A new family of elliptic curves with unbounded rank

TL;DR

This paper constructs a family of elliptic curves over function fields with unbounded Mordell--Weil rank, explicitly computes their $L$-functions, and confirms they satisfy the BSD conjecture, revealing new insights into rank growth and $L$-function behavior.

Contribution

It introduces a new family of elliptic curves over function fields with unbounded rank and provides explicit $L$-function formulas, advancing understanding of rank variation and BSD in this context.

Findings

Rank of $E_d(K)$ is unbounded as $d$ varies.

Explicit $L$-function expressions are derived.

Curves satisfy BSD conjecture, linking rank to $L$-function vanishing.

Abstract

Let $\mathbb{F}_q$ be a finite field of odd characteristic and $K= \mathbb{F}_q(t)$. For any integer $d\geq 2$ coprime to $q$, consider the elliptic curve $E_d$ over $K$ defined by $y^2=x(x^2+t^{2d} x-4t^{2d})$. 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 $d$.