Exotic Units in Group Rings of Rank p^{2}.

by

Klaus Hoechsmann

0. Introduction. Let A be a finite abelian group, ZZA its integral group ring and UZZA = U(A) the units therein. Consider the cokernel Γ_{U}(A) of the natural map α : ∏ U(C) ⟶ U(A), where C ⊆ A runs over all cyclic subgroups: it constitutes the obstruction to reducing the unit problem to the cyclic case (which is hard enough). By [1], this cokernel is known to be finite. If A is a p-group, so is Γ_{U}(A); and it is trivial if p is regular.

The same is true of the analogous cokernel Γ_{Ω}(A) for the subfunctor Ω(A) ⊆ U(A) of circular units, i.e. those which appear as cyclotomic units under every character of A. In fact, if p is semi-regular (as all primes are conjectured to be), the two cokernels are actually isomorphic, but Ω(C) is more concrete than U(C).

If p is irregular, these cokernels are no longer trivial. For instance, if p = 37 (the first irregular prime) and A non-cyclic of order p^{2}, it has been known for some time that Γ_{U}(A) = Γ_{Ω}(A) is of order p^{5} — cf. [4]. However, the nature of the units in U(A) outside of im(α) remained a mystery. Nobody seemed to have ever seen one, and it was not known whether they could be constructed systematically. The present note deals with these “exotic” units, not only for p = 37 but for all irregular p. For groups A of order p^{2}, generators of Γ_{U}(A) will be explicitly constructed.

What makes Ω(A) attractive is its proximity to the subgroup Λ(A) ⊆ Ω(A) of constructible units. Λ(A) = ∏_{C} W (C) is the direct product of canonical subgroups W (C) ⊆ Ω(C) each of which can be presented in terms of explicit generators and relations. The W (C) are maximal in a precise sense and properly contain the units originally constructed by Bass [1].

It appears that Ω(A)/Λ(A), which obviously has Γ_{Ω}(A) as a homomorphic image, is an intrinsically more important invariant than the latter. As long as A is elementary abelian, the two coincide. For |A| > p, the quotient Ω(A)/Λ(A) is trivial if and only if p is regular — even for cyclic A. For irregular p, the question now arises as to the nature of the units in Ω(A) but ouside of Λ(A). Again we shall give an explicit recipe for groups A of order p^{2} (of course, only the cyclic case remains).

In the interest of unobstructed narrative, we have left out a few fine points which must now be mentioned. If R denotes any commutative coefficient ring, consider the so-called augmentation map RA ⟶ R, which takes every element of RA to its coefficient sum. Its kernel ΔRA is of special interest, as is the subgroup U_{1}RA = URA∩(1+ΔRA). Another special item is the involution ⋆ on RA which is defined by ⋆ : z ↦ z^{−1} for z ∈ A. An element of RA which is left fixed by ⋆ is called ⋆-symmetric. For any subset E ⊆ RA, we denote by E^{+} the ⋆-symmetric component (the “+” alluding to the eigenvalue +1 of the involution). If A is an abelian p-group, another important map is the ring epimorphism π : RA ⟶ RA^{p} produced by taking p-th powers of the elements of A.

In this paper, R will sometimes be IF_{p} and rarely ZZ_{p}, the ring of p-adic integers. Most of the time, we shall deal with R = ZZ, in which case we drop the reference to R and write UZZA = U(A), etc. For R = ZZ and |A| odd, it so happens that U_{1}^{+}ZZA is torsion-free and the full unit group is a direct product: UZZA = ±A × U_{1}^{+}ZZA. To avoid torsion, we shall work with the functor U_{1}^{+}(A) = U_{1}^{+}ZZA rather than with all of U(A). The group Ω(A) does not need to be similarly restricted: it is defined as a subgroup of U_{1}^{+}(A) to begin with.

