New York Journal of Mathematics

New York J. Math. 15 (2009) 435–450.

Formal group law homomorphisms over O_{ℂ_{p}}

Kevin Keating

Abstract. Let K be a finite extension of the p-adic field ℚ_{p} and let F (X, Y) and G(X, Y) be one-dimensional formal group laws over the ring of integers O_{K} of K. Let ϕ(X) be a homomorphism from F to G which is defined over the ring of integers O_{ℂ_{p}} of the completion ℂ_{p} of ℚ_{p}^{alg}. In this paper we prove that if ker(ϕ) is finite then there is a discretely valued subfield L ⊂ ℂ_{p} such that ϕ(X) is defined over O_{L}.

Contents

1. Power series over O_{ℂ_{p}} 436 2. Formal group laws 440 3. Homomorphisms of formal group laws 444 References 449

Let ℚ_{p} be the field of p-adic numbers, let ℚ_{p}^{alg} be an algebraic closure of ℚ_{p}, and let ℂ_{p} be the completion of ℚ_{p}^{alg} with respect to the p-adic valuation. Let K be a closed discretely valued subfield of ℂ_{p} and let F (X, Y), G(X, Y) be one-dimensional formal group laws over the ring of integers O_{K} of K. Let L/K be a subextension of ℂ_{p}/K. A homomorphism from F to G over O_{L} is defined to be a power series ϕ(X) ∈ O_{L}[[X]] such that ϕ(0) = 0 and ϕ(F (X, Y)) = G(ϕ(X), ϕ(Y)). Let K^{alg} be the algebraic closure of K in ℂ_{p}. It is well-known (see [3, p. 106]) that every homomorphism from F to G with coefficients in O_{K^{alg}} is defined over O_{M} for some finite subextension M/K of K^{alg}/K. In [10] Schmitz raised the question of whether every homomorphism from F to G with coefficients in O_{ℂ_{p}} is defined over the ring of integers O_{L} of some discretely valued subfield L of ℂ_{p}. In this paper we show that this is true as long as F and G both have height h < ∞. If we allow

435^{ISSN 1076-9803/09}

Received March 14, 2009.

Mathematics Subject Classification. 11S31.

Key words and phrases. p-adic formal group laws, homomorphisms, Newton polygons.

F and G to have different (finite) heights, there can exist homomorphisms from F to G with coefficients in O_{ℂ_{p}} which are not defined over O_{M} for any finite extension M of K (see Example 3.8). Any such homomorphism must have infinite kernel.

We now describe the contents of the various sections. In §1 we recall some basic facts about power series in one variable over O_{ℂ_{p}}. In §2 we extend some results of Lubin [6] on formal group laws over O_{K} to formal group laws over O_{ℂ_{p}}. In §3 we prove our main result, as described in the preceding paragraph.

I would like to thank the referee for suggestions which led to several major improvements in the exposition. I also thank David Schmitz for his careful reading of an earlier version of the paper, and especially for pointing out a fatal flaw in my first attempt to prove Theorem 3.2.

1. Power series over O_{ℂ_{p}}

In this section we review the theory of power series over O_{ℂ_{p}}, in particular the Weierstrass preparation theorem and the theory of Newton polygons. The material in this section is certainly well-known (see for instance [5]), but we include it in order to make the paper more self-contained.

Let p be a prime, let ℚ_{p} be the field of p-adic numbers, and let ℚ_{p}^{alg} be an algebraic closure of ℚ_{p}. The p-adic valuation v_{p} on ℚ_{p} extends uniquely to a valuation on ℚ_{p}^{alg}, which we also denote by v_{p}. Let ℂ_{p} denote the completion of ℚ_{p}^{alg} with respect to v_{p}. Then ℂ_{p} is an algebraically closed field [5, Th. 13, p. 72]. For any subfield K of ℂ_{p} we define the ring of integers of K to be O_{K} = {x ∈ K : v_{p}(x) ≥ 0}. Then O_{K} is a local ring with maximal ideal M_{K} = {x ∈ O_{K} : v_{p}(x) > 0}. We say that K is discretely valued if v_{p}(K^{×}) = \frac{1}{e} · ℤ for some natural number e. The algebraic closure K^{alg} of K in ℂ_{p} is dense in ℂ_{p}, and if K is closed in ℂ_{p} then the action of Gal(K^{alg}/K) on K^{alg} extends uniquely to a continuous action of Gal(K^{alg}/K) on ℂ_{p}.

Let R be a commutative ring with 1 and let R[[X]] denote the ring of formal power series in one variable over R. For ϕ(X), ψ(X) ∈ R[[X]] such that ψ(0) = 0 we define the composition of ϕ with ψ to be (ϕ ∘ ψ)(X) = ϕ(ψ(X)). We say ϕ(X) ∈ R[[X]] is an invertible power series if ϕ(X) = a_{1}X +a_{2}X^{2} +⋯ with a_{1} ∈ R^{×}. In this case there exists a unique ϕ^{−1}(X) ∈ R[[X]] such that (ϕ ∘ ϕ^{−1})(X) = (ϕ^{−1} ∘ ϕ)(X) = X, and the series ϕ^{−1}(X) is itself invertible.

Since O_{ℂ_{p}} is a valuation ring, power series over O_{ℂ_{p}} can be studied using Newton polygons and Newton copolygons. Let ϕ(X) = a_{0}+a_{1}X+a_{2}X^{2}+⋯ be a nonzero element of O_{ℂ_{p}}[[X]]. For each i ≥ 0 with a_{i} ≠ 0 let

(1.1)  Q_{i} = {(x, y) ∈ ℝ^{2} : x ≥ i, y ≥ v_{p}(a_{i})},

Formal group law homomorphisms over O_{ℂ_{p}} 437

and define C to be the convex hull of the union of the collection {Q_{i} : a_{i} ≠ 0}. Let B denote the boundary of C and let m = min{i : a_{i} ≠ 0}. By removing the half-line {(m, y) : y > v_{p}(a_{m})} from B we get the Newton polygon N_{ϕ} of ϕ(X). If there exists k ≥ 0 such that v_{p}(a_{k}) ≤ v_{p}(a_{i}) for all i ≥ 0 then N_{ϕ} is the union of finitely many line segments with negative slope and the halfline {(x, v_{p}(a_{d})) : x ≥ d}, where d = min{j : v_{p}(a_{j}) = v_{p}(a_{k})}. Otherwise, N_{ϕ} is the union of countably many line segments with negative slope. The endpoints of the line segments which make up N_{ϕ} are called the vertices of N_{ϕ}.

The valuation function Ψ_{ϕ} : (0, ∞) → (0, ∞) of ϕ(X) is defined by

(1.2)  Ψ_{ϕ}(x) = min{v_{p}(a_{i}) + ix : i ≥ 0}.

