Local units, elliptic units, Heegner points and elliptic curves
type: Item
instance of: scholarly article
title: Local units, elliptic units, Heegner points and elliptic curves
zbMATH Open document ID: 0623.14006
DOI: 10.1007/BF01388915
author: Karl Rubin
published in: Inventiones Mathematicae
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143460
review text: 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 [\textit{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 \textit{R. Greenberg} [Invent. Math. 72, 241-265 (1983; Zbl 0546.14015)].
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 14K22
zbMATH DE Number: 4010662
zbMATH Keywords: Birch and Swinnerton-Dyer conjecture
zbMATH Keywords: elliptic curve
zbMATH Keywords: complex multiplication
zbMATH Keywords: Iwasawa algebra
zbMATH Keywords: Heegner points
zbMATH Keywords: elliptic units
zbMATH Keywords: Tate- Shafarevich group
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4010662