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

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## 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).

## Mathematics Subject Classification

11G05 11G40## Preview

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