Let ψ(X) ∈ O_{ℂ_{p}}[[X]] be another nonzero power series. One easily verifies that Ψ_{ϕ·ψ} = Ψ_{ϕ} + Ψ_{ψ}, and that Ψ_{ϕ∘ψ} = Ψ_{ϕ} ∘ Ψ_{ψ} if ψ(0) = 0. The graph of Ψ_{ϕ}(x) is known as the Newton copolygon of ϕ, and is denoted N_{ϕ}^{∗}. The Newton copolygon, like the Newton polygon, may be described as a union of line segments and a half line. In fact there is a one-to-one correspondence between the vertices of N_{ϕ} and the segments of N_{ϕ}^{∗}, and a one-to-one correspondence between the finite-length segments of N_{ϕ} and the vertices of N_{ϕ}^{∗}: The vertex (a, b) on N_{ϕ} corresponds to a segment (or half line) on N_{ϕ}^{∗} with slope a and y-intercept b, and a segment on N_{ϕ} with slope −a and y-intercept b corresponds to the vertex (a, b) on N_{ϕ}^{∗}.

Say that the power series

(1.3)  ϕ(X) = a_{0} + a_{1}X + a_{2}X^{2} + ⋯ ∈ O_{ℂ_{p}}[[X]]

has Weierstrass degree d if a_{i} ∈ M_{ℂ_{p}} for 0 ≤ i < d and a_{d} ∉ M_{ℂ_{p}}. We define a distinguished polynomial in O_{ℂ_{p}}[X] to be a monic polynomial

(1.4)  b_{0} + b_{1}X + ⋯ + b_{d−1}X^{d−1} + X^{d}

such that b_{i} ∈ M_{ℂ_{p}} for 0 ≤ i < d.

Let K be a closed subfield of ℂ_{p}. In order to study formal group law homomorphisms defined over O_{K} we need to formulate versions of the Weierstrass preparation theorem for power series defined over O_{K}. Since the field K may not be discretely valued, the ring O_{K} need not be Noetherian. The following proposition is an analog of [1, 10.2.1] for power series over O_{K}, and is proved by essentially the same method.

Proposition 1.1. Let K be a closed subfield of ℂ_{p} and let ϕ(X), ψ(X) be elements of O_{K}[[X]] such that ψ(X) has Weierstrass degree d. Then there exist unique q(X) ∈ O_{K}[[X]] and r(X) ∈ O_{K}[X] such that deg(r) < d and ϕ(X) = ψ(X)q(X) + r(X).

Proof. Write ψ(X) = a_{0} + a_{1}X + a_{2}X^{2} + ⋯ and choose c ∈ M_{K} such that a_{i} ∈ cO_{K} for 0 ≤ i < d. We will inductively construct a sequence (q_{n}(X)) of power series and a sequence (r_{n}(X)) of polynomials of degree < d such that

(1.5)  ϕ(X) ≡ ψ(X)q_{n}(X) + r_{n}(X) (mod c^{n})
(1.6)  q_{n+1}(X) ≡ q_{n}(X)  (mod c^{n})
(1.7)  r_{n+1}(X) ≡ r_{n}(X)  (mod c^{n})

for all n ≥ 0. Clearly (1.5) holds for n = 0 with q_{0}(X) = r_{0}(X) = 0. Let n ≥ 0 and assume (1.5) holds for n. Then

(1.8)  ϕ(X) = ψ(X)q_{n}(X) + r_{n}(X) + c^{n}g(X)

for some g(X) ∈ O_{K}[[X]]. Write g(X) = b_{0}+b_{1}X+b_{2}X^{2}+⋯ and recall that ψ(X) = a_{0}+a_{1}X +a_{2}X^{2}+⋯ with a_{d} ∈ O_{K}^{×}. Let ω(X) be the multiplicative inverse of a_{d} + a_{d+1}X + a_{d+2}X^{2} + · · ·. Then it is straightforward to verify that

(1.9)  q_{n+1}(X) = q_{n}(X) + c^{n}ω(X) · (b_{d} + b_{d+1}X + b_{d+2}X^{2} + ⋯)
(1.10)  r_{n+1}(X) = r_{n}(X) + c^{n}(b_{0} + b_{1}X + ⋯ + b_{d−1}X^{d−1})

satisfy

(1.11)  ϕ(X) ≡ ψ(X)q_{n+1}(X) + r_{n+1}(X) (mod c^{n+1}).

We now define q(X), r(X) to be the limits of the sequences (q_{n}(X)), (r_{n}(X)) with respect to the c-adic topology on O_{K}[[X]]. Then q(X), r(X) satisfy the conditions of the proposition.

To prove uniqueness suppose we also have ϕ(X) = ψ(X)q̃(X)+r̃(X) with deg(r̃) < d. Then

(1.12)  ψ(X)(q(X) − q̃(X)) = r̃(X) − r(X).

If q(X) ≠ q̃(X) let c^{n} be the largest power of c which divides q(X) − q̃(X). Then q(X)−q̃(X) = c^{n}h(X) for some h(X) ∈ O_{K}[[X]] which is not divisible by c. Since

(1.13)  ψ(X) ≡ a_{d}X^{d} + a_{d+1}X^{d+1} + ⋯ (mod c)

with v_{p}(a_{d}) = 0 this implies that the left side of (1.12) has at least one term of degree ≥ d whose coefficient is not divisible by c^{n+1}. Since the right side of (1.12) is a polynomial of degree < d, this is a contradiction. It follows that q(X) = q̃(X), and hence that r(X) = r̃(X). □

The following corollary extends the Weierstrass preparation theorem to power series with coefficients in the (possibly non-Noetherian) local ring O_{K} (cf. [1, 10.2.4]).

Corollary 1.2. Let K be a closed subfield of ℂ_{p} and let ϕ(X) ∈ O_{K}[[X]] be a power series with Weierstrass degree d. Then there exist a unit power series u(X) ∈ O_{K}[[X]]^{×} and a distinguished polynomial f(X) ∈ O_{K}[X] of degree d such that ϕ(X) = u(X)f(X) and f(X) has the same Newton polygon as ϕ(X).

Proof. This is basically the same as the proof of 10.2.4 in [1]: By Proposition 1.1 there are q(X) ∈ O_{K}[[X]] and r(X) ∈ O_{K}[X] such that X^{d} = ϕ(X)q(X) + r(X) and deg(r) < d. By reducing modulo M_{K}[[X]] we see that r(X) ∈ M_{K}[X]. Hence f(X) := X^{d} − r(X) is a distinguished polynomial. Since f(X) = ϕ(X)q(X), the Weierstrass degree of q(X) is 0, so q(X) is a unit in O_{K}[[X]]. Setting u(X) equal to the multiplicative inverse of q(X) we get ϕ(X) = u(X)f(X), as required. Since u(X) is a unit power series we have Ψ_{u}(x) = 0, and hence

(1.14)  Ψ_{ϕ}(x) = Ψ_{u}(x) + Ψ_{f}(x) = Ψ_{f}(x)

for all x > 0. It follows that N_{ϕ}^{∗} = N_{f}^{∗}, and hence that N_{ϕ} = N_{f}. □

The following is another version of the Weierstrass preparation theorem for power series over O_{ℂ_{p}}. A similar result for power series over ℂ_{p} with constant term 1 is given in [5, Th. 14, p. 97].

