Explicit classification of formal groups over local fields

M.V. Bondarko Vostokov S. V. ^{∗}

June 2002

Introduction

The goal of this work is to explicitly classify commutative formal groups defined over the ring of integers of a local field with finite residue field in terms of their logarithm both up to isogeny and up to isomorphism.

We introduce two new invariants of formal groups. The first one classifies formal groups up to isogeny of some special sort. The second invariant takes finitely many values on finite height formal groups of fixed dimension over a fixed field. The two invariants together fix a formal group up to strict isomorphism. Both behave well if one extends the base field or applies to it some automorphisms. This is an important advantage of our classification when compared with the classification of Breuil (cf. [1]).

For simplicity in this introduction we state most results for one-dimensional formal groups over a totally ramified extension of ℚ_{p}. The reason for this is that in the case of residue fields of cardinality more than p one must use some non-commutative objects. Almost all statements to follow can be extended to the general situation (and even to totally ramified extensions of unramified fields with arbitrary residue field). We will formulate most of this generalizations in the main text. Since this text is rather preliminary, a few of the proofs will be given for the most simple situation.

∗The authors thank Max Planck Mathematical Institute for its valuable support. The work is also supported by RFBR (grant no. 00-01-00140).

We use the following notation throughout the paper.

We denote by K a finite totally ramified (in the introduction) extension of ℚ_{p}, by e its absolute ramification index, by π a uniformizing element of K, by 𝔒_{K} its ring of integers, and by k its residue field. By F we denote a commutative m-dimensional formal group law over 𝔒_{K} (m will be often equal to 1). In the present paper, we consider only commutative finite-dimensional formal group laws.

It is well known that any formal group is strictly isomorphic to a p-typical one. Thus we will assume that the logarithm λ of the formal group F has the form Λ(Δ)(x), where Λ(Δ) ∈ K[[Δ]] and Δ(ax^{b}) = ax^{pb}.

We introduce the following two rings:

R = 𝔒_{K}[[Δ]]⊗K = 𝔒_{K}[[Δ]]⊗ℚ_{p} ⊂ K[[Δ]], R_{0} = ℤ_{p}[[Δ]]⊗ℚ_{p} = R∩ℚ_{p}[[Δ]].

Note that f belongs to R if and only if the valuations of its coefficients are bounded from below.

Proposition (A). A1. Λ can be presented in the form v/u, where v ∈ R and u = p − ∑ u_{i}Δ^{i}; here u_{i} ∈ ℤ_{p} and the first invertible u_{i} is u_{h}, where h is the height of F.

A2. If F is isogenious to F′, then u(F′) = uε(Δ) for some ε ∈ 1 + Δℤ_{p}[[Δ]].

This statement can be regarded as an extension of Honda’s classification of formal groups over an unramified field. It is also valid in the most general case if one considers u lying in a certain non-commutative ring. One can make a canonical choice in the equivalence class of u by having all the u_{i} (or their coefficients in the m-dimensional case, in which they are m × m matrices) to be Teichmüller representatives of F_{p} (or F_{q} in the general case). In particular, using this method, we obtain a canonical representative in each strict isomorphism class of formal groups over unramified fields.

We put

r(F) = v(Δ) mod u(Δ) ∈ R mod u(Δ) = K ⊗ ℤ_{p}[[Δ]]/(u(Δ)).

Theorem (B). For finite height formal groups with fixed u, the following statements are valid:

B1. F and F′ are isogenious if and only if there exists an a ∈ K and ε ∈ R_{0}^{∗} such that av ≡ ε(Δ)v′ mod u

B2. F and F′ are isogenious and there exists an isogeny f(x) ≡ ap^{s}x mod deg 2 for a ∈ K and some s ∈ ℤ if and only if av ≡ ε(Δ)v′ mod u for some ε ∈ R_{0}^{∗}.

B3. For a fixed a ∈ K and m ∈ ℤ, we have av ≡ ε(Δ)v′Δ^{m} mod u for some ε ∈ 1 + Δℤ_{p}[[Δ]] if and only if there is an isogeny f = ∑ a_{i}x^{i} ≡ ap^{s}x mod deg 2 from F into F′ for some s ∈ ℤ such that the height of f is equal to sh + m and a_{p^{sh+m}} ≡ u_{h}^{s} mod π.

B4. r = r′ if and only if there is an isogeny f = ∑ a_{i}x^{i} ≡ p^{s}x mod deg 2 from F into F′ for some s ∈ ℤ such that f(x) ≡ [p^{s}]_{F} (x) mod π𝔒_{K}[[x]].

B5. A finite height formal group F is isogenious to a formal group defined over some K′ ⊂ K if and only if v ≡ av′ mod u for some a ∈ K^{∗} and v′ over K′.

Remark (C). C1. Thus, the invariant I(F) = (u(F), r(F)) classifies formal groups up to isogeny as described in the fourth part (“strict isogeny”). This statement is also valid in the most general case. One also can write down the generalizations of all parts, though the third part will become rather long.

C2. One can replace the invariant I(F) by a certain element V (F) of K{{Δ}} mod R (i.e., modulo positive degrees). One defines V (F) as the residue modulo R of the fraction v/u considered in the two-dimensional field K{{Δ}}.

C3. All the statements can also be extended to not necessarily finite height formal groups and morphisms between them.

C4. One can also prove that, for a given u, every r(F) can be realized modulo K^{∗} (modulo Gl_{m}(K) in the general case). Thus all isogeny classes of formal groups can be described explicitly.

C5. Parts 1 and 2 describe Fontaine’s classification results (see the book [3]) explicitly. The invariant I(F) is a modification of Fontaine’s functor. Our definition is much more explicit. The main disadvantage of Fontaine’s functor is that it is formulated in terms of modules, so it gives r(F) only modulo a ℤ_{p}[[Δ]]^{∗} multiplier instead of 1 + Δℤ_{p}[[Δ]] (in the totally ramified case). Thus one cannot recover a_{p^{sh+m}} mod π from Fontaine’s invariant. One can construct non-isomorphic (even non-strictly) formal groups that are glued together by Fontaine’s functor but have distinct I-invariants.

We note also that fixing a base for the formal group law one can formulate substantially more precise results than in the base-independent approach.

Parts B3 - B5 are completely new.

We put

D_{F} = {f ∈ K[[Δ]] : exp_{F} (f(Δ)(x)) ∈ 𝔒_{K}[[x]]}.

This module will be called the invariant (p-typical) Cartier–Dieudonne module for F. In particular, one can recover the theory of Cartier–Dieudonne over local fields by means of the methods of this paper.

Proposition (D). D1. F is strictly isomorphic to F′ if and only if D_{F} = D_{F′}.

D2. More generally, there is a homomorphism f(x) ≡ ax mod deg 2 if and only if aD_{F} ⊂ D_{F′}.

D3. D_{F} is a free ℤ_{p}[[Δ]]-module, and for every base w_{i} of 𝔒_{K} over ℤ_{p} elements λ_{i}(Δ) : λ_{i}(Δ)(x) = λ(w_{i}(x)) form its ℤ_{p}[[Δ]]-base.

D4. D ⊂ K[[Δ]] is equal to D_{F} for some F if and only if it is ℤ_{p}[[Δ]]-free of dimension e, D mod Δ = 𝔒_{K}, p𝔒_{K} ⊂ Δ, and D is ⟨π⟩-stable, where ⟨π⟩(f(Δ))(x) = f(Δ(πx)).

D5. For f ∈ K[[Δ]], the series f(Δ)(x) is a logarithm of a formal group strictly isomorphic to F if and only if f ≡ 1 mod Δ and f ∈ D_{F}.

Part D4 is a version of the Cartier classification theorem. One can also easily prove an exact analog of Cartier’s theorem in this case.

The behavior of modules D with respect to extensions of K and the modules corresponding to formal groups defined over some K′ ⊂ K can be indicated explicitly.

We define

M(F) = D_{F} ℤ_{p}{{Δ}} ⊂ K{{Δ}}.

Theorem (E). E1. If F and F′ are as in the second part of Theorem B, then there is an isogeny ap^{s}x + … if and only if ap^{s}M_{F} ⊂ M_{F′}. In particular, M(F), u(F) and r(F) taken modulo ℚ_{p}^{∗}, and R_{0}^{∗} determine F up to isomorphism.

E2. For an infinite height (additive reduction in the general case) formal group one has D_{F} = M_{F} ∩ R.

E3. Denote by F_{π} the formal group law

F_{π}(X, Y) = \frac{F (πX, πY)}{π}.

If F is a formal group of finite height, then M_{F} = πM_{F_{π}}.

E4. For a fixed K and a fixed dimension of the group, the number of different πM_{F_{π}} is finite.

Remark (F). F1. We have the following algorithm for classifying all formal groups over a fixed field K.

First one calculates (using the universal p-typical formal group law, for example) all possible πD_{F_{π}}. Next for each of them and each u one checks what residues modulo uR have elements v ∈ πM_{F_{π}}, v ≡ p mod Δ such that

⟨π⟩(v/u) ∈ πD_{F_{π}}.  (∗)

After this one recovers V (F), and then, knowing M_{F}, classifies all formal groups. Note that it is sufficient to check (*) modulo a small power of Δ (Δ^{2} for e ≤ p^{2}/2) and that it is “continuous”, so the calculation terminates in finitely many steps. This is fairly simple to see for dimension one and e ≤ p^{2}/2. In particular, Λ(Δ)(x) is a logarithm of a formal group of height > 1 in this case if and only if ((u(F)/p)Λ)(Δ)(x) is a logarithm of a formal group of infinite height.

F2. The M-invariant reflects the obstacle for two (strictly) isogenious formal groups to be isomorphic, i.e., it ’corrects’ Fontaine’s functor. The number of different M is finite for a fixed K but increases with extension of K. In order to handle that one can consider the filtration

πD_{F_{π}} ⊃ π^{2}D_{F_{π^{2}}} ⊃ π^{3}D_{F_{π^{3}}} ⊃ ….

For s ≥ \frac{e}{p−1} (often for much smaller s), one has for D_{s} = π^{s}D_{F_{π^{s}}} the equality D_{s} = π^{s}𝔒_{K}[[Δ]]. In order to recover F it is sufficient to know V (F) and some D_{s} that contains a v with a required residue modulo uR. Note that for an extension L/K with ramification index r one can easily recover D_{sr}^{L}(F) from D_{s}^{K}(F). Thus, for any group it is sufficient to know some finite module information in order to recover it from the V -invariant in any local field.

F3 Using a more general form of E1, one can easily find some bounds for the minimum possible s in Theorem B.

