Sign involutions on para-abelian varieties

TL;DR

This paper investigates sign involutions on twisted forms of abelian varieties, establishing conditions for their existence, and explores geometric properties of the resulting quotients, including their Picard schemes and embeddings of genus-one curves.

Contribution

It characterizes when sign involutions exist on twisted abelian varieties and analyzes the geometric structure of their quotients, including Picard schemes and embeddings.

Findings

Sign involutions exist iff the Weil--Châtelet class is 2-torsion.

The Picard scheme of the quotient is étale and torsion-free.

In dimension one, quotients are Brauer--Severi curves with specific embeddings.

Abstract

We study the so-called sign involutions on twisted forms of abelian varieties, and show that such a sign involution exists if and only if the class in the Weil--Ch\^{a}telet group is annihilated by two. If these equivalent conditions hold, we prove that the Picard scheme of the quotient is \'etale and contains no points of finite order. In dimension one, such quotients are Brauer--Severi curves, and we analyze the ensuing embeddings of the genus-one curve into twisted forms of Hirzebruch surfaces and weighted projective spaces.