Proposition 1.3. Let ϕ(X) ∈ O_{ℂ_{p}}[[X]] and let (d, e) be a vertex on the Newton polygon N_{ϕ} of ϕ. Then there exist a power series ψ(X) ∈ O_{ℂ_{p}}[[X]] and a distinguished polynomial f(X) ∈ O_{ℂ_{p}}[X] of degree d with the following properties:

(1) ϕ(X) = ψ(X)f(X).

(2) N_{f}^{l} = {(x, y) ∈ N_{f} : x ≤ d} is the translation of N_{ϕ}^{l} = {(x, y) ∈ N_{ϕ} : x ≤ d} by (0, −e).

(3) N_{ψ} is the translation of N_{ϕ}^{r} = {(x, y) ∈ N_{ϕ} : x ≥ d} by (−d, 0).

Proof. Let −w_{l}, −w_{r} denote the slopes of the segments of N_{ϕ} immediately to the left and right of (d, e); if (d, e) is the left endpoint of N_{ϕ} let w_{l} = ∞. Then w_{l} > w_{r}, so there exists b ∈ O_{ℂ_{p}} such that w_{r} < v_{p}(b) < w_{l}. Write ϕ(X) = a_{0} + a_{1}X + a_{2}X^{2} + ⋯ and set ρ(X) = ϕ(bX). Then the coefficient of X^{n} in ρ(X) is b^{n}a_{n}, and by the choice of b we have v_{p}(b^{d}a_{d}) < v_{p}(b^{n}a_{n}) for all n ≠ d. Hence a_{d}^{−1}b^{−d}ρ(X) is an element of O_{ℂ_{p}}[[X]] with Weierstrass degree d. By Corollary 1.2 there is a unit power series u(X) and a distinguished polynomial g(X) of degree d such that a_{d}^{−1}b^{−d}ρ(X) = u(X)g(X). Let ψ(X) = a_{d}u(b^{−1}X) and f(X) = b^{d}g(b^{−1}X). Then ψ(X)f(X) = ϕ(X) with f(X) ∈ O_{ℂ_{p}}[X] a distinguished polynomial of degree d. We claim that ψ(X) ∈ O_{ℂ_{p}}[[X]]. By Proposition 1.1 there exist q(X) ∈ O_{ℂ_{p}}[[X]] and r(X) ∈ O_{ℂ_{p}}[X] such that ϕ(X) = f(X)q(X) + r(X) and deg(r) < d. By replacing X with bX we get ρ(X) = g(X)·b^{d}q(bX)+r(bX). Since we also have ρ(X) = g(X) · a_{d}b^{d}u(X), by the uniqueness statement in Proposition 1.1 we get b^{d}q(bX) = a_{d}b^{d}u(X). Hence ψ(X) = a_{d}u(b^{−1}X) = q(X) ∈ O_{ℂ_{p}}[[X]].

It follows from Corollary 1.2 that g(X) has the same Newton polygon as a_{d}^{−1}b^{−d}ρ(X). Hence b^{d}g(X) has the same Newton polygon as a_{d}^{−1}ρ(X). Therefore Ψ_{b^{d}g}(x) = Ψ_{a_{d}^{−1}ρ}(x) for all x > 0. Since f(X) = b^{d}g(b^{−1}X) and

a_{d}^{−1}ϕ(X) = a_{d}^{−1}ρ(b^{−1}X) we get Ψ_{f}(x) = Ψ_{a_{d}^{−1}ϕ}(x) for all x > v_{p}(b). Since v_{p}(b) < w_{l} it follows from the correspondence between Newton polygons and copolygons that N_{f}^{l} = N_{a_{d}^{−1}ϕ}^{l}. Hence N_{f}^{l} is the translation of N_{ϕ}^{l} by

v_{p}(a_{d}) = e units downwards, so property (2) holds.

It follows from the preceding paragraph that the rightmost finite segment of N_{f} has slope −w_{l} and right endpoint (d, 0). Therefore Ψ_{f}(x) = dx for 0 ≤ x ≤ w_{l}. Since ϕ(X) = ψ(X)f(X) we have Ψ_{ψ}(x) = Ψ_{ϕ}(x) − Ψ_{f}(x), and hence Ψ_{ψ}(x) = Ψ_{ϕ}(x) − dx for 0 ≤ x ≤ w_{l}. Since ψ(X) = a_{d}u(b^{−1}X) with u(X) ∈ O_{ℂ_{p}}[[X]]^{×}, for x ≥ v_{p}(b) we have Ψ_{ψ}(x) = v_{p}(a_{d}) = e. Since v_{p}(b) < w_{l}, these two facts determine Ψ_{ψ}(x) and N_{ψ}^{∗} completely. We find that the segments of N_{ψ}^{∗} have the same y-intercepts as the segments of N_{ϕ}^{∗} which lie to the left of x = w_{l}, but the slopes of the segments on N_{ψ}^{∗} are d less than the slopes of the corresponding segments on N_{ϕ}^{∗}. It follows from the correspondence between Newton polygons and Newton copolygons that the vertices of N_{ψ} are the translates of the vertices of N_{ϕ}^{r} by d units to the left. This gives property (3). □

Remark 1.4. In fact the series ψ(X) and the polynomial f(X) in the conclusion of Proposition 1.3 are uniquely determined by ϕ(X) and (d, e). Furthermore, if ϕ(X) ∈ O_{K}[[X]] for some closed subfield K of ℂ_{p} then ψ(X) and f(X) have coefficients in K. We omit the proofs of these facts because they are not needed for our applications.

Corollary 1.5. Let ϕ(X) ∈ XO_{ℂ_{p}}[[X]] and let α be an element of M_{ℂ_{p}} which does not divide ϕ(X) in O_{ℂ_{p}}[[X]]. Then there is β ∈ M_{ℂ_{p}} such that ϕ(β) = α.

Proof. The assumption on α implies that the Newton polygon of ϕ(X) − α has at least one segment with negative slope. It follows from Proposition 1.3 that we can write ϕ(X) − α = ψ(X)f(X), with ψ(X) ∈ O_{ℂ_{p}}[[X]] and f(X) ∈ O_{ℂ_{p}}[X] a distinguished polynomial of degree d ≥ 1. Let β ∈ M_{ℂ_{p}} be any root of f(X). Then ϕ(β) = α. □

2. Formal group laws

In this section we extend some well-known results in the theory of oneparameter formal group laws over a p-adic integer ring to include group laws and homomorphisms which are defined over O_{ℂ_{p}}.

