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

Ricardo P. Conceição (Oxford College of Emory University, Oxford, Georgia, U.S.A.)

Chris Hall (Department of Mathematics, University of Wyoming, Laramie, Wy., U.S.A.)

Douglas Ulmer (School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia, U.S.A.)

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.

Full Text (PDF format)