This survey article summarizes and complements a series of papers which appeared between 1985 and 1995, and all dealt with the problem of describing the unit group U(A) in the integral group ring of a finite abelian group A — or rather, as much of U(A) as possible. Apart from various cosmetic operations (those in Sections 4 and 8 being substantial) it extends the previous results in two ways: the systematic inclusion of groups of even order (especially 2-groups) and the reorganization of everything in terms of “constructible” units, a concept which first surfaced in the 1990 paper with J. Ritter.

The earlier papers were written with a view toward determining the image of the map μ : ∏_{C} U_{∗}(C) → U_{∗}(A) , where C runs over all cyclic subgroups of A, and U_{∗}(A) denotes a canonical free complement of the torsion group ±A of U(A). If A has odd order, U_{∗}(A) is simply the group U_{1}^{+}(A) consisting of units with coefficient sum (“augmentation”) = 1 and fixed under the standard involution coming from z ↦ z^{−1} (z ∈ A). In the case of elementary abelian pgroups A — with which these studies began — the map μ is injective, and its index turns out to equal the order of a certain group of ideal classes in the group ring ZZA , trivial when p is regular, and only then (unless A = C).

In fact, μ continues to play an important role for arbitrary abelian p-groups. But without its injectivity, it is not suitable for defining bases, and its index is not easily related to ideal classes. This leads to the introduction of the subgroup Y(A) ⊆ U_{∗}(A) of constructible units, which is the direct product of very transparent groups W(C) attached to the various cyclic C ⊆ A. Though computationally convenient, Y(A) has the disdadvantage of lying inside the subgroup U_{∗}^{⊕}(A) ⊆ U_{∗}(A) of circular units, i.e., those which map into cyclotomic units under every character of A. If A is elementary abelian, Y(A) is just the image of μ restricted to circular units. The later papers take the factor group Γ(A) = U_{∗}^{⊕}(A)/Y(A) as their principal theme, and so does the article under review.

This retreat to circular units is not as restrictive as might be thought. For instance, if A is a p-group with p semi-regular (i.e., satisfying “Vandiver’s Conjecture”), the index of μ divides the circular index |Γ(A)|, and actually equals it in the elementary abelian case. Moreover, the three main theorems of this survey — numbered I to III in the present review — show how Γ(A) relates to certain canonical objects and properties. In the first two, A is a finite abelian p-group. Theorem I: U_{∗}^{⊕}(A) = Y(A) if and only if |A| = p or p is regular.

For the second theorem, consider the group D^{+}(A) of projective ideal classes in ZZA which are invariant under the standard involution and become trivial in the maximal order of QA. Further consider the incidence matrix whose rows and columns are labelled by the non-trivial cyclic subgroups C and factor-groups K of A, respectively, with the entry “1” meaning that C survives non-trivially in K and “0” that it does not. Let χ(A) denote the absolute value of its determinant; it equals 1 if A is cyclic. Theorem II: |D^{+}(A)| = χ(A) · |Γ(A)| whenever p is semi-regular.

Three quarters of the survey deal directly or indirectly with these two theorems — in other words, with p-groups. More general groups A occur only in the initial set-up and again in the ninth and final section, which takes a stab at cyclic groups of order pq. The resulting Theorem III will be dicussed at the end of this review.

A leading role in this theory is played by the Galois groups H_{n} of the real fields Q(ζ_{n}+ζ_{n}^{−1}), where ζ_{n} is a primitive n-th root of unity. If C ⊆ A is cyclic of order n, the component W(C) ⊆ Y(A) of constructible units belonging to C is isomorphic (as an H_{n}-module) to the square of the augmentation ideal in ZZH_{n}. This isomorphism, which depends only on the choice of a generator in C, is explained in Section 2 of the survey, along with the directness of the product making up Y(A). Some of these results go back more than thirty years to work of Bass and Milnor, but the unit groups constructed here are much larger than their precursors, which tend to have huge indices in U_{∗}^{⊕}(A).

Section 3 prepares a recursive treatment for cyclic p-groups A by deriving a short exact sequence 1 → Γ(A^{p}) → Γ(A) → I(A) → 1, where I(A) is related to the kernel (on circular units) of the coefficient reduction map ZZA → IF_{p}A. More precisely, I(A) is this kernel modulo units of the form u^{π}/u^{p}, where π denotes the endomorphism of ZZA induced by the p-th power map A → A^{p}. By a generalization of Kummer’s Lemma, I(A) will later be shown to be trivial for regular p. For now, it is seen to be a p-group — and hence so is Γ(A).

Section 4 extends this last insight to non-cyclic A. Since U_{∗}^{⊕}(A) has p-power index in the group E^{⊕}(A) of circular 1-units in the maximal order of QA, it suffices to check the index of the map ∏_{C} E^{⊕}(C) → E^{⊕}(A). Unlike U_{∗}^{⊕}(A), the group E^{⊕}(A) is just the direct product of its cyclotomic components, i.e., the QA-value of a certain lattice-valued functor on a certain category of separable Q-algebras. The main result of the section is a determinant criterion for such a functor V to have the property that the map μ_{V} : ∏_{C} V ( QC) → V ( QA) has p-power index. For V = E^{⊕} this criterion can either be laboriously verified — as it was in one of the original papers — or inferred (via its necessity) from the analogous question involving all units, where the p-power index property is well-known from K-theory.

