On Shafarevich-Tate groups and the arithmetic of Fermat Curves

2002/10/31 by William G. McCallum, McCallum, William G., Pavlos Tzermias +1
Mathematics · #11G10 #11G30 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT #msc:11G10 #msc:11G30

paper · pdf · doi:10.48550/arxiv.math/0210483

Added missing bibliography, other minor changes to title and abstract

arxiv created 2002/10/31 · openalex publication_date 2002/10/31 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We extend the results of the first author on nontrivial elements in the Shafarevich-Tate group of the jacobian of a quotient of a Fermat curve of prime degree, and use the methods of the second author to derive a result bounding the quadratic points on the Fermat curve of degree 19.

Related

• Root numbers and Selmer groups for the Jacobian varieties of Fermat curves
• Behaviour of the order of Tate-Shafarevich groups for the quadratic twists of elliptic curves
• The Tate-Shafarevich group for elliptic curves with complex multiplication II
• Large Tate--Shafarevich orders from good abc triples
• A remark on the decomposition of the Jacobian variety of Fermat curves of prime degree