Explicit formula for the action of the automorphism group of a formal group on the moduli space of its deformations.

O. Demchenko and A. Gurevich^{1}

0. Introduction.

There are two different approaches to the study of deformations of a formal group over a finite field or, to be more precise, to the study of ⋆-isomorphism classes of such deformations, where a ⋆-isomorphism of deformations means an isomorphism with identity reduction. First of them is based on the Dieudonné theory (see [2] or [3]) that assigns to any formal group over a finite field a profinite module over the Dieudonné ring of this field providing in this manner an equivalence between the category of formal groups and the category of Dieudonné modules. For any deformation of a formal group over a finite field which is defined over the ring of Witt vectors of this field, Fontaine has constructed a submodule in the Dieudonné module of this formal group in such a way that ⋆-isomorphic deformations correspond to the same submodule (see [3]). Another approach is due to Lubin and Tate who introduced a universal deformation of a formal group over a finite field, i.e. a deformation Γ with a property that any deformation of the given formal group which is defined over any complete local algebra is ⋆-isomorphic to the formal group which results from Γ by applying some homomorphism to its coefficients (see [8]). Our aim is to compare these two approaches and to establish a kind of explicit connection between them.

Let Φ be a formal group over a finite field of finite height. We consider an action of the automorphism group of Φ on the set of ⋆-isomorphism classes of deformations of Φ which is defined by the formula φ[F] = [g^{−1}(F (g, g))], where F is a deformation of Φ, φ is an automorphism of Φ and g is a formal power series with reduction equal to φ. One can easily check that the stabilizer and the orbit of a ⋆-isomorphism class [F] are the set of the reductions of automorphisms of F and the set of ⋆-isomorphism classes of deformations isomorphic to F respectively.

According to the results of Honda [6] whose theory is parallel to the theory of Dieudonne and Fontaine, every endomorphism of Φ is equal to the reduction of the formal power series f_{0}^{−1}(∑_{i=0}^{h−1} β_{i}Δ^{i}f_{0}) for some β_{i} belonging to the ring of Witt vectors, where h is the height of Φ, f_{0} is the logarithm of a fixed deformation F_{0} of Φ over the ring of Witt vectors, Δ operates on the ring of formal power series in the following way: Δ (∑_{i=1}^{∞} c_{i}x^{i}) = ∑_{i=1}^{∞} Frob(c_{i})x^{pi}, and different tuples (β_{0}, ..., β_{h−1}) give rise to different endomorphisms. Moreover, the endomorphism corresponding to such a tuple is the reduction of an endomorphism of F_{0} if and only if β_{i} = 0 for all i ≠ 0. Then the left cosets of the stabilizer of the ⋆-isomorphism class [F_{0}] in the automorphism group of Φ can be represented by the reductions of the formal power series f_{0}^{−1}(f_{0} +∑_{i=1}^{h−1} β_{i}Δ^{i}f_{0}) with β_{i} from the ring of Witt vectors. That gives a parametrization of the orbit of the ⋆-isomorphism class [F_{0}].

1The second author was partially supported by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by Minerva Foundation. He is also thankful to the Max Planck Institute for its hospitality.

According to the results of Lubin and Tate [8], a universal deformation Γ of Φ is defined over the ring of formal power series in h − 1 variables with coefficients in the ring of Witt vectors. A homomorphism from this ring to the ring of Witt vectors is uniquely determined by the images τ_{1}, ..., τ_{h−1} of the variables which have to belong to the maximal ideal. Applying different homomorphisms to the coefficients of Γ, we get deformations of Φ belonging to different ⋆-isomorphism classes. Thus we have another parametrization of the orbit of the ⋆-somorphism class [F_{0}]. In this article, we obtain an explicit expression of one of these two parameters through another.

We use Hazewinkel’s universal p-typical formal group to construct a special universal deformation which suits to further calculations. Then we consider two ⋆-isomorphic deformations of Φ, the first one is the formal group with the logarithm f_{0} + ∑_{i=1}^{h−1} β_{i}Δ^{i}f_{0} and the second one is resulted from the special universal deformation by applying the homomorphism corresponding to a tuple (τ_{1}, ..., τ_{h−1}) to its coefficients, and use Honda’s technique to prove some equalities modulo p involving the coefficients of the logarithms of these deformations. Further we define rigid analytic power series ρ_{0}, ..., ρ_{h−1} that allow us to find exact equalities containing parameters β_{i} and τ_{i}. From them we deduce the formula β_{i} = \frac{ρ_{i}}{ρ_{0}} (τ_{1}, ..., τ_{h−1}) which is our main result (see Theorem 8.3).

Finally, we consider a special case when Φ is the reduction of the Artin-Hasse formal group. It has a benefit that all of the deformations of Φ over the ring of Witt vectors are isomorphic, and thus the orbit of the ⋆-isomorphism class [F_{0}] coincides with the set of all ⋆-isomorphism classes of deformations of Φ. Moreover in this case, the general formula for β_{i} through τ_{i} can be reduced to a simpler one.

1. Lubin-Tate’s moduli space of formal group deformations.

Let k be an algebraic extension of F_{p}, R be the ring of Witt vectors of k and K be the fraction field of R. We consider a one-parameter formal group Φ over k of finite height h. Let A be a complete noetherian local R-algebra with maximal ideal M ⊇ pA and the residue field A/M ⊇ k. A formal group F over A such that its reduction modulo M is equal to Φ is called a deformation of Φ over A. If an isomorphism between the deformations F and G over A has identity reduction modulo M, we say that it is a ⋆-isomorphism. In this case the deformations F and G are called ⋆-isomorphic. The ⋆-isomorphism class of the deformation F will be denoted by [F].

Lubin and Tate constructed a moduli space for ⋆-isomorphism classes of deformations of Φ. We review their main results.

The formal group Φ over k of height h is said to be in normal form if

Φ(x, y) ≡ x + y + a C_{p^{h}}(x, y) mod deg (p^{h} + 1)

holds for some a ∈ k, where C_{p^{i}}(x, y) = ((x + y)^{p^{i}} − x^{p^{i}} − y^{p^{i}})/p.