F4. Using the invariant Cartier–Dieudonne modules, one can easily define the (usual) Cartier–Dieudonne module structure on the kernels of isogenies of formal groups (cf. [6]). Thus, one can use this theory to classify finite (smooth) flat group schemes. If one knows Ker[p^{s}]_{F}, then one can calculate the coefficient at Δ^{i} of a (possible) logarithm modulo p^{s−i−1}. Hence, in order to recover M_{F} it is sufficient to know Ker[p^{s}]_{F} for any s ≥ 2 log_{p}(2e/(p−1)) as a group scheme (usually much smaller is sufficient). Note that if we fix F and change K, then we do not have to increase the exponent. Therefore, one can fix the (strict) isomorphism class of a formal group by fixing the V -invariant and the group scheme corresponding to some (explicitly bounded from above) level of torsion. This statement can be used for proving theorems about good reduction of Abelian varieties. It will be possible to give an explicit answer to the question of Nicholas Katz: whether one can be sure that an Abelian variety that acquires good reduction over K and is defined over K′ ⊂ K has good reduction over K′ knowing that some level of p-torsion for this variety (depending only on e) gives a flat group scheme over 𝔒_{K′}.

Remark (General remarks). G1. One can extend all statements formulated above to formal A-modules, where A is a ring of integers of a local field, by making very natural changes.

G2. It also is almost certain that this theory can be extended to general p-divisible groups. Probably, one can calculate the V -invariant for a dual formal group, thus answering the question of Conrad (cf. [2]).

In the one-dimensional case one can prove a more precise statement about classes of isogenious formal groups. It will not be proved in this work.

Theorem (H). H1. Every one-dimensional formal group is isogenious to a group with logarithm (v(Δ)/u(Δ))(x), where v = p+∑_{i=1}^{h−1} v_{i}Δ^{i} and v_{K}(v_{i}) ≥ e(1 − \frac{i}{h}).

H2. Every (v(Δ)/u(Δ))(x) for v as in H1, is a logarithm of a formal group over 𝔒_{K} (we call it a “nice” group) .

Remark (I). I1. One can also show that if v′ mod u has the form described in the theorem, then the group F′ is strictly isomorphic to F. Moreover, all homomorphisms between “nice” formal groups can be described in terms of V (F). The Newton polygon of a “nice” group and the “residues” of the torsion elements of F (𝔒_{K}^{alg}) can be calculated. Using this, one can calculate the Newton polygons and some invariants of the Tate module T (F) for all groups isogenious to a given “nice” group. Unfortunately, it is quite difficult to determine which of them are defined over 𝔒_{K}.

I2. Theorem 2 can be extended to the case of an arbitrary residue field, thus giving a generalization of the classical result of Laffaille.

I3. Probably these results can be extended to multidimensional formal groups. An analog of the first part is not completely clear yet.

We begin section 1 by recalling the notion of a p-typical formal group. For simplicity, we deal only with formal groups of this type, though our methods work for general formal groups. Next we recall the results of Honda on classification of formal groups over rings of integers of unramified local fields. Their generalization (by Hazewinkel) to a wider class of rings will be applied in this paper. We also describe a canonical representative in each strict isomorphness class of formal group laws. This result seems to be new. We end the section by proving the first part of proposition A in the general case.

In the section 2, we describe the restriction of scalars for formal groups over complete discrete valuation fields.

The goal of the third section is application of Honda’s classification of formal groups over unramified fields to classification of formal groups over general local fields. The main idea is to replace an m-dimensional group over 𝔒_{K} by a me-dimensional one over an unramified ring W by means of restriction of scalars. We illustrate this method by classifying formal groups over the rings of ramification index < p. For simplicity, we describe the proof only in the one-dimensional case. Making slight changes in the formulas, one can obtain the general proof. In fact, authors were able to carry out the corresponding calculations for any e ≤ 2p − 2 (for one-dimensional groups, at least).

In section 4 we define the fractional part invariant. Next we state and prove a generalization of the first four parts of Theorem B.

In section 5, we define the invariant Cartier–Dieudonne modules for formal groups. The difference from the Cartier definition is that we take logarithms of p-typical curves. That gives us a canonical imbedding of our module in K[[Δ]]^{m}. In order to illustrate usefulness of such modification we prove B5.

In section 6, we define the module invariant M_{F}. Next we prove that along with the I-invariant it classifies formal groups up to an isomorphism. We prove some properties of M_{F}. In the end of the section we apply our methods to classification of one-dimensional formal groups of height > 1 for e ≤ p^{2}/2.

Notation. Throughout the paper M_{m}(𝔄) will denote the ring of m × m matrices over a (possibly, non-commutative) ring 𝔄, I_{m} will denote the unit matrix of size m; e_{i} will denote the m-vector (0, 0, …, 1, 0, …, 0) (1 is on the i-th place).

v_{K} is the normalized valuation on K, e = v_{K}(p) is the absolute ramification index of K, 𝔐 is the maxiaml ideal of K.

We will also use the notation from the introduction.

m will usually mean the dimension of a formal group F, X = (X_{i}) = X_{1}, … X_{m} will denote formal variables, as well as x (one variable).

1 Classification over unramified fields and related topics

In the book [4] the results of Honda on classification of formal groups over rings of integers unramified local fields. were generalized to a wider class of rings. In one of the proofs we will apply this classification to formal group laws over ℜ[[t]], where ℜ is an unramified ring.

1.1 p-typical groups

Since throughout this paper we will work with commutative formal group laws over characteristic zero rings, our formal group laws will have logarithms. The following statement is well known.

Proposition 1.1.1. Let 𝔄 be a commutative ℤ_{p}-algebra. Then a formal group law with logarithm λ = (λ_{i}), 1 ≤ i ≤ m, where λ_{i} = ∑ a_{i_{1}...i_{m}}^{i}X_{1}^{i_{1}} … X_{m}^{i_{m}} is strictly isomorphic to a formal group law, whose logarithm is equal to (λ′_{i}), for λ′_{i} = ∑ a′_{i_{1}...i_{m}}^{i}X_{1}^{i_{1}} … X_{m}^{i_{m}} for a′_{i_{1}...i_{m}}^{i}X_{1}^{i_{1}} = a_{i_{1}...i_{m}}^{i}X_{1}^{i_{1}} … X_{m}^{i_{m}} if and only if i_{s} = 0 for all s except one, the remaining i_{s} is a power of p, all other a′_{i_{1}...i_{m}}^{i} are 0.

Thus we can assume that the logarithm λ of the formal group F has the form Λ(Δ)(X), where Λ(Δ) = (Λ_{i}) is in the matrix ring M_{m}(K[[Δ]]) and Δ(ax_{i}^{b}) = ax_{i}^{pb} for any i, 1 ≤ i ≤ m.

1.2 Classification over σ-rings. The results of Honda

We consider a torsion free commutative ℤ_{p}-algebra 𝔄 with an operator σ satisfying σ(x) − x^{p} ∈ p𝔄 for any x ∈ 𝔄.

For any such pair 𝔄, σ we introduce a non-commutative ring 𝔄[[Δ]]′, which is equal to the ring 𝔄[[Δ]] as a right 𝔄-module and where the the relation Δa = σ(a)Δ is fulfilled for any a ∈ 𝔄.

We will often identify 𝔄[[Δ]]′ and 𝔄[[Δ]] as sets and 𝔄-modules.

Theorem 1.2.1. 1. Λ(Δ)(X), Λ(Δ) ∈ ℚ_{p}𝔄[[de]], is a logarithm of a formal group over 𝔄 if and only if Λ = pU^{−1} for some U ∈ M_{m}(𝔄[[Δ]]′), U ≡ pI_{m} mod (Δ).

2. Λ and Λ′ give strictly isomorphic formal group laws if and only if U′ = 𝔈U for some 𝔈 ∈ M_{m}(𝔄[[Δ]]′), ε ≡ I_{m} mod (Δ).

3. More generally, there exists a homomorphism f from F to F′ of dimensions m and m′, f ≡ AX mod deg 2, A is some m × m′ matrix over 𝔄, if and only if there exist C ∈ M_{m×m′}(𝔄[[Δ]]′) such that CU = U′A.

4. If Λ ≡ Λ′ mod p, then Λ′(X) is the logarithm of a formal group F′ over 𝔄 if and only if Λ(X) satisfies the same property. In this case F′ is strictly isomorphic to F.

Following Honda, we will call matrices, satisfying the conditions for U in the theorem, special elements of M_{m}(𝔄[[Δ]]′).

This result can naturally be extended to formal modules. See [4] for details.

1.3 Canonical representatives

The previous theorem gives a classification of formal groups over 𝔄 in terms of equivalence classes of certain matrices U. In the one-dimensional case one can choose in any class a polynomial of the minimal possible degree, thus obtaining a canonical representatives. This method cannot be extended to multi-dimensional groups. However, canonical representatives can be chosen in a completely different way.

We fix a system of representatives θ : 𝔄/p𝔄 → 𝔄.

Proposition 1.3.1. Each equivalence class of ’special’ matrices contains a unique representative U such that (U − pI_{m}) = rΔ where the coefficients of r belong to θ(𝔄/p𝔄).

Proof. 1. We prove the existence of U by induction on the powers of Δ. Thus we assume that the coefficients of r_{s}, where r_{s}Δ = (U_{s} − pI_{m}), at the powers of Δ less than s for some s ≥ 1 belong to θ(𝔄/p𝔄). We put U = ∑ V_{l}Δ^{l} and take

U_{s+1} = I_{m} + θ(\frac{θ(V_{s}) − V_{s}}{p})Δ^{s},

θ of a matrix V denotes a matrix whose coefficients are the representatives of coefficients of V. Obviously, the coefficients of (U_{s} − p)I_{m}Δ at powers of Δ less than s + 2, belong to θ(𝔄/p𝔄). The limit of U_{i} will be the desired U, it will lie in the same equivalence class as all U_{i}.

2. Suppose that U and U′ are equivalent, distinct, and satisfy the condition of the theorem. Let s be the smallest number such that U′ ≢ U mod Δ^{s}. Then U′ = 𝔈U, 𝔈 ≡ I_{m} mod Δ^{s}. Since U′−U = (𝔈−I_{m})U, we have U ≡ U′ mod (Δ^{s}, p). Since the coefficients of U and U′ are representatives, U ≡ U mod Δ^{s}. We obtain a contradiction.

A natural choice of θ is the Teichmüller representative map. This choice is very convenient when one studies reduction of formal group laws.

1.4 General form of Proposition A

Let N be the inertia subfield of a complete discrete valuation field K, and let 𝔒_{N} be its ring of integers. The ring 𝔒_{N} has a natural Frobenius σ. We have e = [K : N].

Let W denote the ring 𝔒_{N}[[Δ]]′ defined as in subsection 1.2, and let W′ denote N[[Δ]]′.

