A new family of elliptic curves with unbounded rank

Abstract: Let

m a t h b b F_{q}

be a finite field of odd characteristic and

K = m a t h b b F_{q} (t)

. For any integer

d g e q 2

coprime to

q

, consider the elliptic curve

E_{d}

over

K

defined by

y^{2} = x (x^{2} + t^{2 d} x - 4 t^{2 d})

. 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

.