THE GROUP ACTION ON THE CLOSED

FIBER OF THE LUBIN-TATE MODULI SPACE

Ching-Li Chai

Abstract. The action of the automorphism group of a 1-dimensional formal Amodule on the closed fiber of the Lubin-Tate deformation space is studied using the Cartier-Dieudonné theory. We obtain a formula for the leading term of this action and give some applications.

0. Introduction

Let G_{0} be a 1-dimensional formal group of height h < ∞ over an algebraically closed field k of characteristic p > 0. It is known that the deformations of G_{0} are unobstructed. The universal deformation space of G_{0} is isomorphic to Spf W (k)[[u_{1}, …, u_{h−1}]], and the equicharacteristic deformation space of G_{0} is isomorphic to Spf k[[u_{1}, …, u_{h−1}]]. The group Aut(G_{0}) of automorphisms of G_{0} is isomorphic to the group of units O_{B}^{×} of the maximal order O_{B} in the central division algebra B = B_{\frac{1}{h}} over ℚ_{p} with invariant \frac{1}{h}. There is a natural action of Aut(G_{0}) on Spf W (k)[[u_{1}, …, u_{h−1}]] and Spf k[[u_{1}, …, u_{h−1}]] “by changing the marking of the closed fiber”. Even though the deformation space looks so simple, the action of the compact p-adic group Aut(G_{0}) on it is extremely complicated. For instance, the equivariant cohomologies of some naturally defined vector bundles contain deep informations about the stable homotopy theory. Very little computation of these equivariant cohomology groups has been done. This action itself remains a mystery in most aspects.

Much insight about this action comes from the p-adic period map

ρ : Spf W (k)[[u_{1}, …, u_{h−1}]] ⊗ frac(W (k)) → ℙ_{frac(W (k))}^{h−1} ,

which was described explicitly in [GH]. Here Spf W (k)[[u_{1}, …, u_{h−1}]] ⊗ frac(W (k)) denotes the rigid open unit ball, thought of as the generic fiber of Spf W (k)[[u_{1}, …, u_{h−1}]]. The p-adic period map comes from comparing the position of the Hodge filtration relative to the space of horizontal sections of the covariant isocrystal attached to G_{0}. The target space is the projective space of 1-dimensional quotients of the covariant isocrystal attached to G_{0}, therefore (End(G_{0})⊗_{ℤ_{p}} ℚ_{p})^{×} operates on it via a natural linear representation on the isocrystal. It is a pleasant fact that the period map ρ is étale and surjective ([GH], [Laf]). The period map ρ is equivariant with respect to the inclusion Aut(G_{0}) ⊂ (End(G_{0}) ⊗_{ℤ_{p}} ℚ_{p})^{×}, so the complicated action of Aut(G_{0}) is ‘linearized’ by the period map ρ. However the period map is far from

Typeset by AMS-TEX

This research is partially supported by grant DMS90-02574 from the National Science Foundation

being defined over W (k). This makes it extremely difficult to extract information about the action of Aut(G_{0}) on the closed fiber Spf k[[u_{1}, …, u_{h−1}]].

This article is a report on my attempt to study the orbit structure of this action, after simplifying the problem by taking the Zariski closure of the orbits. On the one hand this problem is similar to the problem on the deformation of superspecial abelian varieties studied in [Ch], which in turn was motivated by Milne’s sharp formulation of the conjecture on the existence of canonical integral model. On the other hand I was particularly influence by the inspiring paper [GH] by Gross and Hopkins. It is a pleasure to thank Gross, Hopkins and Milne for many useful conversations and correspondences. I would also like to thank the referee for a very careful and thorough reading of the manuscript, and for useful suggestions. The bulk of this work was done during a visit to the Mathematical Sciences Research Institute at Berkeley in the spring of 1993, and I would like to thank the University of Pennsylvania and MSRI for their financial support, and MSRI for hospitality.

The main results of this paper are stated in Theorems 1 and 2 in section 2. They give the main term of the action of an automorphism of G_{0} on k[[u_{1}, …, u_{h−1}]], with respect to two different filtrations on k[[u_{1}, …, u_{h−1}]]. Two applications are given in section 3. We also formulate a naive guess on the possible structure of the Zariski closure of orbits of open subgroups of Aut(G_{0}) (see the “Hope” in section 3). It is well-known that much of the theory and results for one-dimensional commutative formal groups generalize to one-dimensional formal A-modules, where A is a complete discrete valuation ring with finite residue field. We therefore formulate things in this more general context. As a result q takes the place of p. The reader may recover the formal group case by taking A = ℤ_{p} and q = p.

1. The universal group law and its Cartier-Dieudonné module

In this section we set up the notation and the coordinates of the Lubin-Tate space. Let K be a nonarchimedian local field with finite residue field 𝔽_{q}, and let A be the ring of integers in K. (So K is either a finite extension of ℚ_{p} or isomorphic to 𝔽_{q}[[t]] .) Denote by 𝔪 the maximal ideal of A and π a uniformizer of A. Let h be a positive integer. Let G_{1} be the one-dimensional formal A-module with logarithm

g_{1}(x) = ∑_{m≥0} \frac{x^{q^{hm}}}{π^{m}} .

The power series g_{1}(x) satisfies the functional equation

g_{1}(x) = x + \frac{1}{π} g_{1}(x^{q^{h}}) .

It follows from Hazewinkel’s functional equation lemma ([Ha], Chap. 1, sec. 2, especially pp. 9–10) that g_{1}(x) is the logarithm of a formal A-module G_{1} over A. Explicitly, the group law is

G_{1}(x, y) = g_{1}^{−1}(g_{1}(x) + g_{1}(y)) ;

the A-module structure is

[a](x) = g_{1}^{−1}(a · g_{1}(x))

for a ∈ A. The A-module G_{1} reduces modulo 𝔪 to a one-dimensional formal Amodule G_{0} over κ = A/𝔪 of height h. It is well-known that any one-dimensional formal A-module of height h over an algebraically closed overfield k of κ is isomorphic to G_{0} ⊗_{κ} k over k. This A-module G_{1} may be called a “canonical lifting” of G_{0}. (In the sense that it underlies a Lubin-Tate formal group for the unramified extension of K of degree h, and not to be confused with canonical liftings of ordinary abelian varieties. The usual convention is that a canonical lifting of G_{0} is determined by an embedding of the maximal order of a finite separable extension of K of degree h into the ring of endomorphisms of G_{0}.)

The deformation functor Def(G_{0}) of G_{0} assigns to any artinian local A-algebra R the set

Def(G_{0})(R_{/A}) = {(G, α)}/isomorphisms ,

where G is a one-dimensional formal A-module over R and

α : G ⊗_{R} R/𝔪_{R} ⟶^{∼} G_{0} ⊗_{k} R/𝔪_{R}

is an isomorphism of formal A-modules. It is well-known that the functor Def(G_{0}) is formally smooth over A, and is represented by a formal scheme isomorphic to the formal spectrum of a power series ring Spf A[[u_{1}, …, u_{h−1}]]. For any perfect field k containing κ, we can restrict Def(G_{0}) to the category of artinian local Aalgebras whose residue fields are overfields of k, and then this deformation functor is represented by Spf W_{A}(k)[[u_{1}, …, u_{h−1}]], where W_{A}(k) is the ring of ramified Witt vectors soon to be explained in this section.

A beautiful way to describe the universal deformation G of G_{0} is to give a recursive formula for its logarithm, when the coordinates u_{1}, …, u_{h−1} and the coordinate x for G are carefully chosen. Let g\underline{u}(x) be the logarithm of G, so that the group law for G is

G(x, y) = g−1 ; \underline{u}(g\underline{u}(x) + g\underline{u}(y)) .

We can choose good coordinates so that g\underline{u}(x) satisfies the following functional equation

g\underline{u}(x) = x + \frac{u_{1}}{π} g(q) ; \underline{u}(x^{q}) + \frac{u_{2}}{π} g(q^{2}) ; \underline{u}(x^{q^{2}}) + ⋯ + \frac{u_{h−1}}{π} g(q^{h−1}) ; \underline{u}(x^{q^{h−1}}) + \frac{1}{π} g(q^{h}) ; \underline{u}(x^{q^{h}}) .

Here g(q^{i}) ; \underline{u} is obtained from g\underline{u} by the change of rings A[[\underline{u}]] → A[[\underline{u}]], u_{j} ↦ u_{j}^{q^{i}}. This functional equation is really a recursive definition of the coefficients a_{n}(\underline{u}) of

g\underline{u} = ∑_{n≥0} a_{n}(\underline{u}) x^{q^{n}} ,

namely

a_{0}(\underline{u}) = 1, a_{n}(\underline{u}) = 0 ∀n < 0,

a_{n}(\underline{u}) = \frac{u_{1}}{π} a_{n−1}(\underline{u}^{(q)})+ \frac{u_{2}}{π} a_{n−2}(\underline{u}^{(q^{2})})+⋯+ \frac{u_{h−1}}{π} a_{n−h+1}(\underline{u}^{(q^{h−1})})+ \frac{1}{π} a_{n−h}(\underline{u}^{(q^{h})}) ,

∀n ≥ 1 .

From this one can write down a closed formula for the coefficients a_{n}(\underline{u})’s due to Jiu-Kang Yu. The closed formula involves ‘ordered partition’ and will not be reproduced here.

Besides the power series method with functional equations, there is another method of describing formal A-modules using Cartier-Dieudonné theory due to Cartier and Drinfel’d. Here we review some aspects of the theory in order to introduce the relevent notation. For any A-algebra R, let W_{A}(R) be the ring of ramified Witt vectors (r_{0}, r_{1}, …, r_{n}, …), where each r_{n} ∈ R, and the ring structure is given by the family of Witt polynomials (w_{n}^{A})_{n≥0}, where

w_{n}^{A}(\underline{r}) = r_{0}^{q^{n}} + πr_{1}^{q^{n−1}} + ⋯ + π^{n−1}r_{n−1}^{q} + π^{n}r_{n} ,

so that each w_{n}^{A} is a functorial homomorphism of commutative A-algebras. Although the Witt polynomials depend on the choice of the uniformizer π, the rings W_{A}(R) do not. The Teichmüller representative (r, 0, 0, …) ∈ W_{A}(R) will be denoted ⟨r⟩. The Witt ring W_{A}(R) has the following properties:

- The Verschiebung map

υ = υ_{A} : (r_{0}, r_{1}, …, r_{n}, …) ↦ (0, r_{0}, r_{1}, …, r_{n}, …)

is an additive homomorphism.

- There is a unique functorial homomorphism of rings τ = τ_{A} : W_{A} → W_{A} such that w_{n}^{A} ∘ τ_{A} = w_{n+1}^{A}.

- The natural A-algebra structure of W_{A}(R) is given by the composite

A ⟶^{Δ(A)} W_{A}(A) → W_{A}(R) ,

where Δ(A) is the unique homomorphism from A to W_{A}(A) such that w_{n}^{A} ∘ Δ(A) = id_{A}, ∀n ≥ 0.

