A finiteness theorem for the symmetric square of an elliptic curve
type: Item
instance of: scholarly article
title: A finiteness theorem for the symmetric square of an elliptic curve
zbMATH Open document ID: 0781.14022
DOI: 10.1007/BF01232029
author: Matthias Flach
published in: Inventiones Mathematicae
publication date: 1993-06-29T00:00:00Z
full work available at URL: https://eudml.org/doc/144023
review text: Let \(E\) be an elliptic curve and \(M\) be symmetric square of the Tate module \(T_ p(E)\). Bloch and Kato have given a conjectural formula for the value of the motivic \(L\)-function attached to \(M\) at the point \(s=2\). It includes an order of a generalization of the Tate-Shafarevich group for \(M\). If we assume that \(E\) is a modular elliptic curve then the modular parametrisation of \(E\) gives an explicit expression for this value through the degree \(\deg\pi\) of the parametrisation and other invariants of \(E\). In this case it is known that the \(L\)-function is an entire function on the whole \(s\)-plane. The author proves that \(\deg \pi\) annihilates the Tate-Shafarevich group if \(p>3\), \(E\) has good reduction in \(p\) and the image of representation of \(\text{Gal}(\overline\mathbb{Q}/\mathbb{Q})\) on \(E_ p\) equals \(\text{GL}_ 2(\mathbb{F}_ p)\). Thus the author proves the finiteness of this group. The main tool of the proof is an extension of Kolyvagin's technique for the usual Tate-Shafarevich group. This paper is becoming widely known because in the end of June A. Wiles has announced the proof of the Taniyama-Weil conjecture for stable elliptic curves. The key ingredient of his proof is a generalization of the Flach theorem giving an exact value for the Tate-Shafarevich group.
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14G40
zbMATH DE Number: 220739
zbMATH Keywords: motivic \(L\)-function
zbMATH Keywords: Tate module
zbMATH Keywords: Tate-Shafarevich group
zbMATH Keywords: modular elliptic curve
zbMATH Keywords: Taniyama-Weil conjecture
zbMATH Keywords: Flach theorem
reviewed by: A. N. Parshin
MaRDI profile type: Publication
cites work: Gersten's conjecture and the homology of schemes
cites work: Q3972724
cites work: Q3708977
cites work: Iwasawa theory for the symmetric square of an elliptic curve.
cites work: Les Schémas de Modules de Courbes Elliptiques
cites work: Q3972629
cites work: A generalisation of the Cassel-Tate pairing.
cites work: Q3972775
cites work: Congruences of cusp forms and special values of their zeta functions
cites work: Cohomology of Group Extensions
cites work: Continuous étale cohomology
cites work: Mixed motives and algebraic K-theory. (Almost unchanged version of the author's habilitation at Univ. Regensburg 1988)
cites work: Arithmetic Moduli of Elliptic Curves. (AM-108)
cites work: Q3972985
cites work: Cycles in a product of elliptic curves, and a group analogous to the class group
cites work: Q3754095
cites work: Higher algebraic K-theory: I
cites work: Q3491711
cites work: On modular units
cites work: Galois properties of points of finite order of elliptic curves
cites work: Some theorems on the k-theory of coherent sheaves
cites work: The special values of the zeta functions associated with cusp forms
cites work: Modular Parametrizations of Elliptic Curves
description: scientific article; zbMATH DE number 220739