Explicit points on the Legendre curve

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10 Scopus citations

Abstract

We study the elliptic curve E given by y2 = x(x + 1)(x + t) over the rational function field k(t) and its extensions K d = k(μd, t1/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 Sh{cyrillic}(E/K d). When k has characteristic 0, we show that E has rank 0 over K d for all d.

Original languageEnglish (US)
Pages (from-to)165-194
Number of pages30
JournalJournal of Number Theory
Volume136
DOIs
StatePublished - Mar 1 2014
Externally publishedYes

Keywords

  • Conjecture of Birch and Swinnerton-Dyer
  • Elliptic curves
  • Ranks

ASJC Scopus subject areas

  • Algebra and Number Theory

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