EXPLICIT POINTS ON THE LEGENDRE CURVE II

ABSTRACT. Let E be the elliptic curve y2 = x(x + 1)(x + t) over the field Fp(t) where p is an odd prime. We study the arithmetic of E over extensions Fq(t1/d) where q is a power of p and d is an integer prime to p. The rank of E is given in terms of an elementary property of the subgroup of (Z/dZ) × generated by p. We show that for many values of d the rank is large. For example, if d divides 2(pf − 1) and 2(pf − 1)/d is odd, then the rank is at least d/2. When d = 2(pf − 1), we exhibit explicit points generating a subgroup of E(Fq(t1/d)) of finite index, and we bound the index as well as the order of the Tate-Shafarevich group. 1