- For every perfect field k over κ, W_{A}(k) is a complete discrete valuation ring over A with residue field k and π is a uniformizer; moreover W_{A}(k) is characterized by this property. Especially when k is an algebraic extension of κ, W_{A}(k) is the π-adic completion of the unramified extension of A with residue field k.

- (r_{0}, r_{1}, …, r_{n}, …) = ∑_{n≥0} υ^{n}⟨r_{n}⟩. When R is a perfect overfield of κ, we have

(r_{0}, r_{1}, …, r_{n}, …) = ∑_{n≥0} π^{n}⟨r_{n}^{q^{−n}}⟩ .

- ⟨a⟩⟨b⟩ = ⟨ab⟩, ∀a, b ∈ R.

- τ⟨a⟩ = ⟨a^{q}⟩, ∀a ∈ R.

- τυx = πx, ∀x ∈ W_{A}(R).

- (υx)y = υ(x · τy) , ∀x, y ∈ W_{A}(R). Especially υτx = (0, 1, 0, 0, …) · x.

- When π = 0 in R, we have (0, 1, 0, 0, …) = π in W_{A}(R), and

τ(r_{0}, r_{1}, …, r_{n}, …) = (r_{0}^{q}, r_{1}^{q}, …, r_{n}^{q}, …) .

Moreover, ⟨a⟩ + ⟨b⟩ = ⟨a + b⟩, ∀a, b ∈ R.

We defined the ring of ramified Witt vectors W_{A}(R) in order to describe the ramified Cartier ring Cart_{A}(R). For any A-algebra R, Cart_{A}(R) is a noncommutative ring functorial in R. Set theoretically Cart_{A}(R) consists of all elements of the form

∑_{m,n≥0} V_{q}^{m}⟨r_{m,n}⟩F_{π}^{n}

such that r_{m,n} ∈ R for all m, n, and for any m, r_{m,n} = 0 for all but finitely many n’s.

There is an embedding of rings W_{A}(R) ↪ Cart_{A}(R)

(r_{0}, r_{1}, …) ↦ ∑_{n≥0} V_{q}^{n}⟨r_{n}⟩F_{π}^{n} .

This embedding determines the additive structures of Cart_{A}(R). It also gives an embedding of A in Cart_{A}(R). Explicitly, for any a ∈ A, there exisits a uniquely determined element (b_{0}, b_{1}, …) ∈ W_{A}(A) such that w_{n}^{A}(b_{0}, b_{1}, …) = a ∀n ≥ 0. Then the image of a in Cart_{A}(R) is the element ∑_{n≥0} V_{q}^{n}⟨b_{n}⟩F_{π}^{n}. When there is no danger of confusion, we abbreviate V_{q} by V and F_{π} by F. The Cartier ring Cart_{A}(R) satisfies the following commutation relations:

- For any a ∈ A, the image of a in Cart_{A}(R) commutes with both V and F.

- F_{π}⟨r⟩ = ⟨r^{q}⟩F_{π}, ∀r ∈ R.

- ⟨r⟩V_{q} = V_{q}⟨r^{q}⟩, ∀r ∈ R.

- F_{π}V_{q}x = πx = xπ, ∀x ∈ Cart_{A}(R).

The main result in the Cartier-Dieudonné theory, extended to formal A-modules (not necessarily one-dimensional) by Drinfel’d, states that the functor which assigns to each smooth formal A-module its group of q-typical curves, establishes an equivalence of categories between the category of smooth formal A-modules over any A-algebra R and the category of V_{q}-reduced left Cart_{A}(R)-modules, see [Dr2], [Ha]. This translates problems about formal A-modules into linear algebra problems for modules over the Cartier ring.

In order to use this theory, we need to find an explicit form of the Cartier-Dieudonné module of the universal deformation G = G\underline{u}. It turns out that our previous choice of coordinates also gives a very simple presentation of the covariant Dieudonné module of G\underline{u}.

Let u_{1}, …, u_{h−1} be the coordinates for the deformation space of G_{0} and x be the coordinate for the group G. We shall use t as the parameter for curves in G\underline{u} (i.e. elements of G(A[[\underline{u}]][[t]]). The curve γ(t) = t is a q-typical curve in G\underline{u} since the group law for G\underline{u} is q-typical. (Its logarithm involves only powers of q.) The following lemma explictly describes the Dieudonné module of G\underline{u}.

Lemma 1. The Cartier-Dieudonné module for the universal formal A-module G\underline{u} over A[[\underline{u}]] is isomorphic to

Cart_{A}(A[[\underline{u}]])/Cart_{A}(A[[\underline{u}]]) · (F − V^{h−1} − ∑_{1≤i≤h−1} V^{i−1}⟨u_{i}⟩) .

The generator here is the tautological curve γ(t) = t. The logarithm g\underline{u}(x) of the formal group law (in coordinate x) is determined by the functional equation

g\underline{u}(x) = x + \frac{u_{1}}{π} g(q) ; \underline{u}(x^{q}) + \frac{u_{2}}{π} g(q^{2}) ; \underline{u}(x^{q^{2}}) + ⋯ + \frac{u_{h−1}}{π} g(q^{h−1}) ; \underline{u}(x^{q^{h−1}}) + \frac{1}{π} g(q^{h}) ; \underline{u}(x^{q^{h}}) .

The group law for G\underline{u} is

G\underline{u}(x, y) = g−1 ; \underline{u} (g\underline{u}(x) + g\underline{u}(y)) ∈ A[\underline{u}][[x, y]] ⊂ A[[\underline{u}]][[x, y]] .

The A-module structure for G\underline{u} is given by

[a](x) = g−1 ; \underline{u}(a · g\underline{u}(x)) ∈ A[\underline{u}][[x]] ⊂ A[[\underline{u}]][[x]] , a ∈ A .

Lemma 1 is a specialization of [Ha] (29.5.3); see also [Ha] (27.7.15) and (27.7.19 (ii)). It is also possible to prove it directly. □

Let ζ_{h} be a primitive (q^{h} − 1)st root of 1 in \overline{K}. The ring A_{h} = A[ζ_{h}] is the unramified extension of degree h of A with residue field κ_{h} = A[ζ_{h}]/πA[ζ_{h}] = κ(ζ_{h}), the cyclic extension of degree h over κ. Every element of End(G_{0}⊗\overline{κ}) is defined over κ_{h}, and End(G_{0} ⊗ \overline{κ}) is isomorphic to the maximal order in the central division algebra over K with invariant \frac{1}{h}. The ring A[ζ_{h}] is canonically identified with End(G_{1}_{/A_{h}}), such that the action of ζ_{h} on G_{1} sends x to ζ_{h}x. The ring End(G_{0}⊗\overline{κ}) is generated by A_{h} and ϕ = F rob_{q} ∈ End(G_{0}):

End(G_{0} ⊗ \overline{κ}) = A_{h} ⊕ A_{h}ϕ ⊕ ⋯ ⊕ A_{h}ϕ^{h−1} .

The action of an element Φ_{0} ∈ Aut(G_{0} ⊗ κ_{h}) on A_{h}[[\underline{u}]] can be characterized as follows: There is a unique ring isomorphism

h(Φ_{0}) : A_{h}[[\underline{u}]] → A_{h}[[\underline{u}]]

such that there is a homomorphism

Φ : h(Φ_{0})_{∗}G_{A_{h}[[\underline{u}]]} → G_{A_{h}[[\underline{u}]]}

which reduces to Φ_{0} modulo π. It is clear that A^{×} operates trivially on A_{h}[[\underline{u}]], as A ⊆ End(G_{A_{h}[[\underline{u}]]}). The action of an element ζ ∈ μ_{q^{h}−1}(\overline{K}) is easy to compute:

Lemma 2. For ζ ∈ μ_{q^{h}−1}(\overline{K}) ⊆ Aut(G_{0} ⊗ κ_{h}), the corresponding A_{h}-algebra homomorphism h(ζ) : A_{h}[[\underline{u}]] → A_{h}[[\underline{u}]] sends u_{i} to ζ^{q^{i}−1} ·u_{i} for each 1 ≤ i ≤ h−1.

Proof. There is a homomorphism from

Cart_{A}(A_{h}[[\underline{u}]])/Cart_{A}(A_{h}[[\underline{u}]]) · (F_{π} − V_{q}^{h−1} − ∑_{1≤i≤h−1} V_{q}^{i−1}⟨ζ^{q^{i}−1}u_{i}⟩)

to

Cart_{A}(A_{h}[[\underline{u}]])/Cart_{A}(A_{h}[[\underline{u}]]) · (F_{π} − V_{q}^{h−1} − ∑_{1≤i≤h−1} V_{q}^{i−1}⟨u_{i}⟩)

sending the class of ⟨1⟩ to the class of ⟨ζ⟩. This finishes the proof of Lemma 2. It is also possible to prove it using the power series method and Hazewinkel’s functional equation. □

The Cartier-Dieudonné module of G_{0} ⊗ κ(ζ_{h}) is

M_{0} = Cart_{A}(κ_{h})/Cart_{A}(κ_{h}) · (F_{π} − V_{q}^{h−1}) .

M_{0} is a free module of rank h over W_{A}(κ_{h}) = A_{h}; the classes of 1, V, …, V^{h−1} is a free A_{h}-basis of M_{0}. An endomorphism Φ_{0} of G_{0} ⊗ κ(ζ_{h}) corresponds to an endomorphism of M_{0}, which in turn is completely determined by the image of e : = the class of 1 under Φ_{0}, say Φ_{0} : e ↦ ∑_{i=0}^{h−1} a_{i} V^{i} · e, a_{i} ∈ A_{h}. This is just the previous decomposition

End(G_{0} ⊗ κ(ζ_{h})) = ⨁_{i=0}^{h−1} Aϕ^{i} .

The filtration of End(G_{0} ⊗ κ(ζ_{h})) by the powers of its maximal ideal is determined by congruence of the coefficients a_{i}’s of the V^{i}e’s: if m = h · r + s, m, r, s ∈ ℕ, 0 ≤ s ≤ h − 1, then Φ_{0} ≡ 0 modulo the mth power of the maximal ideal of End(G_{0} ⊗ κ(ζ_{h})) iff

a_{0} ≡ ⋯ ≡ a_{s−1} ≡ 0 (mod π^{r+1}) and a_{s} ≡ ⋯ ≡ a_{h−1} ≡ 0 (mod π^{r}) .

In the next section we shall examine the action of an element Φ_{0} of Aut(G_{0} ⊗ κ(ζ_{h})) = End(G_{0} ⊗ κ(ζ_{h}))^{×} which is congruent to the identity modulo a high power of the maximal ideal of End(G_{0} ⊗ κ(ζ_{h})).

2. Computation of the group action

Let Φ_{0} ∈ End(G_{0} ⊗ κ(ζ_{h})), and we assume that

Φ_{0}(e) = e + ξ · V^{ℓ}e, ξ ∈ μ_{q^{h}−1}(\overline{K}), ℓ ∈ ℕ, ℓ ≥ 1 .

