## 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.

### MSC:

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

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