JPAA 47 (1987) 253-264

UNITS AND CLASS-GROUPS IN INTEGRAL ELEMENTARY ABELIAN GROUP RINGS

1. Introduction

If R is a commutative ring and A a finite abelian group, we denote by U_{k}RA those units of the group-ring RA which are congruent 1 modulo the k-th power of the augmentation ideal ΔRA = ker(RA ⟶ R) with A ⟶{1}. It is well-known (cf. [2, Lemma2.6]) that U_{1}ZZA = A × U_{2}ZZA, the product being direct, and that the elements of U_{2}ZZA remain fixed under the involution u ↦ u^{∗} defined by a^{∗} = a^{−1} for a ∈ A. Thus U_{2}ZZA ⊆ U_{1}^{+}ZZA, where the superscript + denotes ∗-invariance. If the group-order |A| is odd, Δ^{+}RA is spanned by elements of the form (a − 1) + (a^{−1} − 1) = (a − 1)(a^{−1} − 1), and therefore Δ^{+} ⊂ Δ^{2} and U_{1}^{+} = U_{2}. For such groups, U_{1}^{+}ZZA thus emerges as a convenient subgroup representing the torsion-free part of UZZA.

Let A be an elementary abelian group of order |A| = p^{n+1} , p > 2. This paper will deal with various groups associated with the diagram

∏_{C} U_{1}^{+}ZZC ⟶^{α} U_{1}^{+}ZZA ⟶^{β} ∏_{K} U_{1}^{+}ZZK
↓λ  ↓λ  ↓λ
∏_{C} U_{1}^{+}ZZ_{p}C ⟶^{α_{p}} U_{1}^{+}ZZ_{p}A ⟶^{β_{p}} ∏_{K} U_{1}^{+}ZZ_{p}K ,

(1)

in which ZZ_{p} denotes the (complete) ring of p-adic integers, and C and K run over all non-trivial subgroups and factor-groups, respectively. The products are direct, and α is simply a product of inclusions, while β is induced by the obvious ring-homomorphisms RA ⟶ RK. The map λ (localization) comes from the ring-inclusions ZZA ⟶ ZZ_{p}A.

It has been shown (cf. [8, Lemma 3]) that the usual power-series defines an isomorphism log : U_{1}^{+}ZZ_{p}A⟶^{∼}Δ^{+}ZZ_{p}A, so that the bottom row of (1) could be rewritten in terms of Δ^{+}. Furthermore it is known that α_{p} is an isomorphism [8, Lemma 3] and that all the other horizontal arrows represent injections of finite p-power index (cf. [9]).

The focus of our attention will be the finite p-group

M = coker(coker(β)⟶^{λ}coker(β_{p}))

derived from (1). In Section 3 below, it will be shown that M = M_{0} × N, where (a) |N| equals the index of α,

(b) 2 · log_{p} |M_{0}| = n · (1 + p + ⋯ + p^{n}) + (1 − p) + ⋯ + (1 − p^{n}),

(c) N = ∏_{k} N_{2k} , where k = 1, …, (p−3)/2, and N_{2k} is trivial unless the Bernoulli number B_{2k} vanishes in IF_{p}.

In particular, N = {0} if p is regular. If p is semi-regular (as all primes are conjectured to be, cf. [10, Chapter 5, §4]), we can explicitly compute the order of N ⊗ IF_{p}, cf. Theorem 3.4, below. The main interest here is in (a), which says that |N| ‘measures’ those units of ZZA which cannot be written as products of ‘cyclic’ units. It will be seen that N ≠ {0}, for irregular, semi-regular p.

Section 4 will make the connection between M and the kernel D of the class-group map cl (ZZA) ⟶ cl (M), M the maximal order in QA. It turns out that M = D^{+}, the ∗-invariant part of D. This complements the computation of D^{−} in [4].

2. Preliminaries

In this section we delve into the more or less well-known and assemble a few useful facts.

Our first task is to describe the image of ZZA under the Wedderburn isomorphism

QA ⟶^{∼} ∏_{φ} E_{φ} ,

where φ runs over Q-irreducible characters of A. Of course, each E_{φ} = Q[ζ_{φ}] is a cyclotomic field, and ZZA winds up in ∏_{φ} Q[ζ_{φ}]. We claim that a ‘vector’ {b_{φ}} in this product lies in the image of ZZA if and only if for each a ∈ A

∑_{φ} Tr_{φ}[φ(a^{−1}) b_{φ}] ≡ 0 (mod |A|) ,

where Tr_{φ} is the absolute trace ZZ[ζ_{φ}] ⟶ ZZ.

