The Artin-Hasse Power Series and p-adic Group Rings. *)

by

KLAUS HOECHSMANN Department of Mathematics, University of British Columbia, Vancouver, B.C., V6T 1Y4, Canada.

and

JÜRGEN RITTER Mathematisches Institut, Universität Augsburg, 8900 Augsburg, Germany.

ABSTRACT:

If p is an odd prime and A an abelian p-group, consider the multiplicative group formed by {1 + ∑_{z∈A} a_{z}(z − 2 + z^{−1}) | a_{z} ∈ ZZ_{p}} in the p-adic group ring. This group is shown to be a ZZ_{p}-module with the basis {E(z − 2 + z^{−1}) | 1 ≠ z ∈ A}, where E(T) denotes the well-known p-adically integral power series exp ( ∑_{n≥0} p^{−n}T^{p^{n}}).

*) Support of this work by DFG (Germany) and NSERC (Canada) is gratefully acknowledged.

1. Introduction.

Let p^{m} be a fixed odd prime power, A a finite abelian group of exponent p^{m}, thus containing a maximal cyclic subgroup C of order p^{m}. This paper will be devoted to the proof of a single theorem concerning the problem of generating the multiplicative ZZ_{p}-module UZZ_{p}A of units in the p-adic group ring ZZ_{p}A. Obviously it is enough to consider the subgroup U_{1}ZZ_{p}A of units whose augmentation (i.e., coefficient sum) equals 1. Moreover, since one is ultimately interested in the global group ring ZZA, one might as well concentrate on the “real” component U_{1}^{+}ZZ_{p}A, consisting of units fixed under the ring involution ⋆ defined by z^{⋆} = z^{−1} for z ∈ A.

In the ring of formal power series Q[[T]] consider the element

E(T) = exp P (T)  where  P (T) = ∑_{n=0}^{∞} p^{−n}T^{p^{n}}.  (1)

It is well known that E(T), which was invented in 1928 by Artin and Hasse (cf. [1]), actually lies in ZZ_{p}[[T]] (cf. [6], for a quick proof). Therefore it makes sense to substitute T ↦ t, if t is in some ZZ_{p}-algebra, and t^{n} converges to 0 in the p-adic topology. This works, in particular, for t = z − 1 ∈ ZZ_{p}A with z ∈ A, because the p^{m}-th power of that element is divisible by p. As an easier example, we may take the finite group ring IF_{p}A, and again put t = z − 1 for any z ∈ A.

Here is our main result.

Theorem 1.1: U_{1}^{+}ZZ_{p}A has a ZZ_{p}-basis consisting of the elements E(z − 1)E(z^{−1} − 1), or alternatively of E(z^{−1} − 2 + z), as (z, z^{−1}) ranges over all reciprocal pairs of non-trivial elements of A.

An earlier paper [4] on “local units” already proved the existence of a polynomial f(X) ∈ ZZ_{p}[X] with f(1) = 0 and such that the set {1 + f(z) + f(z^{−1}) | 1 ≠ z ∈ A} forms a basis of U_{1}^{+}ZZ_{p}A. This was merely inferred from work of BoreviVc based on local class field theory, and gave neither an indication of the shape of these generic local units w(z) = 1 + f(z) + f(z^{−1}) nor any assurance that they could be chosen in a systematic way for m → ∞. In that context, the main point of Theorem 1.1 is to furnish a direct proof and exhibit explicit generators.

As shown in [4], the crux of the problem resides in the cyclic case A = C. Once a basis of the form {w(z) | 1 ≠ z ∈ C} is known to exist for U_{1}^{+}ZZ_{p}C, the transition to more general A is straightforward (cf. loc. cit., §4). We shall not here repeat this transition, but henceforth concentrate entirely on the cyclic group C. With A = C, the present paper can be read as a self-contained essay on the relation between E(T) and the ⋆-symmetric units in the cyclic group ring ZZ_{p}C. Taking the projective limit over m, the reader may wish to reformulate our result in terms of units in the ring ZZ_{p}[[T]] of p-adically integral power series.

The unit problem will be be reduced (in section 4 below) to the modular units U_{1}^{+}IF_{p}C by a kind of Kummer-Nakayama principle. Of course, one cannot expect a basis in the modular case, so we will simply be seeking a set of generators. For later reference, we formulate the result as a separate proposition.

Proposition 1.2: The multiplicative group U_{1}^{+}IF_{p}C = 1 + Δ^{+}IF_{p}C can be generated by either of the sets (a) {E(z − 1)E(z^{−1} − 1) | z ∈ C} or (b) {E(z^{−1} − 2 + z) | z ∈ C}.