This notation will be used throughout the paper except the next subsection (where any other N will be possible).

The ring K[[Δ]] has a natural structure of a left W -module. Note that in order to define it one does not need to extend σ to K.

We consider the ring 𝔄 = 𝔒[[t]] with σ(t) = t^{p} and 𝔄[[Δ]]′. Obviously, 𝔄 satisfies the conditions of Theorem 1.2.1.

Proposition 1.4.1. Suppose that Λ ∈ M_{m}(K[[Δ]]) comes from a formal group F (i.e., Λ(Δ)(X) is the logarithm of a p-typical group F). Then Λ can be presented in the form v/u where vp^{l}π^{−p^{l}} ∈ M_{m}(𝔒_{K}[[Δ]]), l = [log_{p}(e/(p − 1))], and u is a special element of M_{m}(W).

Proof. We fix an epimorphism g : A → 𝔒_{K} by assuming g(t) = π. The existence of a universal p-typical formal group law and its properties (freeness of the corresponding algebra) imply that there exists a p-typical formal group law G over A such that g_{∗}(G) = F.

Theorem 1.2.1 implies that there exists a special matrix U over 𝔄[[Δ]]′ such that the logarithm of G is equal to pU^{−1}(X).

We present U as u − w, where u ∈ M_{m}(W), u ≡ pI_{m} mod Δ, w ∈ tM_{m}(𝔄[[Δ]]′)Δ.

Then we have

U^{−1} = (u − w)^{−1} = (I_{m} − u^{−1}w)^{−1}u^{−1} = (∑_{i}(u^{−1}w)^{i})u^{−1}.

It is easily seen that u^{−1} = ∑_{i=0}^{∞} p^{−i−1}Δ^{i}u_{i} for a certain u_{i} ∈ M_{m}(𝔒) (for example, it follows from the fact that pu^{−1}(X) is a logarithm of a Honda formal group over 𝔒). Applying the equality Δt = t^{p}Δ, we obtain that u^{−1}w can be presented in the form ∑_{i=0}^{∞} p^{−i−1}t^{p^{i}}u_{1i}Δ^{i+1} for some u_{1i} ∈ M_{m}(𝔄[[Δ]]′). One can also easily prove by induction that (u^{−1}w)^{j} can be written in the form

∑_{i=0}^{∞} p^{−i−j}t^{p^{i}+···+p^{i+j−1}}u_{ji}Δ^{i+j}

for some u_{ji} ∈ M_{m}(𝔄[[Δ]]′). It follows that

U^{−1} = ∑_{i=0} p^{−i−1}t^{p^{i}}u^{−1}w_{i}  (1)

for some w_{i} ∈ M_{m}(𝔄[[Δ]]′) (since p^{i} is the smallest possible degree of t at p^{−i−1}).

We checks that p^{−l}π^{p^{l}} | p^{−i}π^{p^{i}} for any i. Indeed, the valuation v_{i} of π^{p^{i}}p^{−i} is equal to p^{i} − ei. The difference v_{i} − v_{i+1} is equal to (p − 1)p^{i} − e, hence v_{l} is the minimum of v_{i}. We obtain v_{i} ≥ p^{l} − el.

Since w is obtained by substitution of π into (1), we obtain the claim.

2 Restriction of scalars and other auxiliary results

2.1 Restriction of scalars

In order to apply the results of the preceding section to ramified local fields we will replace an m-dimensional group over 𝔒_{K} by an me-dimensional one over W. In order to do this we need some properties of the Weil restriction of scalars. Some of them are quite general and well known, so we will not prove them. Other ones are special for the setting of formal group laws. We formulate the results for an arbitrary extension of complete discrete valuation fields. That will allow us to apply them both to formal modules and to formal group laws over multi-dimensional rings (in future).

Let K/N be an extension of complete discrete valuation fields of degree s, and let 𝔒 be the ring of integers of N. As always in this paper, the characteristic of K is 0.

We consider the categories C_{N} and C′_{N} of formal groups over 𝔒, C_{K} and C′_{K} are the same for 𝔒_{K}. The objects of C_{N} and C′_{N} are commutative finitedimensional formal group laws, the morphisms of C_{N} are homomorphisms of formal group laws over 𝔒, the morphisms of C′_{N} are homomorphisms over N. The categories C_{K} and C′_{K} are defined similarly. We have C_{N} ⊂ C′_{N} and C_{K} ⊂ C′_{K}.

We fix a basis w = (w_{1}, …, w_{s}) of 𝔒_{K} over 𝔒.

Consider an operator S = S_{w} : N^{s} → K that maps (n_{i}) into ∑ n_{i}w_{i} and extend it to various polynomial rings over N.

Our goal is to construct a functor ϕ from C′_{K} to C′_{N} that maps C_{K} to C_{N}. Suppose that we have an m-dimensional formal group law F = (F_{l}) over

𝔒_{K}, i.e., an m-tuple of series in variables X_{1}, …, X_{m}, Y_{1}, …, Y_{m}. We introduce the variables X_{l}^{r}, Y_{l}^{r} for 1 ≤ l ≤ m and 1 ≤ r ≤ s. Assume that the new variables ’take their values in N’ and satisfy the relations X_{l} = S((X_{l}^{r})), Y_{l} = S((Y_{l}^{r})). Then standard properties of Weil’s restriction of scalars imply that there exists a unique ms-tuple (F_{l}^{r}) of formal power series in X_{1}, …, X_{m}, Y_{1}, …, Y_{m} over 𝔒 satisfying

F_{i}(S((X_{l}^{r})), S((X_{l}^{r}))) = S((w_{r}F_{i}^{r}(X_{1}, …, X_{m}, Y_{1}, …, Y_{m}))  (2)

for all i, 1 ≤ i ≤ m. Moreover, F_{w} = (F_{l}^{r}) is a formal group law. Applying similar process to homomorphisms of formal group laws, we obtain the desired functor.

Remark 2.1.1. 1. A logarithm of a formal group law is an isomorphism with G_{a}^{m} (the m-th power of the additive group law), hence ϕ(λ) is the logarithm of F.

2. An m × n matrix A can be regarded as a homomorphism from G_{a}^{m} into G_{a}^{n}. Thus it corresponds to a unique ns × ms matrix A′ over N and this correspondence is functorial and compatible with ϕ. A′ is integral if and only if so is A (cf. the reasoning below). We will denote A′ by ϕ(A).

3. If C is a matrix of the base change from (w) into (w′), i.e., C(w) = (w′), then ϕ′(F (X, Y)) = CϕF (C^{−1}X, C^{−1}Y) and ϕ′(f(X)) = CϕF (C^{−1}X) for F being a formal group, where f is a homomorphism of formal groups.

We formulate the main properties of this functor.

Proposition 2.1.2. 1. Let f ∈ C′_{K}(F_{1}, F_{2}). Then ϕ(f) ∈ C_{N}(ϕ(F_{1}), ϕ(F_{2})) if and only if f ∈ C_{K}(F_{1}, F_{2}).

2. There exists a homomorphism f from F_{1} of dimension m_{1} into F_{2} of dimension n_{2}, f ≡ AX mod deg 2, A is an m_{1} × m_{2} matrix over 𝔒_{K}, if and only if there exists a homomorphism f′ ∈ C_{N}(ϕ(F_{1}), ϕ(F_{2})), f′ ≡ A′X mod deg 2.

3. F_{1} and F_{2} are strictly isomorphic if and only if so are ϕ(F_{1}) and ϕ(F_{2}).

Proof. 1. Suppose that the coefficient a^{l}((a_{i})) of f_{l} at ∏ X_{i}^{a_{i}} is non-integral. We choose a j such that w_{j} ∈ 𝔒_{K}^{∗}. We denote the coefficients of ϕ(f) as b. According to (2)

S((b_{v}^{l}((b_{i}^{r}))) = w_{j}^{∑_{i} a_{i}}a^{l}((a_{i})),

where b_{i}^{r} = 0 for r ≠ j and b_{j}^{j} = a_{j}. Thus at least one of b_{v}^{l}((b_{i}^{r})) is also non-integral.

2. If f ≡ AX mod deg 2 then ϕ(f) ≡ A′X mod deg 2 according to the definition of ϕ.

The converse implication. It is well known that there exist homomorphisms g ∈ C′_{K}(F_{1}), F_{2}), g′ ∈ C′_{N}(ϕ(F_{1}), ϕ(F_{2})) satisfying g ≡ AX mod deg 2, g′ ≡ A′X mod deg 2 and they are unique. Thus the statement follows from part 1.

3. This is a special case of part 2 with A equal to I_{m}.

2.2 Fractional parts

We consider the ring R_{X} = 𝔒_{K}[[X_{i}]]×Q_{p}, i.e., the ring of series with bounded denominators. We denote by DR_{X} the K-module of series whose partial derivatives belong to R_{X}.

Definition 2.2.1. We denote the residue of an element f ∈ K[[X_{i}]] modulo R_{X} by {f} and call it the fractional part of f.

Thus {f} = {g} if and only if f − g ∈ R_{X}.

Lemma 2.2.2. Let f, g ∈ DR_{X} and h_{1}, h_{2} ∈ 𝔒_{K}[[X_{i}]]_{0}^{m} (i.e., the constant terms are zero).

1. If {f} = {g} and h_{1} ≡ h_{2} mod (π), then {f(h_{1})} = {g(h_{2})}.

2. Let λ be a logarithm of a finite height formal group. If {λ(h_{1})} = {λ(h_{2}}, then h_{1} ≡ h_{2} mod (π).

3. We introduce the right action of NW on K[[X]] in the obvious way. Then {fr} = {f}r for any r ∈ NW, f ∈ K[[X]].

Proof. 1. The equality {f(h_{1})} = {g(h_{1})} is obvious. Let now h_{2} = (h_{2i}) be equal to (h_{1i}) + π(r_{i}), r_{i} ∈ 𝔒_{K}[[X_{i}]]^{m}.

We decompose

g(h_{2}) = g(h_{1}) + ∑_{(a_{i})} g_{(a_{i})}C_{(a_{i})}π^{∑_{i} a_{i}}r_{1}^{a_{1}} … r_{m}^{a_{m}},

where (a_{i}) are multiindices, g_{(a_{i})} are coefficients of g, C_{(a_{i})} are polynomial coefficients. One can easily check that there exists a constant c (depending on the ramification index of K) such that

π^{c}C_{(a_{i})}π^{∑_{i} a_{i}}/ gcd(a_{i}) ∈ 𝔒_{K}.  (3)

Since g ∈ DR_{X}, the sequence g_{(a_{i})} gcd(a_{i}) has bounded denominators. Applying (3), we obtain {g(h_{1})} = {g(h_{2})}.

