New York Journal of Mathematics

New York J. Math. 12 (2006) 235–247.

Canonical and filling subgroups of formal groups

David J. Schmitz

Abstract. Let F be a one-dimensional full or almost full p-adic formal group. We look for finite subgroups C of F for which the quotient formal group F/C is full. In particular, we investigate the connection between such groups and the congruence-torsion subgroups of F described in Lubin, 1979. In doing so, we prove a conjecture of Jonathan Lubin concerning this relationship when F has height 2.

Contents

1. Introduction and notation 235 2. Filling subgroups 238 3. Canonical subgroups 239 4. Further results concerning canonical subgroups 245 References 246

1. Introduction and notation

Let p be a prime number, and let \overline{ℚ}_{p} be a fixed algebraic closure of the field ℚ_{p} of p-adic numbers. Let v denote the unique extension of the p-adic valuation on ℚ_{p} to a rational valuation on \overline{ℚ}_{p}, normalized so that v(p) = 1. If ℂ_{p} is the completion of \overline{ℚ}_{p} with respect to v, then v extends uniquely, by continuity, to a rational valuation on ℂ_{p}, and we denote this valuation by v as well. Furthermore, we write 𝔒 for the set {x ∈ ℂ_{p} | v(x) ≥ 0} of v-integers, and 𝔐 for the maximal ideal {x ∈ 𝔒 | v(x) > 0} of 𝔒. For any subfield K of ℂ_{p}, we denote by 𝔬_{K} the integer ring of K, i.e., 𝔬_{K} = K ∩ 𝔒.

If F (X, Y) is a p-adic formal group, i.e., a one-dimensional formal group of finite height defined over some 𝔬_{K} with [K : ℚ_{p}] < ∞, then F endows 𝔐 with a group structure, where, for α, β ∈ 𝔐, α +_{F} β = F (α, β). This abelian group, called the points of F, will be denoted F (𝔒). The torsion subgroup of F (𝔒) is

Λ(F) = ⋃_{n∈ℕ} ker [p^{n}]_{F} ,

235^{ISSN 1076-9803/06}

Received February 28, 2006.

Mathematics Subject Classification. 11S31, 14L05.

Key words and phrases. p-adic formal groups, congruence-torsion subgroups.

This work was partially funded by a North Central College Junior Faculty Enhancement Grant.

where [p^{n}]_{F} denotes multiplication-by-p^{n} on the group F (𝔒) [Lu2, §1.0]. If G is another p-adic formal group, then a homomorphism g : F → G (assumed to be defined over 𝔒) induces a group homomorphism (via evaluation) g : F (𝔒) → G(𝔒), and the set Hom(F, G) of all homomorphisms from F to G forms a group. The map c : Hom(F, G) → 𝔒, which assigns to each homomorphism its linear coefficient, is injective [Lu3, 2.1.1]. An easily-proved, yet very useful result is the following:

Proposition 1.1 ([S, 1.2]). If g : F → G is a homomorphism and α ∈ F (𝔒), then v(g(α)) ≥ v(α), with equality if and only if either α = 0 or c(g) ∈ 𝔒^{×}.

A homomorphism g : F → G for which c(g) ∈ 𝔒^{×} is compositionally invertible (over 𝔒) and is called an isomorphism from F to G. We say that g : F → G is an isogeny if g is defined over a complete, discretely-valued subfield of ℂ_{p}; it is enough to check only that c(g) belongs to such a subfield of ℂ_{p} [S, §1]. The set Isog(F, G) of all isogenies from F to G is a group [S, 1.6], and we say that F and G are isogenous if and only if Isog(F, G) ≠ 0. If 0 ≠ g ∈ Isog(F, G), then ker(g) = {α ∈ F (𝔒) | g(α) = 0} is a finite subgroup of F (𝔒) and g maps Λ(F) onto Λ(G) [S, §1].

The absolute endomorphism ring of F, denoted End(F), is the set Hom(F, F). If g ∈ End(F) and c(g) = a, then instead of g we will sometimes write [a]_{F}. The notation [m]_{F} for the multiplication-by-m endomorphism is consistent with this convention since its linear coefficient is m. If F has height h, then [Σ_{F} : ℚ_{p}] divides h, where Σ_{F} (the endomorphism field of F) is the fraction field of c(End(F)) in ℂ_{p} [Lu3, 2.3.2]. In particular, Σ_{F} is algebraic over ℚ_{p}, and so every endomorphism of a p-adic formal group is an isogeny, and c(End(F)) is a ℤ_{p}-order of Σ_{F}. When [Σ_{F} : ℚ_{p}] = h and c(End(F)) = 𝔬_{Σ_{F}} (the maximal order of Σ_{F}), then F is said to be full. When [Σ_{F} : ℚ_{p}] = h but c(End(F)) ≠ 𝔬_{Σ_{F}}, then F is said to be almost full. Finally, we will call quasi-full any p-adic formal group F of height h where End(F) is integrally closed but [Σ_{F} : ℚ_{p}] < h. Lubin and Tate show in [LT] how to construct a full p-adic formal group F over a given p-adic integer ring 𝔬_{K} such that c(End(F)) = 𝔬_{K}. According to [Lu3, 4.3.2], two full p-adic formal groups are isomorphic (via an isogeny) if and only if they have the same endomorphism field. An analogous result holds for almost full p-adic formal groups of height 2.

Theorem 1.2 ([S, 6.4]). Two almost full p-adic formal groups F and G of height 2 are isomorphic via an isogeny if and only if c(End(F)) = c(End(G)).

