Vol. 8, No. 10, 2014

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Explicit points on the Legendre curve III

Douglas Ulmer

Vol. 8 (2014), No. 10, 2471–2522
Abstract

We continue our study of the Legendre elliptic curve y2 = x(x + 1)(x + t) over function fields Kd = Fp(μd,t1d). When d = pf + 1, we have previously exhibited explicit points generating a subgroup V d E(Kd) of rank d 2 and of finite, p-power index. We also proved the finiteness of Ш(EKd) and a class number formula: [E(Kd) : V d]2 = |Ш(EKd)|. In this paper, we compute E(Kd)V d and Ш(EKd) explicitly as modules over p[Gal(KdFp(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