Lemma 1.1 Every formal group of finite height over k is isomorphic to a formal group in normal form.

Proof. See [7], Lemma 6.

Suppose Φ to be in normal form.

Theorem 1.2 There exists a formal group Γ over R[[t_{1}, ..., t_{h−1}]] such that the reduction of Γ(0, ..., 0) modulo p is equal to Φ and

Γ(0, ..., 0, t_{i}, ..., t_{h−1})(x, y) ≡ x + y − t_{i} C_{p^{i}}(x, y) mod deg (p^{i} + 1)

for 1 ≤ i ≤ h − 1.

Proof. See [8], Proposition 1.1.

A formal group satisfying the conditions of Theorem 1.2 is called a generic formal group for Φ. It is not unique, because constructing it, one can make several arbitrary choices.

Theorem 1.3 Let Γ be a generic formal group for Φ and F be a deformation of Φ over A. Then there is a unique (h−1)-tuple (τ_{1}, ..., τ_{h−1}), τ_{i} ∈ M, such that F is ⋆-isomorphic to Γ(τ_{1}, ..., τ_{h−1}) and the ⋆-isomorphism is uniquely defined.

Proof. See [8], Theorem 3.1.

Theorem 1.3 means that Γ is a universal deformation of the formal group Φ, and the set of ⋆-isomorphism classes of the deformations of Φ over A is in one-to-one correspondence with M × ... × M (h − 1 times).

2. Hazewinkel’s p-typical formal groups.

Hazewinkel used a universal p-typical formal group to get a parametrization of a large number of generic formal groups, which in particular allows to choose one of them canonically. We review here his construction.

Denote E = Q_{p}[t_{1}, t_{2}, ...], D = Z_{p}[t_{1}, t_{2}, ...]. Define a Q_{p}-endomorphism σ of E by σ(t_{i}) = t_{i}^{p}. Let σ operate on the ring E[[x]]_{0} of formal power series over E without constant term by the formula

σ (∑_{n=1}^{∞} c_{n}x^{n}) = ∑_{n=1}^{∞} σ(c_{n})x^{pn}.

There exists a unique formal power series f_{T} = ∑_{n=0}^{∞} d_{n}x^{p^{n}} ∈ E[[x]]_{0} which satisfies the functional equation

pf_{T} − ∑_{i=1}^{∞} t_{i}σ^{i}(f_{T}) = px.

It is clear that d_{0} = 1, d_{n} ∈ Q_{p}[t_{1}, ..., t_{n}] ⊂ E and d_{n} ≡ t_{n}/p mod t_{1}, ..., t_{n−1}. Proposition 2.1 The coefficients of f_{T} satisfy the recurrent relation

pd_{n} = ∑_{i=1}^{n} t_{i}^{p^{n−i}}d_{n−i}.

Proof. See [5], formula 3.3.9.

Propositon 2.2

i) f_{T} is the logarithm of a formal group F_{T} defined over D. ii) F_{T} (x, y) ≡ x + y − t_{i} C_{p^{i}}(x, y) mod t_{1}, ..., t_{i−1}, deg (p^{i} + 1).

Proof. The part i) is an immediate consequence of Hazewinkel’s functional equation lemma, see [5], section 2.2.i) and formula 2.3.7. The part ii) follows from the fact that f_{T} ≡ x + (t_{i}/p)x^{p^{i}} mod t_{1}, ..., t_{i−1}, deg (p^{i} + 1).

A formal group F over a Z_{p}-algebra B is called p-typical if there is a homomorphism ξ from D to B such that F = ξ_{∗}F_{T}. Evidently, F_{T} is a universal p-typical formal group.

Proposition 2.3 Every formal group over a Z_{p}-algebra B is isomorphic to a p-typical one.

Proof. See [5], Theorem 15.2.9 or Theorem 21.5.6.i).

Lemma 2.4 Let ξ be a homomorphism from D to k. Then the height of the formal group ξ_{∗}F_{T} is equal to the minimal i satisfying ξ(t_{i}) ≠ 0.

Proof. It follows from f_{T} ≡ x + (t_{i}/p)x^{p^{i}} mod t_{1}, ..., t_{i−1}, deg (p^{i} + 1).

Corollary 2.5 Every p-typical formal group over k is in normal form.

Proof. It follows from Proposition 2.2.ii) and Lemma 2.4.

Now suppose Φ to be a p-typical formal group. By Corollary 2.5, it is automatically in normal form. Take a homomorphism ξ from D to k such that Φ = ξ_{∗}F_{T} and define a Q_{p}-homomorphism Ξ from E to K[[t_{1}, ..., t_{h−1}]] as follows: Ξ(t_{i}) = t_{i} for i < h; Ξ(t_{i}) = r_{i} ∈ R for i ≥ h, where the reduction of r_{i} modulo p is equal to ξ(t_{i}). Then the formal group Γ = Ξ_{∗}F_{T} is defined over R[[t_{1}, ..., t_{h−1}]] and its reduction modulo p, t_{1}, ..., t_{h−1} is equal to Φ. Moreover by Proposition 2.2.ii), Γ is a generic formal group for Φ, and hence by Theorem 1.3, it is a universal deformation of Φ. The choice of r_{i} in the definition of Ξ is not unique. From now on, we fix r_{i} to be the multiplicative representatives of ξ(t_{i}) in R. Then by Lemma 2.4, we have r_{i} = 0 for i < h, r_{h} is invertible. This fixation determines Ξ and Γ uniquely.

It is clear that the formal power series γ = Ξ_{∗}f_{T} is the logarithm of Γ. Put a_{n} = Ξ(d_{n}) ∈ K[t_{1}, ..., t_{h−1}]. Then we have γ(x) = ∑_{n=1}^{∞} a_{n}x^{p^{n}}. Moreover, Proposition 2.1 implies

pa_{n} = ∑_{j=1}^{min(n,h−1)} t_{j}^{p^{n−j}}a_{n−j} + ∑_{i=h}^{n} r_{i}^{p^{n−i}}a_{n−i}.

