The Period and Index of a Generic Geometrically Elliptic Normal Curve

TL;DR

This paper constructs and analyzes genus one curves on certain algebraic varieties, demonstrating that all period and index combinations are achievable for these curves.

Contribution

It introduces a method to construct versal genus one curves on generic Severi--Brauer varieties and computes their period and index, showing all combinations are possible.

Findings

All period and index combinations are realizable for these curves.

Constructs versal genus one curves on Severi--Brauer varieties.

Provides explicit computations of periods and indices.

Abstract

We construct genus one curves on base extensions of generic Severi--Brauer varieties of a given index and period which are versal objects for families of geometrically elliptic normal curves. We also compute the periods and indices of these curves showing that all possible period/index combinations are possible.