Geometry of the $p$-adic special orthogonal group $SO(3)_p$

TL;DR

This paper explicitly characterizes the structure of $p$-adic special orthogonal groups in three dimensions, revealing their rotation properties, subgroup classifications, and the existence of angle parametrizations for odd primes, but not for $p=2$.

Contribution

It provides a detailed structural analysis of $SO(3)_p$ for all primes, including subgroup classifications and angle parametrizations, extending understanding of $p$-adic orthogonal groups.

Findings

Every element of $SO(3)_p$ is a rotation around an axis.

Rotation subgroups are abelian and parametrized by the projective line.

For odd primes, $SO(3)_p$ admits a Cardano angle representation, but no Euler decomposition exists for $p=2$.

Abstract

We derive explicitly the structural properties of the $p$-adic special orthogonal groups in dimension three, for all primes $p$, and, along the way, the two-dimensional case. In particular, starting from the unique definite quadratic form in three dimensions (up to linear equivalence and rescaling), we show that every element of $SO(3)_p$ is a rotation around an axis. An important part of the analyis is the classification of all definite forms in two dimensions, yielding a description of the rotation subgroups around any fixed axis, which all turn out to be abelian and parametrised naturally by the projective line. Furthermore, we find that for odd primes $p$, the entire group $SO(3)_p$ admits a representation in terms of Cardano angles of rotations around the reference axes, in close analogy to the real orthogonal case. However, this works only for certain orderings of the product of rotations around the coordinate axes, depending on the prime; furthermore, there is no general Euler angle decomposition. For $p=2$, no Euler or Cardano decomposition exists.