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# Mathematical Research Letters

## Volume 21 (2014)

### Number 2

### Explicit points on the Legendre curve II

Pages: 261 – 280

DOI: http://dx.doi.org/10.4310/MRL.2014.v21.n2.a5

#### Authors

#### Abstract

Let $E$ be the elliptic curve $y^2=x(x+1)(x+t)$ over the field $\mathbb{F}_p(t)$, where $p$ is an odd prime. We study the arithmetic of $E$ over extensions $\mathbb{F}_q(t^{1/d})$, 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 $(\mathbb{Z} / d\mathbb{Z})^\times$ 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 $E(\mathbb{F}_q(t^{1/d}))$ 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.

#### 2010 Mathematics Subject Classification

Primary 11G40, 14G05. Secondary 11G05, 14G10, 14G25, 14K15.