Elliptic curves of rank zero satisfying the p-part of the Birch and Swinnerton-Dyer conjecture

Abstract

Let \({p \in \{3,5,7\}}\) and \({E/\mathbb{Q}}\) an elliptic curve with a rational point P of order p. Let D be a square-free integer and E D the D-quadratic twist of E. Vatsal (Duke Math J 98:397–419, 1999) found some conditions such that E D has (analytic) rank zero and Frey (Can J Math 40:649–665, 1988) found some conditions such that the p-Selmer group of E D is trivial. In this paper, we will consider a family of E D satisfying both of the conditions of Vatsal and Frey and show that the p-part of the Birch and Swinnerton-Dyer conjecture is true for these elliptic curves E D . As a corollary we will show that there are infinitely many elliptic curves \({E/\mathbb{Q}}\) such that for a positive portion of D, E D has rank zero and satisfies the 3-part of the Birch and Swinnerton-Dyer conjecture. Previously only a finite number of such curves were known, due to James (J Number Theory 15:199–202, 1982).