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Elliptic curves and \({\mathbb{Z}}_ p\)-extensions. (English) Zbl 0599.14028
Let E be an elliptic curve defined over a number field F with a complex multiplication in the ring of integers of an imaginary quadratic \(field\quad K.\) Let \(p>2\) be a rational prime such that E has good reduction at all primes of F above p. The purpose of the paper is to study arithmetic invariants of E (the Mordell-Weil group and Tate- Shafarevich group), using techniques of Iwasawa theory. The conclusions are quite different from the case of splitting p. Some of examples are given in the final \(\sec tion\quad 6\) (the case \(F={\mathbb{Z}}\), cyclotomic and anti-cyclotomic \({\mathbb{Z}}_ p\)-extensions).
Reviewer: K.Katayama

MSC:
14H45 Special algebraic curves and curves of low genus
14K22 Complex multiplication and abelian varieties
11R18 Cyclotomic extensions
14H52 Elliptic curves
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References:
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