Section 5 deals with units in the p-adic group ring ZZ_{p}A, in particular with a canonical torsion-free subgroup U_{∗}^{+}ZZ_{p}A — which is equal to U_{1}^{+}ZZ_{p}A if |A| is odd. It shows that maps of type μ are a priori surjective in this context. The correct p-adic counterpart of U_{∗}^{+}(A) is not U_{∗}^{+}ZZ_{p}A itself but the kernel Ú_{∗}^{+}ZZ_{p}A of the H_{n}-norm (with n chosen so as to trivialize A^{n}, an exact fit not being required in the absence of torsion). The main result is the surjectivity of the natural map μ_{p} : ∏_{C} Ú_{∗}^{+}ZZ_{p}C → Ú_{∗}^{+}ZZ_{p}A. This is harder than its analogue without the acute accent, and occasions some preliminary work on “polarized bases”, i.e., bases adapted to the H_{n}-action, in certain ideals of ZZ_{p}A. These are then linked to the multiplicative structure by a logarithmic map — which, however, works only on small pieces of U_{∗}^{+}ZZ_{p}A, because the exp-series often fails to converge, and the log-series does not always yield homomorphisms.

After these two stand-alone treatises on μ-type maps, Section 6 proves the “if”-part of Theorem I. For an abelian p-group A with p regular, it first establishes the density of U_{∗}^{⊕}(A) in Ú_{∗}^{+}ZZ_{p}A. By the results of the preceding sections, this closes the gap between A and its family of cyclic subgroups C. To deal with the latter, one then applies the aforementioned generalization of Kummer’s Lemma — i.e., the triviality of I(C) — which again uses the density lemma as well as the logarithm from the previous section. The crucial density ultimately flows from the injectivity of \overline{U}ZZ[ζ] → \overline{U}ZZ_{p}[ζ], where ζ is a p^{m}-th root of 1 and the bar means working modulo p-th powers. Regularity enforces this by banning unramified p-extensions of Q(ζ).

For the converse, i.e., |Γ(A)| ≠ 1 whenever |A| > p and p irregular, it suffices to consider the case |A| = p^{2}, because Γ preserves injectivity. Section 7 describes how I(C) can be mapped into Γ(A) via “co-induction” along any epimorphism ε : A → C, where |C| = p. For every w ∈ ker [W(C) → UIF_{p}C], a unique ε̃(w) ∈ U_{∗}^{⊕}(A) is “co-induced”: it maps to w by ε and to 1 by all characters of A that do not factor over ε. Via its well-known connection with Bernoulli numbers, irregularity makes I(C) non-trivial: there exist some w ∈ ker [W(C) → UIF_{p}C] such that w ∉ W(C)^{p}. Their counterparts ε̃(w) ∈ U_{∗}^{⊕}(A) turn out to lie outside Y(A). In other words, ε̃ yields an injection (actually a bijection) of I(C) into Γ(A). The section concludes by exhibiting a ZZH_{n}-generator of Γ(A) for |A| = 37^{2}. So much for Theorem I.

Since p = 2 is regular, p can be assumed odd — whence U_{∗}(A) = U_{1}^{+}(A) — throughout the proof of Theorem II given in Section 8. It begins by establishing an analogy between Ú_{1}^{+}ZZ_{p}A and Y(A), which uses the fact that U_{1}^{+}ZZ_{p}A has a basis of the form {v_{p}(z)}, with (z, z^{−1}) ranging over all non-trivial reciprocal pairs in A; indeed v_{p}(z) can be taken to equal E(z − 2 + z^{−1}), where E(t) ∈ ZZ_{p}[[t]] denotes the Artin-Hasse power series. With w_{p}(z) = v_{p}(z)^{σ−1}, where σ generates H_{n}, this makes Ú_{1}^{+}ZZ_{p}A a direct product of components W_{p}(C) attached to the cyclic subgroups C ⊆ A. Next, this analogy is pushed all the way to the equality [E^{⊕}(A) : Y(A)] = [É_{p}^{+}(A) : Ú_{1}^{+}ZZ_{p}A], which then easily becomes |Γ(A)| = [É_{p}^{+}(A) : E^{⊕}(A) Ú_{1}^{+}ZZ_{p}A]. Finally, semi-regularity — which is known to hold for all p < 4 × 10^{6} — allows the replacement of the ⊕ in the last formula by a simple +. The theorem is now obtained by recalling the standard identification of D^{+}(A) with E_{p}^{+}(A)/E^{+}(A) U_{1}^{+}ZZ_{p}A. The extra factor χ(A) is necessary because of the acute accents in the formula for |Γ(A)| derived above.

To describe the main result of the final section, it is convenient to define an analogue of the finite group I(C) used in Sections 3 and 7: for |C| = p and another prime q, let J_{q}(C) be defined as ker [W(C) → UIF_{q}C] modulo units of the form u^{τ_{q}}/u^{q}, where τ_{q} denotes the automorphism of ZZC induced by the q-th power map C → C^{q}. If A is cyclic of order pq, co-induction yields a map from J_{q}(A_{p}) to Γ(A) as in Section 7. Theorem III: Let A be a cyclic group of order pq, with (p − 1, q − 1) ≤ 2. Then the product of the co-induction maps Ξ : J_{q}(A_{p}) × J_{p}(A_{q}) → Γ(A) is an isomorphism.

This is only the simplest case of the full result, whose statement is too intricate for the present purpose. For (p − 1, q − 1) = 2m > 2, it says that kernel and cokernel of Ξ are small, with orders given explicitly in terms of divisors of m. The order of J_{q}(C), on the other hand is computable only case-by-case and seems highly erratic.