In their joint paper [1], Artin and Hasse also used E(T) to generate a finite multiplicative group associated with ZZ_{p}C — indeed a factor group of U_{1}IF_{p}C — namely the kernel of (ZZ[ζ]/(ζ − 1)^{M}ZZ[ζ])^{×} ⟶ IF_{p}^{×}, where ζ denotes a primitive p^{m}-th root of 1, the arrow means taking classes modulo (ζ − 1), and M equals mp^{m} − (m − 1)p^{m−1} + 1. They used elements of the form E((ζ − 1)^{a}) instead of E((ζ^{a} − 1)) and did not need to limit themselves to a proper subgroup of U_{1}, as we definitely must (cf. Remark 3.7 below). It seems unlikely that the present result could be derived directly form their work.

The proof of Proposition 1.2 encounters two major technical obstacles.

(i) The identity E(T) = exp P (T) obviously requires at least rational coefficients. However, the local ring Q[[T]] does not admit any homomorphisms into a group ring. In fact, any relation of the form (1 + T)^{n} − 1, with n > 0, forces T into triviality.

(ii) There is no satisfactory logarithm connecting the multiplicative group U_{1}IF_{p}C = 1 + ΔIF_{p}C to the additive group ΔIF_{p}C, except in the almost trivial case |C| = p. However, it is only in the additive group that we can see exactly what we have got.

To get around (i), we break the homomorphism ZZ_{p}[[T]] ⟶ IF_{p}C into two stages

ZZ_{p}[[T]] ⟶ ZZ_{p}[Ṫ] ⟶^{ρ} IF_{p}C,  (2)

with Ṫ^{p^{m}} = 0. The intermediate stage can now be embedded in the artinian local ring Q_{p}[Ṫ], in which (1) still holds but certain denominators no longer intrude, because they were attached to higher powers of T.

To overcome (ii), we use induction and the exact sequence

1 ⟶ ΔR(C, C_{p}) ⟶ RC ⟶^{π} RC^{p} ⟶ 1,  (3)

where C_{p} = {z ∈ C | z^{p} = 1}, R denotes some coefficient ring, and π is the ring epimorphism induced by z ↦ z^{p}. Of course R will be either ZZ_{p} or IF_{p}, and we are chiefly concerned with the restriction of (3) to the unit groups, where the kernel of π : U_{1}RC ⟶ U_{1}RC^{p} is precisely U_{1}R(C, C_{p}) = 1 + ΔR(C, C_{p}).

From now on, we shall fix a generator x of C and put t = x − 1. It is easy to see that ΔIF_{p}(C, C_{p}) = ker π = t^{q}IF_{p}[t], where q = p^{m−1}. Since (t^{q})^{p} = 0 in IF_{p}C, the substition of arguments from t^{q}IF_{p}[t] leads to the vanishing of the higher terms (beyond the (p−1)-st power) in the exponential and logarithmic series. This yields an isomorphism exp_{0} : t^{q}IF_{p}[t] ⟶ 1 + t^{q}IF_{p}[t] between additive and multiplicative groups, whose inverse is log_{0}, where

exp_{0} T = ∑_{0≤n<p} T^{n}/n! and log_{0}(1 − T) = − ∑_{1≤n<p} T^{n}/n.  (4)

To recall how this works, we begin with the commutative square

Ṫ^{q}ZZ_{p}[Ṫ] ⟶ Ṫ Q_{p}[Ṫ]
exp_{0} ↓  ↓ exp
1 + Ṫ^{q}ZZ_{p}[Ṫ] ⟶ 1 + Ṫ Q_{p}[Ṫ]

(5)

again with (Ṫ^{q})^{p} = Ṫ^{p^{m}} = 0, the horizontal arrows representing inclusions. In Q_{p}[Ṫ], the familiar exp − log mechanism functions formally and induces all the usual properties and identities on exp_{0} and log_{0} by restriction. These are then transferred to IF_{p}[t] by reduction modulo p.

The automorphism group G = Aut(C) operates naturally on most objects under consideration. Given a natural number c prime to p, we let σ(c) denote the automorphism defined by x ↦ x^{c}. If m > 1, we shall be especially interested in τ = σ(1 + p^{m−1}), which generates the subgroup G_{p} of order p. Note that U_{1}IF_{p}C and ΔIF_{p}C are ZZ_{p}G-modules, whereas U_{1}IF_{p}(C, C_{p}) and ΔIF_{p}(C, C_{p}), being elementary abelian, can be considered as IF_{p}G-modules.

