Generalizing Hurwitz's quaternionic proof of Lagrange's and Jacobi's four-square theorems

TL;DR

This paper extends Hurwitz's quaternionic proof of classical four-square theorems to more general number fields, identifying key algebraic conditions and using recent class number results to derive universality and representation formulas.

Contribution

It generalizes Hurwitz's quaternionic approach to quadratic forms over number fields, providing new criteria and explicit formulas for representations.

Findings

Identified conditions for suborders satisfying the orbit condition

Derived universality results for quadratic forms over number fields

Provided a quaternionic proof of Götzky's four-square theorem

Abstract

A proof of Lagrange's and Jacobi's four-square theorem due to Hurwitz utilizes orders in a quaternion algebra over the rationals. Seeking a generalization of this technique to orders over number fields, we identify two key components: an order with a good factorization theory and the condition that all orbits under the action of the group of elements of norm $1$ acting by multiplication intersect the suborder corresponding to the quadratic form to be studied. We use recent results on class numbers of quaternion orders and then find all suborders satisfying the orbit condition. Subsequently, we obtain universality and formulas for the number of representations by the corresponding quadratic forms. We also present a quaternionic proof of G\"otzky's four-square theorem.