Clearly the Φ_{0}’s of this type form a system of representatives of

F il_{V}^{ℓ} (Aut(G_{0} ⊗ κ(ζ_{h}))) =^{def} id + V^{ℓ} · End(G_{0} ⊗ κ(ζ_{h}))

modulo F il_{V}^{ℓ+1}(Aut(G_{0} ⊗ κ(ζ_{h}))) . The action of Φ_{0} on the closed fiber Spf κ_{h}[[\underline{u}]] corresponds to a homomorphism

h(Φ_{0}) : κ_{h}[[\underline{u}]] → κ_{h}[[\underline{u}]], u_{i} ↦ s_{i} .

We would like to compute the s_{i}’s. Φ_{0} extends to a homomorphism Φ from

Cart_{A}(κ_{h}[[\underline{u}]]) · e′ ≅

Cart_{A}(κ_{h}[[\underline{u}]])/Cart_{A}(κ_{h}[[\underline{u}]]) · (F − V^{h−1} − ∑_{i=1}^{h−1} V^{i−1}⟨s_{i}⟩)

to

Cart_{A}(κ_{h}[[\underline{u}]]) · e ≅

Cart_{A}(κ_{h}[[\underline{u}]])/Cart_{A}(κ_{h}[[\underline{u}]]) · (F − V^{h−1} − ∑_{i=1}^{h−1} V^{i−1}⟨u_{i}⟩)

Write ℓ = n · h + b, 0 ≤ b ≤ h − 1, and

Φ(e′) = e + π^{n} · ξ · V^{b} e + ∑_{m≥0} V^{m}⟨a_{m}(\underline{u})⟩ · e ,

where a_{m}(\underline{u}) are elements of κ_{h}[[\underline{u}]], uniquely determined by Φ_{0}. Computing the action of Φ_{0} hinges largely on calculating the a_{m}(\underline{u})’s. Clearly Φ(e′) must satisfy the equation

(F − V^{h−1} − ∑_{i=1}^{h−1} V^{i−1}⟨s_{i}⟩) · Φ(e′) = 0 .

Using the fact that every element of Cart_{A}(κ_{h}[[\underline{u}]]) · e can be uniquely written in the form

∑_{n≥0} V^{n}⟨x_{n}⟩ · e x_{n} ∈ κ_{h}[[\underline{u}]] ,

the above equation for Φ(e′) can be translated into a system of equations for the s_{i}’s and a_{m}(\underline{u})’s. Then we solve this system of equations modulo some suitable ideal I (which depends on Φ_{0}) to compute the leading term of s_{i}’s. This would give us the leading term of the action of Φ_{0} on κ_{h}[[\underline{u}]]. The computation itself is a bit long and quite messy. Roughly, we first see that for large m’s a_{m}(\underline{u}) is congruent to 0 modulo I. From some m_{0} down a_{m}(\underline{u}) becomes nonzero modulo I. The system of equations give a recursive formula which computes a_{m}(\underline{u}) in terms of the a_{n}(\underline{u})’s with n > m. Going from m_{0} down to 0 , we compute all a_{m}(\underline{u})’s modulo I. The system of equations for the s_{i}’s can then be solved easily modulo I. In fact only a_{0}(\underline{u}), …, a_{h−1}(\underline{u}) are needed, but we had to compute the rest of the a_{m}(\underline{u})’s before we can get to them. Now the computation begins. We use the assumed form of Φ(e′) , the commutation relations in the Cartier ring, and the fact that

F e = V^{h−1}e + ∑_{i=1}^{h−1} V^{i−1}⟨u_{i}⟩ · e

to get

(F − V^{h−1}) · Φ(e′)

= ∑_{i=1}^{h−1} V^{i−1}⟨u_{i}⟩ · e + V^{n+b} ξ^{σ^{n+b+1}} ∑_{i=1}^{h−1} V^{i−1}⟨u_{i}^{q^{n}}⟩ F^{n} · e

+ ∑_{m≥0} V^{m}⟨a_{m}(\underline{u})^{q}⟩ (∑_{i=1}^{h−1} V^{i−1}⟨u_{i}⟩ · e + V^{h−1} · e) − ∑_{m≥0} V^{h−1+m}⟨a_{m}(\underline{u})⟩ · e

= ∑_{i=1}^{h−1} V^{i−1}⟨u_{i}⟩ · e + ∑_{i=1}^{h−1} V^{n+b+i−1}⟨ξ^{σ^{n+b+i}} u_{i}^{q^{n}}⟩ F^{n} · e

+ ∑_{m≥0} ∑_{i=1}^{h−1} V^{m+i−1}⟨a_{m}(\underline{u})^{q^{i}} u_{i}⟩ · e + ∑_{m≥0} V^{m+h−1}⟨a_{m}(\underline{u})^{q^{h}}⟩ · e

− ∑_{m≥0} V^{m+h−1}⟨a_{m}(\underline{u})⟩ · e

Here σ denotes F r_{q}, and the ξ in the second equality denotes the image of ξ in κ_{h}. On the other hand

∑_{i=1}^{h−1} V^{i−1}⟨s_{i}⟩ · Φ(e′)

= ∑_{i=1}^{h−1} V^{i−1}⟨s_{i}⟩ · e + ∑_{i=1}^{h−1} V^{n+i−1+b}⟨ξ^{σ^{n+b}} · s_{i}^{q^{n+b}}⟩ F^{n} · e

+ ∑_{i=1}^{h−1} ∑_{m≥0} V^{i−1+m}⟨s_{i}^{q^{m}} · a_{m}(\underline{u})⟩ · e

Assume now that n ≥ 1. We make some simple estimate:

⟨u_{i}^{q^{n}}⟩ F^{n} · e =⟨u_{i}^{q^{n}}⟩ (V^{h−1} + ∑_{j=1}^{h−1} V^{j−1}⟨u_{j}^{q^{n−1}}⟩) ⋯ (V^{h−1} + ∑_{j=1}^{h−1} V^{j−1}⟨u_{j}^{q}⟩)

· (V^{h−1} + ∑_{j=1}^{h−1} V^{j−1}⟨u_{j}⟩) · e

≡ 0 (mod (\underline{u})^{σ^{n}+···+σ+1}) .

Here we have abused notation and used σ to denote the κ_{h}-algebra homomorphism from κ_{h}[[\underline{u}]] to itself sending each u_{i} to u_{i}^{q}; (\underline{u})^{σ^{i}} denotes the ideal (u_{1}^{q^{i}}, …, u_{h−1}^{q^{i}}) ⊆ κ_{h}[[\underline{u}]], and

(\underline{u})^{σ^{n}+···+σ+1} = (\underline{u})^{σ^{n}} · (\underline{u})^{σ^{n−1}} ⋯ (\underline{u})^{σ} · (\underline{u}) .

Similarly

⟨s_{i}^{q^{n+b}}⟩ F^{n} · e ≡ 0 (mod (\underline{u})^{σ^{n}+···+σ+1}) .

Therefore

\{ \begin{matrix}s_{i} & ≡ u_{i} (mod (\underline{u})^{σ^{n}+···+σ+1}) ∀ 1 ≤ i ≤ h − 1 \\ a_{m}(\underline{u}) & ≡ 0 (mod (\underline{u})^{σ^{n}+···+σ+1}) ∀ m ≥ 0 \end{matrix}

is a solution modulo (\underline{u})^{σ^{n}+···+σ+1}, hence it is the unique solution modulo the ideal (\underline{u})^{σ^{n}+···+σ+1}. Now we ‘peel down the layers’ and try to dig out the first few nonvanishing terms. From the previous expression for ⟨u_{i}^{q^{n}}⟩ F^{n} · e we get

⟨u_{i}^{q^{n}}⟩ F^{n} · e ≡ ⟨u_{i}^{q^{n}} · u_{1}^{q^{n−1}+···+q+1}⟩ · e (mod (\underline{u})^{σ^{n+1}+σ^{n−2}+σ^{n−3}+···+σ+1}) .

Similarly, modulo (\underline{u})^{σ^{n+1}+σ^{n−2}+σ^{n−3}+···+σ+1} we have for 0 ≤ b ≤ h − 1,

⟨s_{i}^{q^{n+b}}⟩ F^{n} · e ≡ \{ \begin{matrix}⟨u_{i}^{q^{n}} · u_{1}^{q^{n−1}+···+q+1}⟩ · e \\ 0 \end{matrix} \begin{matrix}if b = 0 \\ if b ≠ 0 . \end{matrix}

We shall work for a while modulo the ideal (\underline{u})^{σ^{n+1}+σ^{n−2}+σ^{n−3}+···+σ+1}. Observe that s_{i}^{q^{j}} ≡ u_{i}^{q^{j}} if j ≥ 1, and a_{m}(\underline{u})^{q^{j}} ≡ 0 if j ≥ 1. Let us consider first the easier case char(A) = p ≥ 0, i.e. A ≅ 𝔽_{q}[[t]] for some q = p^{r}. Modulo (\underline{u})^{σ^{n+1}+σ^{n−2}+σ^{n−3}+···+σ+1} we have

(F − V^{h−1}) · Φ(e′)

≡ ∑_{i=1}^{h−1} V^{i−1}⟨u_{i}⟩ · e − ∑_{0≤m≤n−h+b} V^{m+h−1}⟨a_{m}(\underline{u})⟩ · e

+ ∑_{i=1}^{h−1} V^{n+b+i−1}⟨ξ^{σ^{n+b+i}} u_{i}^{q^{n}} u_{1}^{q^{n−1}+···+q+1}⟩ · e − ∑_{i=1}^{h−1} V^{n+b+i−1}⟨a_{n+b+i−h}(\underline{u})⟩ · e

− ∑_{m≥n+b+h−1} V^{m}⟨a_{m−h+1}(\underline{u})⟩ · e

≡ ∑_{i=1}^{h−1} V^{i−1}⟨u_{i}⟩ · e − ∑_{h−1≤m≤n+b−1} V^{m}⟨a_{m−h+1}(\underline{u})⟩ · e

+ ∑_{i=1}^{h−1} V^{n+b+i−1}⟨ξ^{σ^{n+b+i}} u_{i}^{q^{n}} u_{1}^{q^{n−1}+···+q+1} − a_{n+b+i−h}(\underline{u})⟩ · e

− ∑_{m≥n+b+h−1} V^{m}⟨a_{m−h+1}(\underline{u})⟩ · e

We also have

∑_{i=1}^{h−1} V^{i−1}⟨s_{i}⟩ · Φ(e′)

≡ ∑_{i=1}^{h−1} V^{i−1}⟨s_{i}⟩ · e + ∑_{1≤i ≤h−1} V^{n+b+i−1}⟨ξ^{σ^{n+b}} · u_{i}^{q^{n+b}} · u_{1}^{q^{n−1}+···+q+1}⟩ · e