3. The action of the automorphism group on the moduli space of deformations.

Let l be an algebraic extension of k such that A/M ⊇ l. The group of automorphisms of Φ over l acts on the set of ⋆-isomorphism classes of deformations of Φ over A in the following way. If φ ∈ Aut_{l}Φ and F is a deformation of Φ over A, then φ[F] = [g^{−1}(F (g, g))], where g ∈ A[[x]]_{0} is a power series without constant term, the reduction of g modulo M is equal to φ. The definition is correct, i.e. it does not depend on the choice of F and g.

Proposition 3.1 Let F be a deformation of Φ over A and A/M = l. Then i) Stab[F] is the set of the reductions of automorphisms of F over A; ii) Orb[F] is the set of the ⋆-isomorphism classes of deformations of Φ which are isomorphic to F over A.

Proof. i) If φ is the reduction of an automorphism g of F, then we have g^{−1}(F (g, g)) = F and φ[F] = [F]. If φ ∈ Stab[F], then there exists a ⋆-iso-morphism f from F to g^{−1}(F (g, g)), where g ∈ A[[x]]_{0}, the reduction of g is equal to φ. Then g(f) is an automorphism of F and its reduction is equal to φ. ii) If φ ∈ Aut_{l}Φ and g ∈ A[[x]]_{0}, the reduction of g is equal to φ, then g provides an isomorphism between g^{−1}(F (g, g)) ∈ φ[F] and F. If g is an isomorphism between a deformation G of Φ over A and F, then the reduction of g is an automorphism of Φ and it maps the class [F] to the class [G].

According to Theorem 1.3, we have an action of Aut_{l}Φ on M × ... × M (h − 1 times). We are going to consider it for the case where A is the ring of Witt vectors of l, and to focus our attention on the orbit of the point (0, ..., 0).

4. Honda’s explicit description of the automorphism group.

Honda described explicitly the automorphism group of a formal group over a finite field. In this section, we review his theory.

From now on, we suppose A to be the ring of Witt vectors of l. Let L be the fraction field of A and Δ denote the Frobenius automorphism of L. The reduction from A to l modulo p will be denoted by overline.

We consider the non-commutative ring A[[Δ]] of formal power series over A with the multiplication rule Δa = a^{Δ}Δ, a ∈ A. This ring has several common properties with the standard power series ring A[[x]], namely

1) a power series s ∈ A[[Δ]] is invertible if and only if the constant term of s is invertible in A;

2) the non-commutative version of Weierstrass’ preparation lemma holds, i.e. for any u ∈ A[[Δ]] which is not divisible by p, there exists a unique invertible s ∈ A[[Δ]] such that su is a polynomial with the greatest coefficient equal to 1 and deg su is equal to the least power of Δ in the power series u having an invertible coefficient; 3) A[[Δ]] admits uniquely defined left division transformation, that means for any w ∈ A[[Δ]] and u ∈ A[Δ] with the greatest coefficient equal to 1, there exist unique q ∈ A[[Δ]] and r ∈ A[Δ] such that w = qu + r and deg r < deg u.

Let Δ operate on the ring L[[x]]_{0} of formal power series over L without constant term by the formula

Δ ∑_{i=1}^{∞} c_{i}x^{i} = ∑_{i=1}^{∞} c_{i}^{Δ}x^{pi}.

That determines an action of A[[Δ]] on L[[x]]_{0}.

Let u ∈ A[[Δ]] be such that u ≡ p mod Δ. A power series f ∈ L[[x]]_{0} is said to be of type u if f(x) ≡ x mod x^{2} and uf ≡ 0 mod p. Evidently, for any u ∈ A[[Δ]], u ≡ p mod Δ and g ∈ L[[x]]_{0}, g(x) ≡ x mod x^{2}, there exists a unique f ∈ L[[x]]_{0} of type u such that uf = pg.

Theorem 4.1 Let u ∈ A[[Δ]], u ≡ p mod Δ. Then

i) if f, g ∈ L[[x]]_{0} are of type u, then g^{−1}(f) has coefficients in A; ii) if f ∈ L[[x]]_{0} is of type u, then f is the logarithm of a formal group over A.

Proof. See [6], Lemma 2.4 and Theorem 2.

Lemma 4.2 Let u_{1}, u_{2} ∈ A[[Δ]], u_{1} ≡ p mod Δ and f ∈ L[[x]]_{0} be of type u_{1}. If u_{2}f ≡ 0 mod p, then there exists s ∈ A[[Δ]] such that u_{2} = su_{1}.

Proof. See [6], Proposition 2.6.

Theorem 4.3 Let a ∈ A, u_{i} ∈ A[[Δ]], u_{i} ≡ p mod Δ and f_{i} ∈ L[[x]]_{0} be of type u_{i} for i = 1, 2. Then f_{2}^{−1}(af_{1}) has coefficients in A if and only if there exists z ∈ A[[Δ]] such that u_{2}a = zu_{1}.

Proof. See [6], Theorem 3.

Theorem 4.4 If F is a formal group over A with the logarithm f, then there exists u ∈ A[[Δ]] such that u ≡ p mod Δ and f is of type u.

Proof. See [6], Proposition 3.3.

Lemma 4.5 Let u ∈ A[[Δ]], u ≡ p mod Δ, f ∈ L[[x]]_{0} be of type u, ψ_{1} ∈ L[[x]]_{0} and ψ_{2} ∈ A[[x]]_{0}. Then f(ψ_{1}) ≡ f(ψ_{2}) mod p if and only if ψ_{1} ≡ ψ_{2} mod p.

Proof. See [6], Lemma 4.2.

