Explicit points on the legendre curve II

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(p^f - 1) and 2(p ^f - 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(t^1/d)) of finite index in the "2-new" part, and we bound the index as well as the order of the "2-new" part of the Tate-Shafarevich group.