Section 2 provides some technical nuts and bolts which are essential to the proof, given in section 3, of the proposition. The latter is then used in section 4 to prove the theorem.

2. Preliminaries.

This section is somewhat technical and should probably be skimmed over at first reading. Though it is logically prior to what follows, it can be properly understood only in retrospect. Most of the work will take place in one of the rational algebras Q[X] ⊂ Q[X, X^{−1}] ⊂ Q[[T]], where T = X − 1.

We begin in the infinite cyclic group algebra Q[X, X^{−1}], on which every integer c yields an algebra endomorphism σ(c) by σ(c) : X ↦ X^{c}. Since Q[X, X^{−1}] is (T)-adically dense in the algebra Q[[T]] of formal power series, we can extend σ(c) to the latter by continuity. Obviously σ(c) can also be restricted to ZZ[X, X^{−1}] and ZZ[[T]], and all these games can be played with p-adic coefficients as well. If c is positive, σ(c) maps the polynomial algebra Q[X] into itself. In the latter part of this section, the automorphism ⋆ = σ(−1) will play a special role.

For now we fix some pair l, n of positive integers and concentrate our attention on the endomorphism τ = σ(1 + q), with q = p^{l}. Our aim is to relate the n-th summand of (τ − 1)P (T) to some reasonable object without p-denominators, namely the polynomial g_{n}(T) ∈ ZZ_{p}[T] defined by

g_{n}(T) = ∑_{h∈J(p^{n})} h^{−1}(1 + T)^{h},  (6)

where J(p^{n}) = {h ∈ ZZ | (h, p) = 1, 0 < h < p^{n}}. For brevity, let us use the notation A ≅ B to mean that two elements A and B of the algebra Q_{p}[[T]] differ by an element of the ZZ_{p}[[T]]-submodule T^{p^{m}} Q_{p}[[T]]+p T ZZ_{p}[[T]], where m = l+1. This is clearly related to the two maps shown in (2): the first of these summands disappears in the transition T ↦ Ṫ, the second is killed by ρ.

Lemma 2.1: For every integer h > 0, there exists a polynomial φ_{h}(T) ∈ ZZ_{p}[T] without constant coefficient such that

(τ − 1) \frac{1}{h}X^{h} ≅ X^{h}φ_{h}(T^{q}).  (7)

Proof: Since τ(X) = X^{1+q}, the left hand side is just h^{−1}X^{h}(X^{hq} − 1). Setting k = hq, we now convert h^{−1}(X^{k} − 1) = qk^{−1}(X^{k} − 1) into a polynomial in T, to see which powers of T are really afflicted by non-integral (∉ ZZ_{p}) coefficients. Equating the derivatives of both sides in

\frac{1}{k} [(1 + T)^{k} − 1] = ∑_{j=1}^{k} \frac{d_{k,j}}{j} T^{j},  (8)

we see that d_{k,j} ∈ ZZ. If we now multiply (8) through by q, the coefficients occurring on the right hand side will lie in pZZ_{p}, ZZ_{p}, or Q, depending on whether ord_{p} j is smaller than, equal to, or greater than l = ord_{p} q, respectively. Accordingly, we can write

h^{−1}(X^{hq} − 1) = p ε_{h}(T) + φ_{h}(T^{q}) + θ_{h}(T^{p^{l+1}}) ≅ φ_{h}(T^{q}),  (9)

where ε_{h}, φ_{h}, and θ_{h} are polynomials without constant terms, the first two having coefficients in ZZ_{p}. This proves our first lemma.

Remark 2.2: For later reference, we must point out the trivial fact that X^{h} can be replaced by (1 + T^{p})^{i} in (7), if it so happens that h = ip.

Lemma 2.3: There is a polynomial f_{n}(T) ∈ T^{q}ZZ_{p}[T^{p}] such that

(τ − 1) p^{−n}T^{p^{n}} ≅ f_{n}(T) + (τ − 1) g_{n}(T).  (10)