Let R be a commutative ring with 1. A (one-parameter) formal group law over R is a power series F (X, Y) ∈ R[[X, Y]] such that F (X, 0) = X, F (X, Y) = F (Y, X), and F (F (X, Y), Z) = F (X, F (Y, Z)). Let F (X, Y) and G(X, Y) be formal group laws over R and let S be a commutative ring which contains R. A homomorphism from F to G over S is a power series ϕ(X) ∈ S[[X]] such that ϕ(0) = 0 and ϕ(F (X, Y)) = G(ϕ(X), ϕ(Y)). Given homomorphisms ϕ(X) and ψ(X) from F to G, we define the sum of ϕ and ψ to be (ϕ +_{G} ψ)(X) = G(ϕ(X), ψ(X)). The series (ϕ +_{G} ψ)(X) is itself a homomorphism from F to G. This operation makes the set of S-homomorphisms from F to G into an abelian group, which we denote by Hom_{S}(F, G). If ϕ(X) ∈ Hom_{S}(F, G) is invertible (with respect to composition of power series) we say that ϕ is an isomorphism from F to G. In this case ϕ^{−1}(X) is an isomorphism from G to F. A homomorphism from F to itself is called an endomorphism of F. The group End_{S}(F) = Hom_{S}(F, F) of S-endomorphisms of F is a ring, with multiplication given by composition of power series. For n ∈ ℤ we define [n]_{F} (X) to be the image of n under the canonical map from ℤ to End_{S}(F). Thus for n positive [n]_{F} (X) is the sum in End_{S}(F) of n copies of the identity map [1]_{F} (X) = X.

Let F (X, Y) be a formal group law over a field k of characteristic p > 0. Then either [p]_{F} (X) = 0 or [p]_{F} (X) = ψ(X^{p^{h}}) for some h ≥ 1 and ψ(X) ∈ k[[X]] such that ψ(0) ≠ 0 (see for instance [4, 18.3.1]). In the first case we say that F has infinite height; in the second case we say that F has height h. Suppose that R is a local ring with maximal ideal M whose residue field R/M has characteristic p. Then the image F (X, Y) of F (X, Y) in (R/M)[[X, Y]] is a formal group law over R/M. We define the height of F (X, Y) to be equal to the height of F (X, Y).

Let E be a finite extension of ℚ_{p} and let R be an O_{E}-algebra, with structure map i : O_{E} → R. A formal O_{E}-module over R is defined to be a pair (F, j), where F (X, Y) is a formal group law over R and

(2.1)  j : O_{E} ⟶ End_{R}(F)

is a ring homomorphism with j(c) = i(c)X + a_{2}X^{2} + ⋯ for every c ∈ O_{E}. We write j(c) = [c]_{F} (X). The height of the formal O_{E}-module (F, j) is defined to be h/[E : ℚ_{p}], where h is the height of the the formal group law F. If h < ∞ then the height of (F, j) is a positive integer [4, 21.8.2]. Suppose R is a complete local ring with maximal ideal M whose residue field R/M has characteristic p. Then R is a ℤ_{p}-algebra and the map n ↦ [n]_{F} (X) can be uniquely extended to a ring homomorphism j : ℤ_{p} → End_{R}(F). The pair (F, j) is then a formal ℤ_{p}-module.

Let F (X, Y) be a formal group law over O_{ℂ_{p}}. We define the point group F (M_{ℂ_{p}}) of F to be the set M_{ℂ_{p}} with the operation α ∗ β = F (α, β). Any formal group law homomorphism ϕ : F → G over O_{ℂ_{p}} induces a group homomorphism α ↦ ϕ(α) from F (M_{ℂ_{p}}) to G(M_{ℂ_{p}}). The kernel of ϕ(X) is defined to be the kernel of the associated homomorphism of point groups. Suppose F (X, Y) has height h < ∞. Then ker([p^{n}]_{F} (X)) is a free (ℤ/p^{n}ℤ)-module of rank h and α ↦ [p]_{F} (α) gives a surjective homomorphism from ker([p^{n+1}]_{F} (X)) to ker([p^{n}]_{F} (X)). By taking the projective limit of ker([p^{n}]_{F} (X)) for n ≥ 1 we get the Tate module T_{p}(F), which is a free ℤ_{p}-module of rank h. If F (X, Y) is defined over O_{K} for some closed subfield K of ℂ_{p} then Gal(K^{alg}/K) acts on ker([p^{n}]_{F} (X)) and on T_{p}(F).

Let 1 ≤ h < ∞. Since ℚ_{p} is a locally compact field it follows from Krasner’s Lemma that there are only finitely many subextensions L/ℚ_{p} of ℚ_{p}^{alg}/ℚ_{p} such that [L : ℚ_{p}] ≤ h. Let J_{h} denote the compositum of all such fields L. Then J_{h} is a finite extension of ℚ_{p}.

Lemma 2.1. Let F (X, Y) be a formal group law of height h < ∞ over O_{ℂ_{p}} and let ϕ(X) = a_{1}X + a_{2}X^{2} + ⋯ ∈ O_{ℂ_{p}}[[X]] be an endomorphism of F. Then a_{1} ∈ O_{J_{h}}.

Proof. Let T_{p}(ϕ) denote the endomorphism of T_{p}(F) induced by ϕ(X) and let g(X) be the minimum polynomial of T_{p}(ϕ) over ℚ_{p}. Then g(X) ∈ ℤ_{p}[X], deg(g) ≤ h, and g(ϕ) ∈ End_{O_{ℂp}}(F) annihilates T_{p}(F). Hence for n ≥ 1 the p^{n}-torsion points of F lie in the kernel of g(ϕ). Let f(X) = ∏_{α}(α − X), where the product is taken over α ∈ ker([p^{n}]_{F} (X)). Then f(X) divides g(ϕ) in O_{ℂ_{p}}[[X]]. It follows that the product of the nonzero p^{n}-torsion points of F divides g(a_{1}), the coefficient of X in g(ϕ). Since the sum of the p-valuations of the nonzero p^{n}-torsion points of F is n, we get p^{n} | g(a_{1}) for every n ≥ 1. Thus g(a_{1}) = 0. Since deg(g) ≤ h, this implies a_{1} ∈ J_{h} ∩ O_{ℂ_{p}} = O_{J_{h}}. □

Proposition 2.2. Let K be a subfield of ℂ_{p}, let F (X, Y) be a formal group law of height h < ∞ over O_{K}, and let ϕ(X) ∈ End_{O_{ℂp}}(F). Then ϕ(X) has coefficients in O_{M}, where M = KJ_{h}.

