A family of elliptic curves of large rank
Let \(k\) be a finite field and \(d\) a positive integer not divisible by the characteristic of \(k\). For each \(a \in k\) let \(E\) be the elliptic curve defined by the equation \(x(x-1)(y-a) = t^dy(y-1)(x-a)\) over \(k(t)\). In this article it is shown that the conjecture of Birch and Swinnerton-Dyer holds for \(E\), and that \(E\) has rank at least \(d\) over \(\mathbb F_q(t)\) if \(d\) divides \(q-1\), and that the rank over \(\overline{\mathbb F_q}(t)\) is at least \(d\) if \(d\) is coprime to \(q\).
scientific article; zbMATH DE number 6015646