Explicit points on the Legendre curve III

Ulmer, Douglas

doi:10.2140/ant.2014.8.2471

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Explicit points on the Legendre curve III

Douglas Ulmer

Vol. 8 (2014), No. 10, 2471–2522

DOI: 10.2140/ant.2014.8.2471

Abstract

We continue our study of the Legendre elliptic curve $y^{2} = x (x + 1) (x + t)$ over function fields $K_{d} = F_{p} (μ_{d}, t^{1 ∕ d})$ . When $d = p^{f} + 1$ , we have previously exhibited explicit points generating a subgroup $V_{d} \subset E (K_{d})$ of rank $d - 2$ and of finite, $p$ -power index. We also proved the finiteness of Ш $(E ∕ K_{d})$ and a class number formula: ${[E (K_{d}) : V_{d}]}^{2} = | Ш(E ∕ K_{d}) |$ . In this paper, we compute $E (K_{d}) ∕ V_{d}$ and Ш $(E ∕ K_{d})$ explicitly as modules over $ℤ_{p} [Gal (K_{d} ∕ F_{p} (t))]$ .

An errata was posted on 31 May 2017 in an online supplement.

Keywords

elliptic curves, function fields, Tate–Shafarevich group

Mathematical Subject Classification 2010

Primary: 11G05, 14G05

Secondary: 11G40, 14K15

Milestones

Received: 26 June 2014

Revised: 20 October 2014

Accepted: 23 November 2014

Published: 31 December 2014

Authors

Douglas Ulmer
School of Mathematics Georgia Institute of Technology 686 Cherry Street Atlanta, GA 30332 United States

Vol. 8, No. 10, 2014