This follows from the fact that any f = ∑_{a∈A} f(a) a ∈ QA can be reconstructed from its Wedderburn components φ(f) by the usual trick,

|A| f(a) = ∑_{φ} Tr_{φ}[φ(a^{−1}) φ(f)] ,

of equating two evaluations of the regular character on a^{−1}f = f(a) · 1 + ∑_{x≠1} f(ax)x. This works for any finite abelian A and, mutatis mutandis, any reasonable coefficient ring instead of ZZ. It shows, for example, that ZZ_{l}A is the maximal order in Q_{l}A if l is prime to |A|.

Now let A be elementary abelian as in the introduction. Apart from the trivial character φ_{0}, there are (1 + p + ⋯ + p^{n}) characters φ, each of which factors over the cyclic group K_{φ} = A/ ker(φ). For f ∈ QA, let f_{φ} denote the image in QK_{φ}. Applying the formula to K_{φ}, we get

p f_{φ}(c) = φ_{0}(f) + Tr_{φ}[φ(c^{−1}) φ(f)] ,

whereas on A, we have

p^{n+1} f(a) = φ_{0}(f) + ∑_{φ≠φ_{0}} Tr_{φ}[φ(a^{−1}) φ(f)] = φ_{0}(f) + ∑_{φ≠φ_{0}} [pf_{φ}(a_{φ}) − φ_{0}(f)] ,

hence

p^{n}f(a) = −(1 + p + ⋯ + p^{n−1}) φ_{0}(f) + ∑_{φ≠φ_{0}} f_{φ}(a_{φ}) .

This means that, given a set {f_{φ} ∈ ZZK_{φ}} of cyclic group-ring elements with a common augmentation f_{0}, there exists an element f ∈ ZZA such that β(f) = {f_{φ}} ∈ ∏_{φ≠φ_{0}} ZZK_{φ} if and only if the congruences

∑_{φ≠φ_{0}} f_{φ}(a_{φ}) ≡ (1 + p + ⋯ + p^{n−1}) f_{0}

are solvable modulo p^{n}. If f_{φ} ∈ U_{1}ZZK_{φ} for each φ, the resulting f ∈ ZZA will also be in U_{1}^{+}ZZA since it is invertible in the maximal order of QA (cf. [12, Proposition 3]). The solvability of the required congruences can be checked in any ring R which detects divisibility by p^{n}, i.e., such that ι (p^{n}ZZ) = ι (ZZ)∩p^{n}R, where ι : ZZ ⟶ R is the canonical homomorphism. We have proved

Lemma 2.1. If R detects divisibility by p^{n}, the diagram

U_{1}ZZA ⟶^{β} ∏_{K} U_{1}ZZK
ι↓  ↓ι

U_{1}RA ⟶^{β} ∏_{K} U_{1}RK

is a fiber product.

We shall apply this lemma with R = ZZ_{p}, but we could also get away with R = ZZ/p^{n}ZZ, for instance.

The rest of this section will be concerned with a description of the very simple representation theory of ZZ_{p}G-modules, where G = IF_{p}^{×} acts canonically and compatibly on all elementary abelian groups and related structures. We first note that the reduction map ZZ_{p}^{×} ⟶ IF_{p}^{×} has a canonical right inverse τ : IF_{p}^{×} ↪ ZZ_{p}^{×} and that the absolutely irreducible characters of G are precisely the maps τ^{d} : g ↦ τ(g)^{d} for d = 1, …, (p − 1). Since (p − 1)^{−1} ∈ ZZ_{p} as well, ZZ_{p}G contains (p − 1) orthogonal idempotents e_{d}, and ZZ_{p}G = ∏_{d} e_{d}ZZ_{p}G. Hence,

(i) every ZZ_{p}G-module M has a canonical decomposition M = ∏_{d} M_{d}, where M_{d} = e_{d}M_{d} and G acts via the character τ^{d}, i.e., multiplication with something in ZZ_{p}^{×}, (ii) every irreducible ZZ_{p}G-module must therefore be indecomposable over ZZ_{p}, i.e., isomorphic to a factor-module of ZZ_{p,d} = e_{d}ZZ_{p}G, which is just ZZ_{p} with a certain G-action.