Whereas there is essentially only one full p-adic formal group having a particular endomorphism field, Lubin showed [Lu2] how to construct an almost full p-adic formal group G for which c(End(G)) is isomorphic to a given nonmaximal ℤ_{p}- order R in a p-adic number field K. Starting with a full p-adic formal group F such that c(End(F)) = 𝔬_{K}, one can associate to R a finite subgroup C of Λ(F) [Lu2, 2.2 and 3.2]. If φ_{C}(T) is the power series

φ_{C}(T) = ∏_{γ∈C} F (T, γ),

then φ_{C} is an isogeny from F to the p-adic formal group F/C given by

(F/C)(X, Y) = φ_{C}(F (φ_{C}^{−1}(X), φ_{C}^{−1}(Y))),

both of which are defined over any p-adic integer ring containing C and all the coefficients of F [Lu2, 1.4]. We refer to F/C as the quotient of F by C and to φ_{C} as the projection homomorphism from F to F/C. If we set G = F/C, then c(End(G)) = R [Lu2, 3.2] and G is almost full because isogenous p-adic formal groups have the same height [Lu3, 2.2.3 and 2.3.1].

Another approach is to ask about the structure of the quotient of a given full or almost full p-adic formal group F by a finite subgroup C of Λ(F). Since F and F/C have the same height and the same endomorphism field [Lu2, 3.0], F/C is either full or almost full as well. A more detailed description of F/C when F has height two is the subject of a couple of recent conjectures of Lubin, one of which is proved in [S]. We first recall that a finite subgroup D of Λ(F) is said to be a deflated subgroup of F if there is no subgroup C of Λ(F) with fewer elements such that F/C is isomorphic to F/D. We note that when F is full, then this is equivalent to D not containing the kernel of any noninvertible F -endomorphism [S, 3.5].

Theorem 1.3 ([S, 6.3]: Lubin’s First Conjecture). Let F be a full p-adic formal group of height 2 and C a deflated (cyclic) subgroup of F of order p^{n}. Then c(End(F/C)) = ℤ_{p} + p^{n} 𝔬_{Σ_{F}}.

In light of Theorem 1.2 and the fact that each ℤ_{p}-order in a quadratic extension K of ℚ_{p} has the form ℤ_{p} +p^{n} 𝔬_{K} for some integer n ≥ 0, Theorem 1.3 shows how an arbitrary almost full p-adic formal group of height 2 can be realized as the quotient of a full p-adic formal group by a cyclic subgroup of a particular order. This is a special case of the fact that any p-adic formal group G is isomorphic to the quotient of a full or quasi-full p-adic formal group by some finite subgroup. Indeed, Lubin [Lu2, 3.2 and 1.6] proved that there exists a p-adic formal group F with End(F) integrally closed and isogenies g : F → G and g̃ : G → F each defined over a p-adic integer ring. If C = ker(g), then the following formal analogue of a standard result from the theory of elliptic curves can be used to show that F/C and G are isomorphic.

Theorem 1.4 ([Lu2, 1.5], [S, 1.3]). Let 𝔬_{K} be the ring of integers of a complete, discretely-valued subfield of ℂ_{p}. If F, G, and H are p-adic formal groups defined over 𝔬_{K}, and if g_{1} : F → G, g_{1} ≠ 0, and g_{2} : F → H are isogenies defined over 𝔬_{K} with ker(g_{1}) ⊆ ker(g_{2}), then there is a unique isogeny j : G → H defined over 𝔬_{K} such that j ∘ g_{1} = g_{2}. If ker(g_{1}) = ker(g_{2}), then j is an isomorphism.

Since C is the kernel of both g : F → G and φ_{C} : F → F/C, Theorem 1.4 implies that G and F/C are isomorphic. The same reasoning shows that G/D is isomorphic to F, where D = ker(g̃). In other words, every p-adic formal group possesses at least one subgroup which yields a full or quasi-full quotient. Lubin’s second conjecture deals with these special subgroups, which we call filling subgroups. He guessed that the quotient of a given height 2 almost full p-adic formal group G by a cyclic subgroup of Λ(G) of a certain order should be full, and that this subgroup should be distinguishable in some (other) way from among all cyclic subgroups of Λ(G) of that order. His precise statement is given below.

Conjecture 1.5 (Lubin’s Second Conjecture). Let G be a height 2 almost full padic formal group with c(End(G)) = ℤ_{p} + p^{n}𝔬, where 𝔬 is the integer ring in a quadratic extension of ℚ_{p}. Then Λ(G) has a cyclic subgroup D of order p^{n}, “canonical” somehow, where G/D is full.

Proving this conjecture and some of its generalizations provides the focus for this paper. We will prove that the subgroup D in the second conjecture is “canonical” in the following sense: it consists of the p^{n} elements in the kernel of [p^{n}]_{G} which have the largest valuations. Furthermore, we will show that this subgroup is the smallest filling subgroup of G. Finally, we will investigate the set of all filling subgroups of an arbitrary almost full p-adic formal group G and explore how these subgroups relate to the canonical subgroups of G.

2. Filling subgroups

We begin by investigating those finite subgroups D of the points of a full or almost full p-adic formal group G which yield full quotients. We will calculate the orders of all such subgroups of Λ(G) and prove that no two of them have the same order.

Definition 2.1. Let G be a p-adic formal group. A finite subgroup D of Λ(G) is a filling subgroup of G if End(G/D) is integrally closed, i.e., if c(End(G/D)) = 𝔬_{Σ_{G}}.