2. We have λ(h_{1}) − λ(h_{2}) = λ(h_{1} −_{F} h_{2}). Thus it is sufficient to prove

λ(h) ∈ R_{X}^{m} ⟹ π | h. It is obvious for m = 1. The proof of the general case using λ(X) = lim_{i} p^{−i}[p^{i}](X) will be given elsewhere.

3. This is obvious since R_{X} · r ⊂ R_{X} for any r ∈ NW.

We introduce the valuation v_{X} on R_{X} as the minimum of the valuations of coefficients. Remark 2.2.3. It is easily seen that (3) also implies that v_{X}(g(h_{1})−g(h_{2})) ≥ min v_{X}(\frac{dg}{dX_{i}}) − c.

3 Logarithmic matrix; classification for small ramification index

The goal of the this section is to apply Honda’s classification of formal groups over unramified fields to the classification of formal groups over general local fields. We apply the functor ϕ and consider a matrix that corresponds to the logarithm of ϕ(F).

3.1 Construction of the matrix

We define an operator ⟨α⟩ on K[[Δ]]: ⟨α⟩(∑ c_{i}Δ^{i}) = ∑ c_{i}α^{p^{i}}Δ^{i}.

Thus for any h ∈ K[[Δ]] we have

⟨α⟩(h)(x) = h(αx).

In what follows, we always assume that N is the inertia subfield of K and w is a 𝔒-base of 𝔒_{K}. Sometimes we will impose some extra restrictions on w.

Definition 3.1.1. For an m-tuple f = (f_{i}) of p-typical series in X_{i} over K we define the matrix T_{w}(f) consisting of the columns (t_{j}^{i}), the coefficients of t_{j}^{i} are t_{jk}^{il} ∈ W′, by the equality

f(w_{i}xe_{j})_{k} = (∑_{l} w_{l}t_{jk}^{il})(xe_{j}).

Thus S_{l}(t_{jk}^{il}) = ⟨w_{i}⟩Λ_{j}.

The following statement is obvious.

Lemma 3.1.2. Let A be an n × m matrix over K, and let A′ = ϕ(A) be the en × em matrix defined in Remark 2.1.1. Then T (Af) = a′T (F).

If Λ(X) is a logarithm of a formal group F, we call T_{w}(Λ) the logarithmic matrix for F.

3.2 Main properties of the logarithmic matrix

We study the connection of ϕ(F) with the formal group obtained from T (Λ).

Proposition 3.2.1. 1. Let Λ ∈ K[[Δ]]^{m}. Then λ = Λ(X) is a logarithm of a formal group F over 𝔒_{K} if and only if λ_{e} = T (Λ)((X_{j}^{i})) is a logarithm of a formal group over 𝔒.

2. If the conditions of the first part are fulfilled, then T (Λ)((X_{j}^{i})) is a logarithm of a formal group law that is strictly isomorphic to ϕ(F).

Proof. Obviously, T (Λ)((X_{j}^{i})) is the p-typical part of ϕλ. Thus all statements follow form Proposition 1.1.1, except the fact that λ is a logarithm of a formal group over 𝔒_{K} if λ_{e} is a logarithm of a formal group over 𝔒.

Suppose that λ ≡ λ′ mod deg p^{s}, where λ′ is a logarithm of some ptypical formal group over 𝔒_{K}. Note that this is always true for s = 1. Then λ′_{e} (i.e., the p-typical part of ϕ(λ)) is a logarithm of a formal group over 𝔒, besides λ′_{e} ≡ λ_{e} mod deg p^{s}. Using the form of the logarithm of universal p-typical group law (see [4]), we obtain λ′_{e} ≡ λ_{e} mod (deg p^{s+1}, 1/p). The same reasoning, as in the proof of Proposition 2.1.2 shows that λ ≡ λ′ mod (deg p^{s}, 1/p). Applying the form of the logarithm of universal p-typical group law again, we obtain that there exists a logarithm λ′′ of some p-typical formal group over 𝔒_{K} such that λ ≡ λ′′ mod deg p^{s}. Hence for any l we have λ ≡ λ_{l} mod deg p^{l}, where λ_{l} is some ’integral’ logarithm. Thus λ gives an integral group law (since the group law is integral modulo any degree).

We also express the condition for the existence of homomorphisms in terms of logarithmic matrix.

Proposition 3.2.2. There exists a homomorphism f from F into F′ of dimensions m and m′ respectively, f ≡ AX mod deg 2, A is some m × m′ matrix over 𝔒_{K}, if and only if there exist C ∈ M_{me×m′e}(W) such that ϕ(A)T (F) = T (F′)C

Proof. According to Proposition 2.1.2 we should find out when there exists a homomorphism g from ϕ(F) to ϕ(F′), g ≡ ϕ(A)X mod deg 2. Besides, we can replace ϕ(F) and ϕ(F′) by F_{1} and F′_{1}, whose logarithms are T (F)(X_{i}^{j}) and T (F′)(X_{i}^{j}).

We should check the condition of Theorem 1.2.1. The corresponding special elements are equal to T (F)^{−1} and T (F′)^{−1}. Multiplying the equality CT (F)^{−1} = T (F′)^{−1}ϕ(A) by T (F) from the right and by T (F′) from the left, we obtain the claim.

Remark 3.2.3. We can apply elementary transformations over W to the columns of the matrix T without changing the strict isomorphism class of the corresponding me-dimensional group.

3.3 Classification for small ramification

We classify formal groups for e < p. In order to make the proof perfectly clear for the reader we consider the one-dimensional case. It can easily be seen that a very similar reasoning works for an arbitrary dimension.

Theorem 3.3.1. 1. λ = Λ(Δ)(X) is a logarithm of a p-typical formal group if and only if Λ = vu^{−1} for some u ∈ M_{m}(W), u ≡ pI_{m} mod Δ and v ∈ M_{m}(𝔒_{K}[Δ]), v ≡ pI_{m} mod πΔ.

2. F corresponding to vu^{−1} is strictly isomorphic to F_{1} corresponding to v_{1}u_{1}^{−1} if and only if u_{1} = εu, v_{1} = v + gu, where ε ∈ I_{m} + ΔM_{m}(W), g ∈ πM_{m}(𝔒_{K}[Δ])Δ.

Proof. We assume that m = 1.

We choose w_{i} = π^{i−1}, 1 ≤ i ≤ e.

1. A p-typical series λ = Λ(x) can be a logarithm of a formal group only if it is equal to 1+∑_{i>0} \frac{c_{i}}{p^{i}} Δ^{i}, c_{i} ∈ 𝔒_{K}. Let c_{r} = ∑ c_{ri}w_{i} be the decomposition of c_{r} in w_{i}, and let T (Δ) = (T_{1}, T_{2}, …, T_{e}) be the matrix corresponding to λ. Then from the definition of this matrix we obtain

(Λ_{1} ; ⋮ ; Λ_{e}) ,

where Λ_{r} = ∑ \frac{c_{ir}}{p^{i}} Δ^{r} for r > 1, Λ_{1} = 1 + ∑ \frac{c_{i1}}{p^{i}} Δ^{r}. Since pπ | π^{p}, the r-th column for r > 1 has the form

( pα_{1,r} ; ⋮ ; pα_{r−1,r}Δ ; ε_{r} ; α_{r+1,r}Δ ; ⋮ ; α_{e,r}Δ) ,

where a_{i,r} ∈ W, ε_{r} ∈ 1 + W Δ.

We replace T_{2}, …, T_{e} by

T′_{r} = ( 0 ; ⋮ ; 0 ; ε_{r} ; α_{r+1,r}Δ ; ⋮ ; α_{e,r}Δ) .

This change adds to T some M ∈ pΔM_{e}[W]. Hence, according to Theorem 1.2.1, the new matrix gives an ’integral’ logarithm if and only if T does, and the new logarithm will give a strictly isomorphic formal group.

Applying elementary transformations to T′_{2}, …, T′_{e} (cf. Remark 3.2.3), we eventually obtain the matrix

T′ = (\begin{matrix}Λ_{1} & 0 & … & 0 \\ Λ_{2} & 1 & … & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ \\ Λ_{e} & 0 & … & 1 \end{matrix}) .

According to Theorem 1.2.1, T′ gives an integral logarithm if and only if there exist a special matrix U over W, UT′ = pI_{e}. Clearly, U should have the form

(\begin{matrix}u(Δ) & 0 & … & 0 \\ r_{1} & p & … & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ \\ r_{e−1} & 0 & … & p \end{matrix}) ,

where u ∈ p + W Δ, r_{i} ∈ W Δ. Matrix U of this form is the inverse to T if and only if Λ_{1} = p, r_{i}Λ_{1} + pΛ_{i} = 0 for i > 1. Hence Λ_{1} = p/u, Λ_{i} = −r_{i+1}/u. We obtain the only if part of the claim.

Now we prove the converse statement. We represent v in the form ∑ Λ_{i}π^{i−1}. Since π | v, we have p | Λ_{1}. We take u′ = pu/Λ_{1}. For this u′ (with the same Λ) we have Λ_{1} = p. Therefore we can construct U and T′, then T and Λ. Hence the if part is also valid.

2. We construct the matrices T′ and T′_{1} in the same way as above, U and U_{1} are their inverses of the form described above.

We want to check when U′ = CU, where C ∈ M_{e}[[W]], C ≡ I_{m} mod Δ. Obviously, this can happen only for

C = (\begin{matrix}ε & 0 & … & 0 \\ g_{1} & 1 & … & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ \\ g_{e−1} & 0 & … & 1 \end{matrix}) ,

where ε ∈ 1 + W Δ, g_{i} ∈ W Δ. Performing the matrix mutliplication, we obtain the claim.

In a similar way one can describe formal groups of height > 1 for e = p. It is possible since in this case we also have a_{12} ∈ W. It is possible also to classify formal groups for e ≤ 2p − 2. To this end one should take into account the second column of T. The method of the last section seems to be more useful, as it easily gives classification of formal groups of height > 1 for e ≤ p^{2}/2. Remark 3.3.2. 1. One can also prove a result about homomorphisms between formal groups similar to part 3 of 1.2.1. One should take v into account in the same way as in the results of the next section.

2. One can choose a canonical representative in each class of strict isomorphism by demanding the coefficients of u at positive degrees of Δ to be Teicmüller representatives (see Proposition 1.3.1), the coefficients of v at positive degrees of Δ to have zero trace into 𝔒.

3. One can easily prove that in the one-dimensional case the height of F is equal to the index of the first invertible coefficient of u.

3.4 Main matrix lemma

We formulate a lemma that will allow us to check the conditions of Theorem 1.2.1. We will use it (later) for different rings R.

