# Quadratic twists of abelian varieties and disparity in Selmer ranks

## TL;DR

This paper investigates the distribution of 2-Selmer rank parities in quadratic twists of abelian varieties, extending previous work on elliptic curves and hyperelliptic Jacobians, and explores how Shafarevich--Tate group properties influence these statistics.

## Contribution

It generalizes existing results on Selmer rank parity distributions to a broader class of abelian varieties, accounting for Shafarevich--Tate group variations.

## Key findings

- Determined the proportion of quadratic twists with odd and even 2-Selmer ranks.
- Identified differences in the statistics due to Shafarevich--Tate group order considerations.
- Described the relationship between parities of 2-Selmer ranks and 2-infinity Selmer ranks.

## Abstract

We study the parity of 2-Selmer ranks in the family of quadratic twists of a fixed principally polarised abelian variety over a number field. Specifically, we determine the proportion of twists having odd (resp. even) 2-Selmer rank. This generalises work of Klagsbrun--Mazur--Rubin for elliptic curves and Yu for Jacobians of hyperelliptic curves. Several differences in the statistics arise due to the possibility that the Shafarevich--Tate group (if finite) may have order twice a square. In particular, the statistics for parities of 2-Selmer ranks and 2-infinity Selmer ranks need no longer agree and we describe both.