Quaternion Orders and Diophantine Equations

The main goal of this talk is to illustrate the role of quaternions in number theory. We propose to do this in two parts: first by studying the concept of a Non-Commutative Principal Ideal Domains (PID) in Quaternion Algebras, and, after, by using quaternions to prove the universality (or not) of some Diophantine equations. More precisely, this talk starts with a few results about factorization in quaternion orders. Then, we present a finite algorithm to determine if a given order is a PID, based on a criterion attributed to Dedekind and Hasse. Lastly, some examples of orders in order are analysed, in order to see how one can use the concepts addressed in the first part of the talk to work with Diophantine equations. Namely, we show that the form t²+2x²+5y²+10z² is universal and that the form t²+x²+7y²+7z² represents all natural numbers except those of a specific form.