Proof: We temporily use the abbreviation p^{n} = N, and first compute

N^{−1}T^{N} = N^{−1}(X − 1)^{N} = ∑_{k=0}^{N} a_{N,k} X^{N−k} = ∑_{h=0}^{N} b_{N,h} X^{h},  (11)

with the coefficients b_{N,h} = a_{N,N−k}, where a_{N,k} = (−1)^{k} \frac{1}{N} \binom{N}{k}. In particular,

a_{N,k} = −\frac{1}{k} ∏_{i=1}^{k−1} (1 − \frac{N}{i} ) = \frac{1}{N − k} ∏_{i=1}^{k} (1 − \frac{N}{i} ),  (12)

for 0 < k < N. Note that each of the factors 1 − N/i occurring in (12) represents an element of ZZ_{p} which is ≡ 1 modulo p. Hence b_{N,h} = h^{−1}c_{N,h} with c_{N,h} ∈ 1 + pZZ_{p}, for h = 1, …, N. Since τ − 1 annihilates constant terms (h = 0), we can write (τ − 1)N^{−1}T^{N} as the sum of (τ − 1)b_{N,h}(1 + T)^{h}, with h ≥ 1. For indices h such that 1/h ∈ ZZ_{p}, we can immediately replace these terms modulo p to get

(τ − 1) N^{−1}T^{N} ≅ (τ − 1)g_{n}(T) + (τ − 1) ∑_{p|h, h≤N} \frac{c_{N,h}}{h} X^{h}  (13).

Substituting (7) into (13), putting h = ip, and applying Remark 2.2, we now define

f_{n}(T) = ∑_{ip≤N} c_{N,ip} (1 + T^{p})^{i}φ_{ip}(T^{q})  (14)

to finish the proof.

Going back to the infinite cyclic group algebra Q[X, X^{−1}], we single out the fundamental ⋆-symmetric element

S = T + T^{⋆} = (X − 1) + (X^{−1} − 1) = X^{−1}(X − 1)^{2}.  (15)

We want to do for P (S) what we just did for P (T). It will be convenient to use the notation f^{+}(T) = f(T) + f(T^{⋆}), for any power series f(T). The proof of the following lemma is a very close imitation of the preceding one.

Lemma 2.4: There is a polynomial v_{n}(T) ∈ T^{q}ZZ_{p}[T^{p}] such that

(τ − 1) p^{−n}S^{p^{n}} ≅ v_{n}^{+}(T) + 2(τ − 1) g_{n}^{+}(T).  (16)

Proof: Again putting N = p^{n}, we binomially compute

N^{−1}S^{N} = ∑_{k=0}^{N} a′_{N,k} (X^{N−k} + X^{k−N}) = ∑_{h=0}^{N} b′_{N,h} X^{h} + ∑_{h=0}^{N} b′_{N,h} X^{−h},  (17)

with the coefficients b′_{N,h} = a′_{N,N−k}, where a′_{N,k} = (−1)^{k} \frac{1}{N} \binom{2N}{k} for k < N and half of that for k = N; in particular,

a′_{N,k} = −\frac{2}{k} ∏_{i=1}^{k−1} (1 − \frac{2N}{i} ) = \frac{2}{N − k} ∏_{i=1}^{k} (1 − \frac{2N}{i} ),  (18)

for 0 < k < N. Again b′_{N,h} = h^{−1}c′_{N,h}, but this time c′_{N,h} ∈ 2 + pZZ_{p}, for h = 1, …, N. We now work with only the first part of the final sum in (17) and get

(τ − 1) ∑_{h=0}^{N} b′_{N,h} X^{h} ≅ (τ − 1) 2g_{n}(T) + v_{n}(T)  (19)

in analogy to (13) and (14), with v_{n}(T) defined like f_{n}(T), but having coefficients c′_{N,ip}.

3. Modular Units.

In this section the proof of Proposition 1.2 will be given after a series of lemmas. The inductive step of the proof will concentrate our attention on the small IF_{p}G-module

V IF_{p}(C, C_{p}) = U_{1}IF_{p}(C, C_{p})/U_{1}IF_{p}(C^{p}, C_{p})  (20)

and its ⋆-symmetric component V^{+}IF_{p}(C, C^{p}).

Lemma 3.1: exp_{0}(τ − 1)t is an IF_{p}G-generator of V IF_{p}(C, C_{p}). Proof: We hasten to point out that exp_{0}(τ − 1)t is not the same as (exp_{0} t)^{τ−1}, because exp_{0}, though compatible with the action of G, is not a homomorphism on all of ΔIF_{p}C. However, since it does induce a G-isomorphism between ΔIF_{p}(C, C_{p})/ΔIF_{p}(C^{p}, C_{p}) and V IF_{p}(C, C_{p}), we are reduced to showing that (τ − 1)t is an IF_{p}G-generator of the first of these modules.

