Explicit points on the Legendre curve II

TL;DR

This paper investigates the rank and explicit points on a specific elliptic curve over function fields, revealing conditions for large rank and providing explicit generators and bounds related to the Tate-Shafarevich group.

Contribution

It offers new explicit constructions of points on the Legendre curve over certain extensions and characterizes the rank in terms of subgroup properties, extending previous results.

Findings

Rank is large for many values of d, especially when d divides 2(p^f-1) with specific parity conditions.

Explicit points generating a subgroup of finite index are constructed for d=2(p^f-1).

Bounds are provided for the index and the Tate-Shafarevich group related to these points.

Abstract

Let $E$ be the elliptic curve $y^2=x(x+1)(x+t)$ 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 $2(p^f-1)$ and $2(p^f-1)/d$ is odd, then the rank is at least $d/2$. When $d=2(p^f-1)$, 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.