Arbitrarily large p-torsion in Tate-Shafarevich groups

2022/09/16 by Flynn, E. Victor, Shnidman, Ari
#14H40 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #Primary 11G30 #Secondary 11G10

paper · doi:10.48550/arxiv.2209.08088

Abstract

We show that, for any prime p, there exist absolutely simple abelian varieties over ℚ with arbitrarily large p-torsion in their Tate-Shafarevich group. To prove this, we construct explicit μ_p-covers of Jacobians of the form y^p = x(x-1)(x-a) which violate the Hasse principle. In the appendix, Tom Fisher explains how to interpret our proof in terms of a Cassels-Tate pairing.

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