We now turn to studying the G-structure of the diagram

U_{1}^{+}ZZK ⟶^{λ} U_{1}^{+}ZZ_{p}K ⟶^{ρ} U_{1}^{+}IF_{p}K
log ↓  ↓ln

Δ^{+}ZZ_{p}K ⟶^{ρ} Δ^{+}IF_{p}K ,

(2)

where K is cyclic of order p. Of course, all these maps are G-maps; λ is injective, ρ (reduction of coefficients) both times surjective, and the vertical arrows are groupisomorphisms: log defined by the usual power-series, and ln by the first (p − 1) terms thereof. We make this notational distinction because, a priori, ρ log(1+t) and lnρ(1+t) differ by the term ρ(t^{p}/p), which luckily vanishes for t ∈ Δ^{+}ZZ_{p}K since Δ^{+} ⊂ Δ^{2} and ord_{p}(t^{p}) ≥ 2. After this remark, we can forget about ‘ln’ and call a log a log.

We begin our study in the lower right corner of (2). Let K = ⟨x⟩, and put t = x − 1, z = log(1 + t). Both t and z are generators of ΔIF_{p}K, but z has the advantage that (ΔIF_{p}K)_{d} = IF_{p} · z^{d}. To see this, write x = exp(z), hence x^{a} = exp(az). Now, x ↦ x^{a} is precisely the way G acts on K, and this multiplies z by a and z^{d} by a^{d}. In particular, z^{∗} = −z, which shows that Δ^{+}IF_{p}K = ∑_{k} IF_{p} · z^{2k} for k = 1, …, (p − 1)/2.

Δ^{+}ZZ_{p}K is a free ZZ_{p}-module with basis {(x^{i} − 1) + (x^{−i} − 1) | i = 1, …, (p − 1)/2}. Since ρ must send (ΔZZ_{p}K)_{d} into IF_{p} · z^{d}, it follows that ΔZZ_{p}K = ∑_{k} ZZ_{p,2k} over G, where k = 1, …, (p − 1)/2. Ditto for U_{1}^{+}ZZ_{p}K.

Finally, U_{1}^{+}ZZK is a free ZZ-module of rank (p − 3)/2 by Dirichlet’s Unit Theorem. Since G acts as the Galois group on U_{1}ZZ[ζ_{p}], into which U_{1}^{+}ZZK injects, and since rational units are scarce, it follows that U_{1}^{+}ZZK has no G-invariant elements, i.e., τ^{p−1} does not occur. However, λ⊗ZZ_{p} : [U_{1}^{+}ZZK]⊗ZZ_{p} ⟶ U_{1}^{+}ZZ_{p}K is still injective and must respect the G-action. Hence ([U_{1}^{+}ZZK] ⊗ ZZ_{p})_{d} ≃ ZZ_{p,d} for d = 2k, k = 1, …, (p − 3)/2, and trivial otherwise. Let us summarize.

Lemma 2.2. As ZZ_{p}G-modules, both U_{1}^{+}ZZ_{p}K and [U_{1}^{+}ZZK] ⊗ ZZ_{p}] are direct products of ZZ_{p,2k}, where k runs from 1 to (p − 1)/2 in the first case and only up (p − 3)/2 in the second.

Convention. Since all groups in diagrams (1) and (2) are ∗-invariant, G acts on them modulo the subgroup {±1}, −1 ∈ G being the famous ∗. Therefore only even characters τ^{2k} occur in the sequel. Whenever we use the notation M_{d}, ZZ_{p,d}, etc. just described, we shall henceforth assume that d = 2k, k = 1, …, (p − 1)/2.

The time has come to get the Bernoulli numbers into the act without making the story too long. The heart of the matter is the following fact relating to diagram (2).

Fact. The range of the composite ρ∘log ∘λ : U_{1}^{+}ZZK ⟶ Δ^{+}IF_{p}K contains the elements B_{2k}z^{2k}, where k = 1, …, (p − 3)/2, and z = log x has the same meaning as in the text preceding Lemma 2.2.

