Explicit points on the Legendre curve

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(µ 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 X(E/K d ). When k has characteristic 0, we show that E has rank 0 over K d for all d.We note that in [Ca09], Conceição gives an equally simple construction of polynomial points on certain isotrivial elliptic curves over F p (t) and uses them to show that the rank is large.

Explicit points on the Legendre curve III

“…Remark. We showed in [Ulm14a,Cor. 4.3] that the group V d appearing in Theorem 1.1 is a cyclic module over…”

Analogue of the Brauer–Siegel theorem for Legendre elliptic curves

“…Remark 2.1 We follow [Ulm14] in calling E d a Legendre curve: see [Ulm14, §2] for more comments on this choice of terminology. We also note the slight change in points of view compared to [Ulm14], [CHU14]: instead of considering a fixed curve E 1 over a varying field F q (t 1/d ), we fix the base field F q (t) and vary the curve E d . This is only a matter of convenience, and has no influence on the results.…”

“…The proof goes roughly as follows (see [Ulm14,§11] and [Ulm13, §7] for more details). We denote by π : E d → P 1 the minimal regular model of E d .…”

“…The proof of (3.2) in [CHU14, §3] hinges on a clever manipulation of character sums. Since the minimal regular model of E d /K is explicitely known (see [Ulm14,§7]), the computation can also be done via cohomological methods. Though less elementary, the latter has the advantage of "explaining" the appearance of Jacobi sums in the L-function.…”

“…To conclude this introduction, let us give the plan of the paper, as well as a rough sketch of the proof of Theorem 1.1. The Legendre elliptic curves E d have been previously studied by Ulmer, Conceição and Hall in a series of papers ( [Ulm14], [CHU14], ...). In particular, they proved that E d satisfies the Birch and Swinnerton-Dyer conjecture (henceforth abbreviated as BSD).…”