Torsion elements of order 𝑝2 in the Nottingham group

Abstract

Abstract Let κ be a characteristic p finite field of q elements and 𝔑 κ {\mathfrak{N}_{\kappa}} the Nottingham group over κ. Lubin associated to every conjugacy class of torsion element of 𝔑 κ {\mathfrak{N}_{\kappa}} a type. We establish an upper bound B ⁢ ( q ; l , m ) {B(q;l,m)} on the number of conjugacy classes of order p 2 {p^{2}} torsion elements u of 𝔑 κ {\mathfrak{N}_{\kappa}} of type 〈 l , m 〉 {\langle l,m\rangle} . In the case where l < p {l<p} , the bound B ⁢ ( q ; l , m ) {B(q;l,m)} is the exact number of conjugacy classes. Moreover, we give a criterion on when u and u n {u^{n}} are conjugate.