  Let $E$ be an elliptic curve defined over a number field $F$. In this paper,
we study the structure of the $p^\infty$-Selmer group of $E$ over $p$-adic Lie
extensions $F_\infty$ of $F$ which are obtained by adjoining to $F$ the
$p$-division points of an abelian variety $A$ defined over $F$. The main focus
of the paper is the calculation of the $\Gal(F_\infty/F)$-Euler characteristic
of the $p^\infty$-Selmer group of $E$. The final section illustrates the main
theory with the example of an elliptic curve of conductor 294.