Lemma 3.4.1. I Let ℜ, 𝔏 be integral domains (not necessarily commutative), gr ⊂ 𝔏, T_{1}, T_{2} be be matrices over 𝔏, T_{1} is non-degenerate.

1. T_{1}^{−1}T_{2} ∈ M_{m}(ℜ) if and only if T_{2}(ℜ^{m}) ⊂ T_{1}(ℜ^{m}).

2. T_{1}^{−1}T_{2} ∈ Gl_{m}(ℜ) if and only if T_{2}(ℜ^{m}) = T_{1}(ℜ^{m}).

II For a matrix T ∈ Gl_{m}(𝔏) we have pT^{−1} ∈ M_{m}(ℜ) if and only if pℜ^{m} ⊂ T (ℜ^{m}).

Proof. I 1. If T_{1}^{−1}T_{2} ∈ M_{m}(ℜ), then T_{2}(ℜ^{m}) = T_{1}(T_{1}^{−1}T_{2}ℜ^{m}) ⊂ T_{1}(ℜ^{m}).

The inverse implication: for any i there exist v_{i} ∈ ℜ^{m} such that T_{2}e_{i} = T v_{i}. We have T_{2} = tI_{m} = T (e_{i}), where (e_{i}) is the matrix consisting of all e_{i}. Hence T_{1}(v_{i}) = T_{2}, where (v_{i}) is the matrix consisting of all v_{i}, we obtain the claim.

2. The second claim follows immediately from the first one applied to the pairs T_{1}, T_{2} and T_{2}, T_{1}.

II This follows immediately from I1 if we take T_{1} = T and T_{2} = pI_{m}.

Now we combine this lemma with Propositions 3.2.1 and 3.2.2.

Proposition 3.4.2. 1. A p-typical m-tuple λ = Λ(X), Λ ≡ I_{m} mod Δ, is a logarithm of a formal group law over 𝔒_{K} if and only if pW^{m} ∈ T (Λ)W^{m}.

2. There exists a homomorphism f from F to F′ of dimensions m and m′ respectively, f ≡ AX mod deg 2, A is some m × m′ matrix over 𝔒_{K}, if and only if ϕ(A)T (F)(W^{m}) ⊂ T (F′)(W^{m}).

Proof. Since T (F) ≡ I_{me} mod Δ and T (F′) ≡ I_{m′e} mod Δ, the matrices T (F) andT (F′) are non-degenerate.

1. According to Proposition 3.2.1, we must check that the matrix pT (λ) is integral (see theorem 1.2.1). Applying part II of Lemma 3.4.1 for ℜ = W, 𝔏 = W′, we obtain the claim.

2. We apply Proposition 2.1.2. The conditions of Lemma 3.4.1, part I1, are obviously fulfilled for ℜ = W, 𝔏 = W′, the matrix T_{1} = T (F′), and T_{2} = ϕ(A)T (F). Hence we obtain the claim.

4 Fractional part invariant: classification up to a strict isogeny

In this section we prove that the fractional part of a logarithm of a formal group classifies formal groups up to isogeny of an explicitly described sort.

We will consider the algebra NW = N × W, which is equal to ∪_{s}p^{−s}W.

4.1 Classification up to rational isogeny; homomorphisms

For a formal group law F we define r(F) to be the residue of Λ mod M_{m}(R). We have r(F) ∈ M_{m}(R)u^{−1}/M_{m}(R), where Λ = vu^{−1}. We always assume that u has the minimum possible rank over W.

First we formulate how can r(F) of two strictly isomorphic formal groups differ.

Proposition 4.1.1. If F is strictly isomorphic to F′, then

1. u′ = uη for η ∈ I_{m} + M_{m}(W)Δ (i.e., the ideals, generated by u and u′ coincide)

2. r′ = rε for ε ∈ I_{m} + M_{m}(W)Δ.

Proof. 1. For totally ramified one-dimensional case the statement follows from Proposition 3.2.2 combined with the fact that det(T (F)) = \frac{p}{u}s, s ∈ 1 + Δℤ_{p}[[Δ]].

The general proof is quite technical and will be given elsewhere.

2. According to Proposition 3.2.2, T (F′) = T (F)𝔈 for some 𝔈 ∈ I_{me} + M_{me}(W).

Then we have T (F′) ≡ T (F)𝔈 mod NW. We assume that w_{1} = 1, π | w_{j} for j > 1. Then we obtain S(T (F)(e_{i}^{1})) = Λ_{i} and S(T (F)(e_{i}^{j}) ∈ NW^{m} for j > 1. Thus vu^{−1} = εv′u′^{−1} for some ε ∈ I_{m}+M_{m}(W), ε is the corresponding block of 𝔈.

Thus one can also define the invariant I(F) as in the introduction. It will be well defined if we take for u(F) the canonical representative. Otherwise one should consider u up to a I_{m} + M_{m}(W)Δ multiplier.

One can easily show that for each F any ε ∈ I_{m} + M_{m}(W)Δ is possible, i.e., the claim of Proposition is the best possible.

We prove the following necessary and sufficient condition for the existence of homomorphisms for ε ∈ Gl_{m′}(NW).

Theorem 4.1.2. 1. There exists a homomorphism f from F into F′, the dimensions of F, F′ are m, m′, f(X) ≡ Ap^{s}x mod deg 2 for a fixed A ∈ M_{m×m′}(K) and some s ∈ ℤ if and only if Ar = r′ε for some ε ∈ M_{m′}(NW), where r = r9F), r′ = r(F′).

2. F and F′ of dimension m and finite height are isogenious if and only if there exists an A ∈ Gl_{m}(K) and ε ∈ M_{m}(NW) such that Ar = r′ε.

Proof. 1. Again we take w_{1} = 1, π | w_{j} for j > 1. Then we obtain S(T (F)(NW^{m})) = vu^{−1}(NW^{m}) + R^{m}. Thus the condition on the fractional parts is equivalent to AT (F)(NW^{me}) ⊂ T (F′)(NW^{m′e}). According to Proposition 3.4.2 for R = NW this is equivalent to the existence of B ∈ M_{m′e}(NW), AT (F) = T (F′)B. Since C ∈ M_{m}(NW) if and only if p^{s}C ∈ M_{m}(W) for some s ∈ ℤ, Theorem 1.2.1 gives us the claim.

2. For finite height formal groups, f is isogeny if and only if f mod deg 2 is an invertible operator. Hence part 2 follows from part 1 immediately.

We call two formal groups rationally isogenious if they satisfy the conditions of the theorem for A = I_{m}.

4.2 Strict isogeny: classification

Now we describe what does it mean that two logarithms have equal fractional parts. Note that this gives us an answer to the question which formal groups are glued together by the r-invariant (we call them strictly isogenious).

Theorem 4.2.1. For finite height formal groups r = r′ if and only if there is an homomorphism f ≡ p^{s}I_{m}X mod deg 2 from F into F′ for some s ∈ ℤ such that f(x) ≡ [p^{s}]_{F} (X) mod π𝔒_{K}[[X]]^{m}.

Proof. Suppose that r = r′. As is proved above, we have a homomorphism f ≡ p^{s}I_{m}X mod deg 2 from F to F′. We have the equality

λ′(f(X)) = p^{s}λ(X) = p^{s}λ([p^{s}]_{F} (X)).  (4)

According to the first part of Lemma 2.2.2 we obtain

{λ(f(X))} = {p^{s}λ([p^{s}]_{F} (X))}.

Now the second part of Lemma 2.2.2 immediately gives us the claim.

The converse implication is even easier: one just calculates fractional parts in equality (4).

Remark 4.2.2. 1. More generally, the same reasoning proves that for fixed r-invariants of F and F′ the knowledge of f mod deg 2 fixes f modulo π.

In particular, we obtain that two strictly isomorphic logarithms have equal fractional parts if and only if the strict isomorphism between them is congruent to X modulo π. Thus our technique can be applied to the study of deformation problems.

2. Since strict isomorphisms multiply r by ε ∈ I_{m} + M_{m}(W)Δ and any such ε is possible, it seems natural to describe homomorphisms between F and F′ fixing r and r′ only modulo I_{m} + M_{m}(W)Δ. We will try to do this in the next subsection.

4.3 Strict isogeny: homomorphisms in the one-dimensi-onal totally ramified case

Now we give a more precise version of Theorem 4.1.2 by considering ε ∈ I_{m} + M_{m}(W)Δ (instead of Gl(NW)). Unfortunately, in order to describe the result for multi-dimensional case one should introduce many technical definitions and results. This will be done in subsequent papers. For simplicity we also assume N = ℚ_{p}. Note that in this case there are no homomorphisms between finite height groups with non-equivalent u. We have W = ℤ_{p}[[Δ]].

We expand u as p − ∑ u_{i}Δ^{i}, u_{i} ∈ ℤ_{p}. We have u_{h} ∈ ℤ_{p}^{∗}, u_{i} ∈ pℤ_{p} for i < h, where h is the height of F (see the remark below).

Theorem 4.3.1. Suppose that two finite height formal groups have the same u, Then for a fixed a ∈ K and m ∈ ℤ we have av ≡ ε(Δ)v′Δ^{m} mod u for some ε ∈ 1 + Δℤ_{p}[[Δ]] if and only if there is an isogeny f = ∑ a_{i}x^{i} ≡ ap^{s}x mod deg 2 from F into F′ for some s ∈ ℤ such that the height of f equals sh + m and a_{p^{sh+m}} ≡ u_{h}^{s} mod π.

Proof. 1. Let the logarithms satisfy the condition on fractional parts. Then, according to Theorem 4.1.2, some isogeny f ≡ ap^{s}x mod deg 2 does exist. Therefore we must check that f satisfies other assertions of the claim.

We denote by P_{f} the series ∏_{t∈Ker f}(x−_{F}t). According to a result of Lubin

(see [4]) f can be presented as a composition of P_{f} and an isomorphism. Hence we can write

f = cx ∘ g(x) ∘ P_{f}(x),

where c ∈ 𝔒_{K}^{∗}, g(x) ≡ x mod deg 2.

This is equivalent to the relation

ap^{s}λ(x) = λ′(cx ∘ g(x) ∘ P_{f}(x)).

Now we calculate how each of this functions affects fractional parts. We denote the height of f by H.

I. We have c^{p^{l}} → θ_{c} for l → ∞, where θ_{c} is the Teichmüller representative of θ mod π. Thus {λ′(cx)} = θ_{c}{λ′(x)}.

II. The series c^{−1}λ′(cx) is a logarithm of a formal group F_{c}(x, y) = c^{−1}F (cx, cy), which is isomorphic to F, the isomorphism is equal to cx. Hence λ′(cx ∘ g(x)) is a logarithm of some formal group, strictly isomorphic to F_{c}.