The proof of this fact is written down explicitly in [7, Paragraph 4] or can be gleaned without much effort from [1, Chapter V, §6.3], provided that one knows how to use the arrow U_{1}^{+}ZZK ↪ UZZ[ζ_{p}] in reverse by raising a unit of ZZ[ζ_{p}] to a certain power (at most p − 1) and adjusting the result by a root of unity in order to get a pre-image in U_{1}^{+}ZZK. In ZZ[ζ_{p}] there are certain standard units 1+ζ +⋯ ζ^{r}, 2 ≤ r ≤ (p−1)/2, whose relatives in U_{1}^{+}ZZK are responsible for catching the B_{2k}z^{2k} in the image of ρ ∘ log ∘λ. These relatives of the standard units generate a subgroup U_{1}^{#}ZZK ⊂ U_{1}^{+}ZZK of index jh_{0}, where j is a product of divisors of (p−1), and h_{0} is the class-number of ZZ[ζ_{p} +ζ_{p}^{−1}]. This is easily derived from the corresponding phenomenon in ZZ[ζ_{p}], cf. [1, Chapter V, §5.2].

The lack of detail in the preceding description will not seriously hurt this exposition since we shall concentrate our attention on semi-regular primes p, i.e., those for which h_{0} is prime to p. Luckily, no primes are known to be non-semi-regular, and most experts seem to believe that all primes are semi-regular (“Vandiver’s Conjecture”, cf. [10], loc. cit.).

If p is semi-regular, [U_{1}^{+}ZZK : U_{1}^{#}ZZK] is prime to p, and the image of ρ ∘ log ∘λ : U_{1}^{+}ZZK ⟶ Δ^{+}IF_{p}K is spanned by {B_{2k}z^{2k}}, k = 1, …, (p − 3)/2. Of course, this homomorphism kills p-th powers and hence factors over a map

μ : U_{1}^{+}ZZK/(U_{1}^{+}ZZK)^{p} ⟶ Δ^{+}IF_{p}K .

Being a p-group, the domain of this map is a ZZ_{p}G-module, and we can slice μ according to the characters τ^{2k} of G.

The fact that (ΔIF_{p}K)_{d} = IF_{p} · z^{d}, as shown in the context of Lemma 2.2, now suffices to prove

Lemma 2.3. im(μ_{2k}) = IF_{p} · B_{2k}z^{2k} for k = 1, …, (p − 3)/2.

In other words, μ_{2k} is the null-map or an isomorphism, depending on whether or not B_{2k} vanishes in IF_{p}.

3. Two cokernels

We begin by copying diagram (1) in the following abbreviated form:

\begin{matrix}X & ⟶^{α} & Y & ⟶^{β} & Z \\ ↓ & & ↓λ & & ↓ \\ X′ & ⟶^{∼} & Y′ & ⟶^{β_{p}} & Z′ . \end{matrix}  (1^{∗})

Since, by Lemma 2.1, the square on the right is a fiber product, we obtain an injection^{1} Z/Y ⟶^{λ}Z′/Y′ whose cokernel we denote by M. Equivalently, we could write an exact sequence

0 ⟶ L ⟶ Γ ⟶ Γ′ ⟶ M ⟶ 0 ,  (3)

where L = Y/X, Γ = Z/X, Γ′ = Z′/X′, the advantage of which is the close relation between Γ and Γ′ expressed by

Proposition 3.1. For each even d < p − 1, the ZZ_{p}-modules Γ_{d} and Γ′_{d} are isomorphic.

Proof. The basic reason is that both groups can be identified with the cokernel of one and the same matrix acting on a free ZZ_{p}-module of the appropriate size.

For this identification, let P ⊂ A be a set containing exactly one non-trivial element from every non-trivial cyclic subgroup C ⊂ A. P serves as an index set for the various injections IF_{p} ⟶ IF_{p}x = C ⟶ A involved in the definition of α. If φ : A × A ⟶ IF_{p} is a non-degenerate pairing, P can also serve as an index set for the projections φ(−, y) : A ⟶ IF_{p} = K_{y} which together define β. If one follows an element of IF_{p} through the composition IF_{p}⟶^{x}A⟶^{y}IF_{p}, one sees that it gets multiplied by φ(x, y).

Now let E be a functor from elementary abelian p-groups to ZZ_{p}-modules such that E(0) = 0. By sheer functoriality, G then acts on all the image-modules E(A). The corresponding map γ_{E} : ∏_{x} E(C_{x}) ⟶ ∏_{y} E(K_{y}) is still given by the matrix (φ(x, y)), whose non-zero entries now have to be interpreted as elements of G acting on E(IF_{p}), while 0 yields the map E(IF_{p}) ⟶ E(0) = 0.

