Finiteness of the Tate-Shafarevich Groups for Elliptic Curves over the Field of Rational Numbers

2013/09/29 by Lan Nguyen, Nguyen, Lan
Mathematics · #11C08 #11D09 #11D25 #11G05 #11G07 #11Y16 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1309.7675

openalex publication_date 2013/09/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let E be an elliptic curve over ℚ. Let \underlineIII(E) be a certain group of equivalence classes of homogeneous spaces of E called its Tate-Shafarevich group. We show in this paper that this group has finite cardinality and discuss its role in the Birch and Swinnerton-Dyer Conjecture. In particular, our result implies the Parity Conjecture, or the Birch and Swinnerton-Dyer Conjecture modulo 2. It also removes the finiteness condition of \underlineIII(E) from previous results in the literature of this subject and makes possible for some computation problems concerning the Strong Birch and Swinnerton-Dyer Conjecture. In addition, we also give an analogue of the Hasse-Minkowski Theorem for cubic plane curves.

Related

• On the vanishing of almost all primary components of the Shafarevich-Tate group of elliptic curves over the rationals
• Elliptic curves with large Tate-Shafarevich groups over \mathbbF_q(t)
• On The Birch and Swinnerton-Dyer Conjecture for CM Elliptic Curves over \BQ
• Reduction of a problem of finiteness of Tate-Shafarevich group to a result of Zagier type
• Birch and Swinnerton-Dyer conjecture in the complex multiplication case and the congruent number problem