In this short note, by assuming the Birch and Swinnerton-Dyer conjecture for all quadratic twists \(E_ D\) of a fixed modular curve \(E\) over \(\mathbb{Q}\), the authors prove that there exist infinitely many \(D\) and \(D'\) such that \(E_ D\) has rank 1 and \(| \text{ Ш}_ D | R_ D \gg_ \varepsilon (N_ D)^{1/4 - \varepsilon}\), where the implied constant depends on \(\varepsilon\); and \(E_{D'}\) has rank 0 and \(| \text{ Ш}_{D'} | R_{D'} \gg N_{D'}^{1/4 - \varepsilon}\). Here \(\text{ Ш} (\text{ Ш}_{D'})\), \(R_ D\), \(N_ D (N_{D'})\) are the Shafarevich-Tate group, the regulator, and the conductor of \(E_ D (E_{D'})\), respectively.
For the entire collection see [Zbl 0788.00052].