+ ∑_{{0≤m≤n \atop 1≤i≤h−1}} V^{i−1+m}⟨s_{i}^{q^{m}} a_{m}(\underline{u})⟩ · e

≡ \{ \begin{matrix}∑_{i=1}^{h−1} V^{i−1}⟨s_{i} + ∑_{1≤j≤i} s_{j}^{q^{i−j}} a_{i−j}(\underline{u})⟩ · e \\ + ∑_{h−1≤m≤n+h−2} V^{m}⟨ ∑_{1≤i≤h−1} s_{i}^{q^{m−i+1}} a_{m−i+1}(\underline{u})⟩ · e \\ (if b ≠ 0), or \\ ∑_{i=1}^{h−1} V^{i−1}⟨s_{i} + ∑_{1≤j≤i} s_{j}^{q^{i−j}} a_{i−j}(\underline{u})⟩ · e \\ + ∑_{h−1≤m≤n−1} V^{m}⟨ ∑_{1≤i≤h−1} s_{i}^{q^{m−i+1}} a_{m−i+1}(\underline{u})⟩ · e \\ + ∑_{1≤i≤h−1} V^{n+i−1}⟨ξ^{σ^{n}} · u_{i}^{q^{n}} · u_{1}^{q^{n−1}+···+q+1} + ∑_{1≤j≤h−1} s_{j}^{q^{n+i−j}} a_{n+i−j}(\underline{u})⟩ · e \\ (if b = 0) \end{matrix}

Now we can compare “the coefficients of the V^{j}’s” to get information. When b ≠ 0, we get modulo (\underline{u})^{σ^{n+1}+σ^{n−2}+σ^{n−3}+···+σ+1} :

a_{m} ≡ 0 ∀ m ≥ n + b

u_{i} ≡ s_{i} + ∑_{1≤j≤i−1} u_{j}^{q^{i−j}} a_{i−j}(\underline{u}) + s_{i} a_{0}(\underline{u}) (1 ≤ i ≤ h − 1)

Thus we need a_{0}(\underline{u}), …, a_{h−1}(\underline{u}) in order to get the functions s_{1}, …, s_{h−1}. They are given recursively as follows:

a_{n+b+i−h}(\underline{u}) ≡ ξ^{σ^{n+b+i}} u_{i}^{q^{n}} · u_{1}^{q^{n−1}+···+q+1} ∀ 1 ≤ i ≤ h − 1 .

This gives a_{n+b−1}(\underline{u}), …, a_{n+b+1−h}(\underline{u}) ; and for h − 1 ≤ m ≤ n + b − 1 we have

a_{m−h+1}(\underline{u}) ≡ − ∑_{1≤i≤h−1} s_{i}^{q^{m−i+1}} a_{m−i+1}(\underline{u}) .

In other words, for 0 ≤ m ≤ n + b − h,

a_{m}(\underline{u}) ≡ − ∑_{1≤i≤h−1} s_{i}^{q^{m+h−i}} a_{m+h−i}(\underline{u}) .

Hence for b ≠ 0 (i.e. 1 ≤ b ≤ h − 1), we get

\{ \begin{matrix}a_{m}(\underline{u}) & ≡ 0  ∀ m ≥ n + b \\ a_{n+b+i−h}(\underline{u}) & ≡ ξ^{σ^{n+b+i}} u_{i}^{q^{n}} · u_{1}^{q^{n−1}+···+q+1}  ∀ 1 ≤ i ≤ h − 1 \\ a_{m}(\underline{u}) & ≡ − ∑_{1≤i≤h−1} u_{i}^{q^{m+h−i}} a_{m+h−i}(\underline{u})  ∀ 0 ≤ m ≤ n + b − h \\ u_{i} & ≡ s_{i} (1 + a_{0}(\underline{u})) + ∑_{1≤j≤i−1} u_{j}^{q^{i−j}} a_{i−j}(\underline{u}) \\ & (mod (\underline{u})^{σ^{n+1}+σ^{n−2}+σ^{n−3}+···+σ+1}) \end{matrix}

When b = 0, again modulo (\underline{u})^{σ^{n+1}+σ^{n−2}+σ^{n−3}+···+σ+1}, we have

a_{m}(\underline{u}) ≡ 0 ∀ m ≥ n

ξ^{σ^{n+i}} u_{i}^{q^{n}} u_{1}^{q^{n−1}+···+q+1} − a_{n+i−j}(\underline{u})

≡ ξ^{σ^{n}} u_{i}^{q^{n}} u_{1}^{q^{n−1}+···+q+1} + ∑_{i+1≤j≤h−1} s_{j}^{q^{n+i−j}} a_{n+i−j}(\underline{u})  (1 ≤ i ≤ h − 1)

This gives a_{n−1}(\underline{u}), …, a_{n−h+1}(\underline{u}). Also

a_{m−h+1}(\underline{u}) ≡ − ∑_{1≤i≤h−1} s_{i}^{q^{m−i+1}} a_{m−i+1}(\underline{u}) (h − 1 ≤ m ≤ n − 1)

This gives a_{n−h}(\underline{u}), …, a_{0}(\underline{u}). Moreover

u_{i} ≡ s_{i} + ∑_{1≤j≤i−1} s_{j}^{q^{i−j}} a_{i−j}(\underline{u}) + s_{i} a_{0}(\underline{u}) (1 ≤ i ≤ h − 1)

In other words we have

\{ \begin{matrix}a_{m}(\underline{u}) ≡ & 0  ∀ m ≥ n \\ a_{n+i−h}(\underline{u}) ≡ & (ξ^{σ^{n+i}} − ξ^{σ^{n}}) u_{i}^{q^{n}} u_{1}^{q^{n−1}+···+q+1} \\ & − ∑_{i+1≤j≤h−1} s_{j}^{q^{n+i−j}} a_{n+i−j}(\underline{u})  (∀ 1 ≤ i ≤ h − 1) \\ a_{m}(\underline{u}) ≡ & − ∑_{1≤i≤h−1} u_{i}^{q^{m+h−i}} a_{m+h−i}(\underline{u})  (∀ 0 ≤ m ≤ n − h) \\ u_{i} ≡ & s_{i} (1 + a_{0}(\underline{u})) + ∑_{1≤j≤i−1} u_{j}^{q^{i−j}} a_{i−j}(\underline{u})  (∀ 1 ≤ i ≤ h − 1) \\ & (mod (\underline{u})^{σ^{n+1}+σ^{n−2}+σ^{n−3}+···+σ+1}) \end{matrix}

Consider now the case when char(A) = 0. Let e be the absolute ramification index of A, q = p^{r}. We shall see need some simple calculation about ramified Witt vectors. We need information about the Witt components of the sum of two Teichmüller representatives. Suppose

⟨x⟩ + ⟨y⟩ = (S_{0}, S_{1}, S_{2}, …) ,

so S_{0} = x + y, S_{1} = \frac{1}{π} (x^{q} + y^{q} − (x + y)^{q}) , …. If e = 1, q = p, i.e. A = ℤ_{p}, then

S_{1} ≡ −x^{p−1}y (mod ∑_{2≤i≤p−1}(x)^{p−i}(y)^{i} + (π))

S_{2} ≡ −x^{p^{2}−1}y (mod ∑_{2≤i≤p^{2}−1}(x)^{p^{2}−i}(y)^{i} + (π))

⋮
S_{m} ≡ −x^{p^{m}−1}y (mod ∑_{2≤i≤p^{m}−1}(x)^{p^{m}−i}(y)^{i} + (π))

⋮

If e > 1, then no matter whether q = p or not,

S_{m} ≡ 0 (mod π) if e ∤ m
S_{m} ≡ − x^{q^{m}−1}y (mod ∑_{2≤i≤q^{m}−1}(x)^{q^{m}−i}(y)^{i} + (π)) if e | m

While if q = p^{r}, r > 1, then

S_{m} ≡ 0 (mod ∑_{p^{r−1}≤i≤q^{m}−p^{r−1}}(x)^{q^{m}−i}(y)^{i} + (π)) ∀ m ≥ 1 .

Moreover, we always have

S_{m} ≡ 0 (mod (x, y)^{q^{m}}) .

With the above estimates, we can go back to our calculation. We have

(F − V^{h−1}) Φ(e′)

≡ ∑_{i=1}^{h−1} V^{i−1}⟨u_{i}⟩ e − ∑_{0≤m≤n−h+b} V^{m+h−1}⟨a_{m}(\underline{u})⟩ e

+ ∑_{i=1}^{h−1} V^{n+b+i−1}⟨ξ^{σ^{n+b+i}} u_{i}^{q^{n}} u_{1}^{q^{n−1}+···+q+1}⟩ e

− ∑_{i=1}^{h−1} V^{n+b+i−1} ⟨a_{n+b+i−h}(\underline{u})⟩ e − ∑_{m≥n+b+h−1} V^{m}⟨a_{m−h+1}(\underline{u})⟩ e

(mod (\underline{u})^{σ^{n+1}+σ^{n−2}+···+σ+1} )

≡ ∑_{i=1}^{h−1} V^{i−1}⟨u_{i}⟩ e − ∑_{h−1≤m≤n+b−1} V^{m}⟨a_{m−h+1}(\underline{u})⟩ e

+ ∑_{i=1}^{h−1} V^{n+b+i−1} ⟨ξ^{σ^{n+b+i}} u_{i}^{q^{n}} u_{1}^{q^{n−1}+···+q+1} − a_{n+b+i−h}(\underline{u})⟩ e

− ∑_{m≥n+b+h−1} V^{m} ⟨a_{m−h+1}(\underline{u})⟩ e

(mod (\underline{u})^{σ^{n+1}+σ^{n−2}+···+σ+1} + (\underline{u})^{q(σ^{n}+σ^{n−1}+···+σ+1)})

Case 1. q = p^{r}, r > 1.

Then modulo (\underline{u})^{p^{r−1}(p−1)+p^{r−1}(σ^{n}+σ^{n−1}+···+σ+1)}, we get

∑_{i=1}^{h−1} V^{i−1} ⟨s_{i}⟩ Φ(e′)

≡ \{ \begin{matrix}if b ≠ 0 : \\ ∑_{i=1}^{h−1} V^{i−1} ⟨s_{i} + ∑_{1≤j≤i} s_{j}^{q^{i−j}} a_{i−j}(\underline{u})⟩ e \\ + ∑_{h−1≤m≤n+h−2} V^{m} ⟨∑_{i=1}^{h−1} s_{i}^{q^{m−i+1}} a_{m−i+1}(\underline{u})⟩ e \\ or, if b = 0 : \\ ∑_{i=1}^{h−1} V^{i−1} ⟨s_{i} + ∑_{1≤j≤i} s_{j}^{q^{i−j}} a_{i−j}(\underline{u})⟩ e \\ + ∑_{h−1≤m≤n−1} V^{m} ⟨∑_{i=1}^{h−1} s_{i}^{q^{m−i+1}} a_{m−i+1}(\underline{u})⟩ e \\ + ∑_{i=1}^{h−1} V^{n+i−1} ⟨ξ^{σ^{n}} · u_{i}^{q^{n}} u_{1}^{q^{n−1}+···+q+1} + ∑_{1≤j≤h−1} s_{j}^{q^{n+i−j}} a_{n+i−j}(\underline{u})⟩ e \end{matrix}

Comparing this with the formula for the case char(A) = p, we see that the formula is the same if read modulo the ideal (\underline{u})^{p^{r−1}(p−1)+p^{r−1}(σ^{n}+σ^{n−1}+···+σ+1)}. Case 2. q = p.

