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Explicit points on the Legendre curve II (1307.4251v2)
Published 16 Jul 2013 in math.NT
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 $\Fq(t{1/d})$ 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 $(\Z/d\Z)\times$ 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 $E(\Fq(t{1/d}))$ 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.