Proof. Let l(X) = X + c_{2}X^{2} + c_{3}X^{3} + ⋯ ∈ K[[X]] be the logarithm of F. Thus l(X) is an isomorphism over K from F (X, Y) to the additive formal group law 𝔾_{a}(X, Y) = X +Y (see [4, 5.4.4]). Since ϕ(X) = a_{1}X +a_{2}X^{2}+⋯ is an endomorphism of F with coefficients in O_{K}, the series l(ϕ(l^{−1}(X))) ∈ K[[X]] is an endomorphism of 𝔾_{a}. Since K is a field of characteristic 0, every endomorphism of 𝔾_{a} over K is of the form i(X) = bX for some b ∈ K (see for instance Corollary 2 on p. 97 of [3]). Since the coefficient of X in l(ϕ(l^{−1}(X))) is a_{1} we get l(ϕ(l^{−1}(X))) = a_{1}X. By Lemma 2.1 we have a_{1} ∈ J_{h}. It follows that ϕ(X) = l^{−1}(a_{1}l(X)) has coefficients in O_{ℂ_{p}} ∩ M = O_{M}. □

Remark 2.3. It is proved in [3, IV §1, Prop. 4] that if K is a closed discretely valued subfield of ℂ_{p} and F (X, Y) is formal group law defined over O_{K} then the conclusion to Proposition 2.2 is valid with J_{h} replaced by the compositum of the extensions L/ℚ_{p} such that [L : ℚ_{p}] divides h. One could obtain this improved version of Proposition 2.2 for general K ⊂ ℂ_{p} using the fact (stated without proof in [7, 2.3.0]) that the reduction map from End_{O_{ℂp}}(F) to End_{𝔽_{p}^{alg}}(F) is injective.

Proposition 2.4. Let F, G be formal group laws over O_{ℂ_{p}} such that F has height h < ∞, and let ϕ : F → G be a nonzero homomorphism over O_{ℂ_{p}}. Then there is no c ∈ M_{ℂ_{p}} such that c divides every coefficient of ϕ(X).

Proof. Write [p]_{F} (X) = a_{1}X +a_{2}X^{2} +⋯ and choose 1 ≤ j < p^{h} such that v_{p}(a_{j}) ≤ v_{p}(a_{i}) for 1 ≤ i < p^{h}. Since a_{1} = p and F (X, Y) has height h we have 0 < v_{p}(a_{j}) ≤ 1. Let S be the set of elements of O_{ℂ_{p}} which divide ϕ(X) in O_{ℂ_{p}}[[X]], and set m = sup{v_{p}(a) : a ∈ S}. If m > 0 then there exists c ∈ O_{ℂ_{p}} such that \frac{1}{2}m < v_{p}(c) < m and v_{p}(c) > m−v_{p}(a_{j}). Since v_{p}(c) < m we have ϕ(X) = cψ(X) for some ψ(X) = b_{1}X +b_{2}X^{2}+⋯ ∈ O_{ℂ_{p}}[[X]]. Since m − v_{p}(c) < v_{p}(a_{j}) and m − v_{p}(c) < v_{p}(c), the b_{i} do not all lie in the ideal I = (c, a_{j}) of O_{ℂ_{p}}. Let n be the smallest positive integer such that b_{n} ∉ I. Since ϕ([p]_{F} (X)) = [p]_{G}(ϕ(X)) we have

(2.2)  cψ([p]_{F} (X)) = [p]_{G}(cψ(X)).

The coefficients on the right side of (2.2) all lie in cI. Since

(2.3)  ψ(X) ≡ b_{n}X^{n} + ⋯ (mod I)
(2.4)  [p]_{F} (X) ≡ a_{p^{h}}X^{p^{h}} + ⋯ (mod I)

with a_{p^{h}} ∈ O_{ℂ_{p}}^{×}, the coefficient of X^{np^{h}} on the left side of (2.2) does not lie in cI. This is a contradiction, so we must have m = 0. Hence ϕ(X) is not divisible by any element of M_{ℂ_{p}}. □

Corollary 2.5. Let F, G be formal group laws over O_{ℂ_{p}} such that F has finite height, and let ϕ : F → G be a homomorphism defined over O_{ℂ_{p}} which has trivial kernel. Then ϕ(X) is an isomorphism.

Proof. Since ϕ(X) = a_{1}X + a_{2}X^{2} + ⋯ is nonzero and O_{ℂ_{p}} has characteristic 0 we have a_{1} ≠ 0. If ϕ(X) is not an isomorphism then v_{p}(a_{1}) > 0. Using Proposition 2.4 we see that ϕ(X) is not divisible by any element of M_{ℂ_{p}}. Therefore the Newton polygon of ϕ(X) has at least one segment with negative slope. Hence by Proposition 1.3 we have ϕ(X) = ψ(X)f(X) with ψ(X) ∈ O_{ℂ_{p}}[[X]] and f(X) ∈ O_{ℂ_{p}}[X] a distinguished polynomial of degree d > 1. It follows that ϕ(X) has a nontrivial zero in M_{ℂ_{p}}, contrary to assumption. Therefore ϕ : F → G is an isomorphism. □

The following proposition is an apparent generalization of Theorem 1.4 in [6]. The result presented here applies to homomorphisms ϕ(X) defined over O_{ℂ_{p}}, rather than over the ring of integers of a finite extension of K as in [6]. However, it will follow from Theorem 3.2 that every ϕ(X) which satisfies the hypotheses of Proposition 2.6 also satisfies the hypotheses of [6, Th. 1.4]. The more general statement given below is needed for the proof of Theorem 3.2.

Proposition 2.6. Let K be a discretely valued subfield of ℂ_{p} and let F, G be formal group laws defined over O_{K} such that F has height h < ∞. Let ϕ : F → G be a homomorphism defined over O_{ℂ_{p}} such that W = ker(ϕ) is finite, and let L be the closure of K(W) in ℂ_{p}. Then L is discretely valued and there is a formal group law G′ defined over O_{L}, a homomorphism ϕ′ : F → G′ defined over O_{L}, and an isomorphism i : G′ → G defined over O_{ℂ_{p}} such that i ∘ ϕ′ = ϕ. Furthermore, G and G′ both have height h.

Proof. Since W is a finite subgroup of F (M_{ℂ_{p}}) we have W ≤ ker([p^{n}]_{F} (X)) for some n ≥ 0. By the Weierstrass preparation theorem the elements of ker([p^{n}]_{F} (X)) are the roots of a distinguished polynomial with coefficients in O_{K}. Hence [K(W) : K] < ∞, so K(W) is discretely valued. It follows that L is discretely valued as well.

By [6, Th. 1.4] there is a formal group law G′ defined over O_{L} such that the power series

(2.5)  ϕ′(X) = ∏_{γ∈W} F (X, γ) ∈ O_{L}[[X]]

is a homomorphism from F to G′ with kernel W. By considering the reduction modulo M_{ℂ_{p}} of the formula ϕ′ ∘ [p]_{F} = [p]_{G′} ∘ ϕ′ we see that G′ has the same height h as F. Set ξ = ϕ′(X) and A_{0} = O_{K}[[ξ]] ⊂ O_{L}[[X]]. Then by [6, Lemma 1.3] the power series

(2.6)  Q(T) = −ξ + ∏_{γ∈W} F (T, γ)

lies in A_{0}[[T]]. Since A_{0} is a complete Noetherian local ring, by the general form of the Weierstrass preparation theorem [1, 10.2.4] we have Q(T) = U(T)H(T), where U(T) ∈ A_{0}[[T]]^{×} and H(T) ∈ A_{0}[T] has degree |W |.

