Symmetric Squares of Elliptic Curves: Rational Points and Selmer Groups

Abstract

We consider the Bloch-Kato conjecture applied to the symmetric square L-function of an elliptic curve over $\QQ$, at $s=2$. In particular, we use a construction of elements of order l in a generalised Shafarevich-Tate group, which works when E has a rational point of infinite order and a rational point of order l. The existence of the latter places us in a situation where the recent theorem of Diamond, Flach, and Guo does not apply, but we find that the numerical evidence is quite convincing.