Genus one curves and Brauer-Severi varieties

TL;DR

This paper constructs explicit genus 1 curves over a field K that split a given central simple algebra, specifically when the associated Brauer-Severi variety has dimension at most 4, advancing understanding of the algebra-geometry connection.

Contribution

It provides an explicit construction of genus 1 curves splitting a central simple algebra for Brauer-Severi varieties of dimension up to 4.

Findings

Constructed explicit genus 1 curves splitting the algebra

Applicable to Brauer-Severi varieties of dimension ≤ 4

Advances the understanding of algebraic structures via genus 1 curves

Abstract

Let K be a field. Let A be a central simple algebra over K and let X be the associated Brauer-Severi variety over K. It has recently been asked if there exists a genus 1 curve C over K such that K(C) splits A. In other words, is there a genus one curve C over K with a morphism to X? In this short note, we explicitly construct such a curve in the case where X has dimension at most 4 (equivalently, when A has degree at most 5).