Metacommutation as a group action on the projective line over \(\mathbb F_p\) (Q2821720)

Let \(\mathcal H\) be the (left and right) principal ideal domain of integers in the quaternion skew-field \(\mathbb H\). The prime elements, or the (left) prime ideals of \(\mathcal H\), are called Hurwitz primes. An element of \(\mathcal H\) is prime if and only if its norm is a rational prime. Factorization into Hurwitz primes is unique up to three phenomena, the most interesting one being metacommutation. This means that the product \(PQ\) of two primes lying over distinct rational primes can be switched to a product \(Q'P'\) with \(P,P'\) and \(Q,Q'\) lying over the same primes, respectively. Some interesting properties of metacommutation were discovered recently by \textit{H. Cohn} and \textit{A. Kumar} [ibid. 143, No. 4, 1459--1469 (2015; Zbl 1309.11080)]. For example, the sign of the metacommutation permutation with respect to distinct rational primes \(p,q\not=2\) is given by the Legendre symbol \(\bigl(\frac{p}{q}\bigr)\).NEWLINENEWLINEIn this nice paper under review, metacommutation is expressed by the standard action of \(\mathrm{PGL}_2(\mathbb F_p)\) on the projective line over the prime field \(\mathbb F_p\), which leads to simplified proofs of Cohn and Kumar's results [loc. cit.].