As explained in the introduction, every p-adic formal group has at least one filling subgroup. In fact, if c(End(G)) = ℤ_{p}, (such formal groups of all heights exist according to [Lu3, 5.2.1]), then every finite subgroup D of Λ(G) is filling; indeed, ℤ_{p} ⊆ c(End(G/D)) ⊆ 𝔬_{Σ_{G}} = ℤ_{p}. On the other hand, only full or almost full p-adic formal groups have filling subgroups that yield full quotients because isogenous padic formal groups have the same endomorphism fields and equal heights. Lubin’s second conjecture is concerned with a specific filling subgroup of an almost full p-adic formal group of height 2, but we aim to classify all filling subgroups of any full or almost full p-adic formal group. The next proposition is a first step in this direction.

Proposition 2.2. Let D be any finite subgroup of the points of a p-adic formal group G. Then G/D ≅ G if and only if D is the kernel of some nonzero G-endo-morphism. In particular, the filling subgroups of a full p-adic formal group are the kernels of its nonzero endomorphisms.

Proof. Assume first that D = ker(g), where 0 ≠ g ∈ End(G). As g : G → G and φ_{D} : G → G/D have the same kernel, it follows from Theorem 1.4 that G is isomorphic to G/D. Conversely, if u : G/D → G is an isomorphism, then u ∘ φ_{D} is a nonzero endomorphism of G with kernel D. The last statement in the proposition follows from the fact that full p-adic formal groups with the same endomorphism field are isomorphic [Lu3, 4.3.2]. □

The endomorphism kernels of a full or quasi-full formal group F are easy to describe because End(F) is a discrete valuation ring.

Proposition 2.3 ([S, §2]). Let F be a full or quasi-full p-adic formal group and let π be a uniformizer of c(End(F)). If g is any nonzero F -endomorphism, then ker(g) = ker [π^{e·v(c(g))}]_{F} where e is the ramification index of Σ_{F} /ℚ_{p}. In particular, for each integer m ≥ 0, ker [p^{m}]_{F} = ker [π^{me}]_{F}.

Suppose F is a full or quasi-full p-adic formal group and that π is a uniformizer of c(End(F)). If X is a finite subset of Λ(F), we will write ℓ(X) = k where k is the smallest nonnegative integer such that X ⊆ ker [π^{k}]_{F}. Thus, ℓ measures the depth of a subset of Λ(F) in the filtration of subgroups {ker [π^{m}]_{F} }_{m≥0}. According to [S, 2.2], ker [π^{m}]_{F} has p^{m(h/e)} elements, where h is the height of F and e is the ramification index of Σ_{F} /ℚ_{p}. Therefore, in light of Proposition 2.2, if F is full, there is for each nonnegative integer m exactly one filling subgroup of F of order p^{mf} (where f is the residue field degree of Σ_{F} /ℚ_{p}), and F has no other filling subgroups. We next prove similar results for an arbitrary p-adic formal group, namely, that the orders of some of its filling subgroups form an arithmetic sequence with common difference equal to h/e, and that no two of these filling subgroups have the same order.

Theorem 2.4. Let D be a finite subgroup of the points of a full or quasi-full padic formal group F with ℓ(D) = n. For each m ≥ 0, let C_{m} = φ_{D}(ker [π^{m}]_{F} ), where π is a uniformizer of c(End(F)). Then {C_{m}}_{m≥n} is the set of distinct filling subgroups of F/D yielding a quotient isomorphic to F. Moreover, if |D| = p^{k}, then for each m ≥ n, |C_{m}| = p^{m(h/e)−k}, where h is the height of F and e is the ramification index of Σ_{F} /ℚ_{p}.

Proof. Let G = F/D. If u : G/C → F is an isomorphism, then u ∘ φ_{C} ∘ φ_{D} is an endomorphism g of F with D ⊆ ker(g) = φ_{D}^{−1}(C). According to Proposition 2.3, ker(g) = ker [π^{m}]_{F} for some integer m ≥ ℓ(D). Using the surjectivity of φ_{D}, it follows that C = φ_{D}(ker [π^{m}]_{F} ) = C_{m}. Conversely, if m ≥ ℓ(D), then using Theorem 1.4 and Proposition 2.2, we have

G/C_{m} ≅ F/(D +_{F} ker [π^{m}]_{F}) = F/ ker [π^{m}]_{F} ≅ F.

To compute the order of C_{m} (m ≥ n), we note that D = ker(φ_{D}) ⊆ ker [π^{m}]_{F} and then refer to the discussion preceding the statement of the theorem. These orders show that the C_{m} are distinct. □

Corollary 2.5. Let D be a finite subgroup of the points of a full p-adic formal group F and let π be a uniformizer of c(End(F)). Then {φ_{D}(ker [π^{m}]_{F} )}_{m≥ℓ(D)} is the set of all filling subgroups of F/D.

Our second corollary follows from Theorem 2.4, plus the fact that {0} is a filling subgroup of a full p-adic formal group.

Corollary 2.6. A full or almost full p-adic formal group has a unique deflated filling subgroup.

By contrast, a full or almost full p-adic formal group can have several nonfilling deflated subgroups which yield isomorphic quotients. For example, let F be a full p-adic formal group of height 2, and assume that Σ_{F} /ℚ_{p} is unramified. Since | ker [p]_{F} | = p^{2}, Λ(F) has more than one (cyclic) subgroup of order p. If C and D are two such subgroups, then c(End(F/C)) = c(End(F/D)) = ℤ_{p} + p 𝔬_{K} by Theorem 1.3, whence F/C ≅ F/D by Theorem 1.2. Furthermore, C and D are both deflated subgroups of F since they are not filling and since {0} is the only subgroup of Λ(F) with fewer elements.

