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Local units, elliptic units, Heegner points and elliptic curves. (English) Zbl 0623.14006
Let E be an elliptic curve over \({\mathbb{Q}}\) with complex multiplication by the ring of integers of an imaginary quadratic field K. This paper gives a possible starting point to extend a result of Yager [R. Y. Yager, Ann. Math., II. Ser. 115, 411-449 (1982; Zbl 0496.12010)] giving the structure of a certain module over the Iwasawa algebra \({\mathbb{Z}}_ p[| Gal(K(E_{p^{\infty}})/K(E_ p)|]\), where p is a prime of \({\mathbb{Q}}\) which splits in K, to the case where this prime does not split. Instead of working over the Iwasawa algebra as above, the author uses a quotient denoted \(V_{\infty}\) of the inverse limit of local units at p, which is a free module of rank 2 over the Iwasawa algebra attached to the anticyclotomic \({\mathbb{Z}}_ p\)-extension of \(K_ p\). The module \(V_{\infty}\) has two natural submodules; conditions on the prime p are given which ensure that the direct sum of these is isomorphic to \(V_{\infty}\). The author conjectures that this direct sum decomposition always holds. The proof that under the conditions on the prime p given in the paper this conjecture is true is far from trivial; it uses global techniques as Heegner points and elliptic units.
As an application, it is shown how some of the results of the paper can be used to prove the following theorem: Let E be an elliptic curve over \({\mathbb{Q}}\) with complex multiplication and suppose that the sign in the functional equation of \(L(E,s)\) is -1. Then either E(\({\mathbb{Q}})\) is infinite or the p-part of the Tate-Shafarevich group of E over \({\mathbb{Q}}\) is infinite for every prime \(p\geq 5\) where E has good, supersingular reduction. With “supersingular” replaced by “ordinary” this was proved by R. Greenberg [Invent. Math. 72, 241-265 (1983; Zbl 0546.14015)].
Reviewer: J.Top

MSC:
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14H25 Arithmetic ground fields for curves
11R18 Cyclotomic extensions
14K22 Complex multiplication and abelian varieties
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