A note on the order of the Tate--Shafarevich group modulo squares

2024/04/25 by Alexandros Konstantinou, Konstantinou, Alexandros
Engineering · Materials Science · Mathematics · #11G10 (Primary) #11G40 (Secondary) #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #Quasicrystal Structures and Properties #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2404.16785

openalex publication_date 2024/04/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We investigate the order of the Tate--Shafarevich group of abelian varieties modulo rational squares. Our main result shows that every square-free natural number appears as the non square-free part of the Tate--Shafarevich group of some abelian variety, thereby validating a conjecture of W. Stein.

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