A new family of elliptic curves with unbounded rank

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 $d$.