If E(A) = (U_{1}^{+}ZZ_{p}A)_{d}, we get Γ′_{d} as the cokernel of the matrix (τ^{d}φ(x, y)) acting on ZZ_{p}^{|P |}, because (U_{1}^{+}ZZ_{p}C)_{d} ≃ ZZ_{p,d} by Lemma 2.2.

In order to deal similarly with Γ, we first note that, being a finite p-group, it is also the cokernel of γ ⊗ ZZ_{p}, where γ = β ∘ α. Now we take E(A) = [(U_{1}^{+}ZZA) ⊗ ZZ_{p}]_{d}, i.e., we turn a free ZZ-module into a free ZZ_{p}-module and again take the ‘slice’ corresponding to τ^{d}. By Lemma 2.2, we again find E(C) = ZZ_{p,d}, and Γ_{d} results from the same matrix (τ^{d}φ(x, y)) acting on ZZ_{p}^{|P |}.

Remark. The order of Γ is computed in [9]. It turns out log_{p} |Γ| = (n/2)|P |(p − 3)/2. By Proposition 3.1, ∑_{k} log_{p} |Γ′_{2k}| with 1 ≤ k ≤ (p − 3)/2 has the same value.

Proposition 3.2. Γ_{p−1} is trivial, and

2 · log_{p} |Γ′_{p−1}| = n · (1 + p + ⋯ + p^{n}) + (1 − p) + ⋯ + (1 − p^{n}) .

Proof. Since the trivial character τ^{p−1} does not occur in the G-structure of X or Z (cf. Lemma 2.2), the first statement is obvious. For the second, we go back to the proof of Proposition 3.1,with E(A) = (U_{1}^{+}ZZ_{p}A)_{p−1} = (U_{1}^{+}ZZ_{p}A)^{G}, and compute the index of X′_{p−1} in Z′_{p−1} as the determinant of the matrix (ψ(x, y)) = (φ(x, y)^{p−1}),

1 The same injectivity will be obtained in Section 4, Sequence 6, by other means. whose entries are 0 or 1 depending on whether or not φ(x, y) is 0. Multiplying it with its transpose, thus squaring the determinant, we obtain the P × P -matrix S(x, y) = ∑_{z∈P} ψ(x, z)ψ(y, z). Since every summand is unchanged when we replace z by az, with a ∈ IF_{p}^{×}, we get (p − 1)S(x, y) = ∑_{z∈A} ψ(x, z)ψ(y, z). By suitably choosing coordinates in A with respect to the linear functionals φ(x, −) and φ(y−), it is easy to see that S(x, x) = p^{n}, while S(x, y) = p^{n−1}(p − 1) for x ≠ y. Evaluating the determinant by row reduction yields the result. Details can be found in the proof of the theorem in [6].

Combining Propositions 3.1 and 3.2 with a contemplation of the exact sequences

0 ⟶ L_{d} ⟶ Γ_{d} ⟶ Γ′_{d} ⟶ M_{d} ⟶ 0 ,  (3d)

resulting from (3) by restriction to irreducible G-modules, we have now established the assertions (a) and (b) of the Introduction. We restate this as

Theorem 3.3. Let L = coker(α) as in Diagram (1). Then |L_{2k}| = |M_{2k}| for k = 1, …, (p − 3)/2. Moreover, M_{p−1} has the order given in Proposition 3.2.

Our attention must now turn to estimating the size of N, i.e., of M_{d} for d < p − 1. Another way of writing Sequence (3), namely

0 ⟶ Y/X ⟶ Z/X ⟶ Z′/Y′ ⟶ M ⟶ 0 ,

(remember X′ ≃ Y′) yields the exact sequence

0 ⟶ Y ⟶ Z × Y′ ⟶ Z′ ⟶ M ⟶ 0 ,  (4)

which has a kind of Mayer-Vietoris flavor arising from the pullback in Diagram (1^{∗}). Reducing modulo p, and slicing according to G-characters, we obtain exact sequences

\overline{Z}_{d} × \overline{Y} ′_{d} ⟶ \overline{Z}′_{d} ⟶ \overline{M}_{d} ⟶ 0 ,

where the bar always means reduction modulo p, i.e., tensoring with IF_{p}. Explicitly, each of these shows \overline{M}_{d} as the cokernel of the map

∏_{K} (\overline{U_{1}^{+}ZZK})_{d} × (\overline{U_{1}^{+}ZZ_{p}A})_{d} ⟶ ∏_{K} (\overline{U_{1}^{+}ZZ_{p}K})_{d}

or, via logarithms,

∏_{K} (\overline{U_{1}^{+}ZZK})_{d} × (Δ^{+}IF_{p}A)_{d} ⟶ ∏_{K} (Δ^{+}IF_{p}K)_{d} ,

which is the difference between ∏_{K} μ_{d} (cf. Lemma 2.3) and (\overline{β_{p}})_{d}.

