Explicit points on the Legendre curve
Douglas Ulmer

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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Algebra and Geometry · Analytic Number Theory Research