3. Canonical subgroups

Lubin’s second conjecture suggests that filling subgroups of an almost full padic formal group can be distinguished somehow from other subgroups of the same order. Unlike with full p-adic formal groups, however, a filling subgroup of an almost full p-adic formal group G cannot be the kernel of an endomorphism since for any 0 ≠ g ∈ End(G), G/ ker(g) is isomorphic to G. We will show eventually that filling subgroups of all full and almost full p-adic formal groups are “canonical” in a sense first described in [Lu1]. We begin here with the basic definitions and some preliminary results.

Definition 3.1. Let G be a p-adic formal group. A subgroup S of Λ(G) is called a congruence-torsion subgroup of G if there exists a positive real number λ such that S = Λ(G)_{λ} = {α ∈ Λ(G) | v(α) ≥ λ}. We say S is a canonical subgroup of G if S = (ker [p^{n}]_{G})_{λ} for some λ ∈ ℝ^{+}, where |S| = p^{n}.

Remarks 3.2.

(i) If λ ∈ ℝ^{+} and C is a subgroup of G(𝔒), then C_{λ} = {α ∈ C | v(α) ≥ λ} is a subgroup of C. This follows from the fact that for any α, β ∈ G(𝔒), v(α +_{G} β) ≥ min{v(α), v(β)}, with equality if v(α) ≠ v(β) [Lu1, §2].

(ii) Every congruence-torsion subgroup of G is finite [Si, IV.6.1]. (iii) If C = ⟨γ⟩ is a cyclic congruence-torsion subgroup of G of order p^{n} and if m

is any integer, then [m]_{G}(γ) generates C if and only if m is prime to p, i.e., if and only if v(m) = 0. Then, according to Proposition 1.1, the generators of C are those elements having the smallest valuation. It follows inductively that every subgroup of C is also a congruence-torsion subgroup of G.

In the next proposition, we show that the congruence-torsion subgroups of G and the canonical subgroups of G are actually the same. This generalizes a result in [Lu1, §4], where C is assumed to be cyclic.

Proposition 3.3. Let G be a p-adic formal group, and let C be a subgroup of Λ(G) of order p^{n}. Then C is a congruence-torsion subgroup of G if and only if C is a canonical subgroup of G.

Proof. Assume first that C is a congruence-torsion subgroup of G, say C = Λ(G)_{λ}. Because |C| = p^{n}, we know that C ⊆ ker [p^{n}]_{G}. Clearly, C = (ker [p^{n}]_{G})_{λ}.

Conversely, assume C is a canonical subgroup of G. Set λ = min {v(α) | α ∈ C}, so that C = (ker [p^{n}]_{G})_{λ}. If C is not a congruence-torsion subgroup of G, then there exists some β ∈ Λ(G) −C such that v(β) ≥ λ. Since β ∉ ker [p^{n}]_{G}, β has order p^{m} for some m > n. Then γ = [p^{m−n}]_{G}(β) has order p^{n} and therefore belongs to ker [p^{n}]_{G}. From Proposition 1.1 it follows that v(γ) > v(β) ≥ λ, and hence γ ∈ C. But then γ is a generator of C since |C| = p^{n}. Because the generators are the elements of smallest valuation in a cyclic subgroup of Λ(G) (Remark 3.2(iii)), we see that v(γ) = λ, a contradiction. □

In [Lu1] and [Kl], the term “canonical subgroup” is more restrictive than our definition allows. There, a canonical subgroup of a p-adic formal group G is a congruence-torsion subgroup of G which is the kernel of a homomorphism defined on G that reduces to the frobenius homomorphism T ↦ T^{p} in characteristic p. For a one-dimensional p-adic formal group G, the kernel of any lifting of T ↦ T^{p} has p elements; thus, only canonical subgroups of order p are considered in [Lu1]. Here, there is no assumed connection between canonical subgroups and lifts of frobenius or any other homomorphism in characteristic p.

The canonical subgroups of a full p-adic formal group F are easily determined using an observation made in the proof of [S, 2.7]. There it is shown that if π is a uniformizer of c(End(F)), then the elements of ker [π^{m}]_{F} − ker [π^{m−1}]_{F} have the same valuation, which is smaller than the valuation of any of the elements of ker [π^{m−1}]_{F}. We thus obtain the following.

Proposition 3.4. Let F be a full p-adic formal group and let π be a uniformizer of c(End(F)). Then for each m ≥ 0, ker [π^{m}]_{F} is the canonical subgroup of F of order p^{mf}, where f is the residue field degree of Σ_{F} /ℚ_{p}. These account for all of the canonical subgroups of F.

In light of Proposition 2.3, we see that the canonical subgroups of a full padic formal group are precisely the kernels of its nonzero endomorphisms. This fact, together with Proposition 2.2, provides the first clue of a connection between canonical subgroups and filling subgroups.

Corollary 3.5. A subgroup of a full p-adic formal group is filling if and only if it is canonical.

We can use this characterization of the canonical subgroups of a full p-adic formal group in order to explore the connection between canonical and filling subgroups of almost full p-adic formal groups. In particular, we will determine the extent to which Corollary 3.5 is true for such formal groups and also prove Lubin’s second conjecture concerning almost full p-adic formal groups of height 2.