Theorem 4.6 Let w, w′ ∈ A[[Δ]], u_{i} ∈ A[[Δ]], u_{i} ≡ p mod Δ, f_{i} ∈ L[[x]]_{0} be of type u_{i} and F_{i} be the formal group with the logarithm f_{i} for i = 1, 2, 3. Then

i) f_{2}^{−1}(wf_{1}) has coefficients in A if and only if there exists z ∈ A[[Δ]] such that u_{2}w = zu_{1}; ii) if f_{2}^{−1}(wf_{1}) has coefficients in A, then \overline{f_{2}^{−1}(wf_{1})} ∈ Hom_{l}(\overline{F_{1}}, \overline{F_{2}}); iii) if f_{2}^{−1}(wf_{1}) and f_{3}^{−1}(w′f_{2}) have coefficients in A, then f_{3}^{−1}(w′wf_{1}) has coefficients in A and \overline{f_{3}^{−1}(w′wf_{1})} = \overline{f_{3}^{−1}(w′f_{2})} ∘ \overline{f_{2}^{−1}(wf_{1})}; iv) if φ ∈ Hom_{l}(\overline{F_{1}}, \overline{F_{2}}), then there exists w ∈ A[[Δ]] such that f_{2}^{−1}(wf_{1}) has coefficients in A and \overline{f_{2}^{−1}(wf_{1})} = φ.

Proof. See [6], Theorem 5 and Theorem 6.

Let u ∈ A[[Δ]], u ≡ p mod Δ, f ∈ L[[x]]_{0} be of type u and F be the formal group with the logarithm f. Denote by A[[Δ]]^{u} the set of power series w ∈ A[[Δ]] satisfying uw = zu for some z ∈ A[[Δ]]. Evidently A[[Δ]]^{u} is a subring of A[[Δ]], and A[[Δ]]u is a two-sided ideal in A[[Δ]]^{u}. Define a map μ from the ring A[[Δ]]^{u}/A[[Δ]]u to End_{l}\overline{F} by the formula μ(w + A[[Δ]]u) = \overline{f^{−1}(wf)}. Due to Lemma 4.5, the definition is correct.

Proposition 4.7 μ is a ring isomorphism.

Proof. By Theorem 4.6.iii), μ is a homomorphism. Theorem 4.6.iv) implies that μ is surjective. If \overline{f^{−1}(wf)} = 0 holds for some w ∈ A[[Δ]]^{u}, then due to Lemma 4.5, we have wf ≡ 0 mod p. Now Lemma 4.2 implies that there exists s ∈ A[[Δ]] such that w = su, i.e. w belongs to the zero coset in A[[Δ]]^{u}/A[[Δ]]u. Thus μ is an isomorphism.

Now suppose that u is not divisible by p and denote by η the least power of Δ in the power series u having an invertible coefficient. According to Weierstrass’ preparation lemma, there exists a unique s ∈ A[[Δ]] such that su is a polynomial of degree η with the greatest coefficient equal to 1. Since A[[Δ]] admits uniquely defined left division transformation, for any w ∈ A[[Δ]], there exist unique q ∈ A[[Δ]] and r ∈ A[Δ] such that w = qsu + r and deg r < η. Moreover if w ∈ A[[Δ]]^{u}, then r ∈ A[[Δ]]^{u}; and if w is invertible, then r is also invertible. It means that the polynomials over A of degree less than η which belong to A[[Δ]]^{u} can be choosen as representatives of the cosets in A[[Δ]]^{u}/A[[Δ]]u, and the invertible cosets are represented by the polynomials with invertible constant term.

5. Correspondence between Honda’s and Lubin-Tate’s parameters.

Let F_{0} = Γ(0, ..., 0) ∈ R[[x, y]] and f_{0} = γ(0, ..., 0) ∈ K[[x]]_{0} be its logarithm. Evidently f_{0}(x) = ∑_{n=0}^{∞} a_{n}(0, ..., 0)x^{p^{n}}. Define a Q_{p}-homomorphism θ from E to K ⊆ L by the formula θ(t_{i}) = r_{i}. Then F_{0} = θ_{∗}F_{T}, f_{0} = θ_{∗}f_{T} and a_{n}(0, ..., 0) = θ(d_{n}) = d_{n}(r_{1}, ..., r_{n}). Moreover we have

pf_{0} − ∑_{i=1}^{∞} r_{i}Δ^{i}(f_{0}) = pf_{0} − ∑_{i=1}^{∞} ∑_{n=0}^{∞} r_{i}a_{n}(0, ..., 0)^{Δ^{i}}x^{p^{i+n}} =

= pf_{0}−∑_{i=1}^{∞} ∑_{n=0}^{∞} r_{i}d_{n}(r_{1}^{p^{i}}, ..., r_{n}^{p^{i}})x^{p^{i+n}} = θ_{∗}(pf_{T} −∑_{i=1}^{∞} ∑_{n=0}^{∞} t_{i}d_{n}(t_{1}^{p^{i}}, ..., t_{n}^{p^{i}})x^{p^{i+n}}) =

= θ_{∗}(pf_{T} − ∑_{i=1}^{∞} t_{i}σ^{i}(f_{T})) = px,

i.e. f_{0} is of type u_{0} = p − ∑_{i=1}^{∞} r_{i}Δ^{i}.

Since \overline{F_{0}} = Φ, r_{i} = 0 for i < h and r_{h} is invertible, Proposition 4.7 and the subsequent remark imply that Aut_{l}Φ is in one-to-one correspondence with the set of polynomials from A[[Δ]]^{u_{0}} of degree less than h with invertible constant term. Let w be such a polynomial. Then w corresponds to the reduction of an automorphism of F_{0} if and only if it is a constant. Indeed, if w is a constant, then by Theorem 4.3, f_{0}^{−1}(wf_{0}) is an automorphism of F_{0} and μ(w + A[[Δ]]u_{0}) = \overline{f_{0}^{−1}(wf_{0})}. Conversely, if μ(w + A[[Δ]]u_{0}) = \overline{g} for some automorphism g of F_{0}, we denote by a the first coefficient of g and observe that μ(a + A[[Δ]]u_{0}) = \overline{f_{0}^{−1}(af_{0})} = \overline{g}. Then by Proposition 4.7, w ∈ a + A[[Δ]]u_{0}, and hence w = a. Now applying Proposition 3.1.i), we get that the set of polynomials from A[[Δ]]^{u_{0}} of degree less than h with constant term equal to one is in one-to-one correspondence with the set of left cosets of Stab[F_{0}] in Aut_{l}Φ which is in one-to-one correspondence with Orb[F_{0}]. Taking into account Theorem 1.3 and Proposition 3.1.ii), we get

Theorem 5.1 There is a canonical correspondence between the set of the polynomials of the form 1 + ∑_{κ=1}^{h−1} β_{κ}Δ^{κ}, β_{κ} ∈ A, which belong to A[[Δ]]^{u_{0}} and the set of the (h − 1)-tuples (τ_{1}, ..., τ_{h−1}), τ_{i} ∈ pA, such that Γ(τ_{1}, ..., τ_{h−1}) is isomorphic to F_{0} over A.

