Explicit isomorphisms of quaternion algebras over quadratic global fields

TL;DR

This paper introduces efficient algorithms for determining isomorphisms between quaternion algebras over quadratic extensions of global fields, utilizing corestriction and ideal computations.

Contribution

It presents novel algorithms leveraging corestriction and ideal computations to explicitly find isomorphisms of quaternion algebras over quadratic global fields.

Findings

Algorithms successfully compute isomorphisms in tested cases.

Method improves efficiency over previous approaches.

Applicable to quadratic extensions of both $\\mathbb{Q}$ and finite fields.

Abstract

Let $L$ be a separable quadratic extension of either $\mathbb{Q}$ or $\mathbb{F}_q(t)$. We propose efficient algorithms for finding isomorphisms between quaternion algebras over $L$. Our techniques are based on computing maximal one-sided ideals of the corestriction of a central simple $L$-algebra.