Infinitely many pairs of non-isomorphic elliptic curves sharing the same BSD invariants

Abstract

Let \(E/\mathbb{Q}\) be an elliptic curve. The Birch and Swinnerton--Dyer (BSD) conjecture relates the leading coefficient of the Taylor expansion of the \(L\)-function of \(E/\mathbb{Q}\) at \(s=1\) to arithmetic invariants of \(E\), such as its Mordell--Weil group, its Tate--Shafarevich group, its Tamagawa numbers, its regulator, and its real period. We call two non-isomorphic elliptic curves over \(\mathbb{Q}\) BSD twins if they have the same \(L\)-function and the same arithmetic data underlying the BSD invariants appearing in the BSD conjecture, with the Mordell--Weil group and the Tate--Shafarevich group compared as groups. We exhibit a family of BSD twins for which the corresponding pairs of \(j\)-invariants are pairwise distinct. We further prove that, even after imposing equality of the Kodaira symbols at every prime and equality of the minimal discriminants, infinitely many BSD twins still exist.