The p-part of Tate-Shafarevich groups of elliptic curves can be arbitrarily large

2003/03/12 by Remke Kloosterman, Kloosterman, Remke
Mathematics · #11G05 (Primary) #11G18 (Secondary) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Number Theory (math.NT) #math.NT #msc:11G05 #msc:11G18

paper · pdf · doi:10.48550/arxiv.math/0303143

Second version; The final section has been changed to correct a mistake in the first version. Some reference are added

openalex publication_date 2003/03/12 · arxiv created 2003/05/12 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper it is shown that for every prime p>5 the dimension of the p-torsion in the Tate-Shafarevich group of E/K can be arbitrarily large, where E is an elliptic curve defined over a number field K, with [K:Q] bounded by a constant depending only on p. From this we deduce that the dimension of the p-torsion in the Tate-Shafarevich group of A/Q can be arbitrarily large, where A is an abelian variety, with dim A bounded by a constant depending only on p.

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• Elliptic curves with large Tate-Shafarevich groups over \mathbbF_q(t)
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