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A finiteness theorem for the symmetric square of an elliptic curve. (English) Zbl 0781.14022
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.

14H52 Elliptic curves
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
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