We introduce an additional assumption that we will call assumption (∗): There exists an invertible v ∈ A[[Δ]] such that vu_{0} has non-zero coefficients

only at the powers of Δ which are multiples of the degree of l over F_{p}.

The assumption (∗) is satisfied if, for example, l = F_{p} or if u_{0} = p − Δ^{h} and l = F_{p^{h}}.

If the assumption (∗) is satisfied, then vu_{0} belongs to the center of A[[Δ]]. It implies that A[[Δ]]^{u_{0}} = A[[Δ]], because for any w ∈ A[[Δ]], we have u_{0}w = v^{−1}vu_{0}w = v^{−1}wvu_{0}. Moreover in this case, any deformation F of Φ over A is isomorphic to F_{0}. Indeed, by Theorem 4.4, the logarithm f of F is of type u for some u ∈ A[[Δ]], u ≡ p mod Δ. Then by Theorem 4.6.iv), the identity morphism from \overline{F} = Φ to \overline{F_{0}} = Φ is equal to \overline{f_{0}^{−1}(wf)} for some invertible w, z ∈ A[[Δ]] satisfying u_{0}w = zu. We have zu = v^{−1}vu_{0}w = v^{−1}wvu_{0}. Hence u_{0}f = v^{−1}w^{−1}vzuf ≡ 0 mod p, i.e. f is of type u_{0}, and by Theorem 4.1.i), F is isomorphic to F_{0}. Thus we get

Theorem 5.2 If the assumption (∗) is satisfied, there is a canonical correspondence between the set of the polynomials of the form 1 + ∑_{κ=1}^{h−1} β_{κ}Δ^{κ}, β_{κ} ∈ A, and pA × ... × pA (h − 1 times).

Our aim is to calculate the correspondence mentioned in Theorem 5.1 and Theorem 5.2 explicitly.

6. The ring of rigid analytic functions.

We define the ring of rigid analytic function L{{t_{1}, ..., t_{h−1}}} as the subring of L[[t_{1}, ..., t_{h−1}]] consisting of the formal power series

f = ∑_{I=(i_{1},...,i_{h−1})∈N^{h−1}} f_{I}t^{I}

such that lim_{|I|→∞} ν(f_{I}) + |I| = ∞, where f_{I} = f_{i_{1},...,i_{h−1}}, t^{I} = t_{1}^{i_{1}}t_{2}^{i_{2}}...t_{h−1}^{i_{h−1}},

|I| = i_{1} + ... + i_{h−1} and ν is the normalized valuation of L. Such power series converge if the variables t_{i}, 1 ≤ i ≤ h − 1, take value in pA, i.e. rigid analitic functions can be evaluated on the set pA × ... × pA (h − 1 times).

Define a norm on L{{t_{1}, ..., t_{h−1}}} by

∥f∥ = p^{− min_{I∈N^{h−1}} {ν(f_{I})+|I|}}.

This norm provides to L{{t_{1}, ..., t_{h−1}}} the structure of a Banach L-algebra (see [1], section 6.1.5, Proposition 1). If a sequence of rigid analitic functions converges to f ∈ L{{t_{1}, ..., t_{h−1}}}, then the sequence of their values at a point in pA × ... × pA (h − 1 times) converges to the value of f at this point.

Proposition 6.1 Let f ∈ L{{t_{1}, ..., t_{h−1}}}, f ≡ 1 mod t_{1}, ..., t_{h−1} and ∥f − 1∥ < 1, i.e. ν(f_{J}) + |J| > 0 for any J ∈ N^{h−1}, J ≠ (0, ..., 0). Then f is invertible in L{{t_{1}, ..., t_{h−1}}}.

Proof. We have f = 1 + ∑_{J∈N^{h−1},J≠(0,...,0)} f_{J}t^{J}. We are going to prove that

the formal power series g = 1−∑ f_{J}t^{J} +(∑ f_{J}t^{J})^{2}−... ∈ L[[t_{1}, ..., t_{h−1}]] belongs

to L{{t_{1}, ..., t_{h−1}}}. If we write g in the form g = 1 + ∑_{I∈N^{h−1},I≠(0,...,0)} g_{I}t^{I}, then

we have

g_{I} = ∑_{J_{1}+...+J_{n}=I,J_{i}≠(0,...,0)}(−1)^{n}f_{J_{1}}f_{J_{2}}...f_{J_{n}},

and hence

ν(g_{I}) + |I| ≤ min_{J_{1}+...+J_{n}=I,J_{i}≠(0,...,0)}{∑_{i=1}^{n}(ν(f_{J_{i}}) + |J_{i}|)}.

Let M > 0 and choose N such that ν(f_{J}) + |J| > M for any J ∈ N^{h−1} with |J| > N. Consider I ∈ N^{h−1} satisfying |I| > MN and let J_{1}, ..., J_{n} ∈ N^{h−1} be such that J_{1} + ... + J_{n} = I, J_{i} ≠ (0, ..., 0). If n ≤ M, then there exists j such that |J_{j}| > N and hence ∑_{i=1}^{n}(ν(f_{J_{i}}) + |J_{i}|) ≥ ν(f_{J_{j}}) + |J_{j}| > M. If n > M, then ∑_{i=1}^{n}(ν(f_{J_{i}}) + |J_{i}|) ≥ n > M. Thus we get that ν(g_{I}) + |I| > M holds for any I ∈ N^{h−1} with |I| > MN, i.e. g belongs to L{{t_{1}, ..., t_{h−1}}}.

