Torsion elements of the Nottingham group of order p^2

TL;DR

This paper provides explicit bounds on the number of conjugacy classes of order p^2 torsion elements in the Nottingham group over a prime field, and characterizes conjugacy relations among these elements.

Contribution

It introduces an explicit upper bound B(p,l,m) for conjugacy classes of certain torsion elements and offers criteria for conjugacy between elements and their powers.

Findings

Bound B(p,l,m) precisely counts conjugacy classes when l < p

Criteria established for conjugacy of elements and their powers

Explicit classification of torsion elements of order p^2

Abstract

We establish an explicit upper bound B(p,l,m), depending on p,l,m, on the number of conjugacy classes of order p^2 torsion elements u of type <l,m> of the Nottingham group defined over the prime field of characteristic p >0. In the cases where l < p, the number of conjugacy classes of type <l,m> coincides with B(p,l,m). Moreover, we give a criterion on when u and u^n are conjugate.