This time we get a slightly different formula:

∑_{i=1}^{h−1} V^{i−1} ⟨s_{i}⟩ Φ(e′)

≡ ∑_{i=1}^{h−1} V^{i−1}⟨s_{i}⟩ e + ∑_{i=1}^{h−1} V^{i−1} ⟨ ∑_{1≤j≤i} s_{j}^{q^{i−j}} a_{i−j}(\underline{u})⟩ e

+ \{ \begin{matrix}if b ≠ 0 : \\ ∑_{h−1≤m≤n+h−2} V^{m} ⟨∑_{i=1}^{h−1} s_{i}^{q^{m−i+1}} a_{m−i+1}(\underline{u})⟩ e \\ or, if b = 0 : \\ ∑_{h−1≤m≤n−1} V^{m} ⟨∑_{i=1}^{h−1} s_{i}^{q^{m−i+1}} a_{m−i+1}(\underline{u})⟩ e \\ + ∑_{i=1}^{h−1} V^{n+i−1} ⟨ξ^{σ^{n}} · u_{i}^{q^{n}} u_{1}^{q^{n−1}+···+q+1} + ∑_{1≤j≤h−1} s_{j}^{q^{n+i−j}} a_{n+i−j}(\underline{u})⟩ e \end{matrix}

modulo the ideal (\underline{u})^{q(σ^{n}+···+σ+1)}.

Modulo the ideal (\underline{u})^{p^{e}+2(σ^{n}+···+σ)} + (\underline{u})^{p^{e}−1} ∑_{i=0}^{h−2} (\underline{u})^{σ^{i}} (a_{i}(\underline{u})) , we still have

∑_{i=1}^{h−1} V^{i−1} ⟨s_{i}⟩ Φ(e′)

≡ ∑_{i=1}^{h−1} V^{i−1}⟨s_{i}⟩ e + ∑_{i=1}^{h−1} V^{i−1} ⟨ ∑_{1≤j≤i} s_{j}^{q^{i−j}} a_{i−j}(\underline{u})⟩ e

+ \{ \begin{matrix}if b ≠ 0 : \\ ∑_{h−1≤m≤n+h−2} V^{m} ⟨∑_{i=1}^{h−1} s_{i}^{q^{m−i+1}} a_{m−i+1}(\underline{u})⟩ e \\ or, if b = 0 : \\ ∑_{h−1≤m≤n−1} V^{m} ⟨∑_{i=1}^{h−1} s_{i}^{q^{m−i+1}} a_{m−i+1}(\underline{u})⟩ e \\ + ∑_{i=1}^{h−1} V^{n+i−1} ⟨ξ^{σ^{n}} · u_{i}^{q^{n}} u_{1}^{q^{n−1}+···+q+1} + ∑_{1≤j≤h−1} s_{j}^{q^{n+i−j}} a_{n+i−j}(\underline{u})⟩ e \end{matrix}

In the case q = p^{r} > p, we conclude that the following formulae (∗) for the case b ≠ 0 (resp. (∗)′ for the case b = 0) hold modulo (\underline{u})^{p^{r−1}(p−1)+p^{r−1}(σ^{n}+···+σ+1)}:

(∗) \{ \begin{matrix}a_{m}(\underline{u}) ≡ & 0  ∀ m ≥ n + b \\ a_{n+b+i−h}(\underline{u}) ≡ & ξ^{σ^{n+b+i}} u_{i}^{q^{n}} u_{1}^{q^{n−1}+···+q+1}  1 ≤ i ≤ h − 1 \\ a_{m}(\underline{u}) ≡ & − ∑_{i=1}^{h−1} u_{i}^{q^{m+h−i}} a_{m+h−i}(\underline{u})  0 ≤ m ≤ n + b − h \\ u_{i} ≡ & s_{i} (1 + a_{0}(\underline{u})) + ∑_{j=1}^{i−1} u_{j}^{q^{i−j}} a_{i−j}(\underline{u})  1 ≤ i ≤ h − 1 \end{matrix}

(∗)′ \{ \begin{matrix}a_{m}(\underline{u}) ≡ & 0  ∀ m ≥ n \\ a_{n+i−h}(\underline{u}) ≡ & (ξ^{σ^{n+i}} − ξ^{σ^{n}}) u_{i}^{q^{n}} u_{1}^{q^{n−1}+···+q+1} \\ & − ∑_{i+1≤j≤h−1} s_{j}^{q^{n+i−j}} a_{n+i−j}(\underline{u})  1 ≤ i ≤ h − 1 \\ a_{m}(\underline{u}) ≡ & − ∑_{i=1}^{h−1} u_{i}^{q^{m+h−i}} a_{m+h−i}(\underline{u})  0 ≤ m ≤ n − h \\ u_{i} ≡ & s_{i} (1 + a_{0}(\underline{u})) + ∑_{j=1}^{i−1} u_{j}^{q^{i−j}} a_{i−j}(\underline{u})  1 ≤ i ≤ h − 1 \end{matrix}

Notice that (∗) and (∗)′ differ only in how a_{n+b−1}, …, a_{n+b−h+1} are defined. When q = p, although the congruence becomes a little worse, we can still con-

clude that the formulae for the s_{i}’s given by (∗) and (∗)′ at least give the correct initial terms of s_{i} − u_{i} in the (\underline{u})-adic filtration of k[[\underline{u}]]:

Theorem 1. Let Φ_{0} ∈ Aut(G_{0}) be an automorphism of G_{0} such that Φ_{0}(e) = e + ξ V^{ℓ}e, where ξ ∈ μ_{q^{h}−1} (W_{A}(𝔽_{q^{h}})), ℓ = nh + b, with 0 ≤ b ≤ h − 1, n ≥ 1. Let s_{i} = h(Φ_{0})(u_{i}) for i = 1, …, h − 1.

(1) If char(A) = p > 0 (i.e. A ≅ 𝔽_{q}[[t]]), then modulo (\underline{u})^{σ^{n+1}+σ^{n−2}+···+1}, the s_{i}’s are given recursively by the formulae (∗) and (∗)′.

(2) In the case char(A) = 0, when q = p^{r} > p, the s_{i}’s are given recursively by the formulae (∗) and (∗)′ modulo (\underline{u})^{p^{r−1}(p−1)+p^{r−1}(σ^{n}+···+σ+1)}. When q = p, the same formulae give at least the correct initial term of s_{i} − u_{i}’s in the (\underline{u})-adic filtration of k[[\underline{u}]].

Remarks. (1) When n = 0, some slight modification is necessary. If 1 ≤ b ≤ h − 1, the formula which gives the leading term of s_{i} − u_{i}’s is:

\{ \begin{matrix}a_{0}(\underline{u}) ≡ & ξ u_{h−b} \\ u_{i} ≡ & s_{i} (1 + a_{0}(\underline{u}) ) 1 ≤ i ≤ b \\ u_{i} + ξ^{σ^{i}} u_{i−b} ≡ & s_{i} (1 + a_{o}(\underline{u}) ) b + 1 ≤ i ≤ h − 1 \end{matrix}

We leave the proof to the patient reader. When n = 0, b = 0, then Φ_{0} is congruent to a (q^{h} − 1)st root of unity modulo V, and the main term of the action is given by lemma 2.

(2) These formulae relate the π-adic filtration of Aut(G_{0}) with the filtration

F il^{n} = \{ϕ ∈ Aut(G_{0}) | \begin{matrix}ϕ extends to an automorphism of \\ G ×_{Spec κ_{h}[[\underline{u}]]} Spec κ_{h}[[\underline{u}]]/(\underline{u})^{n+1} \end{matrix} \} .

(3) Similar results can be obtained for the action of Aut(G_{0}) on

Spf κ_{h}[[u_{1}, …, u_{h−1}]]/(u_{1}, …, u_{h_{1}−1}) ,

the maximal subscheme of Spf κ_{h}[[\underline{u}]] over which the height of the universal deformation is at least h_{1}. K. Keating has investigated the case h_{1} = h − 1 in [Ke].

We shall reformulate theorem 1 using matrix notation so that the role of the Hasse invariant u_{1} becomes apparent. Let \underline{u} and \underline{s} denote the column vectors

\underline{u} = ( u_{h−1} ; ⋮ ; u_{1} ) , \underline{s} = ( s_{h−1} ; ⋮ ; s_{1} )

respectively. Denote by R the (h − 1) × (h − 1) matrix

R = ( \begin{matrix}u_{1}^{q^{h−2}} & u_{2}^{q^{h−3}} & … & u_{h−2}^{q} & u_{h−1} \\ 0 & u_{1}^{q^{h−3}} & … & u_{h−3}^{q} & u_{h−2} \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & … & u_{1}^{q} & u_{2} \\ 0 & 0 & … & 0 & u_{1} \end{matrix} ) ,

and let R′ be the (h − 1) × (h − 1) matrix

R′ = ( \begin{matrix}u_{1}^{q^{h−2}} & u_{2}^{q^{h−3}} & … & u_{h−2}^{q} & s_{h−1} \\ 0 & u_{1}^{q^{h−3}} & … & u_{h−3}^{q} & s_{h−2} \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & … & u_{1}^{q} & s_{2} \\ 0 & 0 & … & 0 & s_{1} \end{matrix} ) .

Let C_{m} denote the (h − 1) × (h − 1) matrix

C_{m} = ( \begin{matrix}0 & 1 & 0 & … & 0 & 0 \\ 0 & 0 & 1 & … & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & 0 & … & 1 & 0 \\ 0 & 0 & 0 & … & 0 & 1 \\ −u_{1}^{q^{m+h−1}} & −u_{2}^{q^{m+h−2}} & −u_{3}^{q^{m+h−3}} & … & −u_{h−2}^{q^{m+2}} & −u_{h−1}^{q^{m+1}} \end{matrix} ) .

Notice that det(R) = u_{1}^{q^{h−2}+···+q+1} , and det(C_{m}) = (−1)^{h−1} u_{1}^{q^{m+h−1}} . For b = 1, …, h − 1, let \underline{d}_{b,n,ξ} be the (h − 1) × 1 column vector

\underline{d}_{b,n,ξ} = ( ξ^{σ^{n+b+h−1}} · u_{h−1}^{q^{n}} · u_{1}^{q^{n−1}+···+q+1} ; ξ^{σ^{n+b+h−2}} · u_{h−2}^{q^{n}} · u_{1}^{q^{n−1}+···+q+1} ; ⋮ ; ξ^{σ^{n+b+2}} · u_{2}^{q^{n}} · u_{1}^{q^{n−1}+···+q+1} ; ξ^{σ^{n+b+1}} · u_{1}^{q^{n}} · u_{1}^{q^{n−1}+···+q+1} )

= ( \begin{matrix}u_{h−1}^{q^{n}} & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & u_{1}^{q^{n}} \end{matrix} ) · ( ξ^{σ^{n+b+h−1}} ; ⋮ ; ξ^{σ^{n+b+2}} ; ξ^{σ^{n+b+1}} ) · u_{1}^{q^{n−1}+···+q+1} .

