Explicit points on the Legendre curve

TL;DR

This paper explicitly constructs points on a specific elliptic curve over function fields, proves their subgroup's properties, and confirms the Birch and Swinnerton-Dyer conjecture in certain cases, revealing deep arithmetic insights.

Contribution

It provides explicit points on the Legendre elliptic curve over function fields and proves the BSD conjecture for these cases using elementary and advanced methods.

Findings

Explicit points generate a subgroup of rank d-2 with finite index

BSD conjecture holds for the curve over certain extensions

Rank of the curve is zero in characteristic zero cases

Abstract

We study the elliptic curve E given by y^2=x(x+1)(x+t) over the rational function field k(t) and its extensions K_d=k(\mu_d,t^{1/d}). When k is finite of characteristic p and d=p^f+1, we write down explicit points on E and show by elementary arguments that they generate a subgroup V_d of rank d-2 and of finite index in E(K_d). Using more sophisticated methods, we then show that the Birch and Swinnerton-Dyer conjecture holds for E over K_d, and we relate the index of V_d in E(K_d) to the order of the Tate-Shafarevich group \sha(E/K_d). When k has characteristic 0, we show that E has rank 0 over K_d for all d.