If G is an arbitrary almost full p-adic formal group, then G is isomorphic to the quotient of a full p-adic formal group F by a finite subgroup D of Λ(F). Since isomorphisms map canonical subgroups to canonical subgroups (Proposition 1.1) and filling subgroups to filling subgroups (Theorem 1.4), we may assume without loss of generality that G = F/D. We will use the homomorphism φ_{D} : F → G to study the valuations of the elements of Λ(G). (We note here that the following discussion applies to any p-adic formal group F.) According to the explicit definition of φ_{D}, we see that for any α ∈ Λ(F),

v(φ_{D}(α)) = ∑_{γ∈D} v(α +_{F} γ) = ∑_{β ∈ α +_{F} D} v(β) . (1)

In order to more easily compare the valuations of the images under φ_{D} of different elements of Λ(F), we introduce the following notation: if α ∈ Λ(F), then α̃ will denote an element in the coset α +_{F} D having maximum valuation. If γ ∈ D and v(γ) < v(α̃), then v(α̃ +_{F} γ) = v(γ); on the other hand, if v(γ) ≥ v(α̃), then v(α̃ +_{F} γ) = v(α̃) according to how α̃ is defined. These observations follow immediately from Remark 3.2(i). As φ_{D}(α) = φ_{D}(α̃), we infer from Equation (1) that

v(φ_{D}(α)) = ∑_{γ∈D} v(α̃ +_{F} γ) = ∑_{\begin{matrix}γ∈D \\ v(γ)<v(α̃) \end{matrix}} v(γ) + ∑_{\begin{matrix}γ∈D \\ v(γ)≥v(α̃) \end{matrix}} v(α̃) . (2)

If α̃ = 0, then both sides of (2) are infinite; otherwise, both sides are finite since φ_{D}(α) = 0 if and only if α̃ = 0.

Lemma 3.6. Let D be a finite subgroup of the points of a p-adic formal group F. If α, β ∈ Λ(F), then v(φ_{D}(β)) ≥ v(φ_{D}(α)) if and only if v(β̃) ≥ v(α̃). In particular, v(φ_{D}(α)) = v(φ_{D}(β)) if and only if v(α̃) = v(β̃).

Proof. We may assume without loss of generality that β̃ ≠ 0. If v(α̃) = v(β̃), then it is clear from Equation (2) that v(φ_{D}(α̃)) = v(φ_{D}(β̃)). If v(β̃) > v(α̃), then

v(φ_{D}(β̃)) = ∑_{\begin{matrix}γ∈D \\ v(γ)<v(β̃) \end{matrix}} v(γ) + ∑_{\begin{matrix}γ∈D \\ v(γ)≥v(β̃) \end{matrix}} v(β̃)

= ∑_{\begin{matrix}γ∈D \\ v(γ)<v(α̃) \end{matrix}} v(γ) + ∑_{\begin{matrix}γ∈D \\ v(α̃)≤v(γ)<v(β̃) \end{matrix}} v(γ) + ∑_{\begin{matrix}γ∈D \\ v(γ)≥v(β̃) \end{matrix}} v(β̃)

≥ ∑_{\begin{matrix}γ∈D \\ v(γ)<v(α̃) \end{matrix}} v(γ) + ∑_{\begin{matrix}γ∈D \\ v(α̃)≤v(γ)<v(β̃) \end{matrix}} v(α̃) + ∑_{\begin{matrix}γ∈D \\ v(γ)≥v(β̃) \end{matrix}} v(β̃)

> ∑_{\begin{matrix}γ∈D \\ v(γ)<v(α̃) \end{matrix}} v(γ) + ∑_{\begin{matrix}γ∈D \\ v(γ)≥v(α̃) \end{matrix}} v(α̃)

= v(φ_{D}(α̃)).

We note that the second inequality is strict because D always contains an element (e.g., 0) of valuation larger than v(β̃). The lemma now follows easily. □

We are now in a position to identify the canonical subgroups of any almost full p-adic formal group G. We do so in the next theorem, which is stated in more generality than we need now so as also to be of use in the next section.

Theorem 3.7. Let F be a full or quasi-full p-adic formal group, and let π be a uniformizer of c(End(F)). Assume that {ker [π^{m}]_{F} }_{m≥0} is the set of canonical subgroups of F. If D is any finite subgroup of Λ(F), then {φ_{D}(ker [π^{m}]_{F} )}_{m≥0} is the set of canonical subgroups of F/D.

Proof. Let G = F/D and C_{m} = φ_{D}(ker [π^{m}]_{F} ) = φ_{D}(D +_{F} ker [π^{m}]_{F}). We first show that for every m ≥ 1, C_{m} is a canonical subgroup of G (the case m = 0 is trivial). For any γ ∈ C_{m} and any δ ∈ Λ(F/D) −C_{m}, there exists α ∈ ker [π^{m}]_{F} and β ∈ Λ(F) −(D +_{F} ker [π^{m}]_{F}) such that γ = φ_{D}(α) and δ = φ_{D}(β). Since α̃ ∈ ker [π^{m}]_{F} (because ker [π^{m}]_{F} is a congruence-torsion subgroup of F) and since β̃ ∉ ker [π^{m}]_{F}, it follows that v(α̃) > v(β̃). Therefore by Lemma 3.6, v(γ) > v(δ), and so C_{m} is a congruence-torsion (i.e., canonical) subgroup of G.

