Prime Torsion in the Brauer Group of an Elliptic Curve

TL;DR

This paper presents an algorithm to explicitly compute the q-torsion elements of the Brauer group of an elliptic curve over fields with certain roots of unity, providing insights into their structure and triviality conditions.

Contribution

It introduces a new algorithm for determining q-torsion elements in the Brauer group of elliptic curves and establishes bounds on their symbol length.

Findings

Algorithm for q-torsion elements in Brauer group

Conditions for triviality of Brauer classes

Upper bounds on symbol length of prime torsion

Abstract

We give an algorithm to explicitly determine all elements of the $q$-torsion (for $q$ an odd prime) of the Brauer group of an elliptic curve over any base field of characteristic different from $q$, containing a primitive $q$-th root of unity. These elements of the Brauer group are given as tensor products of symbol algebras over the function field of the elliptic curve. We give sufficient conditions to determine if the Brauer classes that arise are trivial. Using our algorithm, we derive an upper bound on the symbol length of the prime torsion of $\mathrm{Br}(E)/\mathrm{Br}(k)$.