In the perspective of K-theory, the inclusion U(C) ↪ U(A) (which takes place in K_{1}) is related to the K_{0}-operation of forming an induced representation. Therefore the image of this inclusion may be thought of as the units induced from C. The main device used here in the construction of exotic units is a kind of “co-inclusion” of units from a factor group. Such units will therefore be called co-induced. This process will be explained in Section 1 below. Sections 2 and 3 recall and rearrange known material for the present purpose. In Section 4, we describe generators for Γ_{Ω}(A) in case A is non-cyclic of order p^{2}. In Section 5, the same is done for Ω(A)/Λ(A) in the cyclic case.

1. Co-induced Units. Let ε : A ⟶ A′ be an epimorphism of finite abelian groups, with kernel H ⊆ A of order h. If S_{H} denotes the sum, in ZZA, over all elements of H, we have an induced bijection ε_{0} : S_{H} · ΔZZA ⟶^{∼} h · ΔZZA′, which is an isomorphism of ZZAmodules (ideals) in the obvious sense. Therefore it yields an isomorphism of multiplicative semi-groups ε_{1} : 1+S_{H} ·ΔZZA ⟶^{∼} 1+h·ΔZZA′, which finally can be restricted to invertible elements, to produce a group isomorphism

ε_{1} : (1 + S_{H} · ΔZZA) ∩ UZZA ⟶^{∼} (1 + h · ΔZZA′) ∩ UZZA′  (3)

between the groups of units which are ≡ 1 modulo S_{H} and h, respectively. We are mainly interested in the inverse ε̃ of this group isomorphism, which is explicitly given as follows:

ε̃: 1 + h ∑_{z∈A′} a_{z} · z ⟼ 1 + ∑_{x∈A} a_{ε(x)} · x = 1 + S_{H} ∑_{y∈R(H)} a_{ε(y)} · y,  (4)

where R(H) ⊂ A is any system of representatives of A/H. Obviously, if u = 1+h ∑ a_{z} ·z ∈ U_{1}^{+}ZZA′ then ε̃(u) ∈ U_{1}^{+}ZZA. Moreover, if ψ : A ⟶ F^{×} is any homomorphism from A to the multiplicative group of a field F, we have ψ(S_{H}) = h or = 0 depending on whether or not H ⊂ ker ψ, or equivalently, whether or not ψ = ε ∘ ψ′ for suitable ψ′ : A′ ⟶ F^{×}. Therefore

ψ(ε̃(u)) = \{ \begin{matrix}ψ′(u) & if ψ(H) = 1, \\ 1 & otherwise, \end{matrix}  (5)

if u = 1 + h ∑ a_{z} · z ∈ U_{1}ZZA′ as above. In particular, if u is circular, so is ε̃(u). Let us summarize what we have found.

Proposition 1: The map ε̃ given by (4) injects the group U_{h}ZZA′ of units ≡ 1 mod h into UZZA, preserving ⋆-symmetry and circularity.

We remark in passing that the commutativity of A and A′ has so far not been used: if H is a normal subgroup in any finite group A, the element S_{H} will be central in ZZA, and everything works out verbatim. Of course, the part about ψ : A ⟶ F^{×} may not always be terribly relevant.

Remark: In the present note, we will always take H to be of prime order h = p. The domain of ε̃ will then consist of ker ρ_{A′}, where ρ_{A′} : UZZA′ ⟶ UIF_{p}A′ denotes reduction (mod p) of coefficients. Our task is to study the image of the injection ε̃: ker ρ_{A′} ↪ UZZA. This will be done for A of order p^{2}.

2. Circular and Constructible Units. This section is a partial synopsis of [5]. Let q = p^{m} for some m ≥ 1, and let ζ = ζ_{q} denote a primitve q-th root of unity. Then

G = (ZZ/qZZ)^{×} acts as the Galois group on Q(ζ), and G_{+} = G/{±1} plays a similar role for the maximal real subfield Q(ζ + ζ^{−1}). Abbreviating by Δ(G) the augmentation ideal ΔZZG, one easily checks that

Ω(ζ) := [ζ^{(q−1)/2}(1 − ζ)]^{Δ(G)}  (6)

is a subgroup of the unit group U^{+}(ζ) of this real subfield. By a classical result of analytic number theory, the index of Ω(ζ) in U^{+}(ζ) is a finite number h_{q}^{+} (cf. [8], Theorem 8.2). It is conjectured that h_{q}^{+} is always prime to p (“Vandiver’s Conjecture”).