Now we observe, for any coefficient ring R, that the kernel ΔR(C, C_{p}) of R[C] ⟶^{π} R[C^{p}] has a basis of the form {z(y − 1)}, where 1 ≠ y ∈ C_{p} and z runs over a system of representatives of C modulo C_{p}. Of these elements, the ones with z ∈ C^{p} form a basis of ΔR(C^{p}, C_{p}). On the other hand, if z ∉ C^{p}, then z = x^{a} and y = z^{bq}, with a, b prime to p, and q = p^{m−1}. Thus z(y − 1) = z^{1+bq} − z = (τ^{b} − 1)σ(a)x. Since (τ − 1)x = (τ − 1)t, we finally have z(y − 1) = γ_{a,b} · (τ − 1)t, where γ_{a,b} = (1 + τ + ⋯ + τ^{b−1})σ(a) ∈ ZZG, proving the lemma.

Remark 3.2: The main step in proving Proposition 1.1 (cf. 3.5 below) will involve the comparison of exp_{0}(τ − 1)t and E(t)^{τ−1}. This will be foreshadowed by Lemmas 3.3 and 3.4, which are transplants into IF_{p}C of their counterparts 2.3 and 2.4 from the last section.

Lemma 3.3: In IF_{p}C we have

E(t)^{τ−1} ≡ ∏_{n=0}^{m−1} exp_{0} ((τ − 1)g_{n}(t)) modulo U_{1}IF_{p}(C^{p}, C_{p}),  (21)

where g_{0}(T) = T.

Proof: Applying (10) to the nilpotent element Ṫ and summing over n < m, we can immediately conclude that (τ − 1)P (Ṫ) ∈ ZZ_{p}[Ṫ]. More precisely,

(τ − 1)P (Ṫ) = p e(Ṫ) + f(Ṫ) + (τ − 1)g(Ṫ),  (22)

where e(T) ∈ T ZZ_{p}[T], f(T) ∈ T^{q}ZZ_{p}[T^{p}], and g(T) = T +∑_{n<m} g_{n}(T). We still abbreviate p^{m−1} = q.

In Q_{p}[[T]] it is true that E(T)^{τ−1} = exp ((τ − 1)P (T)). From (22) we therefore get

E(Ṫ)^{τ−1} = exp (p e(Ṫ)) exp (f(Ṫ)) ∏_{n=0}^{m−1} exp ((τ − 1)g_{n}(Ṫ)),  (23)

to be read in Q_{p}[Ṫ]. Since p^{n}/n! ∈ ZZ_{p} for all n ≥ 0, the first of these factors is an invertible element of ZZ_{p}[Ṫ]. So is the second factor, which actually equals exp_{0} f(Ṫ), because f(Ṫ) ∈ Ṫ^{q}ZZ_{p}[T_{m}^{p}]. In fact, if we go into IF_{p}C by mapping Ṫ ↦ t, the first two factors together clearly yield nothing more than exp_{0} f(t) ∈ U_{1}IF_{p}(C^{p}, C_{p}).

Modulo that subgroup, all of E(t)^{τ−1} is therefore concentrated in the third factor of (23). To study its n-th constituent, remember that g_{n}(T) is the sum of X^{h}/h over h ∈ J(p^{n}), with 1/h ∈ ZZ_{p}. Now (τ −1)X^{h}/h = (X^{hq}−1)X^{h}/h is of the form p a_{h}(T)+T^{q}b_{h}(T), with a_{h}(T), b_{h}(T) ∈ ZZ[T]. Again the exponential of the first term is in 1 + pT ZZ_{p}[[T]] and that of the second term (with Ṫ instead of T) becomes exp_{0} (Ṫ^{q}b_{h}(Ṫ)). Therefore

exp ((τ − 1)g_{n}(Ṫ)) = ∏_{h∈J(p^{n})} exp (p a_{h}(Ṫ)) exp_{0} (Ṫ^{q}b_{h}(Ṫ)) ∈ ZZ_{p}[[Ṫ]].  (24)

Clearly this maps to the product of the exp_{0} (t^{q}b_{h}(t)) in IF_{p}C. Since (τ − 1)x^{h} = t^{q}b_{h}(t) in IF_{p}C, it gives exactly what we wanted.

Lemma 3.4: Putting s = t + t^{⋆} ∈ IF_{p}C, we have

E(s)^{τ−1} ≡ exp_{0} ((τ − 1)s) ∏_{n=1}^{m−1} exp_{0} ((τ − 1) 2g_{n}^{+}(t)) modulo U_{1}IF_{p}(C^{p}, C_{p}). (25)

Proof: This argument is a close parallel to the preceding one, with equation (16) replacing (10). The details of it are left to the reader. The strategy is to use the factorization (2), taking great care to get rid of all p-denominators before trying to apply the reduction modulo p.

