AN ELEMENT OF ORDER 4 IN THE NOTTINGHAM GROUP AT THE PRIME 2

T. Chinburg and P. Symonds

Draft of Jan. 22, 2008

1. Introduction

The Nottingham group N (k) over a field k of characteristic p > 0 is the group of continuous automorphisms of the ring k[[t]] which are equal to the identity modulo t^{2}. The object of this paper is to construct for p = 2 an explicit element of order 4 in N (k). After giving the proof we will discuss the relation of this construction to some of the literature concerning elements of order p^{2} in N (k).

Theorem 1.1. Suppose k has characteristic p = 2. An automorphism t ↦ σ(t) of order 4 of k[[t]] is given by setting

σ(t) = t + t^{2} + (t^{6}) + (t^{12} + t^{14}) + (t^{24} + t^{26} + t^{28} + t^{30}) + (t^{48} + ⋯ + t^{62}) + ⋯
(1.1)  = t + t^{2} + ∑_{j=0}^{∞} ∑_{ℓ=0}^{2^{j}−1} t^{6·2^{j}+2ℓ}

Proof. Let A = k[[t, w]]/(w + (1 + t)w^{2} + t^{3}). We may define a continuous automorphism σ of A over k by

(1.2)  σ(t) = (t + w)/(1 + t) and σ(w) = w/(1 + t).

It is easily verified that σ has order 4.

Set v = w(1 + t) so that A = k[[t, v]]/(v^{2} + v + t^{3} + t^{4}) and let

(1.3)  s = ∑_{i=0}^{∞}(t^{3} + t^{4})^{2^{i}} = ∑_{i=0} t^{3·2^{i}} + t^{4·2^{i}}.

Then s^{2} +s = t^{3} +t^{4}, so v^{2} +v +t^{3} +t^{4} = (v +s)(v +s+1). The second factor is invertible, so A = k[[t, v]]/(v + s). Thus v = s ∈ A and w = s/(1 + t). Substituting this into the expression for σ(t) in (1.2) leads to the formula in the Theorem.

Remark 1.2. Let (x : y : z) be homogeneous coordinates for the projective space ℙ_{k}^{2}. We may define an order 4 automorphism σ of ℙ_{k}^{2} by σ(x) = x + y, σ(y) = y and σ(z) = x + z. This σ stabilizes the curve E with homogeneous equation z^{2}y + (z + x)y^{2} + x^{3} = 0 as well as the point Q = (0 : 0 : 1) on E. By [8, Appendix A], E is a supersingular elliptic curve with origin Q and j-invariant 0. The completion of the local ring of E at Q is isomorphic to the ring A ≅ k[[t]] above when we let t = x/z and v = (1 + x/z)(y/z), and the action of σ on A results from the action of σ on E.

Date: June 19, 2008.

We now discuss some literature pertaining to Theorem 1.1.

Camina has shown in [2] that N (k) contains every countably based pro-p group as a subgroup [2], so in particular N (k) contains every finite p-group. In [6], Klopsch shows that representatives for the conjugacy classes of elemens of order p are given by the automorphisms t ↦ t(1 − at^{m})^{−1/m} as m ranges over positive integers prime to p and a ranges over elements of k^{∗}. In [3], Camina wrote concerning explicitly described subgroups of N (k) that “An element of order p^{2} is still not known.” Order p^{2} automorphisms of k[[t]] over k have been extensively studied by Green and Matignon [4], and their work contains implicit formulas for elements of N (k) of order p^{2}. Barnea and Klopsch mention in the introduction of [1] that the subgroups of the Nottingham group they consider contain elements of arbitrarily large p-power order. In [7], Lubin uses formal groups of height 1 to give an explicit construction “well suited to machine computation” of an element

(1.4)  t → σ(t) = ∑_{i=1} a_{i}t^{i}

of N (k) of order p^{n} for each integer n ≥ 2. Related constructions of such elements have been considered recently by Green in [5].

The formula in Theorem 1.1 is of interest because it does not require an iterative procedure to produce the a_{i} in (1.4), and because the height of the formal group associated to the elliptic curve E in Remark 1.2 is 2. It would be very interesting if formal groups of height greater than 1 lead to similar formulas for all primes p.

References

[1] Barnea, Y. and Klopsch, B.: Index-subgroups of the Nottingham group, Adv. Math. 180 no. 1, 187-221 (2003).

[2] Camina, R.: Subgroups of the Nottingham group, J. Algebra 196, 101-113 (1997).

[3] Camina, R.: The Nottingham group, in New horizons in pro-p groups, Progr. Math. 184, Birkhäuser Boston, Boston, Mass. (2000).

[4] Green, B. and Matignon, M.: Liftings of Galois covers of smooth curves Compositio Mathematica 113, 237272,= (1998).

[5] Green, B.: Realizing deformations of curves using Lubin-Tate formal groups. Israel J. Math. 139 (2004), 139–148.

[6] Klopsch, B.: Automorphisms of the Nottingham group, J. Algebra 223, 37-56 (2000).

[7] Lubin, J.: Explicitly constructed torsion elements of the Nottingham group, unpublished manuscript (2001).

[8] Silverman, J.: The Arithmetic of Elliptic Curves. Springer, New York (1986).