If A is a finite abelian group such that A^{q} = 1, an element u ∈ U_{1}^{+}(A) is called circular if ψ(u) ∈ Ω(ζ) for every character ψ : A ⟶ Q(ζ)^{×}. The group Ω(A) of circular units obviously has finite index in U_{1}^{+}(A), and if Vandiver’s Conjecture holds for p, this index is also prime to p. In that case, the p-adic completions of these two free ZZ-modules coincide; i.e. from the p-adic point of view one is as good as the other.

Of course, Ω(A) contains the product ∏_{C} Ω(C), where C ⊆ A runs over all cyclics, but we get a more interesting object Λ(A) = ∏_{C} W (C) by constraining the factors to certain canonical subgroups W (C) ⊆ Ω(C). Each W (C) consists of all u ∈ Ω(C) satifying the following uniformity condition: for every character ψ of C, the unit ψ^{p}(u) is the Q(ψ)/ Q(ψ^{p})-norm of ψ(u). The product Λ(A) of the W (C) is then direct, and the index c(A) = [Ω(A) : Λ(A)] is a p-power. Details on this are contained in [5].

For our present purposes, the main properties of these groups are as follows.

(a) If ψ : C ↪ Q(ζ)^{×} is injective, then ψ : W (C) ⟶^{∼} Ω(ζ) ∩ ψ(U_{1}^{+}ZZC) is bijective.

(b) If |C| = q, then W (C) is G-isomorphic to ΔZZG_{+}.

(c) W (C^{p}) = W (C)^{π}, and the product Λ(A) = ∏_{C} W (C) is direct. Roughly speaking (a) claims a kind of maximality for W (C): there is no larger sub-

group of U_{1}^{+}(C) which injects into Ω(ζ). In Theorem 2.3 of [5], (b) is expressed as an isomorphism of W C with ZZG_{+} modulo the sum over all elements of G_{+}; this amounts to the same, because G_{+} is cyclic. A good way of paraphrasing (b) and (c) is to say that there is a polynomial w(x) such that {w(z) | z ∈ A} generates the ZZ-module Λ(A), the only relation between these generators, apart from w(z) = w(z^{−1}), being that the product over each G-orbit is trivial. In [5] an explicit recipe is given for writing down such generic units w(x). Accordingly Λ(A) is called the group of constructible units.

3. Regularity and Kummer Units. In this section we present the usual regularity conditions regarding p, but we give them a group ring slant. If X is a finitely generated ZZ-module, let \overline{X} stand for the IF_{p}-space X ⊗ IF_{p} = X/X^{p}. Likewise, if φ : X ⟶ Y is a homomorphism, \overline{φ} : \overline{X} ⟶ \overline{Y} is its IF_{p}-linear counterpart.

For a group K of order p, the story told in the last section becomes very simple: the uniformity condition defining W (K) is vacuous, and we have W (K) = Ω(K) = Λ(K). Moreover, the group G = Aut(K) is cyclic of order p − 1, and hence Ω(K) and U_{1}^{+}(K) have rank (p − 3)/2. Consider the obvious maps

\overline{Ω}(K) ⟶^{\overline{ι}} \overline{U}_{1}^{+}(K) ⟶^{\overline{ρ}} Ú_{1}^{+}IF_{p}K,  (7)

where the accent on the U means restriction to units whose G-norm is 1. Since each of these IF_{p}-spaces has the same dimension (p − 3)/2, injectivity and surjectivity of each of the maps \overline{ι}, \overline{ρ}, or \overline{ρ} ∘ \overline{ι} are logically equivalent. The prime p is said to be regular if the composite \overline{ρ} ∘ \overline{ι} is injective, semi-regular if \overline{ι} is.