For the sequel, it is essential to reformulate Lemmas 3.3 and 3.4 in terms of certain elements γ_{n} of the group ring IF_{p}G, defined as follows:

γ_{n} = ∑_{h∈J(p^{n})} h^{−1}σ(h),  (26)

where J(p^{n}) is the set of integers {h | (h, p) = 1, 0 < h < p^{n}} first encountered in (6). To make the transition, observe that (τ − 1)x^{h} = (τ − 1)(x^{h} − 1) = σ(h)(τ − 1)t and similarly (τ − 1)(x^{h} + x^{−h}) = σ(h)(τ − 1)s.

Corollary 3.5: In U_{1}IF_{p}(C, C_{p}) we have the following congruences mod U_{1}IF_{p}(C^{p}, C_{p}):

(a) E(t)^{τ−1} ≡ ( exp_{0}(τ − 1)t)^{α}  and  (b) E(s)^{τ−1} ≡ ( exp_{0}(τ − 1)s)^{β} (27)

where α = 1 + ∑_{n<m} γ_{n} and β = 1 + ∑_{n<m} 2γ_{n} in IF_{p}G.

Before we can finally present the proof of Proposition 1.2, we still need to look into the nature of γ_{n}. With p^{m−1} = q and m > 1, we consider its q-th power and find that

γ_{n}^{q} = ∑_{h∈J(p^{n})} h^{−q}σ(h^{q}) = p^{n−1} ∑_{k∈J(p)} k^{−1}σ(k^{q}),  (28)

because the q-th power induces a group epimorphism G ⟶ G_{p−1} onto the subgroup of order p − 1, which gives every element of G_{p−1} exactly p^{n−1} pre-images in J(p^{n}). Since this argument involves the numerical set J(p^{n}) and not a group of residue classes, we should examine it more carefully. For any two integers a and b, we observe that a ≡ b modulo p if and only if a^{p^{l}} ≡ b^{p^{l}} modulo p^{l+1}. Hence, representing elements h ∈ J(p^{n}) in the form h = h_{0} + h_{1}p + ⋯ + h_{n−1}p^{n−1} with 0 ≤ h_{i} ≤ p − 1, we see that σ(h^{q}) ∈ G depends only on the value of h_{0} ∈ J(p) and thus occurs p^{n−1} times — whence (28). In particular, γ_{n}^{q} = 0, for all n > 1; hence α^{q} = 1 + γ_{1}^{q} and β^{q} = 1 + 2γ_{1} always.

(We note in passing that γ_{1}^{q} still depends on m, since G does.)

Using the symmetry k ↦ p − k on J(p) and the fact that σ((p − k)^{q}) = σ(−k^{q}), we can rewrite the summands on the right hand side of (28) as (p − k)^{−1}σ(−k^{q}). Together with the obvious identity σ(−h) = σ(−1)σ(h) this produces the observation

σ(−1) γ_{1}^{q} = ∑_{k∈J(p)} k^{−1}σ(−k^{q}) = −γ_{1}^{q},  (29)

which immediately implies the following result.

Lemma 3.6: 1 + σ(−1) is annihilated by γ_{1}^{q} and left fixed by α^{q} and β^{q}. Therefore α and β act as automorphisms on any IF_{p}G- module left fixed by ⋆ = σ(−1).

Remark 3.7 The preceding lemma will be applied to the ⋆-symmetric part D^{+} of the module D = ΔIF_{p}(C, C_{p})/ΔIF_{p}(C^{p}, C_{p}). Since E(x − 1)^{τ−1} falls into the image of α, one can easily show that the E(z − 1) would generate all of U_{1}IF_{p}C if and only if αD were all of D. The necessity of restricting Proposition 1.2 to U_{1}^{+}IF_{p}C is due to the inequality αD ≠ D, which can be seen as follows.

First we compute

(γ_{1}^{q})^{2} = ∑_{h} ∑_{k} \frac{1}{hk} σ(h^{q}k^{q}) = (p − 1)γ_{1}^{q},  (30)

with h and k running over J(p). This says (γ_{1}^{q})^{2} = −γ_{1}^{q}, and therefore α^{q}γ_{1}^{q} = 0. Now, if it were true that αD equals D, we would have γ_{1}^{q}D = 0. However, this is easily shown to be false, by checking that the G_{p−1}-orbit of the element x(x^{q} − 1) is IF_{p}-linearly independent modulo ΔIF_{p}(C^{p}, C_{p}).

