Selmer Groups over p-adic Lie Extensions I

Let $E$ be an elliptic curve defined over a number field $F$. In this paper, we study the structure of the $p^{\mathrm{\infty }}$-Selmer group of $E$ over $p$-adic Lie extensions $F_{\mathrm{\infty }}$ 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 $\text{Gal}(F_{\mathrm{\infty }}\mathrm{/}F)$-Euler characteristic of the $p^{\mathrm{\infty }}$-Selmer group of $E$. The final section illustrates the main theory with the example of an elliptic curve of conductor 294.