Torsion in Tate-Shafarevich groups of abelian varieties

Abstract

We say that an abelian variety satisfies the Hasse principle if it has a QQ-rational point whenever it has \RR-rational point and a QQ_p-rational point for all primes p. For every abelian variety it is possible to construct its Tate-Shaferevich group. The elements of this group correspond with twists of the abelian variety that violate the Hasse principle. So the study of abelian varieties that violate the Hasse principle is equivalent to the study of their Tate-Shafarevich group.

In a recent paper by Flynn and Shnidman (2022) it is shown that for any prime p>3 there exist absolutely simple abelian varieties over QQ with arbitrarily large -p-torsion in their Tate-Shafarevich group. In this thesis we will examine how Flynn and Shnidman achieved this result by constructing explicit mu_p-covers of certain Jacobians that violated the Hasse principle. Furthermore, we will explore how we can generalize these results to arbitrary integers n and what the mu_n-covers of such Jacobians look like in this case.

Additionally, we examine the case p=2 by approaching a paper by Lemmermeyer and Mollin (2003) using the language of mu_2-covers and contrasting the results we acquire with the results of the paper.