We now show G has no other canonical subgroups. Suppose C_{m} ≠ C_{m+1}. For any γ ∈ C_{m+1}−C_{m}, take α ∈ ker [π^{m+1}]_{F} −(D +_{F} ker [π^{m}]_{F} ) such that γ = φ_{D}(α). Using reasoning similar to that above, we see that α̃ ∈ ker [π^{m+1}]_{F} − ker [π^{m}]_{F}. This, in light of Lemma 3.6 plus the fact that {ker [π^{m}]_{F} }_{m≥0} is the set of canonical subgroups of F, implies that all elements of C_{m+1} − C_{m} have the same valuation, and so there can be no canonical subgroups between C_{m} and C_{m+1}. □

The following corollary is immediate.

Corollary 3.8. If G is an almost full p-adic formal group, then every filling subgroup of G is canonical. Furthermore, every canonical subgroup of G containing the deflated filling subgroup of G is itself filling.

Remark 3.9. In the proof of Theorem 3.7, it is possible for C_{m} and C_{m+1} to be equal, even if D is deflated. (However, according to Corollaries 2.5 and 3.8, this cannot happen if C_{m+1} is filling.) For example, when F is full, Σ_{F} /ℚ_{p} is totally ramified of degree at least 2, and D is the (deflated) cyclic subgroup of Λ(F) of order p generated by an element δ_{0} ∈ ker [π^{2}]_{F} − ker [π]_{F}, then, by considering orders, we see that D +_{F} ker [π]_{F} = ker [π^{2}]_{F} since D ∩ ker [π]_{F} = {0}. Thus φ_{D}(ker [π]_{F} ) = φ_{D}(ker [π^{2}]_{F} ). On the other hand, when F is full, Σ_{F} /ℚ_{p} is unramified of any degree greater than 1, and D is any deflated subgroup of F, then C_{m} is always a proper subset of C_{m+1}. Indeed, C_{m} = C_{m+1} if and only if ker [p^{m+1}]_{F} ⊆ D +_{F} ker [p^{m}]_{F}, in which case we would have ker [p]_{F} = [p^{m}]_{F} (ker [p^{m+1}]_{F} ) ⊆ [p^{m}]_{F} (D) ⊆ D, contradicting the assumption that D is deflated.

We now prove Lubin’s Second Conjecture, which we restate as:

Theorem 3.10. Let G be an almost full p-adic formal group with height 2 and c(End(G)) = ℤ_{p} + p^{n}𝔬, where 𝔬 is the integer ring in a quadratic extension of ℚ_{p}. Then the deflated filling subgroup of G is cyclic of order p^{n} and canonical.

Proof. Using Theorems 1.2 and 1.3, we may assume that G = F/D where F is a full p-adic formal group with 𝔬_{Σ_{F}} = 𝔬 and where D is a deflated cyclic subgroup of F of order p^{n}. The deflated filling subgroup C of F/D is cyclic [S, 6.1] and canonical (Corollary 3.8), and so we have only to determine its order. If Σ_{F} /ℚ_{p} is unramified, then ℓ(D) = n. If Σ_{F} /ℚ_{p} is totally ramified and π is a uniformizer of c(End(F)), then ℓ(D) = 2n, for if ℓ(D) = 2n − 1 then ker [π]_{F} = [p^{n−1}]_{F} (D) ⊂ D, which contradicts our assumption that D is deflated. In either case, Corollary 2.5 says that C = φ_{D} (ker [p^{n}]_{F}). Therefore, |C| = | ker [p^{n}]_{F} |/|D| = p^{2n}/p^{n} = p^{n}. □

Corollary 2.5 and Theorem 3.7 together show that, unlike full p-adic formal groups, almost full p-adic formal groups may have nonfilling canonical subgroups, namely (using the notation of Corollary 2.5) the subgroups φ_{D}(ker [π^{m}]_{F} ) where m < ℓ(D). However, as long as D is deflated, all of these subgroups (except for {0}) are at least “almost filling” as described in the next proposition.

Proposition 3.11. Let F be a full p-adic formal group and D a deflated subgroup of F with ℓ(D) = n > 1. If G = F/D and C_{m} = φ_{D}(ker [π^{m}]_{F} ) where π is a uniformizer of c(End(F)), then for every 1 ≤ m < n,

c(End(G)) ⫋ c(End(G/C_{m})) ⫋ c(End(F)) .

Proof. The second strict containment follows from the fact that F is full and C_{m} is a nonfilling subgroup of G. To derive the first containment, we first note that the kernel of φ_{C_{m}} ∘ φ_{D} is D +_{F} ker [π^{m}]_{F}, and so G/C_{m} ≅ F/(D +_{F} ker [π^{m}]_{F}) by Theorem 1.4. Then, according to [S, 4.1],

c(End(G)) = {ζ ∈ 𝔬_{Σ_{F}} | [ζ]_{F} (D) ⊆ D}

and

c(End(G/C_{m})) = {ζ ∈ 𝔬_{Σ_{F}} | [ζ]_{F} (D +_{F} ker [π^{m}]_{F} ) ⊆ D +_{F} ker [π^{m}]_{F} }.

