Elliptic curves with large Shafarevich-Tate group
The main result of the paper is the existence of infinitely many rational elliptic curves with big Tate-Shafarevich group, under strong hypothesis. More precisely, suppose that the rank zero case of the Birch and Swinnerton-Dyer conjecture holds, and that the the Goldfeld-Szapiro conjecture holds. Then for any real number \(\varepsilon>0\) there are infinitely many elliptic curves \(E\) defined over \(\mathbb Q\) such that the order of their Shafarevich-Tate group is bigger than a constant depending on \(\varepsilon\) only times the absolute value of the discriminant of \(E\) raised to the power \(1/2-\varepsilon\); in a compact form, the statement is \[ |\text{Ш}(E)|\geq h(\varepsilon)\cdot |\Delta(E)|^{1/2-\varepsilon} \] where \(\text{Ш}(E)\) is the Shafarevich-Tate group of \(E\), \(h(\varepsilon)\) is a constant depending only on \(\varepsilon\), and \(\Delta(E)\) is the global minimal discriminant of \(E\). As a corollary, the authors improve a result of \textit{B. M. M. de Weger} [Q. J. Math., Oxf. II. Ser. 49, No. 193, 105--128 (1998; Zbl 0917.11020)].
scientific article; zbMATH DE number 6124169