According to Proposition 4.1.1 for strictly isomorphic formal groups the fractional part of logarithms differ by a η′ ∈ 1 + W Δ. Hence

λ′(cx ∘ g(x) = {λ′(x)}θ_{c}η′.

III. Obviously, P_{f} ≡ x^{p^{H}} mod π. Hence Lemma 2.2.2 implies

{λ′(cx ∘ g(x) ∘ P_{f}(x))} = λ′(cx ∘ g(x))Δ^{H} = {λ′(x)}θ_{c}ηΔ^{H}.

Thus we obtain the relation

ap^{s}r(F) = r(F′)ηθ_{c}Δ^{H}.

On the other hand, since P_{f} ≡ x^{p^{H}} mod π, we have c ≡ a_{p^{H}} mod π.

We can replace u by its canonical representative (with respect to Teichmüller representatives). Then in the ring R_{u} = R_{0}/u (note that we are in the commutative situation!) we have p = Δ^{h} ∑_{i≥0} u_{h+i}Δ^{i}. Therefore, in R_{u} we have p^{s} = u_{h}^{s}η(Δ)Δ^{sh} for some η ∈ 1 + W Δ. Hence we have the congruence

av′Δ^{hs+m}u_{h}^{s}η ≡ η′θ_{a_{p^{H}}} Δ^{H}v′Δ^{m} mod u.

If H ≠ hs + m or a_{p^{sh+m}} ≢ u_{h}^{s} mod π, then the valuation of the element d = Δ^{hs+m}u_{h}^{s}η − θ_{a_{p^{H}}} Δ^{H} in the field ℚ_{p}W/(u) is equal to min(H, hs + m). Therefore, d is invertible which is impossible since dr = 0 and r(F) ≠ 0. Thus we obtain the claim.

2. If f satisfies all conditions of our theorem, then the calculations of fractional parts carried out above show immediately that the relation av ≡ ε(Δ)v′Δ^{m} mod u for some ε ∈ 1 + Δℤ_{p}[[Δ]] is fulfilled.

Remark 4.3.2. 1. In particular, applying the proposition to f = [p]_{F}, F′ = F, we obtain that h defined in terms of u coincides with the height of F (we did not use that in the proof).

2. In all cases one can replace the invariant r(F) by a certain element V (F) of K{{Δ}} mod R (i.e., modulo positive degrees). One defines V (F) to be the residue modulo R of the fraction vu^{−1} considered in the matrix ring over the two-dimensional field K{{Δ}}.

5 Invariant Cartier–Dieudonne modules

5.1 Category of D-modules

Definition 5.1.1. We denote by 𝔇 the category of (left) W -submodules D of K[[Δ]]^{m}, satisfying the following conditions.

1. D is a free W -module of rank em.

2. D mod Δ = 𝔒_{K}^{m}.

3. p𝔒_{K}^{m} ⊂ D.

Lemma 5.1.2. The elements v_{1}, …, v_{me} ∈ D form a W -base of D ∈ 𝔇 if and only if v_{i} mod Δ form a 𝔒-base of 𝔒_{K}^{m}.

Proof. The only if part is obvious.

Now we prove that v_{i} satisfying the conditions of Lemma form a base of D.

Suppose that v′_{i} is a W -base of D, 1 ≤ i ≤ me. Then we have (v_{i}) = P (v′_{i}) for some P ∈ M_{m}(W). Since (v_{i}) mod Δ = (v′_{i}) mod Δ and the dimension of D mod Δ over W mod Δ is equal to the number of v′_{i}, P mod Δ is invertible. Thus P is invertible, hence v_{i} also form a base of D.

Lemma 5.1.3. If D ⊂ D′ for D, D′ ∈ 𝔇, then D = D′

Proof. Choose a basis v_{i} of D. Since v_{i} mod Δ generate 𝔒_{K}^{m}, v_{i} is also a W -base of D′.

Sometimes it can be useful to replace the axioms of 𝔇 (especially the first one) by slightly different ones.

Proposition 5.1.4. I If D, satisfying conditions 2 and 3 of 5.1.1 satisfies also one of the following conditions:

1. D is generated by ≤ me elements.

2. Any v ∈ W^{m}Δ ∩ D also belongs to DΔ, then it belongs to 𝔇.

II Vice versa, any D ∈ 𝔇 satisfies 1 and 2.

Proof. I 1. Let v_{i} ∈ D, 1 ≤ i ≤ r be generating elements. Since the dimension of D mod Δ over 𝔒 = W mod Δ is greater that or equal to s, the elements v_{i} are W -independent. Since this dimension is equal to me, we also obtain s = me.

2. We choose v_{i}, 1 ≤ i ≤ me such that v_{i} mod Δ generate D mod Δ. For v ∈ D we can choose c_{i} ∈ 𝔒 such that v ≡ ∑ v_{i}c_{i} mod Δ. Hence v = ∑ v_{i}c_{i} +v_{1}Δ for some v_{1} ∈ D. Similarly v_{1} ≡ ∑ v_{i}c_{i1} mod Δ. Applying this procedure successively and passing to the limit, we obtain that v lies in the W -envelope of v_{i}. Thus D satisfies condition 1 in this lemma.

II The first condition is obvious.

Let now v belong to W^{m}Δ ∩ D. We choose a base v_{i} for D; v_{i} mod Δ are 𝔒-independent. We have v = ∑ v_{i}w_{i}. Taking this equality modulo Δ, we obtain that w_{i} ∈ W Δ, hence v ∈ DΔ.

5.2 Two definitions of D_{F}

Definition 5.2.1. For a formal group F with logarithm λ = Λ(X) one defines

D_{F} = S(T_{F} (W^{m})) = ⟨⟨w_{i}⟩Λ_{j}⟩.

Proposition 5.2.2. For formal groups F_{1} and F_{2} of dimensions m_{1} and m_{2}, whose D-modules are equal to D_{1} and D_{2}, respectively, the following statements are valid.

1. Let A be a m_{1} × m_{2} matrix over 𝔒_{K}. There exists a homomorphism f from F_{1} into F_{2}, F (X) ≡ AX mod deg 2 if and only if AD_{1} ⊂ D_{2}.

2. For m_{1} = m_{2} the groups F_{1} and F_{2} are strictly isomorphic if and only if D_{1} = D_{2}.

3. For any F D_{F} ∈ 𝔇.

Proof. 1. According to part 2 of Proposition 3.4.2 it is sufficient to check A′(T_{1}(W^{m_{1}})) ⊂ T_{2}(W^{m_{2}}). This is equivalent to the statement of the proposition since S is a bijection.

2. This follows immediately from part 1 for A = I_{m_{1}}.

3. Properties 1 and 2 are obvious (cf. Proposition 5.1.4). Property 3 follows from part 1 of Proposition 3.4.2.

Now we prove that D_{F} is the logarithm of the module of p-typical curves (i.e., of the classical Cartier–Dieudonne module).

Proposition 5.2.3. D_{F} = D′_{F}, where

D′_{F} = {f ∈ K[[Δ]]^{m} : exp_{F} (f(x)) ∈ 𝔒_{K}[[x]],

here exp_{F} is the composition inverse to λ_{F}.

Proof. First we check D_{F} ⊂ D′_{F}.

We have exp_{F} (⟨w_{i}⟩Λ(xe_{i})) = w_{i}xe_{i}, hence the generators of D_{F} belong to D′_{F}.

Next we check that D′_{F} is an W -module. For c_{i} ∈ ℤ_{p} and h ∈ K[[Δ]]^{m} one has

exp_{F} (h ∑ c_{i}Δ_{i})(x) = ∑_{(F)}[c_{i}]_{F} exp_{F} (h(X^{p^{i}})),

therefore D′_{F} is an ℤ_{p}{{Δ}}-module. It remains to check that for any Teichmüller representative θ we have D′F θ ⊂ D′F. This assertion follows immediately from the relation exp_{F} (hθ(x)) = θ exp_{F} (x).

Now, according to Lemma 5.1.3, it is sufficient to prove D′_{F} ∈ 𝔇. We have exp_{F} (h(x)) ≡ h(x) mod deg 2, hence D′_{F} mod Δ ⊂ 𝔒_{K}^{m}. Thus properties 2 and 3 of the definition of 𝔇 follow from the fact D_{F} ⊂ D′_{F}. Obviously, D′_{F} also satisfies condition 2 of Proposition 5.1.4. Therefore D′_{F} ∈ 𝔇 and the proposition is proved.

Remark 5.2.4. Combining the two descriptions of D_{F}, one easily sees that for any base t_{i} of 𝔒_{K} over ℤ_{p} the elements v_{ijl} = ⟨w_{i}t_{j}⟩Λ_{l} form a ℤ_{p}[Δ]-base of D_{F}.

Indeed, v_{i} belong to D_{F} and v_{ijl} mod Δ are a ℤ_{p}-base of D_{f} mod Δ.

5.3 Basic properties of D_{F}

We prove that a module D ∈ 𝔇 corresponds to a formal group over 𝔒_{K} if and only if it is ⟨π⟩-stable.

Proposition 5.3.1. For D ∈ 𝔇 the following conditions are equivalent 1. D = D_{F} for some F.

2. For any a ∈ 𝔒_{K} we have ⟨a⟩D ⊂ D

3. ⟨π⟩D ⊂ D

4. There exist Λ_{i} ∈ D, Λ_{i} ≡ e_{i} mod Δ such that ⟨w_{j}⟩Λ_{i} ∈ D, where w_{i} is some 𝔒-base of 𝔒_{K}.

Proof. 1 ⟹ 2: We have exp_{F} (⟨a⟩(f(Δ))(x)) = exp_{F} (f(Δ)(ax)). It remains to apply Proposition 5.2.3.

2 ⟹ 3 is obvious.

3 ⟹ 4: We choose Λ_{i} ∈ D : Λ_{i} ≡ e_{i} mod Δ. We have ⟨π^{s}⟩Λ_{i} ∈ D. Thus we can take w_{j} = π^{j−1}.

4 ⟹ 1: We have ⟨w_{j}⟩Λ_{i} ≡ w_{j}e_{i} mod Δ. Hence, according to Lemma 5.1.2, the elements ⟨w_{j}⟩Λ_{i} generate D.

We take a formal group F over K whose logarithm is equal to (Λ_{i}(e_{i}X_{i})). The module D satisfies property 3 of the definition, therefore, according to Proposition 3.4.2, F is a p-typical formal group over 𝔒_{K}. We have D ⊂ D_{F}, thus D = D_{F}

Instead of π we could take any other polynomial generator of 𝔒_{K} over 𝔒, for example, 1 + π.

Next we prove that the logarithms of formal groups isomorphic to F can be described in terms of module D_{F}.

