Explicit points on the Legendre curve II

الأصل · EN

Let E be the elliptic curve y²=x(x+1)(x+t) over the field (t) where p is an odd prime. We study the arithmetic of E over extensions (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 (/d)× 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((t¹/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.