Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication
type: Item
instance of: scholarly article
title: Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication
zbMATH Open document ID: 0628.14018
DOI: 10.1007/BF01388984
author: Karl Rubin
published in: Inventiones Mathematicae
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143493
review text: Let K be a number field and \(E/K\) an elliptic curve. The conjecture of Birch and Swinnerton-Dyer gives a relation between the behavior of the L- series \(L(E_{/K},s)\) around \(s=1\) and (among other quantities) the order of the Tate-Shafarevich group Russian{Sh} = Russian{Sh}\((E_{/K})\). As described by \textit{J. T. Tate} [Invent. Math. 23, 179-206 (1974; Zbl 0296.14018)], ``this remarkable conjecture relates the behavior of a \(function\quad L\) at a point where it is not at present known to be defined to the order of a group {Russian{Sh}} which is not known to be finite!'' In this important paper the author gives the first examples of elliptic curves for which it can be proved that the Tate- Shafarevich group is finite. His first result gives a relation between the value of \(L(E_{/K},1)\) and the order of {Russian{Sh}}; and his second relates the order of vanishing of \(L(E_{/K},s)\) at \(s=1\) to the rank of the Mordell-Weil group E(K). Both of these results provide additional evidence for the truth of the Birch and Swinnerton-Dyer conjecture. We now describe the author's results in more detail. Let \(E/K\) be an elliptic curve with complex multiplication by an order \({\mathfrak O}\) in the imaginary quadratic field K, let \({\mathfrak O}_ K\) be the ring of integers of K, and let \(\Omega\) be an \({\mathfrak O}\)-generator of the period lattice of a minimal model for E. Let \(\psi\) be the Hecke character of K attached to E. The L-function of \(E/K\) satisfies \(L(E_{/K},s)=L(\psi,s)L({\bar \psi},s)\), and \(L({\bar\psi},1)/\Omega\in K.\) Theorem A. (a) If \(L(E_{/K},1)\neq 0\), then {Russian{Sh}} is finite. (b) Let \({\mathfrak p}\) be a prime of K not dividing \(| {\mathfrak O}^*_ K|\). If \(| E(K)_{tors}| L({\bar\psi},1)/\Omega \not\equiv 0\) (mod \({\mathfrak p})\), then the \({\mathfrak p}\)-part of {Russian{Sh}} is trivial. Theorem B. Let E be an elliptic curve defined over \({\mathbb{Q}}\) with complex multiplication. If \(\text{rank}_{{\mathbb{Z}…
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 14K22
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14H52
zbMATH DE Number: 4021325
zbMATH Keywords: finite Tate-Shafarevich group
zbMATH Keywords: elliptic curves
zbMATH Keywords: evidence for the truth of the Birch and Swinnerton-Dyer conjecture
zbMATH Keywords: ideal class annihilators
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4021325