Let \underline{d}′_{0,n,ξ} be the (h − 1) × 1 column vector

\underline{d}′_{0,n,ξ} = ( (ξ^{σ^{n+h−1}} − ξ^{σ^{n}}) · u_{h−1}^{q^{n}} · u_{1}^{q^{n−1}+···+q+1} ; (ξ^{σ^{n+h−2}} − ξ^{σ^{n}}) · u_{h−2}^{q^{n}} · u_{1}^{q^{n−1}+···+q+1} ; ⋮ ; (ξ^{σ^{n+2}} − ξ^{σ^{n}}) · u_{2}^{q^{n}} · u_{1}^{q^{n−1}+···+q+1} ; (ξ^{σ^{n+1}} − ξ^{σ^{n}}) · u_{1}^{q^{n}} · u_{1}^{q^{n−1}+···+q+1} )

= ( \begin{matrix}u_{h−1}^{q^{n}} & … & 0 \\ ⋮ & ... & ⋮ \\ 0 & … & u_{1}^{q^{n}} \end{matrix} ) · ( ξ^{σ^{n+h−1}} − ξ^{σ^{n}} ; ⋮ ; ξ^{σ^{n+2}} − ξ^{σ^{n}} ; ξ^{σ^{n+1}} − ξ^{σ^{n}} ) · u_{1}^{q^{n−1}+···+q+1} .

For i = 1, …, h − 1, let D_{i} be the (h − 1) × (h − 1) matrix

D_{i} =  ( \begin{matrix}& & & & & i^{th} \\ & 1 & 0 & … & 0 & 0 & 0 & 0 & … & 0 \\ & 0 & 1 & … & 0 & 0 & 0 & 0 & … & 0 \\ & ⋮ & ⋮ & ... & ⋮ & ⋮ & ⋮ & ⋮ & … & ⋮ \\ & 0 & 0 & … & 1 & 0 & 0 & 0 & … & 0 \\ i^{th} & 0 & 0 & … & 0 & 1 & 0 & 0 & … & 0 \\ & −u_{h−i}^{n−1} & −u_{h−i+1}^{n−2} & … & −u_{h−2}^{q^{n−i+1}} & −u_{h−1}^{q^{n−i}} & 1 & 0 & … & 0 \\ & 0 & 0 & … & 0 & 0 & 0 & 1 & … & 0 \\ & ⋮ & ⋮ & ... & ⋮ & ⋮ & ⋮ & ⋮ & ... & ⋮ \\ & 0 & 0 & … & 0 & 0 & 0 & 0 & … & 1 \end{matrix} ) ,

and let \underline{d}_{0,n,ξ} be the (h − 1) × 1 column vector

\underline{d}_{0,n,ξ} = D_{h−1} ⋯ D_{2} · D_{1} · \underline{d}′_{0,n,ξ} .

With the above notation, Theorem 1 says that

−R′ · C_{0} · C_{1} ⋯ C_{n+b−h} · \underline{d}_{b,n,ξ} ≡ \underline{s} − \underline{u} (mod (\underline{u})^{σ^{n+1}+σ^{n−2}+···+σ+1})

if char(A) = p > 0. If char(A) = 0 and q = p^{r} > p, this congruence holds modulo (\underline{u})^{p^{r−1}(p−1)+p^{r−1}(σ^{n}+···+σ+1)}. Replacing σ by q in the above congruence, we get a cruder congruence, and one sees that the column vector −R·C_{0}·C_{1} ⋯ C_{n+b−h}·\underline{d}_{b,n,ξ} gives the initial form of \underline{s} − \underline{u} with respect to the (\underline{u})-adic filtration on k[[\underline{u}]].

The matricial reformulation above makes it clear that theorem 1 is not well-suited for many applications: for each row of the matrix R, the entries are monomials of different degrees, and similarly for the C_{m}’s. However a natural remedy suggests itself. Instead of giving u_{1}, …, u_{h−1} weight 1, we decree that u_{i} has weight q^{i−1}, and let

w(\underline{u}^{I}) = i_{1} + i_{2} q + ⋯ + i_{h−1} q^{h−2}

for a multi-index I = (i_{1}, …, i_{h−1}) ∈ ℕ^{h−1} . Define

F il^{w} = \{ ∑_{w(I)≥w} a_{I} · \underline{u}^{I} ∈ κ_{h}[[\underline{u}]] | a_{I} ∈ κ_{h} \} ⊆ κ_{h}[[\underline{u}]] .

Clearly each F il^{w} is an ideal of κ_{h}[[\underline{u}]]. Now we shall re-examine the calculation using this filtration. Computing as before:

⟨u_{i}^{q^{n}}⟩ F^{n} e = ⟨u_{i}^{q^{n}}⟩ (V^{h−1} + ∑_{i=1}^{h−1} V^{i−1}⟨u_{i}^{q^{n−1}}⟩ ) ⋯

⋯ (V^{h−1} + ∑_{i=1}^{h−1} V^{i−1}⟨u_{i}^{q}⟩ ) (V^{h−1} + ∑_{i=1}^{h−1} V^{i−1}⟨u_{i}⟩ )
≡ 0 (mod F il^{q^{n+i−1}+q^{n−1}+···+q+1}) .

Similarly,

⟨s_{i}^{q^{n+b}}⟩ F^{n} e ≡ 0 (mod F il^{q^{n+i+b−1}+q^{n−1}+···+q+1}) .

Therefore

\{ \begin{matrix}s_{i} ≡ & u_{i} \\ a_{m}(\underline{u}) ≡ & 0 ∀ m ≥ 0 \end{matrix} (mod F il^{q^{n}+···+q+1})

Moreover,

⟨u_{i}^{q^{n}}⟩ F^{n} e ≡ ⟨u_{i}^{q^{n}} · u_{1}^{q^{n−1}+···+q+1}⟩ e (mod F il^{q^{n+i}+q^{n−2}+···+q+1}) , ⟨s_{i}^{q^{n+b}}⟩ F^{n} e ≡ ⟨u_{i}^{q^{n+b}} · u_{1}^{q^{n−1}+···+q+1}⟩ e (mod F il^{q^{n+b+i}+q^{n−2}+···+q+1}) . We get

(F − V^{h−1}) Φ(e′)

≡ ∑_{i=1}^{h−1} V^{i−1} ⟨u_{i}⟩ e + V^{n+b} ⟨ξ^{σ^{n+b+1}} u_{1}^{q^{n}+q^{n−1}+···+1}⟩ e

− ∑_{m≥0} V^{m+h−1} ⟨a_{m}(\underline{u})⟩ e ( mod F il^{q^{n+1}+q^{n−2}+···+q+1} ) ,

∑_{i=1}^{h−1} V^{i−1} ⟨s_{i}⟩ Φ(e′)

≡ ∑_{i=1}^{h−1} V^{i−1} ⟨s_{i}⟩ e + V^{n+b} ⟨ξ^{σ^{n+b}} s_{1}^{q^{n}} u_{1}^{q^{n−1}+···+q+1}⟩ e

+ ∑_{{1≤i≤h−1 \atop m≥0}} V^{m+i−1} ⟨s_{i}^{q^{m}} a_{m}(\underline{u})⟩ e ( mod F il^{q^{n+1}+q^{n−2}+···+q+1} ) .

If b ≠ 0, then

∑_{i=1}^{h−1} V^{i−1} ⟨s_{i}⟩ Φ(e′)

≡ ∑_{i=1}^{h−1} V^{i−1} ⟨s_{i}⟩ e + ∑_{m≥o} V^{m} ⟨ ∑_{i=1}^{h−1} s_{i}^{q^{m−i+1}} a_{m−i+1}(\underline{u}) ⟩ e

≡ ∑_{i=1}^{h−1} V^{i−1} ⟨ s_{i} + ∑_{1≤j≤i} s_{j}^{q^{i−j}} a_{i−j}(\underline{u}) ⟩ e

+ ∑_{h−1≤m≤n} V^{m} ⟨ ∑_{i=1}^{h−1} s_{i}^{q^{m−i+1}} a_{m−i+1}(\underline{u}) ⟩ e

( mod F il^{q^{n+1}+q^{n−2}+···+q+1} ) .

While if b = 0, then

∑_{i=1}^{h−1} V^{i−1} ⟨s_{i}⟩ Φ(e′)

≡ ∑_{i=1}^{h−1} V^{i−1} ⟨ s_{i} + ∑_{1≤j≤i} s_{j}^{q^{i−j}} a_{i−j}(\underline{u}) ⟩ e

+ ∑_{h−1≤m≤n−1} V^{m} ⟨ ∑_{i=1}^{h−1} s_{i}^{q^{m−i+1}} a_{m−i+1}(\underline{u}) ⟩ e

+ V^{n} ⟨ ξ^{σ^{n}} u_{1}^{q^{n}+···+q+1} + ∑_{j=1}^{h−1} s_{j}^{q^{n+1−j}} a_{n+1−j}(\underline{u}) ⟩ e

( mod F il^{q^{n+1}+q^{n−2}+···+q+1} ) .

When b ≠ 0,

(F − V^{h−1}) Φ(e′) ≡ ∑_{i=1}^{h−1} V^{i−1} ⟨u_{i}⟩ e − ∑_{h−1≤m≤n+b−1} V^{m} ⟨a_{m−h+1}(\underline{u}) ⟩ e

+ V^{n+b} ⟨ ξ^{σ^{n+b+1}} u_{1}^{q^{n}+q^{n−1}+···+q+1} − a_{n+b−h+1}(\underline{u}) ⟩ e

− ∑_{m≥n+b+1} V^{m} ⟨a_{m−h+1}(\underline{u})⟩ e

( mod F il^{q^{n+1}+q^{n−2}+···+q+1} ) .

Comparing this with the expression for ∑_{i=1}^{h−1} V^{i−1} ⟨s_{i}⟩ Φ(e′), we get

\{ \begin{matrix}u_{i} ≡ & s_{i} + ∑_{j=1}^{i} s_{j}^{q^{i−j}} a_{i−j}(\underline{u})  1 ≤ i ≤ h − 1 \\ a_{m}(\underline{u}) ≡ & 0  ∀m ≥ n + b − h + 2 \\ a_{n+b−h+1}(\underline{u}) ≡ & ξ^{σ^{n+b+1}} u_{1}^{q^{n}+q^{n−1}+···+q+1} \\ a_{m−h+1}(\underline{u}) ≡ & 0  m = n + b − 1, …, n + 1 \\ a_{m−h+1}(\underline{u}) ≡ & − ∑_{i=1}^{h−1} s_{i}^{q^{m−i+1}} a_{m−i+1}(\underline{u})  h − 1 ≤ m ≤ n \\ ( ⇔ a_{m}(\underline{u}) ≡ & − ∑_{i=1}^{h−1} s_{i}^{q^{m+h−i}} a_{m+h−i}(\underline{u})  0 ≤ m ≤ n − h + 1 ) \\ & ( mod F il^{q^{n+1}+q^{n−2}+···+q+1} ) \end{matrix}

From the above recursive formulae we can extract the leading terms of the a_{m}(\underline{u})’s:

(1) a_{n+b−h+1}(\underline{u}) ∈ F il^{q^{n}+q^{n−1}+···+q+1}, and its leading term is

ξ^{σ^{n+b+1}} u_{1}^{q^{n}+q^{n−1}+···+q+1} .