Now let p be semi-regular. Then, by Lemma 2.3, μ_{d} is the null map for all d = 2k with B_{2k} = 0 in IF_{p}. In these cases, \overline{M}_{d} = coker(\overline{β_{p}})_{d} = coker(\overline{γ_{p}})_{d}, α_{p} being an isomorphism. By the main result of [6], the rank of (\overline{γ_{p}})_{d} is the binomial coefficient C(d + n, n), the number of monomials of degree d in n + 1 indeterminates. If, however, B_{2k} ≠ 0 in IF_{p}, then μ_{d} is an isomorphism and \overline{M}_{d} is trivial. Thus we have

Theorem 3.4. Let p be semi-regular. Then, for d = 2k < p − 1, dim M_{d} ⊗ IF_{p} equals either (1 + p + ⋯ + p^{n}) − C(d + n, n) or 0, depending on whether or not B_{2k} vanishes in IF_{p}.

We can give a coarser but prettier estimate, based solely on the number δ_{p} of B_{2k} with 2k < p − 1 which vanish in IF_{p}.

Corollary 3.5. Let p be semi-regular and N = ∑_{d<p−1} M_{d}. Then

(1 + p + ⋯ + p^{n−1}) δ_{p} ≤ dim N ⊗ IF_{p} ≤ (1 + p + ⋯ + p^{n}) δ_{p} .

Proof. This simply results from the inequality C(d + n, n) ≤ p^{n}, which can be most easily seen as follows. For d < p − 1, C(d + n, n) is the number of integral (n + 1)-tuples (v_{0}, …, v_{n}) such that

(i) 0 ≤ v_{j} ≤ p − 1  and  (ii) v_{0} + ⋯ + v_{n} = d ,

while p^{n} is the number of such objects satisfying (i) and (iii) v_{0} + ⋯ + v_{n} ≡ d mod p.

Remark. If |A| = p^{2}, i.e., n = 1, the cokernel Γ of γ is elementary abelian by [6, Paragraph 4], and so is L. The same is true of ∏_{k} Γ′_{2k}, k = 1, …, (p − 3)/2, and hence of M. Therefore the number appearing in Theorem 3.4, namely p − d, actually equals log_{p} |M_{d}| = log_{p} |L_{d}|. For instance, if p = 37, the Bernoulli number B_{32} is the only one which dies modulo p, and log_{p} |M| = log_{p} |L| = 5. Thus the group of units in U_{1}^{+}ZZA which can be obtained from cylic subgroups has index 37^{5} instead of 1, as had been hoped (cf. [7]).

If n > 1, the equality |L_{d}| = |N_{d}| together with Theorem 3.4 still gives estimates of |L|. In this connection it is interesting to observe that the number of cycles of any non-trivial L^{d} is at least p − d, since A contains subgroups of order p^{2}.

4. Class Groups

Let H be a finite dimensional, separable, commutative Q-algebra with an ‘augmentation’ κ : H ⟶ Q, i.e., a retraction of the canonical map τ : Q ⟶ H. We shall consider orders A of H. Since κ(A) is a subring of Q which is finitely generated over ZZ, we have κ : A ⟶ ZZ.

Following [3], we consider the group cl(A) = J(H)/U(H)W (H), where J(H) denotes the idèle group of H, U(H) the units of H looked at as principal idèles (i.e., diagonally embedded), and W (A) those idèles (… u_{l} …) such that u_{l} ∈ U(A_{l}) for all l. Incidentally, the set S = {l} of places over which our idèles form a restricted product may or may not include the infinite place, but it must include all finite ones. Under these circumstances, Fröhlich shows that cl(A) is canonically isomorphic to the ideal class group of A (cf. [3, Theorem 1]).