We now construct i(X) by imitating the proof of [6, Lemma 1.3]. Set A = O_{ℂ_{p}}[[ξ]], B = O_{ℂ_{p}}[[X]], K = Frac(A), and L = Frac(B). Then L = K(X) and X is a root of H(T), so we have [L : K] ≤ deg(H) = |W |. On the other hand, for γ ∈ W and Δ(X) ∈ L define (γ · Δ)(X) = Δ(F (X, γ)); the right side of this equation makes sense because v_{p}(γ) > 0. This gives a faithful ℂ_{p}-linear action of W on L. By (2.5) we have γ · ξ = ξ for every γ ∈ W. Thus K is contained in the fixed field L^{W} of W, so we have [L : K] ≥ |W |. Hence [L : K] = |W | and K = L^{W}. Since B is free of rank |W | over A, with basis {1, X, X^{2}, …, X^{|W |−1}}, we have A = K ∩ B = L^{W} ∩ B = B^{W}. Since ϕ(X) ∈ B^{W} = A there is i(X) ∈ O_{ℂ_{p}}[[X]] such that ϕ(X) = i(ξ). Thus ϕ = i ∘ ϕ′.

Since ϕ′ : F → G′ and ϕ : F → G are formal group law homomorphisms, i(X) is a homomorphism from G′ to G. If α ∈ ker(i) then by Corollary 1.5 there is β ∈ M_{ℂ_{p}} such that ϕ′(β) = α. Hence ϕ(β) = i(ϕ′(β)) = i(α) = 0, so β ∈ W. Since ker ϕ′ = W we get α = ϕ′(β) = 0, and hence ker(i) = {0}. It follows from Corollary 2.5 that i(X) is an isomorphism. Since G′ has the same height h as F and G is isomorphic to G′ over O_{ℂ_{p}} we deduce that G also has height h. □

3. Homomorphisms of formal group laws

Let K be a discretely valued subfield of ℂ_{p} and let F, G be formal group laws over O_{K} with the same finite height. In this section we prove that there is a closed discretely valued subfield M of ℂ_{p} such that every homomorphism from F to G is defined over O_{M}.

To begin we assume that K is closed in ℂ_{p} as well as discretely valued. Since O_{K} is an integral domain of characteristic 0 we may identify End_{O_{K}}(F) with a subring of O_{K} using the map D_{F} : End_{O_{K}}(F) → O_{K} which maps ϕ(X) = a_{1}X + a_{2}X^{2} + ⋯ onto D_{F} (ϕ) = a_{1}. If D_{F} (ψ) = b we write ψ(X) = [b]_{F} (X). Let A be a ℤ_{p}-subalgebra of End_{O_{K}}(F); then D_{F} identifies A with a subring of O_{K}. Hence E = Frac(A) is a subfield of K which is a finite extension of ℚ_{p}. The Tate module T_{p}(F) is a module over A, and V_{p}(F) = T_{p}(F) ⊗_{ℤ_{p}} ℚ_{p} is a vector space over E. Let K^{alg} be the algebraic closure of K in ℂ_{p}, and let K^{ab}/K, K^{nr}/K be the maximum abelian subextension and the maximum unramified subextension of K^{alg}/K. The action of Galois on V_{p}(F) gives a representation

(3.1)  ρ_{F} : Gal(K^{alg}/K) ⟶ Aut_{E}(V_{p}(F)).

For σ ∈ Gal(K^{alg}/K^{nr}) let σ̃ denote the restriction of σ to Gal(E^{ab}/E^{nr}), and let uσ̃ be the element of O_{E}^{×} which corresponds to σ̃ under the reciprocity isomorphism Gal(E^{ab}/E^{nr}) ≅ O_{E}^{×} of local class field theory. Then the map

(3.2)  ψ_{E}^{K} : Gal(K^{alg}/K^{nr}) ⟶ O_{E}^{×}

defined by ψ_{E}^{K}(σ) = uσ̃ is a homomorphism.

The following is a slight generalization of a theorem proved by Strauch [11, Th. 3.3] in the case A = O_{E}. The case A = ℤ_{p} was proved by Raynaud [9, Prop. 4.2].

Proposition 3.1. Let K be a closed discretely valued subfield of ℂ_{p} and let F (X, Y) be a formal group law over O_{K} with finite height. Let A be a ℤ_{p}-subalgebra of End_{O_{K}}(F), set E = Frac(A), and let

ρ_{F} : Gal(K^{alg}/K) ⟶ Aut_{E}(V_{p}(F))

be the Galois representation associated to F. Then det(ρ_{F} (σ)) = ψ_{E}^{K}(σ)^{−1} for every σ ∈ Gal(K^{alg}/K^{nr}), where ψ_{E}^{K} is induced by the class field theory isomorphism Gal(E^{ab}/E^{nr}) ≅ O_{E}^{×}.

Proof. By [6, p. 301] there exist a formal O_{E}-module (F′, j′) and a nonzero homomorphism π : F → F′, both defined over O_{K}. The map π induces a Gal(K^{alg}/K)-equivariant ℤ_{p}-module homomorphism

(3.3)  T_{p}(π) : T_{p}(F) ⟶ T_{p}(F′)

which is one-to-one with finite cokernel. Thus the induced map

(3.4)  V_{p}(π) : V_{p}(F) ⟶ V_{p}(F′)

is a Gal(K^{alg}/K)-equivariant isomorphism which is E-linear since

(3.5)  π ∘ [a]_{F} = [a]_{F′} ∘ π

for a ∈ A. It follows that det(ρ_{F} (σ)) = det(ρ_{F′}(σ)) for σ ∈ Gal(K^{alg}/K). Since (F′, j′) is a formal O_{E}-module, by Strauch’s result [11, Th. 3.3] we have det(ρ_{F′}(σ)) = ψ_{E}^{K}(σ)^{−1} for every σ ∈ Gal(K^{alg}/K^{nr}). Hence det(ρ_{F} (σ)) = ψ_{E}^{K}(σ)^{−1} for σ ∈ Gal(K^{alg}/K^{nr}). □

We now use Propositions 2.6 and 3.1 to prove our main result.

Theorem 3.2. Let K be a discretely valued subfield of ℂ_{p}, let F, G be formal group laws over O_{K} such that F has height h < ∞, and let ϕ : F → G be a homomorphism over O_{ℂ_{p}} with finite kernel. Then there is a closed discretely valued subfield M of ℂ_{p} such that ϕ(X) has coefficients in O_{M}.