(2) a_{n−h+b}(\underline{u}) ≡ ⋯ ≡ a_{n−h+2}(\underline{u}) ≡ 0 ( mod F il^{q^{n+1}+q^{n−2}+···+q+1} )

(3) a_{n−h+1}(\underline{u}) ∈ F il^{2q^{n}+q^{n−1}+···+q+1}, with leading term

−ξ^{σ^{n+b+1}} u_{h−b}^{q^{n+b−h+1}} u_{1}^{q^{n}+q^{n−1}+···+q+1} ;

a_{n−h}(\underline{u}) has leading term

−ξ^{σ^{n+b+1}} u_{h−b−1}^{q^{n+b−h+1}} u_{1}^{q^{n}+q^{n−1}+···+q+1} ;

and similarly for a_{n−h−1}(\underline{u}), …, a_{n−2h+b+2}(\underline{u}). For instance, a_{n−2h+b+2}(\underline{u}) has leading term

−ξ^{σ^{n+b+1}} u_{1}^{q^{n+b−h+1}} u_{1}^{q^{n}+q^{n−1}+···+q+1} .

(4) a_{n−2h+b+1}(\underline{u}) has leading term

ξ^{σ^{n+b+1}} u_{h−1}^{q^{n−2h+b+2}} u_{1}^{q^{n+b−h+1}+q^{n}+q^{n−1}+···+q+1} ;

a_{n−2h+b}(\underline{u}) has leading term

ξ^{σ^{n+b+1}} u_{h−2}^{q^{n−2h+b+2}} u_{1}^{q^{n+b−h+1}+q^{n}+q^{n−1}+···+q+1} ;

similarly for a_{n−2h+b−1}(\underline{u}), …, a_{n−3h+b+3}(\underline{u}). For instance, a_{n−3h+b+3}(\underline{u}) has leading term

ξ^{σ^{n+b+1}} u_{1}^{q^{n−2h+b+2}} u_{1}^{q^{n+b−h+1}+q^{n}+q^{n−1}+···+q+1} .

The pattern for the leading terms of the a_{m}(\underline{u})’s should be very clear by now. We shall give the leading term of the first few a_{m}(\underline{u})’s; they appear in the expression for the s_{i} − u_{i}’s.

(5) Write n+b = (h−1)·d+c, with c, d ∈ ℤ, 0 ≤ c ≤ h−2. Then n+b−(h−1)(d−1) = c + h − 1, and a_{c+h−1}(\underline{u}) has leading term

(−1)^{d} ξ^{σ^{n+b+1}} u_{1}^{q^{c+2(h−1)}+···+q^{n+b−2(h−1)}+q^{n+b−(h−1)}+q^{n}+q^{n−1}+···+q+1} .

a_{c+h−2}(\underline{u}) has leading term

(−1)^{d−1} ξ^{σ^{n+b+1}} u_{h−1}^{q^{c+h−1}} u_{1}^{q^{c+2(h−1)}+···+q^{n+b−2(h−1)}+q^{n+b−(h−1)}+q^{n}+q^{n−1}+···+q+1} ;

similarly for a_{c+h−3}(\underline{u}), …, a_{c}(\underline{u}). For instance, a_{c}(\underline{u}) has leading term

(−1)^{d−1} ξ^{σ^{n+b+1}} u_{1}^{q^{c+(h−1)}+q^{c+2(h−1)}+···+q^{n+b−2(h−1)}+q^{n+b−(h−1)}+q^{n}+q^{n−1}+···+q+1} .

For the rest of the a_{m}(\underline{u})’s: a_{c−1}(\underline{u}) has leading term

(−1)^{d} ξ^{σ^{n+b+1}} u_{h−1}^{q^{c}} u_{1}^{q^{c+(h−1)}+q^{c+2(h−1)}+···+q^{n+b−2(h−1)}+q^{n+b−(h−1)}+q^{n}+q^{n−1}+···+q+1} , a_{c−2}(\underline{u}) has leading term (−1)^{d} ξ^{σ^{n+b+1}} u_{h−2}^{q^{c}} u_{1}^{q^{c+(h−1)}+q^{c+2(h−1)}+···+q^{n+b−2(h−1)}+q^{n+b−(h−1)}+q^{n}+q^{n−1}+···+q+1} , similarly for a_{c−3}(\underline{u}), …, a_{0}(\underline{u}). Especially, a_{0}(\underline{u}) has leading term (−1)^{d} ξ^{σ^{n+b+1}} u_{h−c}^{q^{c}} u_{1}^{q^{c+(h−1)}+q^{c+2(h−1)}+···+q^{n+b−2(h−1)}+q^{n+b−(h−1)}+q^{n}+q^{n−1}+···+q+1} . For 1 ≤ i ≤ h − 1,

u_{i} ≡ s_{i} + ∑_{1≤j≤i} s_{j}^{q^{i−j}} a_{i−j}(\underline{u}) ( mod F il^{q^{n+1}+q^{n−2}+···+q+1} ) .

Hence s_{h−1} − u_{h−1} has leading term (−1)^{d} ξ^{σ^{n+b+1}} u_{h−c−1}^{q^{c}} u_{1}^{q^{c+(h−1)}+q^{c+2(h−1)}+···+q^{n+b−2(h−1)}+q^{n+b−(h−1)}+q^{n}+q^{n−1}+···+q+1} , similarly for s_{h−2} − u_{h−2}, …, s_{c+1} − u_{c+1}, with the leading term of s_{c+1} − u_{c+1} being (−1)^{d} ξ^{σ^{n+b+1}} u_{1}^{q^{c}} u_{1}^{q^{c+(h−1)}+q^{c+2(h−1)}+···+q^{n+b−2(h−1)}+q^{n+b−(h−1)}+q^{n}+q^{n−1}+···+q+1} . Next, s_{c} − u_{c} has leading term (−1)^{d−1} ξ^{σ^{n+b+1}} u_{c} u_{h−c}^{q^{c}} u_{1}^{q^{c+(h−1)}+q^{c+2(h−1)}+···+q^{n+b−2(h−1)}+q^{n+b−(h−1)}+q^{n}+q^{n−1}+···+q+1} , s_{c−1} − u_{c−1} has leading term (−1)^{d−1} ξ^{σ^{n+b+1}} u_{c−1} u_{h−c}^{q^{c}} u_{1}^{q^{c+(h−1)}+q^{c+2(h−1)}+···+q^{n+b−2(h−1)}+q^{n+b−(h−1)}+q^{n}+q^{n−1}+···+q+1} , similarly for s_{c−2} − u_{c−2}, …, s_{1} − u_{1}. The leading term of s_{1} − u_{1} is (−1)^{d−1} ξ^{σ^{n+b+1}} u_{1} u_{h−c}^{q^{c}} u_{1}^{q^{c+(h−1)}+q^{c+2(h−1)}+···+q^{n+b−2(h−1)}+q^{n+b−(h−1)}+q^{n}+q^{n−1}+···+q+1} .

Notice that the leading terms of s_{h−1} − u_{h−1}, …, s_{1} − u_{1} have different weights; the weight of s_{c+1} − u_{c+1} is the lowest. So the leading terms of the matrix

( s_{h−1} − u_{h−1} ; ⋮ ; s_{c+1} − u_{c+1} ; ⋮ ; s_{1} − u_{1} )

is

( 0 ; ⋮ ; 0 ; (c+1)^{st} (−1)^{d} ξ^{σ^{n+b+1}} u_{1}^{q^{c}+q^{c+(h−1)}+q^{c+2(h−1)}+···+q^{n+b−(h−1)}+q^{n}+q^{n−1}+···+q+1} ; 0 ; ⋮ ; 0 )

This finishes our discussion of the case b ≠ 0.

When b = 0,

(F − V^{h−1}) Φ(e′)

≡ ∑_{i=1}^{h−1} V^{i−1} ⟨u_{i}⟩ e − ∑_{h−1≤m≤n−1} V^{m} ⟨a_{m−h+1}(\underline{u})⟩ e

+ V^{n} ⟨ ξ^{σ^{n+1}} u_{1}^{q^{n}+q^{n−1}+···+q+1} − a_{n−h+1}(\underline{u}) ⟩ e

− ∑_{m≥n+1} V^{m} ⟨a_{m−h+1}(\underline{u})⟩ e ( mod F il^{q^{n+1}+q^{n−2}+···+q+1} )

Again by comparing with the expression for ∑_{i=1}^{h−1} V^{i−1} ⟨s_{i}⟩ Φ(e′), we get

\{ \begin{matrix}a_{m}(\underline{u}) ≡ & 0  ∀ m ≥ n − h + 2 \\ a_{n−h+1}(\underline{u}) ≡ & (ξ^{σ^{n+1}} − ξ^{σ^{n}}) u_{1}^{q^{n}+q^{n−1}+···+q+1} \\ a_{m}(\underline{u}) ≡ & − ∑_{i=1}^{h−1} s_{i}^{q^{m+h−i}} a_{m+h−i}(\underline{u})  0 ≤ m ≤ n − h \\ u_{i} ≡ & s_{i} + ∑_{1≤j≤i} s_{j}^{q^{i−j}}a_{i−j}(\underline{u})  1 ≤ i ≤ h − 1 \end{matrix}

It follows from the above inductive formulae that

a_{n−h}(\underline{u}) ≡ −(ξ^{σ^{n+1}} − ξ^{σ^{n}}) u_{h−1}^{q^{n−h+1}} u_{1}^{q^{n}+q^{n−1}+···+q+1} ;

a_{n−h−1}(\underline{u}) has leading term

−(ξ^{σ^{n+1}} − ξ^{σ^{n}}) u_{h−2}^{q^{n−h+1}} u_{1}^{q^{n}+q^{n−1}+···+q+1} ;

similarly for a_{n−h−2}(\underline{u}), …, a_{n−2h+2}(\underline{u}); a_{n−2h+2}(\underline{u}) has leading term

−(ξ^{σ^{n+1}} − ξ^{σ^{n}}) u_{1}^{q^{n−h+1}} u_{1}^{q^{n}+q^{n−1}+···+q+1} .

