groups.
Assume that \(\mathcal E\) admits complex multiplication by the ring of integers \({\mathcal O}_ K\) of an imaginary quadratic field \(K\), that \(F\) is abelian over \(K\) and that \(\mathcal E\) has good ordinary reduction at \(p>3\). Let \(F_ \infty\) be the cyclotomic \(\mathbb{Z}_ p\)-extension of \(F\) and let \({\mathcal O}_ \infty\) be its ring of integers. The \(p\)-adic height pairing of the Selmer group of \(\mathcal E\) is induced by the perfect pairing obtained from the global flat duality theorem of Artin and Mazur.
In the function field case the pairing can be interpreted as a pairing of Galois cohomology groups of the fundamental group of the curve \(\bar S\) (\(S\) is the associated projective smooth curve of \(F\)).
The purpose of this paper is to define also in the number field situation a natural Galois extension \(\tilde F\) of \(F\) such that \(G(\tilde F/F_ \infty)\) is a Poincaré group and \(H^ 1({\mathcal O}_{K_ \infty},T_ p(\mathcal E))\) can be interpreted as a Galois cohomology group of \(G(\tilde F/K_ \infty)\) such that the cup-product induces the height pairing for \({\mathcal E}/K_ \infty\).

MSC:

11R32 Galois theory
11R58 Arithmetic theory of algebraic function fields
14H05 Algebraic functions and function fields in algebraic geometry
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