It is now clear that c(End(G)) ⊆ c(End(G/C_{m})). To show the containment is strict, we use the fact that D is deflated to find an α ∈ ker [π]_{F} −D. If β ∈ D is an element of minimum valuation, then v(β) < v(α) since ℓ(D) > 1, and so by [S, 2.5] there is some ζ ∈ c(End(F)) such that [ζ]_{F} (β) = α ∉ D. Thus ζ ∉ c(End(G)). On the other hand, the fact that ℓ(D) = n > 1 implies that β ∈ ker [π^{n}]_{F} − ker [π^{n−1}]_{F}, whence ζ = ϵπ^{n−1} where ϵ is a unit in 𝔬_{Σ_{F}} (since α is a nonzero element of ker [π]_{F}). Therefore, for any 1 ≤ m < n,

[ζ]_{F} (D +_{F} ker [π^{m}]_{F} ) ⊆ [ζ]_{F} (ker [π^{n}]_{F} ) ⊆ [ϵ]_{F} (ker [π]_{F} ) ⊆ D +_{F} ker [π^{m}]_{F} .

We conclude that ζ ∈ c(End(G/C_{m})) − c(End(G)). □

To illustrate this result, let G = F/D where F is a full height 2 p-adic formal group and D is a deflated (cyclic) subgroup of F of order p^{n}. Then c(End(F/D)) = ℤ_{p} + p^{n}𝔬, where 𝔬 = 𝔬_{Σ_{F}}. The deflated filling subgroup of G, C = φ_{D} (ker [p^{n}]_{F}), is cyclic of order p^{n} (see the proof of Theorem 3.10), and therefore, in light of Remark 3.2(iii), G has a nonfilling canonical (cyclic) subgroup C_{m} of order p^{m} for every 0 ≤ m < n. In particular, C_{m} = [p^{n−m}]_{G}(C) = φ_{D} (ker [p^{m}]_{F}). To determine the absolute endomorphism ring of G/C_{m} ≅ F/(D +_{F} ker [p^{m}]_{F}), we first note that D +_{F} ker [p^{m}]_{F} = [p^{m}]_{F}^{−1}([p^{m}]_{F} (D)), and so G/C_{m} ≅ F/ [p^{m}]_{F} (D) [S, 3.1]. But [p^{m}]_{F} (D) is a deflated (cyclic) subgroup of F of order p^{n−m}, whence c(End(G/C_{m})) = ℤ_{p} + p^{n−m}𝔬 by Theorem 1.3.

Our results provide an easy algebraic method for counting the number of elements in the congruence-torsion subgroups of any almost full p-adic formal group. Furthermore, the fundamental relationship in Corollary 3.8 is useful in establishing results such as the following.

Proposition 3.12. Let C be a canonical subgroup of a full or almost full p-adic formal group G, and let 0 ≠ g ∈ End(G). Then

(i) g(C) is a canonical subgroup of G; (ii) g^{−1}(C) is canonical subgroup of G if C is filling.

Proof. (i): As before, we may assume that G = F/D where F is a full p-adic formal group and D is a finite subgroup of Λ(F). Then C = φ_{D}(ker [π^{m}]_{F} ) for some m ≥ 0, where π is a uniformizer of c(End(F)) (Theorem 3.7). If j = [c(g)]_{F} ∈ End(F), then g ∘ φ_{D} = φ_{D} ∘j by the injectivity of c. Thus, g(C) = φ_{D}(j(ker [π^{m}]_{F})). Write j = [ϵπ^{k}]_{F} where ϵ is a unit in 𝔬_{Σ_{F}} and k ≥ 0. If k ≥ m, then g(C) = {0}, while if k < m, then g(C) = φ_{D}(ker [π^{m−k}]_{F} ). In either case, g(C) is a canonical subgroup of G by Theorem 3.7.

(ii): This follows from Corollary 3.8 since G/g^{−1}(C) ≅ G/C by [S, 3.1]. □

Remark 3.13. We note that the preimage of a canonical subgroup under an endomorphism may be noncanonical. For example, if G is a height 2 almost full p-adic formal group with Σ_{G} /ℚ_{p} unramified and End(G) ≅ ℤ_{p} + p^{2} 𝔬_{Σ_{G}}, then G has a nonfilling canonical subgroup C of order p, but no canonical subgroup of order p^{3}. Therefore [p]_{G}^{−1}(C), which has p^{3} elements, cannot be canonical.

4. Further results concerning canonical subgroups

If G is a p-adic formal group which is neither full nor almost full, it is natural to ask whether there is some relationship between the filling subgroups and the canonical subgroups of G. Unfortunately, the situation is not as simple as that described in the previous section. We can show without much difficulty that it is not true in general that all filling subgroups are canonical. For example, if G has height h > 1 and c(End(G)) = ℤ_{p}, then we’ve already seen that every finite subgroup of Λ(G) is filling. However, not every finite subgroup of Λ(G) is canonical since | ker [p]_{G} | > p and there can be at most one canonical subgroup of G of any order.

On the other hand, the canonical subgroups of some p-adic formal groups which are neither full nor almost full do behave in part like the canonical subgroups of full or almost full p-adic formal groups. For example, suppose F is a quasifull p-adic formal group of height h > 1 defined over 𝔬_{K}, where K is a finite unramified extension of Σ_{F}; this happens, in particular, when F is a nonfull padic formal group defined over an unramified extension of ℚ_{p} [Lu2, 3.3]. If π is a uniformizer of c(End(F)), then {ker [π^{m}]_{F} }_{m≥0} is the set of kernels of all nonzero F -endomorphisms, and [π]_{F} is defined over 𝔬_{K}, a complete discrete valuation ring in which π is a uniformizer. Therefore, P (T), the distinguished polynomial factor of [π]_{F} (T)/T coming from the Weierstrass Preparation Theorem, is irreducible over 𝔬_{K} by Eisenstein’s criterion. This implies that the nonzero elements of ker [π]_{F}, i.e., the roots of P (T), are conjugate over K, whence they have the same valuation λ. For any α ∈ ker [π^{m+1}]_{F} − ker [π^{m}]_{F} (m ≥ 1), v(α) < λ by Proposition 1.1, and so