The leading terms for the rest of the a_{m}(\underline{u})’s follow the same pattern, descending in blocks of h-1 steps. When going down from one block to another, an additional factor has to be added. Within the same block, most factors stay the same. The changing factor is the extra factor which was inserted when stepping one block down and has the form −u_{i}^{q^{n−(h−1)k}}. Here k is a natural number fixed for the block and increases by one when going down from one block to the next, and i varies within the block downward from h − 1 to 1. We write n as n = (h − 1) · d + c, with c, d ∈ ℕ, 0 ≤ c ≤ h − 2. Then n − (h − 1) · (d − 1) = c + h − 1, and the leading term of a_{c+h−1}(\underline{u}) is

(−1)^{d} (ξ^{σ^{n+1}} − ξ^{σ^{n}}) u_{1}^{q^{c+2(h−1)}+q^{c+3(h−1)}+···+q^{n−(h−1)}+q^{n}+q^{n−1}+···+q+1} .

a_{c+h−2}(\underline{u}) has leading term

(−1)^{d−1} (ξ^{σ^{n+1}} − ξ^{σ^{n}}) u_{h−1}^{q^{c+(h−1)}} u_{1}^{q^{c+2(h−1)}+q^{c+3(h−1)}+···+q^{n−(h−1)}+q^{n}+q^{n−1}+···+q+1} ;

similarly for a_{c+h−3}(\underline{u}), …, a_{c}(\underline{u}); a_{c}(\underline{u}) has leading term

(−1)^{d−1} (ξ^{σ^{n+1}} − ξ^{σ^{n}}) u_{1}^{q^{c+(h−1)}} u_{1}^{q^{c+2(h−1)}+q^{c+3(h−1)}+···+q^{n−(h−1)}+q^{n}+q^{n−1}+···+q+1} .

a_{c−1}(\underline{u}) has leading term

(−1)^{d} (ξ^{σ^{n+1}} − ξ^{σ^{n}}) u_{h−1}^{q^{c}} u_{1}^{q^{c+(h−1)}+q^{c+2(h−1)}+···+q^{n−(h−1)}+q^{n}+q^{n−1}+···+q+1} ;

similarly for a_{c−2}(\underline{u}), …, a_{0}(\underline{u}); the leading term of a_{0}(\underline{u}) is

(−1)^{d} (ξ^{σ^{n+1}} − ξ^{σ^{n}}) u_{h−c}^{q^{c}} u_{1}^{q^{c+(h−1)}+q^{c+2(h−1)}+···+q^{n−(h−1)}+q^{n}+q^{n−1}+···+q+1} .

s_{h−1} − u_{h−1} has leading term

(−1)^{d} (ξ^{σ^{n+1}} − ξ^{σ^{n}}) u_{h−c−1}^{q^{c}} u_{1}^{q^{c+(h−1)}+q^{c+2(h−1)}+···+q^{n−(h−1)}+q^{n}+q^{n−1}+···+q+1} ,

similarly for s_{h−2}, …, s_{c+1} − u_{c+1}; the leading term of s_{c+1} − u_{c+1} is

(−1)^{d} (ξ^{σ^{n+1}} − ξ^{σ^{n}}) u_{1}^{q^{c}} u_{1}^{q^{c+(h−1)}+q^{c+2(h−1)}+···+q^{n−(h−1)}+q^{n}+q^{n−1}+···+q+1} .

The leading term of s_{c} − u_{c} is

(−1)^{d−1} (ξ^{σ^{n+1}} − ξ^{σ^{n}}) u_{c} u_{h−c}^{q^{c}} u_{1}^{q^{c+(h−1)}+q^{c+2(h−1)}+···+q^{n−(h−1)}+q^{n}+q^{n−1}+···+q+1} ;

the leading term of s_{c−1} − u_{c−1} is

(−1)^{d−1} (ξ^{σ^{n+1}} − ξ^{σ^{n}}) u_{c−1} u_{h−c}^{q^{c}} u_{1}^{q^{c+(h−1)}+q^{c+2(h−1)}+···+q^{n−(h−1)}+q^{n}+q^{n−1}+···+q+1} ;

similarly for s_{c−2} − u_{c−2}, …, s_{1} − u_{1}. For instance, the leading term of s_{1} − u_{1} is

(−1)^{d−1} (ξ^{σ^{n+1}} − ξ^{σ^{n}}) u_{1} u_{h−c}^{q^{c}} u_{1}^{q^{c+(h−1)}+q^{c+2(h−1)}+···+q^{n−(h−1)}+q^{n}+q^{n−1}+···+q+1} .

Notice that the leading terms of the matrix

( s_{h−1} − u_{h−1} ; ⋮ ; s_{c+1} − u_{c+1} ; ⋮ ; s_{1} − u_{1} )

is

( 0 ; ⋮ ; 0 ; (c+1)^{st} (−1)^{d} (ξ^{σ^{n+1}} − ξ^{σ^{n}}) u_{1}^{q^{c}+q^{c+(h−1)}+q^{c+2(h−1)}+···+q^{n−(h−1)}+q^{n}+q^{n−1}+···+q+1} ; 0 ; ⋮ ; 0 )

In the case when char(A) = 0, the calculation is more complicated, but the conclusion about the leading term still holds. We leave this assertion to the patient reader to verify. We summarize our discussion above as

Theorem 2. Let ( F il^{w} )_{w∈ℕ} be the decreasing filtration of κ_{h}[[\underline{u}]] by

F il^{w} = \{ ∑_{w(I)≥w} a_{I} \underline{u}^{I} ∈ κ_{h}[[\underline{u}]] | a_{I} ∈ κ_{h} \} ,

where

w(I) = ∑_{j=1}^{h−1} i_{j} · q^{j−1} for I = (i_{1}, …, i_{h−1}) ∈ ℕ .

For Φ_{0} ∈ Aut(G_{0}) such that Φ_{0}(e) = e + ξ V^{ℓ} · e, where ξ ∈ μ_{q^{h}−1}(W_{A}(𝔽_{q^{h}})), ℓ = nh + b, 0 ≤ b ≤ h − 1, n ≥ 1, n, b ∈ ℕ. Write n + b = (h − 1)d + c, with c, d ∈ ℕ, 0 ≤ c ≤ h − 2. Then for 1 ≤ i ≤ h − c − 1, the leading term of s_{c+i} − u_{c+i} with respect to ( F il^{w} )_{w∈ℕ} is

(−1)^{d} · η · u_{i}^{q^{c}} · u_{1}^{∑_{j=1}^{d−1} q^{c+j(h−1)}+∑_{a=0}^{n} q^{a}} ,

where

η = \{ \begin{matrix}ξ^{σ^{n+b+1}} & if 1 ≤ b ≤ h − 1 \\ (ξ^{σ^{n+1}} − ξ^{σ^{n}}) & if b = 0 \end{matrix} .

For 1 ≤ i ≤ c, the leading term of s_{i} − u_{i} with respect to ( F il^{w} )_{w∈ℕ} is

(−1)^{d} · η · u_{i} · u_{h−c}^{q^{c}} · u_{1}^{∑_{j=1}^{d−1} q^{c+j(h−1)}+∑_{a=0}^{n} q^{a}} ,

and η is the same as before. Especially, the leading term of the vector

(s_{1} − u_{1}, …, s_{h−1} − u_{h−1})

is

(0, …, 0, (−1)^{d} · η · u_{1}^{∑_{j=0}^{d−1} q^{c+j(h−1)}+∑_{a=0}^{n} q^{a}}, 0, …, 0) ,

where only the (c + 1)st entry is nonzero.

3. Consequences of the calculation

The first application of Theorem 2 is

Proposition 3. Let k be an overfield of κ_{h}. If a function f ∈ k[[\underline{u}]] is invariant under a compact open subgroup of Aut(G_{0}), then f is a constant, i.e. f ∈ k.

Corollary. Let k be an overfield of κ_{h}. If W ⊆ k[[\underline{u}]] is a finite dimensional k-vector subspace stable under an open subgroup of Aut(G_{0}), then W ⊆ k.

Remarks.

(1) Heuristically, the quotient of Spec (k[[\underline{u}]]) by an open subgroup H of Aut(G_{0}) in the category of affine schemes should be the spectrum of the H-invariants in k[[\underline{u}]]. From this point of view Proposition 3 indicates that “the generic orbits” of any compact open subgroup of Aut(G_{0}) operating naturally on Spec (k[[\underline{u}]]) (or Spf (k[[\underline{u}]]) ) are likely to be big in the sense that they are Zariski dense in Spec (k[[\underline{u}]]).

(2) The infinite dimensional continuous representation (\underline{u}) · k[[\underline{u}]] of Aut(G_{0}) is a very poorly understood representation. One can view it as being “topologically nilpotent”. It is highly reducible yet indecomposable.

Proof of Corollary. Suppose that W is a finite dimensional k-vector subspace of k[[\underline{u}]] stable under a compact open subgroup H ⊆ Aut(G_{0}). Since the decreasing sequence of vector spaces

W ∩ (\underline{u})^{n}

stablizes as n → ∞, and

⋂_{n} (\underline{u})^{n} = 0 ,

there exists n_{0} ∈ ℕ such that W ∩ (\underline{u})^{n_{0}} = 0. There exists an open subgroup H′ ⊆ H such that

ϕ(f) − f ∈ (\underline{u})^{n_{0}} ∀ ϕ ∈ H′, ∀ f ∈ k[[\underline{u}]] .

Hence all elements of W are invariant under H′, therefore W ⊆ k by Proposition 3. □

Proof of Proposition 3. Suppose that f ∈ (\underline{u})k[[\underline{u}]] is fixed by an open subgroup H ⊆ Aut(G_{0}). There exists a positive integer n_{0} such that id + π^{n_{0}}End(G_{0}) ⊆ H. There exists a unique s ∈ ℕ and a unique f_{0} ∈ (\underline{u})k[[\underline{u}]] such that f = f_{0}^{p^{s}} and

∇f_{0} := (\frac{∂f_{0}}{∂u_{1}} , …, \frac{∂f_{0}}{∂u_{h−1}} ) ≠ 0 .

f_{0} is fixed by H since f is. Therefore we may and do assume that ∇f is nonzero, hence for a suitable choice of a positive integer ℓ (subject to some congruence condition) and Φ_{0} ∈ Aut(G_{0}) with Φ_{0}(e) = e + ξV^{ℓ}e, ξ ∈ μ_{q^{h}−1} \ μ_{q−1} , the pairing of the initial form of ∇f with the initial form of

( h(Φ_{0})(u_{1}) − u_{1}, …, h(Φ_{0})(u_{h−1}) − u_{h−1} )

is not 0. Note that the condition ξ ∉ μ_{q−1} ensures that the element η defined as in Theorem 2 is not equal to zero. The Taylor expansion of f gives a contradiction if ℓ is sufficiently large to force Φ_{0} ∈ H. □

The same argument using the Taylor expansion also proves

Proposition 4. Suppose that k is an overfield of κ_{h}. Let I ⊆ k[[\underline{u}]] be an ideal stable under an open subgroup of Aut(G_{0}). Suppose that the image of u_{1} in gr_{F il^{w}}^{•} (k[[\underline{u}]]/I) is not a zero-divisor of gr_{F il^{w}}^{•} (k[[\underline{u}]]/I). Then I = (0).

The two propositions above are only pretty weak statements concerning the orbit structure of Spf k[[u]] with respect to the action of Aut(G_{0}) or an open subgroup of Aut(G_{0}). These orbits under the compact group Aut(G_{0}) are very complicated, but we can take the Zariski closure of the orbits to simplify the problem. Then an optimist may harbor the following

Hope. Any reduced irreducible formal subscheme of Spf (k[[\underline{u}]]) stable under an open subgroup of Aut(G_{0}) is of the form

Spf (k[[\underline{u}]]/(u_{1}, u_{2}, …, u_{i−1})) for some i , 1 ≤ i ≤ h .

(Notice that Spf (k[[\underline{u}]]/(u_{1}, u_{2}, …, u_{i−1})) is the locus over which the universal deformation has height at least i.)

Proposition 4 can be viewed as a weak evidence for the hope. To prove the hope by computation, one would need much better control of the error term than what is provided by this paper.

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Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, U.S.A.

E-mail address: chai@math.upenn.edu