This can be said more pointedly in terms of the IF_{p}G-structure of the underlying spaces and of characters of G. Every homomorphism G ⟶ IF_{p}^{×} is of the form τ^{d} for suitable 1 ≤ d ≤ p − 1, where τ denotes the canonical identification of G = Aut(ZZ/pZZ) with IF_{p}. Accordingly, every IF_{p}G-module V is a direct sum of characteristic submodules V_{d}, and every G-map μ : V ⟶ V′ of such modules decomposes into “slices” μ_{d} : V_{d} ⟶ V′_{d}. Since in our case G acts via G_{+} = G/{±1}, only even characters are involved, that is, d = 2k with k = 1, …, (p − 3)/2 (the trivial character, d = p − 1, being absent). The following lemma summarizes what we need to know.

Lemma 1: Each of the three IF_{p}G-modules shown in (7) above decomposes into exactly (p − 3)/2 characteristic submodules of dimension 1. Thus, each of the maps \overline{ι}_{2k} and \overline{ρ}_{2k}, for 1 ≤ k ≤ (p − 3)/2, is either null or an isomorphism. Proof: We shall see that each of the modules is isomorphic to ΔIF_{p}G_{+}. For the first one this given by (b) in the last section. For the third one, it is shown by the logarithmic isomorphism

U_{1}IF_{p}K = 1 + ΔIF_{p}K ⟶^{∼} ΔIF_{p}K  (8)

available at characteristic p. In fact, if K =< x >, the standard basis {x^{σ} − 1 | σ ∈ G} of ΔIF_{p}K is clearly normal with respect to G, so that U_{1}IF_{p}K is isomorphic to IF_{p}G. The restriction to G-norm 1 knocks out the G-invariant component and hence produces an isomorphism with ΔIF_{p}G. It follows that U_{1}^{+}IF_{p}K is isomorphic to ΔIF_{p}G_{+}.

For the middle term of (7) we can apply the following general argument, with X playing the role of Ω(K), and Y standing in for U_{1}^{+}(K). Any injection X ↪ Y between ZZG-modules of the same rank clearly induces a G-isomorphism X ⊗ F ⟶^{∼} Y ⊗ F for any field F of characteristic 0, e.g. F = Q_{p}. Hence we get the chain of implications

X ⊗ Q_{p} ≃ Y ⊗ Q_{p} ⇒ X ⊗ ZZ_{p} ≃ Y ⊗ ZZ_{p} ⇒ X ⊗ IF_{p} ≃ Y ⊗ IF_{p},  (9)

where ≃ stands for G-isomorphism. The second of these is trivial, while the first comes from the fact that the idempotents

e_{d} = \frac{1}{p − 1} ∑_{σ∈G} θ^{d}(σ^{−1}) · σ  (10)

of Q_{p}G happen to lie in ZZ_{p}G; here θ : G ⟶ ZZ_{p}^{×} is the obvious lift of τ onto the torsion subgroup of ZZ_{p}^{×}. This finishes the proof.

Definition: The prime p is said to be regular at d if (\overline{ρ} ∘ \overline{ι})_{d} is injective, semi-regular at d if \overline{ι}_{d} is injective. An element u ∈ Ω(K) which is not in Ω(K)^{p} but falls into the kernel of (\overline{ρ}∘\overline{ι})_{d} is called a Kummer unit at d. Note that such a unit is unique, modulo p-th powers, up to G-conjugates.

If p is irregular at d, we fix a Kummer unit u_{d} ∈ Ω(K) for future use. It is explicitly manufactured from a ZZG-generator w of Ω(K) and a pre-image η_{d} ∈ ZZG of the d-th idempotent e_{d} of IF_{p}G by setting u_{d} = w^{η_{d}}.