7. The rigid analytic function ρ.

Let L{y}_{p} denote the subring of L[[y]] consisting of the formal power series ∑_{n=0}^{∞} c_{n}y^{p^{n}} such that lim c_{n} = 0. Let Δ operate on L{y}_{p} by the formula

Δ ∑_{n=0}^{∞} c_{n}y^{p^{n}} = ∑_{n=0}^{∞} c_{n+1}^{Δ}y^{p^{n}}.

That determines an action of A[[Δ]] on L{y}_{p}. Because of Weierstrass’ preparation lemma, there exists an invertible element s_{0} ∈ A[[Δ]] such that s_{0}u_{0} = Δ^{h} − π_{h−1}Δ^{h−1} − ... − π_{1}Δ − π_{0}, where π_{i} ∈ pA. Take arbitrary elements ζ_{0}, ..., ζ_{h−1} ∈ Z_{p} and define ζ_{n} ∈ A for n ≥ h by the recurrent formula

ζ_{n} = ∑_{i=1}^{h} π_{h−i}^{Δ^{−h}}ζ_{n−i}^{Δ^{−i}}.

It is clear that ν(ζ_{n}) ≥ [n/h]. In paticular, lim ζ_{n} = 0.

Proposition 7.1 pζ_{n} = ∑_{i=1}^{∞} r_{i}ζ_{n+i}^{Δ^{i}}.

Proof. Denote ψ = ∑_{n=0}^{∞} ζ_{n}y^{p^{n}} ∈ L{y}_{p}. The definition of ζ_{n} implies

ζ_{n+h}^{Δ^{h}} − π_{h−1}ζ_{n+h−1}^{Δ^{h−1}} − ... − π_{1}ζ_{n+1}^{Δ} − π_{0}ζ_{n} = 0

for any n ≥ 0. It follows that (Δ^{h} −π_{h−1}Δ^{h−1} −...−π_{1}Δ−π_{0})ψ = 0. Therefore u_{0}ψ = s_{0}^{−1}s_{0}u_{0}ψ = 0, i.e. pζ_{n} − ∑_{i=0}^{∞} r_{i}ζ_{n+i}^{Δ^{i}} = 0 holds for any n ≥ 0.

Define α_{m}^{(n)} ∈ A for m, n ∈ N as follows: α_{0}^{(n)} = pζ_{n}, α_{m}^{(n)} = α_{m−1}^{(n)} −r_{m}ζ_{n+m}^{Δ^{m}} for m ≥ 1. Proposition 7.1 implies that α_{m}^{(n)} = ∑_{i=m+1}^{∞} r_{i}ζ_{n+i}^{Δ^{i}}, and hence ν(α_{m}^{(n)}) ≥ [(m + n)/h]. Define

b_{m}^{(n)} = ∑_{i=0}^{m} α_{m−i}^{(n+i)Δ^{i}}a_{i} ∈ L[t_{1}, ..., t_{h−1}].

Proposition 7.2 b_{m}^{(n)} = pζ_{n} + ∑_{i=1}^{m} ∑_{j=1}^{min(i,h−1)} ζ_{n+i}^{Δ^{i}}t_{j}^{p^{i−j}}a_{i−j}.

Proof. Evidently b_{0}^{(n)} = α_{0}^{(n)}a_{0} = pζ_{n}. Moreover calculating the difference b_{m}^{(n)} − b_{m−1}^{(n)}, we get

b_{m}^{(n)} − b_{m−1}^{(n)} = α_{0}^{(n+m)Δ^{m}}a_{m} + ∑_{i=1}^{m} (α_{i}^{(n+m−i)} − α_{i−1}^{(n+m−i)})^{Δ^{m−i}} a_{m−i} =

= pζ_{n+m}^{Δ^{m}}a_{m} − ∑_{i=1}^{m} (r_{i}ζ_{n+m}^{Δ^{i}})^{Δ^{m−i}} a_{m−i} = ζ_{n+m}^{Δ^{m}} (pa_{m} − ∑_{i=1}^{m} r_{i}^{Δ^{m−i}}a_{m−i}) =

= ζ_{n+m}^{Δ^{m}} (pa_{m} − ∑_{i=h}^{m} r^{p^{m−i}}a_{m−i}) = ζ_{n+m}^{Δ^{m}} ∑_{j=1}^{min(m,h−1)} t_{j}^{p^{m−j}}a_{m−j}.

Summing up, we obtain the required equality.

Proposition 7.3 The sequence of polynomials b_{m}^{(0)} converges in L{{t_{1}, ..., t_{h−1}}} to the formal power series

ρ = pζ_{0} + ∑_{i=0}^{∞} ∑_{j=1}^{min(i,h−1)} ζ_{i}^{Δ^{i}}t_{j}^{p^{i−j}}a_{i−j} = pζ_{0} + ∑_{i=0}^{∞} ∑_{j=1}^{h−1} ζ_{i+j}^{Δ^{i+j}}t_{j}^{p^{i}}a_{i}.

Proof. The recurrent formula for a_{m} implies that p^{m}a_{m} ∈ A[t_{1}, ...t_{h−1}]. Therefore ∥ ∑_{j=1}^{min(m,h−1)} ζ_{m}^{Δ^{m}}t_{j}^{p^{m−j}}a_{m−j}∥ ≤ p^{m−p^{m−h+1}}, i.e b_{m}^{(0)} is a Cauchy sequence.

