Torsion points on Elliptic Curves and the Jackson Space

TL;DR

This paper explores the use of a non-commutative scheme to analyze the ramification and reduction properties of p-torsion points on elliptic curves, and their relation to the Brauer group.

Contribution

It introduces a novel non-commutative geometric framework to study p-torsion points and their applications in classifying elements of the Brauer group.

Findings

Characterizes ramification properties of p-torsion points

Shows parametrization of Brauer group classes using p-torsion points

Establishes reduction behavior of elliptic curve torsion points

Abstract

In this paper we will use a particular non-commutative scheme to, among other things, study the ramification properties of the field of $p$-torsion points on an elliptic curve and its reduction properties. Also, we show that this non-commutative space also allow us to use the $p$-torsion points to ``parametrise'' classes in the $p$-torsion of the Brauer group of the base field.