Proof of 1.2: We proceed by induction on m; the case m = 1 is nearly trivial. In fact, we have P (Ṫ) = Ṫ, because Ṫ^{p} = 0. In Q_{p}[[Ṫ]]] we therefore obtain E(Ṫ) = exp Ṫ = exp_{0} Ṫ. Since log_{0} : 1 + tIF_{p}[t] ⟶ tIF_{p}[t] is a G-isomorphism, and since log_{0}(exp_{0} t) = t is an IF_{p}Ggenerator of tIF_{p}[t], it follows that E(t) = exp_{0} t is an IF_{p}G-generator of U_{1}IF_{p}C. This first case does not yet require the restriction to U_{1}^{+}.

If m > 1, we use the multiplicative version of the exact sequence (3), but restrict it to ⋆-invariant units, and we abbreviate E(T)E(T^{⋆}) by E^{+}(T).

(a) For u ∈ U_{1}^{+}IF_{p}C, the induction hypothesis says that u^{p} = ∏_{z} E^{+}(z^{p} −1)^{a_{z}}, with z running over a system of representatives of C modulo C_{p}. Dividing u by ∏_{z} E^{+}(z −1)^{a_{z}}, if necessary, we may assume that u^{p} = 1. Thus it suffices to show that all of U_{1}^{+}IF_{p}(C, C_{p}) is contained in the (multiplicative) ZZG-module U_{E}^{+}IF_{p}C generated by E^{+}(t), E^{+}(πt), …, E^{+}(π^{m−1}t), where again t = x − 1 for our fixed generator x of C.

The induction hypothesis now assures us that at least U_{1}^{+}IF_{p}(C^{p}, C_{p}) ⊂ U_{E}^{+}IF_{p}C^{p}. Hence it suffices to generate the quotient V^{+}IF_{p}(C, C_{p}). By Lemma 3.1 this can be accomplished using exp_{0}(τ − 1)t · exp_{0}(τ − 1)t^{⋆}. Rewriting it as ( exp_{0}(τ − 1)t)^{1+σ(−1)} and remembering that α(1+σ(−1))α′ = (1+σ(−1)), where α′ = α^{q−1}, this generator becomes

( exp_{0}(τ − 1)t)^{α(1+σ(−1))α′} ≡ E^{+}(t)^{(τ−1)α′} modulo U_{1}^{+}IF_{p}(C^{p}, C_{p}),  (31)

by Corollary 3.5 (a). Part (b) is proved in an almost identical pattern, which we leave to the reader to write down.

4. Proof of Theorem.

In this section we derive a principle (cf. 4.3 below) which turns Proposition 1.2 into Theorem 1.1 (for A = C). It depends heavily on the existence of the eponential isomorphism

Δ^{+}ZZ_{p}(C, C_{p}) ⟶^{exp} U_{1}^{+}ZZ_{p}(C, C_{p}),  (32)

and its logarithmic inverse. Both exp and log are of course given by the standard power series. The convergence of the exp-series is essentially due to the fact that Δ^{+}ZZ_{p}(C, C_{p}) is contained in the ideal product ΔZZ_{p}(C, C_{p}) · ΔZZ_{p}C — cf. [5] for details. The failure of this exp − log machine to work for the whole of ΔZZ_{p}(C, C_{p}) provides the second reason for restricting our attention to ⋆-invariant units, the first being given by the quirks of α discussed in the last section.

The following easy lemma has already appeared in [3], and is repeated here for the sake of completeness. It may be regarded as a p-adic analog of Kummer’s lemma concerning cyclotomic units for regular primes.

Lemma 4.1: The sequence

U_{1}^{+}ZZ_{p}C ⟶^{π−p} U_{1}^{+}ZZ_{p}C ⟶^{ρ} U_{1}^{+}IF_{p}C  (33)

is exact. Here ρ as usual means reduction of coefficients, and u^{π−p} stands for u^{π}/u^{p}. Proof: We shall use induction on m, this time starting at m = 0, where there is nothing to prove. If m > 0 consider the more familiar exact sequence

1 ⟶ U_{1}^{+}ZZ_{p}(C, C_{p}) ⟶ U_{1}^{+}ZZ_{p}C ⟶^{π} U_{1}^{+}ZZ_{p}C^{p} ⟶ 1,  (34)

and take u ∈ U_{1}^{+}ZZ_{p}C such that ρ(u) = 1. Applying the induction hypothesis to u^{π}, we find a w ∈ U_{1}^{+}ZZ_{p}C with u^{π} = (w^{π})^{π−p}. Hence u_{1} = u · w^{p−π} is in U_{1}^{+}ZZ_{p}(C, C_{p}), the kernel of π.

