A NEW FAMILY OF ELLIPTIC CURVES WITH UNBOUNDED RANK

摘要

Let F-q be a finite field of odd characteristic and K = Fq(t). For any integer d >= 1, consider the elliptic curve E-d over K defined by y(2) = x center dot (x(2) + t(2d) center dot x - 4t(2d)). We show that the rank of the Mordell-Weil group E-d(K) is unbounded as d varies. The curve Ed 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 Ed(,) and use it to study this order of vanishing in terms of d.