Now let B be another order of H with A ⊂ B. The definition of cl(A) then leads to an exact sequence

1 ⟶ U(B)/U(A) ⟶ W (B)/W (A) ⟶ cl(A) ⟶ cl(B) ⟶ 1 ,  (5)

whose derivation, via Noether’s various isomorphism theorems and easy formulas like U(A) = W (A) ∩ U(H), is purely formal. [3, Corollary 1] asserts the exactness of (5) for the case where B is a maximal order; but the proof makes no use of this restriction. We split (5) into two exact sequences

1 ⟶ U(B)/U(A) ⟶ W (B)/W (A) ⟶ D(A, B) ⟶ 1 ,  (5a)
1 ⟶ D(A, B ⟶ cl(A) ⟶ cl(B) ⟶ 1 ,  (5b)

and concentrate on (5a). The term W (B)/W (A) is easily replaced by ∏_{l} U(B_{l})/U(A_{l}), where l runs over S(A, B) = {l | U(A_{l}) ≠ U(B_{l})}. This is a finite set, because at almost all places U(A_{l}) = U(M_{l}), where M denotes the maximal order of H.

So far, we have not used the augmentation κ. We do so now in order to see that the group U_{1} = {u ∈ U | κ(u) = 1} splits off as a direct factor in each case, whether it be U(A), U(A_{l}), U(B), or U(B_{l}). Hence we may replace U by U_{1} wherever U occurs in (5a), even when hidden as a factor of W.

Finally we assume that, in our case, U(A_{l}) = U(M_{l}) for all l except a single finite prime p. Then (5a) takes the form

1 ⟶ U_{1}(B)/U_{1}(A) ⟶ U_{1}(B_{p})/U_{1}(A_{p}) ⟶ D(A, B) ⟶ 1 ,  (6)

which, in analogy to Sequence (4) in Section 3, we write as

1 ⟶ U_{1}(A) ⟶ U_{1}(B) × U_{1}(A_{p}) ⟶ U_{1}(B_{p}) ⟶ D(A, B) ⟶ 1 .  (7)

Now we come back to group-rings, taking H = QA and A = ZZA. If A is an abelian p-group, we have S(A, B) = {p} because A ⊂ M ⊂ |A|^{−1}A, as the reader may look up in [11, Theorem 41.1] or deduce from the character relation |A|f(a) = ∑_{φ} Tr_{φ}φ(a^{−1}f) used in the proof of Lemma 2.1.

From now on, A will again be the elementary abelian p-group of order p^{n+1}. For B we now take the subring of ∏_{K} ZZK consisting of all K-tuplets (… v_{K} …) for which the augmentations κ(v_{K}) are independent of K. Obviously U_{1}(B) = ∏_{K} U_{1}ZZK, and here we connect with the first part of our story. The maximal order M is ZZ × ∏_{K} ZZ[ζ_{K}], as given by Wedderburn’s decomposition of QA, and U_{1}(M) = ∏_{K} UZZ[ζ_{K}]. By a lucky coincidence, ZZ_{p}K = ZZ_{p} ×ZZ_{p}[ζ_{K}] decomposes just like ZZK, and hence U_{1}(M_{p}) = ∏_{K} UZZ_{p}[ζ_{K}].

Proposition 4.1. D(A, B) = D(A, M).

Proof. We have cl(A) ⟶ cl(B) ⟶ cl(M) and D(A, B) ⊂ D(A, M) by their definition through Sequence (5b). From Sequence (7), we obtain the following diagram with exact rows:

∏_{K} U_{1}^{+}ZZK × U_{1}ZZ_{p}A ⟶ ∏_{K} U_{1}^{+}ZZK ⟶ D(A, B) ⟶1
↓  ↓  ↓
∏_{K} U_{1}^{+}ZZ[ζ_{K}] × U_{1}ZZ_{p}A ⟶ ∏_{K} U_{1}^{+}ZZ_{p}[ζ_{K}] ⟶ D(A, B) ⟶1

where the vertical arrows result from the inclusion B ⟶ M, and the bottom row is another copy of (7) in which M replaces B. The elementary exact sequence

1 ⟶ U_{1}ZZK ⟶ UZZ[ζ] ⟶ IF_{p}^{×} ⟶ 1

and its analogue over ZZ_{p} show that [UZZ[ζ] : U_{1}ZZK] = [UZZ_{p}[ζ] : U_{1}ZZ_{p}K]. Hence the orders of D(A, B) and D(A, M) are equal.

