Explicit points on the Legendre curve III

Abstract

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

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