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 + \varepsilon p )(x + \varepsilon 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 $ \delta (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 $ \Z_{l}-$extension via Iwasawa theory. These results provide some useful evidence toward verifying the BSD for a family of elliptic curves.