Remarks:

1. Using the logarithmic isomorphism (8), a classical computation due to Kummer (cf. [2], Ch. V, §6.3) shows that im(\overline{ρ} ∘ \overline{ι})_{d} = B_{d}IF_{p}z^{d}, where B_{d} stands for the d-th Bernoulli number, and z^{d} = log x = log(1 + t) for some generator x of K. This gives an effective way of deciding regularity at d. Incidentally, it is probably the only calculation in this game which cannot be restricted to the ⋆-symmetric part of ΔIF_{p}K (which contains neither t nor z). Hence it is not easily lifted to characteristic 0, where an unrestricted log is not available.

2. The surjectivity of (\overline{ι})_{d} is obviously equivalent to the vanishing of the d-th component of [U_{1}^{+}(K)/Ω(K)]⊗IF_{p}. For this to happen at all d, it is necessary and sufficient that the order of U_{1}^{+}(K)/Ω(K) be prime to p. This is easily seen to be equivalent to h_{p}^{+} being prime to p, i.e. to Vandiver’s Conjecture.

4. The Elementary Case. Let A be non-cyclic of order p^{2}. In order to treat U_{1}^{+}(A) and Ω(A) side-by-side, we let Φ denote any functor from the category of elementary abelian groups to finite free ZZ-modules such that Φ(0) = 0. Our object is to study the cokernels of α and γ = β ∘ α from

∏_{C} Φ(C) ⟶^{α} Φ(A) ⟶^{β} ∏_{K} Φ(K),  (11)

where C and K run over non-trivial cyclic subgroups and factor-groups of A, respectively. By functoriality, all Φ(−) are G-modules. If Φ(C) has no G-invariant elements, it follows from [7] and [3] that

(i) coker γ has order p^{(p+1)r/2}, where r is the rank of Φ(C); (ii) coker \overline{γ}_{d} has IF_{p}-dimension (p + 1) − m_{d}(d + 1), where m_{d} is the multiplicity of τ^{d}

in \overline{Φ}(C).

Lemma 2: If \overline{Φ}(C) is isomorphic to Δ^{+}IF_{p}C, the cokernel of γ is elementary abelian, and the IF_{p}-dimension of its τ^{d} component (coker γ)_{d} is p − d for d = 2, …, p − 3 even, and 0 otherwise. Proof: We repeat a simple argument already implicit in [3]. By (ii) and the present hypothesis, coker \overline{γ}_{d} has the dimension p − d or 0 as indicated above. Since coker \overline{γ} = (coker γ) ⊗ IF_{p}, the p-group (coker γ) has ∑_{d}(p − d) = (p + 1)(p − 3)/4 = (p + 1)r/2 generators. Because of (i), each of these has order p, as was to be shown.

So much for coker γ. In the two scenarios of interest here, namely Φ = Ω and Φ = U_{1}^{+}, the map β is obviously injective, and therefore coker α ⊆ coker γ is also elementary abelian. Its d-th component will be large or small, depending on the regularity of p at d. Suppose it is irregular, and let u_{d} ∈ Ω(K) be the Kummer unit defined in the last section. Then, if ε_{K} : A ⟶ K is one of the epimorphisms underlying β, the relation (5) shows that ε_{K′} maps the co-induced unit ε̃_{K}(u_{d}) either to u_{d} itself (if K′ = K) or to 1 (otherwise).

Now if Φ = Ω, this says that the image of ε̃_{K}(u_{d}) under \overline{β}_{d} has a single non-trivial component in the “slot” corresponding to K; thus in particular \overline{β}_{d} is surjective. The same holds true if Φ = U_{1}^{+} and p is semi-regular at d. In either case coker \overline{α}_{d} = coker \overline{γ}_{d}. Since the cokernels of α and γ are already known to be p-groups, this translates to (coker α)_{d} = (coker γ)_{d}. We summarize this in the following proposition.

