Explicit points on the Legendre curve III

TL;DR

This paper extends the study of explicit points on Legendre elliptic curves over certain function fields, providing detailed module computations of the quotient groups and class groups, and deepening understanding of their algebraic structure.

Contribution

It explicitly computes the structure of the quotient group and the Tate-Shafarevich group for Legendre elliptic curves over specific function fields, building on previous explicit point constructions.

Findings

Explicit description of E(K_d)/V_d as a module over the Galois group

Explicit computation of the Tate-Shafarevich group III(E/K_d)

Confirmation of the class number formula in this setting

Abstract

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