Proposition 5.3.2. For f ∈ M_{m}K[[Δ]] the series f(Δ)(X) is a logarithm of a formal group strictly isomorphic to F if and only if f ≡ I_{m} mod Δ and f(Δ)(xe_{i}) ∈ D_{F}.

Proof. If F′ is strictly isomorphic to F, we have D_{F′} = D_{F} and everything is clear.

Vice versa, for a sequence of Λ_{i} ∈ d : Λ_{i} ≡ e_{i} mod Δ we can construct a formal group F′ as is done in the previous proof. Again we have D = D_{F} ⊂ D_{F′}.

Thus we can choose any set of Λ_{i} ≡ e_{i} mod Δ in D.

We also note that the intersection of D_{F} with K^{m} always contains a module that is slightly larger than p𝔒_{K}^{m}. Remark 5.3.3. 1. For s = −[e/(1 − p)] we have π^{s}𝔒_{K}W^{m} ⊂ D_{F}.

2. Let a ∈ 𝔒_{K}, v_{K}(a) = l, l ≤ s. Then for a formal group law F_{a} = a^{−1}F (aX, aY) one has π^{s−l}𝔒_{K}W^{m} ⊂ D_{F_{a}}.

Proof. 1. This is a special case of the second part for a = 1.

2. We have

D_{F_{a}} = {f ∈ K[[Δ]]^{m} : exp_{F} (af(x)) ∈ a𝔒_{K}[[x]].

Hence it is sufficient to check that π^{l} | exp_{F} (aX).

Since all partial derivarives of λ are integral, the coefficients of λ at X^{a} are divisible by \frac{1}{a}. Hence one can easily proof by induction on the degree of monomials that the coefficients of exp_{F} at ∏ X_{i}^{a_{i}} are divisible by 1/ ∏ a_{i}!. Moreover, exp_{F} (X) ≡ X mod deg p. It follows that for (a_{i}) ≠ e_{i} the valutation of the coefficients of exp_{F} at ∏ X_{i}^{a_{i}} are > −\frac{∑ a_{i}}{p−1} . Thus for (x_{i}) ∈ π^{s}𝔒_{K}^{m} one has min v(x_{i}) < min v(exp_{F} ((x_{i}X_{i}))_{i} − x_{i}X_{i}).

5.4 The change of the base field

We describe how the module D_{F} behaves if we replace K by L ⊃ K. We denote the ring of integers of L by 𝔒_{L}.

Proposition 5.4.1. Let F be a formal group over 𝔒_{K}, s_{i} be an 𝔒_{K}-base of 𝔒_{L}. Then D_{L}(F) = ⟨⟨s_{i}⟩D_{K}(F)⟩ (the envelope can be understood in the Abelian group sense).

Proof. From the exponential definition of D_{F} (see Proposition 5.2.3) it is obvious that ⟨s_{i}⟩D_{K}(F) ⊂ D_{L}(F) for all i.

On the other hand, D′ = ⟨s_{i}⟩D_{K}(F) contains ⟨s_{i}⟩Λ_{j}w for all i and j and all w ∈ W. Applying Remark 5.2.4 for t_{i} = s_{j}w_{l}θ_{r}, where θ_{r} is a ℤ_{p}-base of 𝔒 consisting of Teichmüller representatives, we obtain that D′ contains a ℤ_{p}-base of D_{L}(F).

Now suppose that K′ is a subfield of K and π′ is a uniformizing element of K′.

Proposition 5.4.2. A finite height formal group F is isogenious to a formal group defined over K′ if and only if r = Ar′ε for some r′ defined over K′, A ∈ Gl_{m}(K), and ε ∈ Gl_{m}(W).

Proof. The only if part follows immediately from Theorem 4.1.2.

Now suppose that r = Ar′ε. We consider the module D_{1} = A^{−1}D ∩ K′[[Δ]]. We have D_{1} mod Δ ⊂ A^{−1}𝔒_{K}^{m} ∩ K′^{m} and pA^{−1}𝔒_{K}^{m} ∩ K′^{m} ⊂ D′. Hence if we choose A′ ∈ Gl_{m}(K′) so that A′D_{1} mod Δ = 𝔒′_{K}^{m}, then the module A′D_{1} satisfies properties 2 and 3 of the definition of 𝔇 (over K′). Obviously, D′ satisfies propery 2 of Proposition 5.1.4. Thus D ∈ 𝔇′. Since for any a ∈ 𝔒′_{K} we have ⟨a⟩D′ ⊂ D′, it corresponds to some formal group F′ over 𝔒′_{K}.

It remains to prove that F′ is isogenious to F. Using Proposition 5.4.1, we obtain AA′^{−1}D_{F′} ⊂ D_{F}. Hence there exist a homomorphism f ≡ AA′^{−1}X mod deg 2 from F′ into F.

On the other hand, the condition on the fractional parts implies that there exists an s ∈ ℤ such that for all i, 1 ≤ i ≤ m, p^{s}Λ_{i} ∈ AA′^{−1}D_{F′}. Hence the inverse homomorphism also exists.

Note that when the conditions of the proposition are not fulfilled one still obtains a formal group F′ over K′ and a canonical (up to an isomorphism) homomorphism from F into F′.

6 The module invariant

The goal of this section is an explicit description of a module invariant M_{F}. It completes the V -invariant to a classification of formal groups up to a strict isomorphism; furthermore, the set of possible M_{F} for F of finite height formal groups of fixed dimension over a fixed field K is finite.

6.1 Definition and main properties

As proved above, the module D_{F} belongs to u^{−1}(R^{m}). We can consider u^{−1} as a matrix over K{{Δ}}. Therefore we can embed D_{F} in K{{Δ}}.

Note that we can define on K{{Δ}} the natural right action of

Ω = {∑_{i∈ℤ}a_{i}Δ^{i}, a_{i} ∈ 𝔒, a_{i} → 0 for i → −∞},

the multiplication, as in W, is defined by the relation Δa = σ(a)Δ for any a ∈ 𝔒. We have w ⊂ Ω.

Definition 6.1.1. For a formal group F we define M_{F} as D_{F} Ω ⊂ K{{Δ}}.

Note that M_{F} is a Ω-module.

We describe the main properties of M_{F} that allows to use it in the classification questions as a complement of the invariant r(F).

Theorem 6.1.2. 1. Suppose that F and F′ satisfy the conditions of part 1 of Theorem 4.1.2, i.e., that for some A ∈ M_{m×m′}(K) and ε ∈ M_{m′}(NW) we have Ar = r′ε. Then there exists a homomorphism f from F to F′, f(X) ≡ Ap^{s}x mod deg 2 for a fixed s ∈ ℤ if and only if Ap^{s}M_{F} ⊂ M_{F′}.

2. Suppose that F is a finite height group and for some A ∈ M_{m×m′}(K) and ε ∈ M_{m′}(NW) we have Ar = r′ε. If p^{s}A ∈ M_{m×m′}(𝔐^{l}), where l = p^{[log_{p}(e/(p−1))]} − e[log_{p}(e/(p − 1))] − [e/(1 − p)], ε ∈ M_{m′}(NW), then there exists a homomorphism f from F to F′, f(X) ≡ Ap^{s}x mod deg 2.

3. F and F′ are strictly isomorphic if and only if they are rationally isogenious and M_{F} = M_{F′}.

Proof. 1. If such an f exists, then we have AD_{F} ⊂ D_{F′}. It follows that AD_{F} N ⊂ D_{F′}N, which is equivalent to the condition on the fractional parts. The condition on M_{F} follows immediately from the definition of M_{F}.

Vice versa, let the conditions on F and F′ be fulfilled. Then for the W -modules Y = AD_{F} and Z = D_{F′} we have Y N = ZN and Y Ω = ZΩ. Let z_{i} be a W -base of Z (recall that D_{F′} is W -free). For any r ∈ ZΩN the coefficients r_{i} ∈ ΩN such that r = ∑ z_{i}r_{i} are unique. for y ∈ Y, y = ∑ z_{i}y_{i} we have y_{i} ∈ NW ∩ Ω = W. Hence Y ⊂ Z and Proposition 5.2.2 gives us the claim.

2. It is sufficient to check that p^{s}AM_{F} ⊂ M_{F′}. According to Remark 5.3.3, we have (𝔐^{−[e/(1−p)]}Ω)^{m′} ⊂ M′_{F}. It remains to check that M_{F} ⊂ (𝔐^{r}Ω)^{m} for r = p^{[log_{p}(e/(p−1))]} −e[log_{p}(e/(p−1))]. It is sufficient to check that ⟨w_{i}⟩Λ_{i} ∈ (𝔐^{r}Ω)^{m}. We can take the base w_{i} such that w_{1} = 1 and w_{i} ∈ 𝔐 for i > 1. Since for finite height groups u ∈ Gl_{m}(Ω), Proposition 1.4.1 implies that Λ_{i} ∈ (𝔐^{r}Ω)^{m}. For s ∈ 𝔐 (in particular, s = w_{i}, i > 1) we obtain λ(sxe_{i}) ∈ 𝔐^{r}𝔒_{K}[[X]] directly from the fact that the partial derivatives of λ are integral, thus Λ_{ij} = ∑ w_{r}Δ^{r} for w_{r} ∈ p^{−r}W.

3. This is a special case of part 1 for A = I_{m}.

6.2 M_{F} for finite height groups. Algorithm for classification of formal groups

Proposition 6.2.1. 1. If the logarithm of a formal group law F belongs to R_{X}^{m}, then D_{F} = M_{F} ∩ R^{m}.

2. If F is a finite height formal group, then M_{F} = πM_{F_{π}}, where F_{π} is the formal group law π^{−1}F (πX, πY).

3. If F is a finite height formal group, then D_{F} ∩ R^{m} = πD_{F π}.

Proof. 1. We have D_{F} = ⟨⟨w_{i}⟩Λ_{i}⟩. Hence M_{F} = ⨁ v_{i}Ω, where v_{i} ∈ R^{m}, 1 ≤ i ≤ me. Thus D_{F} ⊂ M_{F} ∩ R^{m}.

On the other hand, p𝔒_{K}^{m} ⊂ D_{f} mod Δ implies that v_{i} form an NW -base of R. Hence ∑ v_{i}r_{i} ∈ R^{m} for r_{i} ∈ Ω if and only if r_{i} ∈ NW. Thus for v ∈ M_{F} ∩ R^{m} we have v = ∑ v_{i}u_{i} for u_{i}ıΩ ∩ NW = W.