Proposition 2: Suppose that p is irregular a d. Then if Φ = Ω, the d-th component of coker α is an IF_{p}-space of dimension p − d generated by the co-induced units ε̃_{K}(u_{d}), as K runs through the (p + 1) factor groups of A. This also holds if Φ = U_{1}^{+} and p is semi-regular (irregular) at d.

To get the complete picture, one has to remember that (coker α)_{d} = coker \overline{α}_{d} = 0 whenever p is regular at d, irrespective of whether Φ is Ω or all of U_{1}^{+}. Since source and target of \overline{α}_{d} both have rank (p + 1), this is equivalent to the injectivity of \overline{α}_{d}, which can be read off the following commutative square

\begin{matrix}∏_{C} \overline{Φ}(C)_{d} & ⟶^{\overline{α}_{d}} & \overline{Φ}(A) \\ ↓ & & ↓ \\ ∏_{C}(U_{1}^{+}IF_{p}C)_{d} & ⟶^{∼} & (U_{1}^{+}IF_{p}A)_{d} \end{matrix}  (12)

where the left vertical is injective by regularity. The isomorphism shown on the bottom line is most easily deduced from its counterpart involving U_{1}^{+}ZZ_{p}A, which in turn follows from the existence of a bijective logarithm U_{1}^{+}ZZ_{p}A ⟶^{∼} Δ^{+}ZZ_{p}A — cf.[6]. In view of the obvious bijectivity ∏_{C} ΔIF_{p}C ⟶ ΔIF_{p}A this might seem rather roundabout, but a more direct argument may be hard to find. For instance, without the restriction to ⋆-symmetric elements, the bottom arrow in (12) would no longer be injective: taking < x_{1} >= C_{1} ≠ C_{2} =< x_{2} > and C_{3} =< x_{3} > with x_{1}x_{2}x_{3} = 1, one easily constructs an element in its kernel.

Summing over all d, we can now draw a general conclusion.

Corollary: The group Ω(A) of circular units is generated by constructible units and the co-induced units ε̃_{K}(u_{d}). Likewise, if p is semi-regular, all of U_{1}^{+}(A) is generated by these co-induced units together with units induced from cyclic subgroups C ⊂ A.

Finally we must justify the term “exotic” applied to the units ε̃_{K}(u_{d}). If one of these were in the image of α, then all of them would be, because Aut(A) permutes them transitively. Therefore all would show up trivial in coker α, contradicting Proposition 2.

Thus, each ε̃_{K}(u_{d}) is circular without being constructible and, for p semi-regular, cannot be written as a product of cyclically induced units.

5. The Cyclic Case. If A = C itself is cyclic of order p^{2}, it makes little sense to ask for the part of U_{1}^{+}(A) induced by cyclic subgroups. The correct question is at once simpler and subtler than in the last section, namely: how does one generate the factor group Ω(C)/Λ(C) ? The answer will involve units co-induced by the p-th power map π : C ⟶ K. Even though K = C^{p} is also the kernel of π, we denote it by H when it occurs in that capacity.

We fix an injection ψ : C ⟶ Q(ζ)^{×}, where ζ is a q-th root of 1. The ring homomorphism from ZZC to ZZ[ζ], induced by ψ, has for its kernel exactly the ideal generated by S_{H}. Elements of U_{1}ZZC therefore lie in ker ψ if and only if they are in 1 + S_{H} · ΔZZC. In other words, if we set ε = π in Section 1, the map ε_{1} shown in (3) yields the bijection

ker [ Ω(C) ⟶^{ψ} Ω(ζ)] ⟶^{∼} ker [ Ω(K) ⟶^{ρ} IF_{p}(K)^{×}]  (13)

induced by π. Thus the kernel appearing on the left is just the set of all π̃(u) with u ∈ Ω(K) such that ρ(u) = 1. More succinctly, we can write: π̃( ker ρ) = ker ψ.

Remember that ker ψ consists of all circular units whose only non-trivial Wedderburn component is the one belonging to character ψ^{p}. From (a) of Section 2 we easily deduce the direct product Ω(C) = ker ψ×W (C). On the other hand, we have Λ(C) = W (C)^{π} ×W (C) by (c) of the same section. To compare the two, we must now determine the constructible part of ker ψ.

