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

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p \in \{3,5,7\}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${E/\mathbb{Q}}$$\end{document} an elliptic curve with a rational point P of order p. Let D be a square-free integer and ED the D-quadratic twist of E. Vatsal (Duke Math J 98:397–419, 1999) found some conditions such that ED has (analytic) rank zero and Frey (Can J Math 40:649–665, 1988) found some conditions such that the p-Selmer group of ED is trivial. In this paper, we will consider a family of ED 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 ED. As a corollary we will show that there are infinitely many elliptic curves \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${E/\mathbb{Q}}$$\end{document} such that for a positive portion of D, ED 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).