2. The map πX is a homomorphism from F_{π} into F. Thus πD_{F_{π}} ⊂ D_{F} and we obtain πM_{F_{π}} ⊂ M_{F}.

We present Λ as vu^{−1}. Since F is a finite height group, u is invertible in M_{m}(Ω). Therefore M_{F} = ⟨v(e_{i})⟩. Since v(e_{i}) ∈ D_{F}, we have exp_{F} (v(xe_{i})) ∈ 𝔒[[x]]^{m}. As v(e_{i}) ∈ R^{m}, we also obtain exp_{F} (v(xe_{i})) ∈ π𝔒[[x]]^{m}. Thus, according to Proposition 5.2.3 for the group F_{π}, π^{−1}v(e_{i}) ∈ D_{F π}. Hence M_{F} ⊂ πM_{F_{π}}.

3. Since D_{F π} ∈ R^{m}, we have πD_{F π} ⊂ D_{F} ∩ R^{m}. On the other hand, D_{F} ∩ R^{m} ⊂ M_{F} ∩ R^{m} and it remains to apply part 2.

One can easily check that λ ∈ R_{X} if the reduction of the formal group law F modulo π is equal to G_{a}^{m}.

Now we prove a result that is crucial for the construction and classification of formal group laws.

Theorem 6.2.2. 1. λ = Λ(X) is a logarithm of a formal group law F over 𝔒_{K} if and only if ⟨π⟩λ_{i} ∈ πD_{F′} for 1 ≤ i ≤ m and (Λu)_{i} ∈ πD_{F′} for some special u and a formal group F′ over 𝔒_{K} satisfying π^{s−1}𝔒_{K}W^{m} ⊂ D_{F′}.

If this happens then F′ is strictly isomorphic to F_{π}.

2. Let s = −[e/(1 − p)], a ∈ 𝔒_{K}, and v_{K}(a) = l ≤ s. Suppose that for some Λ, a special u, and a formal group law F′ over 𝔒_{K} satisfying a^{−1}π^{s}𝔒_{K}W^{m} ⊂ D_{F′} we have ⟨a⟩Λ_{i} ∈ aD_{F′} and (Λu)_{i} ∈ π^{s}D_{F′}. Then Λ is a logarithm of a formal group law F over 𝔒_{K} and F′ is strictly isomorphic to F_{a}.

Proof. 1. According to the previous statement, we have ⟨π⟩λ_{i} ∈ πD_{F_{π}} and (Λu(F))_{i} ∈ πD_{F_{π}}. On the other hand, suppose that the conditions on Λ are fulfilled for some F′ and u. The elements v_{ij} = ⟨π^{j}⟩λ_{i} for 1 ≤ j ≤ e form a W -base of πD_{F_{π}}. Since v_{i1} belong to πD_{F′}, v_{ij} belong to πD_{F′} for any j > 1. Thus F′ is strictly isomorphic to F_{π}.

Now it remains to prove assertion 3 for a = π.

2. According to Remark 5.3.3, we have a^{−1}π^{s}𝔒_{K}W^{m} ⊂ D_{F′}. Next, for any b ∈ 𝔒_{K}^{∗} from Proposition 5.2.3 one immediately obtains the equality D_{F_{b}} = b^{−1}D_{F}. Thus we can assume a = π^{s}.

We prove by induction on r ≥ 0 that the module D_{r} = π^{−r}⟨D, ⟨π^{i}⟩⟩W, i ≥ r corresponds to some formal group.

Suppose that this statement is fulfilled for r = n + 1, n ≥ 0. We have Λu = v ∈ D′^{m}, where u = pI_{m} − ∑ u_{i}Δ^{i}. If n > 0, we have

p⟨π^{n}⟩Λ = ∑⟨π^{p^{i}n}⟩(Λ)u_{i} + ⟨π^{n}⟩v.  (5)

Hence for any n ≥ 0 we have ⟨π^{n}⟩Λu′ ∈ π^{n+1}D_{n+1}^{m} for a special u′.

According to Lemma 5.1.2 we can complete the elements r_{i} = (⟨π^{n}⟩Λu′)_{i}, 1 ≤ i ≤ m to a W -base r_{j}, 1 ≤ j ≤ me, of π^{n+1}D_{n+1}. It follows that π^{n}D_{n} is generated over W by (π^{n}Λ)_{i} and r_{j} for j > m. Hence D_{n} ∈ D, since the other properties are obvious. Since D_{n} contains π^{−n}⟨π^{j}⟩Λ_{i} for j ≥ n, we apply Proposition 5.3.1 for w_{j} = π^{j−1} and obtain that D_{n} is equal to D_{F_{n}} for some F_{n} over 𝔒_{K}.

Now suppose that the conditions on Λ are fulfilled for some F′ and u. As in the proof of the first part, we see that v_{ij} = ⟨π^{j}⟩λ_{i} ∈ π^{s}D_{F′} for

l ≤ j < l + e form a W -base of πD_{F_{π}}. Hence F′ is strictly isomorphic to F_{π^{s}}.

Remark 6.2.3. In particular, the conditions of assertion 3 are fulfilled if ⟨a⟩Λ_{i} ∈ a𝔒_{K}^{m}W and Λu_{i} ∈ a𝔒_{K}^{m}W, since in this case one can take F′ equal to the m-th power of the additive formal group law.

In particular, one can obtain all Honda formal group laws (Lubin–Tate ones, for example) by means of this statement. The canonical representatives of the isogeny classes of formal groups presented in [5] for the case of algebraically closed residue field can also be constructed (see also Theorem H in the introduction).

Now we describe a general algorithm for classifying formal group laws of fixed dimension m over a fixed field K.

First one describes all possible πD_{F_{π}}. To this end one can can use the universal p-typical formal group law to construct the logarithms of the form π^{−1}λ(πX). Since D_{F} depends only on the residues of the coefficients of Λ modulo π^{l}, where l = −[e/1 − p] and only the first few coefficients of ⟨π⟩Λ may not divide π^{l}, the number of different πD_{F_{π}} is finite.

Next one fixes πD_{F_{π}}, and for each u describes Λ satisfying the conditions of the previous proposition. Thus one obtains the description of all p-typical logarithms of formal group laws.

In order to check which of them give strictly isomorphic formal group laws, one can check what residues modulo R^{m}u have elements (v_{i}) ∈ πM_{F_{π}}^{m}, v ≡ pI_{m} mod Δ such that

⟨π⟩(v/u) ∈ πD_{F_{π}}^{m}.  (∗)

After doing this one recovers r(F). Next it is possible to calculate M_{F} (using Proposition 6.2.1 for finite height groups). Hence one obtains the pair r(F), M(F) and is able to apply Theorem 6.2.2.

Note that the condition (*) depends only on the first few coefficients of u and v. Thus if we take u equal to a canonical representative, calculation will terminate in finitely many steps.

6.3 Classification of formal groups for e ≤ p^{2}/2

We illustrate the method described in the previous subsection by classifying one-dimensional formal groups of height > 1 for e ≤ p^{2}/2. This result can easily be extended to multidimensional groups if one defines the height as a certain vector.

6.3.1 Calculation of πD_{F_{π}}

According to Remark 5.3.3, it is sufficient to calculate λ_{π} = π^{−1}λ(πx) = Λ_{π}(Δ)(x) modulo π^{1−[e/(1−p)]}.

The universal p-typical group law is obtained by inverting the power series u_{c} = 1 − ∑ c_{i}Δ^{i}p^{−i} in 𝔄[[Δ]]′, where 𝔄 = ℤ_{p}[c_{i}], i > 0, σ(c_{i}) = c_{i}^{p}, and applying it to x.

We have

u_{c}^{−1} = 1 + \frac{c_{1}}{p} Δ + (\frac{c_{2}}{p} + \frac{c_{1}^{p+1}}{p^{2}} )Δ^{2} + ….  (6)

For the formal group λ_{π} we have π^{p^{i}−1} | c_{i}. Hence for e ≤ \frac{p^{2}}{2} we obtain Λ_{π} ≡ 1 + \frac{c}{p}Δ mod π^{1−[e/(1−p)]}, π^{p−1} | c = c_{1}. Therefore,

πD_{F π} = ⟨π^{i} + \frac{π^{(i−1)p+1}c}{p}⟩, i ≥ 1.

We denote v_{K}(c) by d.

From this one easily obtains that a ∈ πD_{F_{π}} (which is equivalent to aΔ ∈ πD_{F_{π}}) for a ∈ 𝔐 if and only if v_{K}(a) ≥ 1 + \frac{e−d}{p−1}.

6.3.2 Checking (*)

If h > 1 then, according to Proposition 1.3.1, one can assume u ≡ p mod Δ^{2}. For such u and e ≤ \frac{p^{2}}{2} one easily sees

⟨π⟩(v/u) ≡ ⟨π⟩(v/p) mod π^{−[e/(p−1)]}.

Next, the module πD_{F_{π}} was constructed for

Λ = ∑_{i≥0} \frac{c}{π^{p−1}}^{1+···+p^{i−1}}(\frac{Δ}{p})^{i}.

Since h > 1, we have \frac{c}{π^{p−1}} ∈ 𝔐, hence v′ = p + c\frac{Δ}{pπ^{p−1}} ∈ πD_{F_{π}} and v′ satisfies (*) for u = p. Thus we obtain that v satisfies (*) if and only if ⟨π⟩\frac{v−v′}{p} ∈ D_{F}. On the other hand, we have ⟨π⟩\frac{Δ^{2}}{p} D_{F_{π}} ∈ D_{F_{π}}. Therefore, v satisfies (*) for v ∈ πD_{F_{π}} if and only if v ≡ v′ + v_{1}Δ mod deg 2, where v_{1} ∈ 𝔐, v_{K}(v_{1}) ≥ e − p + 1 + \frac{e−d}{p−1}. Furthermore, we obtain

v_{1}Δ + \frac{v_{1}^{p}cΔ^{2}}{pπ^{p−1}} = ⟨v/π⟩v′Δ ∈ πD_{F π}.

Hence the final answer for e > p is d ≥ p,

v = p + c\frac{Δ}{pπ} + v_{1}Δ + \frac{v_{1}^{p}cΔ^{2}}{pπ^{p−1}} + v_{2}Δ^{2}

for any v_{2} ∈ πD_{F π} and v_{1} ∈ 𝔐 satisfying v_{K}(v_{1}) ≥ e − p + 1 + \frac{e−d}{p−1}. One can easily calculate r(F).

For a finite height group one has M_{F} = πD_{F_{π}}Ω. Since πD_{F_{π}} is generated over W [⟨π⟩] by π + \frac{πc}{p}, we obtain that πD_{F_{π}}Ω corresponding to c′ coincides with πD_{F π} for c if and only if v(c−c′) ≥ max(e+1, e+1+ \frac{e−d}{p−1}) (see subsection 6.3.1). For an infinite height group law one can calculates M_{F} directly using D_{F} ⊂ R.

References

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[6] F. Oort, Dieudonne modules of finite local group schemes, Indag. Math. 37, 1975, 103–123