Note that G = Aut(C) contains a group Γ of order p, and so does G_{+}. Letting s ∈ ZZG_{+} denote the sum over all the elements of Γ, we recall by the definition of W (C) that ψ^{p}(u) = ψ(u^{s}) for all u ∈ W (C). Since ψ^{p}(u) = ψ(u^{π}) for any u, this allows us to derive the relation

ker ψ ∩ Λ(C) = W (C)^{π−s}.  (14)

Indeed, taking a typical element w = u^{π} × v ∈ Λ(C), we get ψ(w) = ψ(u^{s}v). By (a) again, this equals 1 if and only if v = u^{−s}, i.e. w = u^{π−s} as was to be shown.

Since Λ(C) can obviously be rewritten as W (C)^{π−s} × W (C) we immediately get an isomorphism ker ψ/W (C)^{π−s} ⟶^{∼} Ω(C)/Λ(C). Now, since Γ acts trivially on C^{p} = K and π trivializes it altogether, the image of W (C)^{π−s} under the map (13) is just W (C^{p})^{π−s} = W (K)^{p} = Ω(K)^{p}. Thus we have a composite of bijections

ker ρ/Ω(K)^{p} ⟶^{∼} ker ψ/W (C)^{π−s} ⟶^{∼} Ω(C)/Λ(C),  (15)

with the first stage given by π̃ and the second induced by the inclusion ker ψ ↪ Ω(C).

The group on the left of (15) is clearly the kernel of the map (\overline{ι}∘\overline{ρ}) : \overline{Ω}(K) ⟶ U_{1}IF_{p}K shown in (7). It is an IF_{p}-space of dimension δ_{p}, the number of d at which p is irregular.

Now, G = Γ × G_{0} is a direct product, with G_{0} cyclic of order p − 1. On K the action of Γ is trivial, and G_{0} acts as Aut(K). As before we can therefore restrict (15) to the submodules belonging to the non-trivial even characters τ^{d} of G_{0}, with d = 2, …, (p − 3). Here is the result.

Proposition 3: Ω(K)/Λ(K) is an IF_{p}-space of dimension δ_{p}. Its d-th component is nontrivial if and only if p is irregular at d, in which case it is generated by the co-induced unit π̃(u_{d}).

It remains to point out that each of these units is “exotic” in that it is circular without being constructible. Though this is clear from the structural context, it is worth a direct argument. By its nature, π̃(u_{d}) is in ker ψ; if it were constructible, it would be in W (C)^{π−s} according to (14); but then u_{d} itself would be a p-th power — which it is not.

6. An Example. As an illustration, we shall demonstrate how these calculations work out for p = 37. If K =< z > with z^{37} = 1, we have G = Aut(K) = (IF_{37})^{×}. For every integer r prime to 37, let us denote by σ(r) the corresponding element of G. Since G is generated by σ(2) as well as by σ(5), the prescriptions of [5] show that w(z) = z^{−2} − z^{−1} + 1 − z − z^{2} is a ZZG-generator of W (K) = Ω(K).

It is well-known that p = 37 is regular for all even d = 2, …, 34 except d = 32. Therefore the critical character is τ^{d} : σ(a) ↦ a^{32} = a^{−4} (values to be read in IF_{p}), and the corresponding idempotent of IF_{p}G is

e_{32} = − ∑_{r=1}^{36} r^{4}σ(r) = − ∑_{n∈N} n^{4}λ(n),  (16)

where N = {1, 2, 3, 3, 4, 5, 8, 9, 10, 15} is a set of representatives modulo the multiplicative subgroup G_{4} = {6, −1, −6, 1} and λ(n) = σ(n) + σ(−n) + σ(6n) + σ(−6n).