Proof. Let W = ker(ϕ) and let L be the closure of K(W) in ℂ_{p}. Then by Proposition 2.6 there is a formal group law G′(X, Y) over O_{L}, a homomorphism ϕ′ : F → G′ over O_{L}, and an isomorphism i : G′ → G over O_{ℂ_{p}} such that i ∘ ϕ′ = ϕ. Furthermore, L is discretely valued and G, G′ both have height h. Recall that J_{h} is the compositum of all the extensions of ℚ_{p} of degree ≤ h, and let N be the closure in ℂ_{p} of the maximum unramified extension of LJ_{h}. Since J_{h}/ℚ_{p} is finite, N is discretely valued. To prove the theorem it suffices to show that there is a finite extension M/N such that i is defined over O_{M}.

Let A = {a_{1} : a_{1}X + a_{2}X^{2} + ⋯ ∈ End_{O_{ℂp}}(G)} be the image of D_{G}, and let E be the field of fractions of A. By Lemma 2.1 we have A ⊂ O_{J_{h}}, which implies that E is contained in N. Therefore for σ ∈ Gal(K^{alg}/N) we have i^{σ} ∘ i^{−1} ∈ Aut_{O_{ℂp}}(G) ≅ A^{×}, and hence i^{σ} ∘ i^{−1} = [a_{σ}]_{G} for some a_{σ} ∈ A^{×}. Let T_{p}(i) : T_{p}(G′) → T_{p}(G) be the isomorphism induced by i. Then for v ∈ T_{p}(G′) we have

(3.6) σ(T_{p}(i)(v)) = T_{p}(i^{σ})(σ · v) = T_{p}([a_{σ}]_{G} ∘ i)(σ · v) = a_{σ}T_{p}(i)(σ · v).

Let

(3.7)  ρ_{G} : Gal(K^{alg}/N) ⟶ Aut_{E}(V_{p}(G))
(3.8)  ρ_{G′} : Gal(K^{alg}/N) ⟶ Aut_{E}(V_{p}(G′))

be the representations associated to G and G′. It follows from (3.6) that

(3.9)  ρ_{G}(σ) ∘ T_{p}(i) = a_{σ}T_{p}(i) ∘ ρ_{G′}(σ).

Since T_{p}(i) induces an isomorphism V_{p}(i) : V_{p}(G′) → V_{p}(G) we get

(3.10)  det(ρ_{G}(σ)) = det(a_{σ}ρ_{G′}(σ)) = a_{σ}^{d} det(ρ_{G′}(σ)),

where d = dim_{E}(V_{p}(G′)) = h/[E : ℚ_{p}]. Hence by Proposition 3.1 we have a_{σ}^{d} = 1 for every σ ∈ Gal(K^{alg}/N). One easily verifies that the map

(3.11)  λ : Gal(K^{alg}/N) ⟶ A^{×}

defined by λ(σ) = a_{σ} is a continuous homomorphism. Hence the fixed field M of the kernel of λ is an extension of N of degree ≤ d. Since i^{σ}(X) = i(X) for all σ ∈ Gal(K^{alg}/M), the coefficients of i(X) lie in the subfield of ℂ_{p} fixed by Gal(K^{alg}/M). Since the completion of K^{alg} is ℂ_{p}, it follows from Ax’s theorem [2] that the fixed field of Gal(K^{alg}/M) acting on ℂ_{p} is M. Hence i(X) has coefficients in M ∩ O_{ℂ_{p}} = O_{M}. □

Remark 3.3. It is well-known that there need not exist a finite extension M/K such that the coefficients of ϕ(X) lie in O_{M}. Indeed, let K be a finite extension of ℚ_{p}, let K^{nr} ⊂ ℂ_{p} be the maximum unramified extension of K, and let N be the closure in ℂ_{p} of K^{nr}. Let F be a formal group law over O_{K} and let α(X) be an automorphism of F which is defined over O_{K} and has infinite order. Then by imitating the proof of [3, III §3, Prop. 3] we see that there is an invertible power series ϕ(X) ∈ O_{N}[[X]] such that ϕ^{σ}(X) = ϕ(α(X)), where σ is the continuous automorphism of N induced by the Frobenius element of Gal(K^{nr}/K). Define G(X, Y) = ϕ(F (ϕ^{−1}(X), ϕ^{−1}(Y))). Then G(X, Y) is a formal group law over O_{K} and ϕ(X) ∈ O_{N}[[X]] is an isomorphism from F to G. But ϕ(X) is not defined over O_{M} for any finite extension M of K.

We now consider the properties of homomorphisms ϕ : F → G over O_{ℂ_{p}} with infinite kernel. In Example 3.8 we will construct an example of such a homomorphism.

Lemma 3.4. Let F, G be formal group laws of finite height over O_{ℂ_{p}} and let ϕ : F → G be a nonzero homomorphism defined over O_{ℂ_{p}}. Then for every α ∈ ker(ϕ) there is n ≥ 0 such that [p^{n}]_{F} (α) = 0.

Proof. Let α ∈ ker(ϕ). For any β ∈ M_{ℂ_{p}} we have

(3.12)  v_{p}([p]_{F} (β)) ≥ min{v_{p}(β) + 1, 2v_{p}(β)}.

Hence if α is not p^{n}-torsion for any n ≥ 0 then (v_{p}([p^{n}]_{F} (α)))_{n≥0} is a sequence of rational numbers which increases without bound. By considering the Newton polygon of ϕ(X) we see that the valuations of the nonzero elements of ker(ϕ) are bounded above. Therefore for n sufficiently large we have [p^{n}]_{F} (α) ∉ ker(ϕ), and hence α ∉ ker(ϕ). This is a contradiction, so [p^{n}]_{F} (α) = 0 for some n ≥ 0. □

Proposition 3.5. Let F, G be formal group laws over O_{ℂ_{p}} whose heights h_{F}, h_{G} are finite. Suppose there exists a nonzero homomorphism ϕ : F → G over O_{ℂ_{p}}. Then h_{F} ≥ h_{G}. Furthermore, if ker(ϕ) is infinite then h_{F} > h_{G}.

