On Twists of A Family of Elliptic Curves and Their $L-$Function

Abstract

Let $E$ be an elliptic curve defined over a number field, the conjecture of Birch and Swinnerton-Dyer (BSD, for short) asserts a deep relation between the group $E(K)$ of rational points and the $L-$function $L(E/K,s)$ of $E$ at $s=1.$ Very few explicit results about $E(K)$ and $L(1)$ are known, even no general method is known to determine $L(1)$ vanishing or not for a given elliptic curve. In this paper, we study some quantities related to BSD of a special class of elliptic curves, more precisely, we study the arithmetic of quadratic twists of elliptic curves $y^2=x(x+\epsilon p)(x+\epsilon q)$ and their $L-$function. Based on some classical works, especially those of Greenberg, Kramer-Tunnell, Kato-Rohrlich, Manin and Mazur, under some conditions, we obtain results about the vanishing of the value at $s=1$ of the $L$-function, and explicitly determine the following quantities: the norm index $ δ(E, \Q, K), $ the root numbers, the set of anomalous prime numbers, a few prime numbers at which the image of Galois representation are surjective. We also study the relation between the ranks of the Mordell-Weil groups, Selmer groups and Shafarevich-Tate groups, and the structure about the $l^{\infty }-$Selmer groups and the Mordell-Weil groups over ${\mathbb{Z}}_l-$extension via Iwasawa theory. These results provide some useful evidence toward verifying the BSD for a family of elliptic curves.

38 pages