If we now choose a preimage η_{32} ∈ ZZG of e_{32} by replacing the n^{4} in (16) by suitable integers a_{n} ≡ n^{4} mod 37, the resulting Kummer unit will have the form

u_{32}(z) = w(z)^{η_{32}} = 1 + 37 ∑_{n∈N} c_{n}(z^{n} + z^{−n} + z^{6n} + z^{−6n} − 4),  (17)

with c_{n} ∈ ZZ, because it must be invariant under G_{4}. For a group A =< x, y > of order p^{2} and an epimorphism ε : A ⟶ K with ker ε =< y >, we can finally write down the coinduced unit ε̃(u_{32}) as

u(x, y) = 1 + \frac{u_{32}(x) − 1}{37} · S_{<y>}.  (18)

For u_{32}(x) it is understood that the z in (17) must be replaced by x, exponents to be read modulo 37. This makes sense in the elementary case (x^{p} = 1) as well as in the cyclic one (x^{p} = y), since the powers of x which occur always form a system of representatives modulo < y >. In either case the units u(x, y) together with the w(z) generate Ω(A). If A is elementary, they even generate all of U_{1}^{+}(A) because it so happens that h_{37}^{+} = 1.

The fly in this ointment is that the coefficients c_{n} (which were kindly computed by Marc Conrad at the University of Saarbrücken) are uniformly big. Of course they depend on the lift η_{32} chosen for e_{32}. If one picks the a_{n} to have minimal absolute value, the c_{n} turn up with 23 to 25 decimal digits. Sporadic experimentation with other liftings did not improve this situation. For the next irregular prime p = 59, analogous calculations produced coefficients ten times longer yet. This is somewhat surprising in view of the lean shape enjoyed by the constructible generators. It remains to be seen whether a basis of Ω(A) for irregular primes must neccessarily include such unwieldy elements.

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Details not included in the version submitted for publication.

w(z)^{−1} = z^{2}−z^{7}+z^{12}−z^{17}+z^{−15}−z^{−10}+z^{−5}−1+z^{5}−z^{10}+z^{15}−z^{−17}+z^{−12}−z^{−7}+z^{−2}.

e_{32} = −λ() − 16λ(2) − 7λ(3) + 3λ(4) + 4λ(5) + 11λ(8) − 12λ(9) − 10λ(10) − 9λ(15).

c_{1} = −1147557105040667341442808  c_{2} = +681732566824536933894968
c_{3} = −54182301438440608270800  c_{4} = −559998713102984680018224
c_{5} = +995966533834975557490332  c_{8} = −610334679098778334688974
c_{9} = +120977500856010523455712  c_{10} = +324974773291376818267889

c_{15} = −81386880130124246971003

References

[1]: H. Bass, The Dirichlet Unit Theorem, Induced Characters, and Whitehead Groups

of Finite Groups, Topology, 4, (1966), 391 – 410.. [2]: Z.I. BoreviVc and I.R. Shafarevich, Number Theory, Acad. Press, N.Y., (1966). [3]: K. Hoechsmann, Functors on Finite Vector Spaces and Units in Abelian Group Rings,,

Can. Math. Bull. 29, (1986), 79 – 83. [4]: K. Hoechsmann, Units and Class-Groups in Integral Elementary Abelian Group

Rings, J. Pure and Appl. Alg. 47, (1987), 253 – 264. [5]: K. Hoechsmann and J. Ritter, Constructible Units in Abelian p-Group Rings, J. of

Pure and Appl. Alg., (to appear 1990). [6]: K. Hoechsmann and S.K. Sehgal, Units in Regular Elementary Abelian Group Rings,

Arch. d. Math. 47, (1986), 413 – 417 . [7]: K. Hoechsmann, S.K. Sehgal, and A. Weiss, Cyclotomic Units and the Unit Group

of an Elementary Abelian Group Ring, Arch. d. Math. 45, (1985), 5 – 7. [8]: L. Washington, Introduction to Cyclotomic Fields, Springer, New York, (1984).

Department of Mathematics University of British Columbia Vancouver, B.C., Canada