Proof. Let F (M_{ℂ_{p}})^{tor}, G(M_{ℂ_{p}})^{tor} denote the torsion subgroups of the point groups F (M_{ℂ_{p}}), G(M_{ℂ_{p}}). We claim that ϕ maps F (M_{ℂ_{p}})^{tor} onto G(M_{ℂ_{p}})^{tor}. Let α ∈ G(M_{ℂ_{p}})^{tor}; then [p^{n}]_{G}(α) = 0 for some n ≥ 0. It follows from Proposition 2.4 that α does not divide ϕ(X) in O_{ℂ_{p}}[[X]]. Therefore by Corollary 1.5 there is β ∈ M_{ℂ_{p}} such that ϕ(β) = α. Hence

(3.13)  ϕ([p^{n}]_{F} (β)) = [p^{n}]_{G}(ϕ(β)) = [p^{n}]_{G}(α) = 0,

so we have [p^{n}]_{F} (β) ∈ ker(ϕ). It follows from Lemma 3.4 that [p^{n}]_{F} (β) is p^{m}-torsion for some m ≥ 0. Therefore β is p^{m+n}-torsion, and hence β ∈ F (M_{ℂ_{p}})^{tor}. By Lemma 3.4 we have ker(ϕ) ⊂ F (M_{ℂ_{p}})^{tor}, and hence G(M_{ℂ_{p}})^{tor} ≅ F (M_{ℂ_{p}})^{tor}/ ker(ϕ). Since

(3.14)  F (M_{ℂ_{p}})^{tor} ≅ (ℚ_{p}/ℤ_{p})^{h_{F}} , G(M_{ℂ_{p}})^{tor} ≅ (ℚ_{p}/ℤ_{p})^{h_{G}},

we get h_{F} ≥ h_{G}, and h_{F} > h_{G} if ker(ϕ) is infinite. □

Remark 3.6. Assume ker(ϕ) is infinite and write ϕ(X) = a_{1}X +a_{2}X^{2}+· · ·. It follows from Proposition 2.4 that there is no c ∈ M_{ℂ_{p}} such that a_{i} ∈ cO_{ℂ_{p}} for every i ≥ 1. On the other hand, if a_{i} ∈ O_{ℂ_{p}}^{×} for some i then ϕ(X) has finite Weierstrass degree, and hence finite kernel, contrary to assumption. Therefore the coefficients a_{1}, a_{2}, … of ϕ(X) generate the ideal M_{ℂ_{p}} in O_{ℂ_{p}}, and the Newton polygon of ϕ(X) is asymptotic to the x-axis.

Using Proposition 3.5 we get a stronger version of Theorem 3.2.

Corollary 3.7. Let K be a discretely valued subfield of ℂ_{p} and let F, G be formal group laws over O_{K} with the same finite height h. Then there is a closed discretely valued subfield M of ℂ_{p} such that every element of Hom_{O_{ℂp}}(F, G) has coefficients in O_{M}.

Proof. Let N be the closure in ℂ_{p} of the maximum unramified extension of K. It follows from Proposition 3.5 and Theorem 3.2 that for every ϕ(X) ∈ Hom_{O_{ℂp}}(F, G) there is a finite extension M_{ϕ}/N such that ϕ(X) ∈ O_{M_{ϕ}}[[X]]. Hence by [3, IV §1, Prop. 3] the reduction map from Hom_{O_{ℂp}}(F, G) to Hom_{𝔽_{p}^{alg}}(F , G) is injective. Since Hom_{𝔽_{p}^{alg}}(F , G) is a free ℤ_{p}-module of rank h^{2} [3, IV §1, Lemma 1] it follows that Hom_{O_{ℂp}}(F, G) is a free ℤ_{p}-module of finite rank. For a ℤ_{p}-basis {ϕ_{1}, …, ϕ_{r}} of Hom_{O_{ℂp}}(F, G), let M be the compositum of the fields M_{ϕ_{1}}, …, M_{ϕ_{r}}. Then M is a closed discretely valued subfield of ℂ_{p} and every element of Hom_{O_{ℂp}}(F, G) has coefficients in O_{M}. □

We conclude by giving an example of a nonzero homomorphism whose kernel is infinite. We first recall some definitions. Let K be a subfield of ℂ_{p} and let F (X, Y) be a formal group law of height h < ∞ over O_{K}. The p^{n}-torsion group scheme F (p^{n}) of F is defined to be the affine scheme Spec(O_{K}[[X]]/([p^{n}]_{F} (X))) with the group operation induced by F (X, Y). Thus F (p^{n}) is a connected group scheme of order p^{nh} over O_{K}. The pdivisible group F (p^{∞}) of F is defined to be the injective system of the p^{n}-torsion group schemes F (p^{n}) for n ≥ 1.

I thank Jonathan Lubin for pointing out the following example to me.

Example 3.8. Let K be a discretely valued subfield of ℂ_{p} and let E be an elliptic curve over O_{K} whose special fiber E ⊗ (O_{K}/M_{K}) is supersingular. By choosing a parameter we get a formal group law Ê for E. Since E has supersingular reduction, the p^{n}-torsion subgroup scheme Ê(p^{n}) of Ê can be identified with the p^{n}-torsion subgroup scheme E(p^{n}) of E. Hence the Weil pairing on E(p^{n}) induces a nondegenerate bilinear map of group schemes over O_{K}

(3.15)  ( , )_{p^{n}} : Ê(p^{n}) × Ê(p^{n}) ⟶ 𝔾_{m}(p^{n}),

where 𝔾_{m}(p^{n}) is the p^{n}-torsion subgroup scheme of the multiplicative formal group law 𝔾_{m}. Let Q_{n} ∈ Ê(M_{ℂ_{p}}) be a point of order p^{n}. Then

(3.16)  (Q_{n}, ∗)_{p^{n}} : Ê(p^{n}) ⟶ 𝔾_{m}(p^{n})

is a surjective group scheme homomorphism defined over O_{ℂ_{p}}. By choosing a sequence (Q_{n})_{n≥1} of points of order p^{n} such that Q_{n} = [p]_{E}(Q_{n+1}) for all n ≥ 1 we get a surjective homomorphism ϕ_{p} : Ê(p^{∞}) → 𝔾_{m}(p^{∞}) of p-divisible groups over O_{ℂ_{p}}.

Let R be a complete Noetherian ring whose residue field has characteristic p. In [12, Prop. 1] Tate showed that the functor F ↦ F (p^{∞}) gives an equivalence from the category of formal group laws of finite height over R to the category of connected p-divisible groups over R. The proof that the induced map from Hom_{R}(F, G) to Hom_{R}(F (p^{∞}), G(p^{∞})) is a bijection is valid for non-Noetherian rings R as long as there is a Noetherian subring R_{0} of R such that the formal group laws F (X, Y) and G(X, Y) are both defined over R_{0}. Hence we can apply Tate’s result to show that ϕ_{p} is induced by a nonzero homomorphism ϕ : Ê → 𝔾_{m} which is defined over O_{ℂ_{p}}. The kernel of ϕ is the infinite subgroup of Ê(M_{ℂ_{p}}) generated by the set {Q_{n} : n ≥ 1}. It follows from Remark 3.6 that ϕ(X) is not defined over any discretely valued subfield of ℂ_{p}. Therefore the conclusion to Theorem 3.2 does not hold if we allow ker(ϕ) to be infinite.

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450 Kevin Keating

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Department of Mathematics, University of Florida, Gainesville, FL 32611, USA keating@ufl.edu

This paper is available via http://nyjm.albany.edu/j/2009/15-25.html.