8. Explicit calculation of the correspondence.

Let w = 1 + ∑_{κ=1}^{h−1} β_{κ}Δ^{κ} ∈ A[[Δ]]^{u_{0}}. Then the power series wf_{0} is of type u_{0}, and, by Theorem 4.1.ii), it is the logarithm of the formal group F_{w} defined over A. Due to Theorem 4.6.i),ii), the power series f_{0}^{−1}(wf_{0}) has coefficients in A, and its reduction is an automorphism of Φ. By the definition of the action of the automorphism group of Φ on the set of ⋆-isomorphism classes of deformations of Φ, this automorphism maps the class [F_{0}] to the class [F_{w}]. According to Theorem 1.3, there exists a unique (h − 1)-tuple (τ_{1}, ..., τ_{h−1}), τ_{i} ∈ pA, such that the formal group F = Γ(τ_{1}, ..., τ_{h−1}) belongs to the same ⋆-isomorphism class that F_{w} does. Denote by f the logarithm of F and by ε the first coefficient of the ⋆-isomorphism from F_{w} to F. Then we obtain f^{−1}(εwf_{0}) ≡ x mod p. Now Lemma 4.5 implies that f ≡ εwf_{0} mod p. We have f(x) = ∑_{i=0}^{∞} a_{i}(τ_{1}, ..., τ_{h−1})x^{p^{i}}. On the other hand wf_{0}(x) = ∑_{i=0}^{∞}(a_{i}(0, ..., 0) + ∑_{κ=1}^{min(i,h−1)} β_{κ}a_{i−κ}(0, ..., 0)^{Δ^{κ}})x^{p^{i}}. Thus we get that a_{i}(τ_{1}, ..., τ_{h−1}) ≡ ε(a_{i}(0, ..., 0) + ∑_{κ=1}^{min(i,h−1)} β_{κ}a_{i−κ}(0, ..., 0)^{Δ^{κ}}) mod p holds for any i.

Proposition 8.1 ρ(τ_{1}, ..., τ_{h−1}) = pε(ζ_{0} + ∑_{κ=1}^{h−1} β_{κ}ζ_{κ}).

Proof. Since ν(α_{m}^{(n)}) ≥ [(m + n)/h], we can deduce that

α_{m−i}^{(i)Δ^{i}}a_{i}(τ_{1}, ..., τ_{h−1}) ≡ εα_{m−i}^{(i)Δ^{i}}(a_{i}(0, ..., 0)+ ∑_{κ=1}^{min(i,h−1)} β_{κ}a_{i−κ}(0, ..., 0)^{Δ^{κ}}) mod p^{[m/h]+1}

holds for any i ≤ m. Then we get

b_{m}^{(0)}(τ_{1}, ..., τ_{h−1}) = ∑_{i=0}^{m} α_{m−i}^{(i)Δ^{i}}a_{i}(τ_{1}, ..., τ_{h−1}) ≡

≡ ε ∑_{i=0}^{m} α_{m−i}^{(i)Δ^{i}} (a_{i}(0, ..., 0) + ∑_{κ=1}^{min(i,h−1)} β_{κ}a_{i−κ}(0, ..., 0)^{Δ^{κ}} ) =

= ε (∑_{i=0}^{m} α_{m−i}^{(i)Δ^{i}}a_{i}(0, ..., 0) + ∑_{κ=1}^{h−1} ∑_{i=0}^{m−κ} β_{κ}α_{m−i−κ}^{(i+κ)Δ^{i+κ}}a_{i}(0, ..., 0)^{Δ^{κ}}) =

= ε (b_{m}^{(0)}(0, ..., 0) + ∑_{κ=1}^{h−1} β_{κ}b_{m−κ}^{(κ)}(0, ..., 0)^{Δ^{κ}}) mod p^{[m/h]+1}.

Proposition 7.2 implies b_{m−κ}^{(κ)}(0, ..., 0) = pζ_{κ}. Therefore

b_{m}^{(0)}(τ_{1}, ..., τ_{h−1}) ≡ pε(ζ_{0} + ∑_{κ=1}^{h−1} β_{κ}ζ_{κ}) mod p^{[m/h]+1}.

It means that the sequence b_{m}^{(0)}(τ_{1}, ..., τ_{h−1}) ∈ L converges to pε(ζ_{0}+∑_{κ=1}^{h−1} β_{κ}ζ_{κ}). On the other hand, it converges to ρ(τ_{1}, .., τ_{h−1}). Thus we obtain the required equality.

Now we are going to specify the sequence ζ_{n}. For 0 ≤ κ ≤ h − 1, define the sequences ζ_{n,κ} by fixing their first h elements as follows: ζ_{n,κ} = δ_{n,κ}, 0 ≤ n ≤ h − 1, where δ_{n,κ} is Kronecker’s delta. Then we can define α_{m,κ}^{(n)} ∈ A, b_{m,κ}^{(n)} ∈ L[t_{1}, ..., t_{h−1}] and ρ_{κ} ∈ L{{t_{1}, ..., t_{h−1}}} for these specified sequences ζ_{n,κ}.

Lemma 8.2 ρ_{0} is invertible in L{{t_{1}, ..., t_{h−1}}}.

Proof. ρ_{0} ≡ pζ_{0,0} = p mod t_{1}, ..., t_{h−1}. We will prove that ρ_{0}/p is invertible in L{{t_{1}, ..., t_{h−1}}}. By Proposition 6.1, it is enough to show that ∥ρ_{0}/p − 1∥ < 1. Since ζ_{n,0} = 0 for 1 ≤ n ≤ h − 1 and ν(ζ_{n,o}) ≥ [n/h], we obtain ν(ζ_{n,o}) ≥ 1 for any n ≥ 1. From the recurrent formula for a_{i}, it follows that ∥a_{i}∥ ≤ p^{i}. Therefore ∥ ∑_{j=1}^{h−1} ζ_{i+j,0}^{Δ^{i+j}}t_{j}^{p^{i}}a_{i}∥ ≤ p^{−1−p^{i}+i} ≤ p^{−2} for any i ≥ 0, and thus ∥ρ_{0} − p∥ ≤ p^{−2}, i.e. ∥ρ_{0}/p − 1∥ ≤ p^{−1} < 1.