Since any w^{p−π} is obviously in the kernel of ρ, we have ρ(u_{1}) = 1. Therefore u_{1} ∈ 1 + p Δ^{+}ZZ_{p}(C, C_{p}) and log u_{1} ∈ p Δ^{+}ZZ_{p}(C, C_{p}) is a multiple of p. By the isomorphism (32) we conclude that u_{1} is a p-th power, say u_{1} = v^{p}, with v ∈ U_{1}^{+}ZZ_{p}(C, C_{p}). Since v^{π} = 1, we have u_{1} = v^{p−π}.

Remark 4.2: It must be emphasized that the present log is very different from the log_{0} used earlier in this paper. The diagram

U_{1}^{+}ZZ_{p}(C, C_{p}) ⟶^{log} Δ^{+}ZZ_{p}(C, C_{p})
ρ ↓  ↓ ρ
U_{1}^{+}IF_{p}(C, C_{p}) ⟶^{log_{0}} Δ^{+}IF_{p}(C, C_{p})

(35)

is NOT commutative, unless m = 1. The difference between ρ log(1 + a) and log_{0} ρ(1 + a) is just a^{p}/p, and it is an amusing exercise to verify that 1 + a ↦ a^{p}/p yields in fact a homomorphism from U_{1}ZZ_{p}(C, C_{p}) to ΔIF_{p}(C, C_{p}). Its non-triviality (even for a = a^{∗}) can easily be checked by taking p^{2} = 9 and trying a = x(x^{3} − 1) + x^{−1}(x^{−3} − 1).

Principle 4.3: Let F (T) ∈ ZZ_{p}[[T]] be such that F (0) = 1 and F (T^{⋆}) = F (T). Then the set (F | C) = {F (z − 1) | 1 ≠ z ∈ C} forms a ZZ_{p}-basis of U_{1}^{+}ZZ_{p}C if and only if ρ(F | C) generates the finite group U_{1}^{+}IF_{p}C. Proof: The “only if” being obvious, let us concentrate on the “if”. The proof will go by induction on m with nothing to prove if m = 0. Now take m > 0 and assume that ρ(F | C) generates U_{1}^{+}IF_{p}C. Since U_{1}^{+}IF_{p}C ⟶^{π} U_{1}^{+}IF_{p}C^{p} is surjective and πF (z − 1) = F (z^{p} − 1), this assumption carries over to U_{1}^{+}IF_{p}C^{p}. By induction hypothesis, we conclude that (F | C^{p}) generates U_{1}^{+}ZZ_{p}C^{p} over ZZ_{p}. This will be used presently.

For any u ∈ U_{1}^{+}ZZ_{p}C we now write ρ(u) = ∏_{z} ρF (z − 1)^{a_{z}} in terms of ρ(F | C), with a_{z} ∈ ZZ. Putting v = ∏_{z} F (z − 1)^{a_{z}}, we get uv^{−1} into the kernel of ρ, and by Lemma 4.1, u = v · w^{π}/w^{p}. Recalling that w^{π} is known to be expressible in terms of (F | C^{p}) ⊂ (F | C), we see that u · w^{p} = ∏_{z} F (z − 1)^{c_{z}} with c_{z} ∈ ZZ_{p}. Since u was arbitrary, this says that (F | C) generates the ZZ_{p}-module U_{1}^{+}ZZ_{p}C modulo p-th powers — which suffices by Nakayama’s Lemma.

To see that (F | C) is actually a ZZ_{p}-basis of U_{1}^{+}ZZ_{p}C, we need only check that the cardinality (p^{m} − 1)/2 of the former is equal to the ZZ_{p}-rank of the latter (which clearly is a torsion-free module), i.e. one less than the rank of the full ⋆-symmetric unit group U^{+}ZZ_{p}C. By elementary general principles (cf. [7], II.2.9, p. 49), this rank is equal to that of the multiplicative group of the maximal order in the ⋆-symmetric part ( Q_{p}C)^{+} of the algebra Q_{p}C, which, by standard results of local number theory (cf. [2], §15.5), is the same as the ZZ_{p}-rank of the corresponding additive group, namely 1 + (p^{m} − 1)/2, Q.E.D.

References

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