Moscow Mathematical Journal

Volume 20, Issue 2, April–June 2020 pp. 343–374.

A New Family of Elliptic Curves with Unbounded Rank

Authors: Richard Griffon (1)
Author institution:(1) Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, 4051 Basel, Switzerland

Summary:

Let $\mathbb{F}_q$ be a finite field of odd characteristic and $K= \mathbb{F}_q(t)$ . For any integer $d\geq 1$ , consider the elliptic curve $E_d$ over $K$ defined by $y^2=x\cdot\big(x^2+t^{2d}\cdot x-4t^{2d}\big)$ . We show that the rank of the Mordell–Weil group $E_d(K)$ is unbounded as $d$ varies. The curve $E_d$ satisfies the BSD conjecture, so that its rank equals the order of vanishing of its $L$ -function at the central point. We provide an explicit expression for the $L$ -function of $E_d$ , and use it to study this order of vanishing in terms of $d$ .

2010 Math. Subj. Class. 11G05, 11M38, 11G40, 14G10, 11L05.

Keywords: Elliptic curves over function fields, explicit computation of L-functions, BSD conjecture, unbounded ranks, explicit Jacobi sums.

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