Theorem 8.3 β_{κ} = \frac{ρ_{κ}}{ρ_{0}} (τ_{1}, ..., τ_{h−1}). The element that corresponds to the tuple (τ_{1}, ..., τ_{h−1}) ∈ pA × ... × pA in the sense of Theorem 5.1 is equal to 1 + ∑_{κ=1}^{h−1} \frac{ρ_{κ}}{ρ_{0}} (τ_{1}, .., τ_{h−1})Δ^{κ}.

Proof. Applying Proposition 8.1 to the sequence ζ_{n,0}, we get the equality pε = ρ_{0}(τ_{1}, ..., τ_{h−1}), applying it to the sequence ζ_{n,κ} for 1 ≤ κ ≤ h − 1, we get pεβ_{κ} = ρ_{κ}(τ_{1}, ..., τ_{h−1}). That implies β_{κ} = ρ_{κ}(τ_{1}, ...τ_{h−1})/ρ_{0}(τ_{1}, ..., τ_{h−1}).

9. The reduction of the Artin-Hasse formal group.

In this section, we consider a special case when Φ is the reduction of the Artin-Hasse formal group over Z_{p} of height h. We put Φ = ξ_{∗}F_{T}, where the Q_{p}-homomorphism ξ from D to F_{p} is defined by ξ(t_{i}) = δ_{i,h}. Then we obtain r_{i} = δ_{i,h}, and the Q_{p}-homomorphism Ξ from D to Q_{p}[[t_{1}, ..., t_{h−1}]] is defined as follows: Ξ(t_{i}) = t_{i} for i < h, Ξ(t_{h}) = 1 and Ξ(t_{i}) = 0 for i > h. The recurrent formula on the coefficients a_{n} of the logarithm γ of the formal group Γ = Ξ_{∗}F_{T} turns to be pa_{n} = ∑_{j=1}^{n} t_{j}^{p^{n−j}}a_{n−j} for n < h and

pa_{n} = ∑_{j=1}^{h−1} t_{j}^{p^{n−j}}a_{n−j} + a_{n−h} for n ≥ h. It implies that a_{mh}(0, ..., 0) = p^{−m}, a_{n}(0, ..., 0) = 0 for n ≠ 0 mod h. Then the logarithm f_{0} of the formal group F_{0} = Γ(0, ..., 0) is equal to ∑_{n=0}^{∞} a_{n}(0, ..., 0)x^{p^{n}} = ∑_{m=0}^{∞} p^{−m}x^{p^{mh}}, i.e. F_{0} is the Artin-Hasse formal group of height h.

In this case, it makes sense to consider l = F_{p^{h}}, because all of the automorphisms of Φ over F_{p}^{alg} are defined over F_{p^{h}} (that can be easily proven with the aid of Theorem 4.6.i),iv). Then the assumption (∗) is satisfied. Furthermore we note u_{0} = p − Δ^{h}, s_{0} = −1, π_{i} = pδ_{i,0} and ζ_{n} = pζ_{n−h}^{Δ^{−h}} if n ≥ h. In particular for 0 ≤ κ ≤ h − 1, it implies ζ_{mh+κ,κ} = p^{m}, ζ_{n,κ} = 0 if n ≠ κ mod h. Now calculating b_{mh+κ,κ}^{(0)}, we get

b_{mh+κ,κ}^{(0)} = pζ_{0,κ} + ∑_{i=1}^{mh+κ} ∑_{j=1}^{min(i,h−1)} ζ_{i,κ}^{Δ^{i}}t_{j}^{p^{i−j}}a_{i−j} =

= pδ_{0,κ} + ∑_{j=1}^{κ} t_{j}^{p^{κ−j}}a_{κ−j} + ∑_{i=1}^{m} ∑_{j=1}^{h−1} p^{i}t_{j}^{p^{ih+κ−j}}a_{ih+κ−j} =

= pδ_{0,κ} + (1 − δ_{0,κ})pa_{κ} + ∑_{i=1}^{m} p^{i}(pa_{ih+κ} − a_{(i−1)h+κ}) =

= pa_{κ} + p^{m+1}a_{mh+κ} − pa_{κ} = p^{m+1}a_{mh+κ}.

Then ρ_{κ} = lim_{m} p^{m+1}a_{mh+κ} and β_{κ} = lim_{m} \frac{a_{mh+κ}}{a_{mh}} (τ_{1}, ..., τ_{h−1}). Thus we obtain

Theorem 9.1 If Φ is the reduction of the Artin-Hasse formal group of height h and l = F_{p^{h}}, then β_{κ} = lim_{m} \frac{a_{mh+κ}}{a_{mh}} (τ_{1}, ..., τ_{h−1}) and the element that corresponds to the tuple (τ_{1}, ..., τ_{h−1}) ∈ pA × ... × pA in the sence of Theorem 5.2

is equal to 1 + ∑_{κ=1}^{h−1} lim_{m} \frac{a_{mh+κ}}{a_{mh}} (τ_{1}, ..., τ_{h−1})Δ^{κ}.

This result can be derived from the considerations of Gross and Hopkins [4].

References.

[1] S. Bosch, U. Güntzer, R. Remmert, Non-Archimedian Analysis, Springer-Verlag, 1984.

[2] J. Dieudonné, Groupes de Lie et hyperalgèbres de Lie sur un corps de caractéristique p > 0, Math. Ann. 134, 1957, pp. 114-133.

[3] J.-M. Fontaine, Groupes p–divisibles sur les corps locaux, Astérisque 47-48, 1977.

[4] B.H. Gross, M.J. Hopkins, Equivariant vector bundles on the Lubin-Tate moduli space, Contemp. Math. 158, 1994, pp. 23-88.

[5] M. Hazewinkel, Formal groups and aplications, Academic Press, 1978.

[6] T. Honda, On the theory of commutative formal groups, Jour. Math. Soc. Japan 22, No. 2, 1970, pp.213-246.

[7] M. Lazard, Sur les groupes de Lie formels à un paramètre, Bull. Soc. math. France 83, 1955, pp. 251-274.

[8] J. Lubin, J. Tate, Formal moduli for one-parameter formal Lie group, Bull. Soc. math. France 94, 1966, pp. 49-60.