Remark. The last-mentioned exact sequence gives an isomorphism between U_{1}ZZ and the group U_{1}ZZ[ζ] of units of ZZ[ζ] which are congruent to 1 modulo (ζ −1). Analogously, one gets U_{1}ZZ_{p}K ≃ U_{1}ZZ_{p}[ζ].

We can now forget M and work with the inclusion A ⟶ B. The next move will be to split the terms of (7) into odd and even parts with respect to the ∗-involution. U_{1}ZZA = A × U_{1}^{+}ZZA is such a splitting for ZZA. In the p-adic case, it would suffice to point out that U_{1}ZZ_{p}A is uniquely divisible by 2, as can be deduced from local number theory (cf. [5, §15.5]), since the inclusion A ⟶ B reduces the the question to U_{1}ZZ_{p}K and thence to UZZ_{p}[ζ].

However, it is more appropriate to stay on the level of group-rings and use the formula

U_{1}ZZ_{p}A = A × U_{2}ZZ_{p}A ,

which is valid for any abelian p-group. It is proved, just as in [2, Lemmma 2.6], that for p-adic integer coefficients v_{a} = m_{a} + v′_{a}p^{r}, m_{a} ∈ ZZ, one gets 1 + ∑_{a} v_{a}(a − 1) ≡ ∏_{a} a^{m_{a}} +∑_{a} v′_{a}(a^{p^{r}} −1) modulo Δ^{2}, and the last term can be killed by choosing r high enough. The well-known isomorphism A⟶^{∼}Δ/Δ^{2} induced by a ↦ a − 1 (and tensored with ZZ_{p}) shows that A ∩ U_{2}ZZ_{p}A = {1}.

Next one can copy the proofs of [8, Lemma 2 and Lemma 3(i)] to conclude that, for elementary abelian A, there is an isomorphism log : U_{2}ZZ_{p}A⟶^{∼}Δ^{2}ZZ_{p}A onto a free ZZ_{p}-module. The main difference between U_{2}ZZA and U_{2}ZZ_{p}A is that the former lies in U^{+} while the latter splits, U_{2}ZZ_{p}A = (U_{2}ZZ_{p}A)^{−} × (U_{2}ZZ_{p}A)^{+}, into two substantial chunks.

For the record, we copy the even part of (7),

1 ⟶ U_{1}^{+}(A) ⟶ U_{1}^{+}(B) × U_{1}^{+}(A_{p}) ⟶ U_{1}^{+}(B_{p}) ⟶ D^{+} ⟶ 1 ,

only to find that we have seen this before as Sequence (4) in Section 3, all of which proves the following result:

Theorem 4.2. Let D = ker (cl(A) ⟶ cl(M)). Then D^{+} = M.

Corollary 4.3.

log_{p} |D^{+}| ≥ \frac{n}{2} · \frac{p^{n+1} − 1}{p − 1} + \frac{(1 − p) + ⋯ + (1 − p^{n})}{2} .

with equality if p is regular.

Remark. For |D^{−}|, Fröhlich [4] has obtained the larger number

log_{p} |D^{−}| = \frac{n}{4} · (p^{n+1} − 1) + (1 − p) + ⋯ + (1 − p^{n}) .

References

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[3] Fröhlich, A., On the classgroup of integral grouprings of finite abelian groups. Mathematika 16 (1969), 143 - 152

[4] Fröhlich, A., On the classgroup of integral grouprings of finite abelian groups II. Mathematika 19 (1972), 51 - 56

[5] Hasse, H., Zahlentheorie. Akademie-Verlag, Berlin (1963)

[6] Hoechsmann, K., Functors on finite vector spaces and units in abelian group rings. Can. Math. Bull. 29(1), 1986, 79 - 83

[7] Hoechsmann, K., Sehgal, S.K., Integral group rings without proper units. Can. Math. Bull. 30(1) (1987), 36 - 41

[8] Hoechsmann, K., Sehgal, S.K., Units in regular elementary abelian group rings. Arch. d. Math. 47 (1986), 413 - 417

[9] Hoechsmann, K., Sehgal, S.K., Weiss, A., Cyclotomic units and the unit group of an elementary abelian group ring. Arch. d. Math. 45 (1985), 5 - 7

[10] Lang, S., Cyclotomic Fields II. Springer Verlag, N.Y. (1982)

[11] Reiner, I., Maximal Orders. Academic Press, N.Y. (1975)

[12] Ritter, J., Sehgal, S.K., Integral group rings of some p-groups. Can. J. Math. 32 (1982), 233 - 246