v([π]_{F} (α)) = v(α · P (α)) = ∑_{ξ∈ ker [π]_{F}} v(α − ξ) = p^{s}v(α),

where p^{s} = | ker [π]_{F} |. By induction, we now see that v(α) = p^{−ms}λ. Therefore, the kernels of the nonzero F -endomorphisms are the canonical subgroups of F. Since the kernels of the nonzero endomorphisms of a quasi-full p-adic formal group are filling by Proposition 2.2, we see that every canonical subgroup of F is filling. Furthermore, if D is any deflated subgroup of F, then according to Theorem 3.7, the set of canonical subgroups of F/D is {φ_{D}(ker [π^{m}]_{F} )}_{m≥0}. When m ≥ ℓ(D), Theorem 2.4 shows that φ_{D}(ker [π^{m}]_{F} ) is also filling. Thus, every canonical subgroup of F/D of large enough order is filling.

We can establish other results, weaker than their analogues in the previous section, describing properties of canonical subgroups of arbitrary p-adic formal groups.

Proposition 4.1. If G is any p-adic formal group and C is any canonical subgroup of G, then c(End(G)) ⊆ c(End(G/C)).

Proof. According to [S, 4.2],

c(End(G/C)) = {ζ ∈ 𝔬_{Σ_{G}} | [p^{n}ζ]_{G}([p^{n}]_{G}^{−1}(C)) ⊆ C},

where n is an integer large enough so that p^{n} 𝔬_{Σ_{G}} ⊆ c(End(G)). Therefore, for any ζ ∈ c(End(G)), we see that [p^{n}ζ]_{G}([p^{n}]_{G}^{−1}(C)) = [ζ]_{G}([p^{n}]_{G}([p^{n}]_{G}^{−1}(C))) = [ζ]_{G}(C). But as C is a canonical subgroup of G, [ζ]_{G}(C) ⊆ C by Proposition 1.1. Thus ζ ∈ c(End(G/C)). □

Proposition 4.2. Let C be a canonical subgroup of a p-adic formal group G. If C contains the kernel of some 0 ≠ g ∈ End(G), then D = g^{−1}(C) is also a canonical subgroup of G.

Proof. Let γ_{0} be an element having the smallest valuation among the elements of C. Because C is a canonical subgroup of G, we know that C = Λ(G)_{v(γ_{0})}. Since v(g(x)) ≥ v(x) for all x ∈ Λ(G), it follows that C ⊆ D, with equality if and only if g ∈ Aut(G). Therefore, we may assume that g ∉ Aut(G), i.e., that ker(g) ≠ {0}. If κ = min{v(ξ) | ξ ∈ ker(g)}, then v(γ_{0}) ≤ κ because ker(g) ⊆ C. If we use the Weierstrass Preparation Theorem to factor

g(T) = U(T) ∏_{ξ∈ ker(g)}(T − ξ)

where the constant term of U(T) ∈ 𝔒[[T]] is a unit, then we see that for any x ∈ Λ(G) with v(x) < κ,

v(g(x)) = v(U(x)) + ∑_{ξ∈ ker(g)} v(x − ξ) = p^{m}v(x)

where p^{m} = | ker(g)|.

Now choose any α_{0} ∈ Λ(G) with g(α_{0}) = γ_{0}. Since g is not an automorphism of G, v(α_{0}) < v(g(α_{0})) = v(γ_{0}) ≤ κ, whence v(α_{0}) = p^{−m}v(γ_{0}). Furthermore, α_{0} has the minimum valuation among the elements of D. Indeed, if there were some δ ∈ D such that v(δ) < v(α_{0}) < κ, then v(g(δ)) = p^{m}v(δ) < p^{m}v(α_{0}) = v(γ_{0}), which would contradict our choice of γ_{0}. Therefore, we need to show that D = Λ(G)_{v(α_{0})}. If not, then there would exist some β ∈ Λ(G) −D such that v(β) ≥ v(α_{0}). We note that β ∉ C (because C ⊂ D), and so v(β) < v(γ_{0}) ≤ κ. Hence, v(g(β)) = p^{m}v(β) ≥ p^{m}v(α_{0}) = v(γ_{0}), which implies that g(β) ∈ C. This, contradicts the fact that β ∉ D. Therefore, D is a canonical subgroup of G. □

We are less likely to find a simple characterization of the filling subgroups of p-adic formal groups which are neither full nor almost full because they are more “plentiful” than are the filling subgroups of full and almost full p-adic formal groups. This stems primarily from the fact that [Lu3, 4.3.2] does not hold for quasi-full padic formal groups. In other words, if F is a quasi-full p-adic formal group, then there may be nonisomorphic quasi-full p-adic formal groups G such that Σ_{F} = Σ_{G}. As the evidence presented earlier in this section could suggest, perhaps those filling subgroups which are canonical yield quasi-full quotients which possess some special property, such as having their coefficients (or those of an isomorphic formal group) belonging to “nice” (e.g., unramified) extension fields of their endomorphism fields.

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Department of Mathematics, North Central College, 30 N Brainard St, Naperville, IL 60540 djschmitz@noctrl.edu

This paper is available via http://nyjm.albany.edu/j/2006/12-14.html.