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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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