September 1997 (superseding May - August 1988)

The El Ab Fantasy Revisited and Clarified.

The following good old diagram of Z_{p}G-modules relates to an el ab group A:

\begin{matrix}∏_{C} Φ̂C & ⟶ & Φ̂A & ⟶ & ∏_{K} Φ̂K \\ ↓ & & ↓ & & ↓ \\ ∏_{C} Φ_{p}C & ⟶ & Φ_{p}A & ⟶ & ∏_{K} Φ_{p}K. \end{matrix}  (1)

Its verticals are collectively labelled λ (“localization” and/or “logarithm”). The western horizontals are denoted α, α′, the eastern ones β, β′, and the composites γ, γ′, respectively. We make the following assumptions (justified in JPAA1, for instance):

(i) all arrows are injective, with α′ bijective; (ii) the eastern square is a pull-back (“intersection”) ^{∗)}; (iii) both Φ̂C and Φ_{p}C are monogenic over Z_{p}G.

Dumping everything into the south-eastern corner, we have by (i) and (ii):

λβ(Φ̂A) = γ′( ∏_{C} Φ_{p}C) ∩ ∏_{C} λ(Φ̂C),  (2)

where the C’s and K’s have been identified (as usual) by means of a bilinear form. Given Z_{p}G-generators w ∈ Φ̂C and v ∈ Φ_{p}C, we have

λ(w) = v^{θ} and λ(Φ̂C) = (Φ_{p}C)^{θ},  (3)

for some θ ∈ Z_{p}G. The cokernel of α can be computed as imβ/imγ ≃ imλβ/imλγ = imλβ/imγ′λ. Using (2) and (3) to evaluate imλβ and imγ′λ, respectively, we get

coker(α) ≃ \frac{γ′(X′) ∩ X′^{θ}}{γ′(X′)^{θ}} ,  (4)

where X′ stands for the direct product of the Φ_{p}C. Remember that γ′ is representable as an m × m-matrix with entries in Z_{p}G (actually only 0’s and elements of G). Letting C_{p} ⊆ (Z_{p}G)^{m} denote the submodule spanned by its columns, our new equation becomes

coker(α) = \frac{C_{p} ∩ θ · (Z_{p}G)^{m}}{θ · C_{p}} .  (5)

There is hope of computability in this direction. In fact, if we slice (5) according to the non-trivial even characters τ^{d} : G → Z_{p}^{×} as in JPAA1, everything depends on θ_{d} and the

∗) That this carries over from ΦA to Φ̂A is due to a simple fact about subsets E, F of a topological space: if F is closed and open, the closure of E ∩ F equals Ê ∩ F. matrix M_{d} associated with γ′ on a typical “slice”. Since G acts via τ^{d}, the coefficient ring is Z_{p} ^{∗∗)}. Left and right multiplication by Gaussian matrices turns M_{d} into a diagonal matrix Δ_{d}, say, with diagonal entries p^{r_{1}} ≥ … ≥ p^{r_{m}}. With ord_{p}(θ_{d}) = t_{d} and μ_{i} = max(r_{i}, t_{d}), we then have

coker(α_{d}) ≃ \frac{C_{p}(Δ_{d}) ∩ p^{t_{d}} · Z_{p}^{m}}{p^{t_{d}} · C_{p}(Δ_{d})} = ∑_{i=1}^{m} p^{μ_{i}}Z_{p}/p^{t_{d}+r_{i}}Z_{p} .  (6)

With s chosen so that r_{i} ≥ t_{d} ⟺ i ≤ s, this turns into

coker(α_{d}) ≃ ∑_{i=1}^{s} Z/p^{t_{d}}Z + ∑_{i=s+1}^{m} Z/p^{r_{i}}Z .  (7)

This whole calculation works also for t_{d} = 0, i.e., when d is “good”, and yields a trivial cokernel in that case. Let us, however, point out that working over Z_{p} only determines the p-primary component of coker(α_{d}) — which is all there is by HSW. So, we cannot get around using the latter, nor can we operate without p-adic logarithms: how else would we know that α′ is an isomorphism?

By JPAA1, the second sum in (7) runs non-trivially only up to m − C(d + n, n), and this number cannot be less than s if t_{d} is non-zero. For more insight, we would need a handle on the sequence r_{1}, …, r_{m}, and — above all — the value of t_{d}. It seems likely that t_{d} will usually be ≤ 1, in which case coker(α_{d}) is elementary abelian.

Let us come down to earth with ΦA = U_{1}^{⊕}ZA and Φ_{p}A = Δ^{+}Z_{p}A. Then λ is the composite

W(C) ⟶^{ι} U_{1}^{+}Z_{p}C ⟶^{log} Δ^{+}Z_{p}C .  (8)

For u ∈ W(C), we then have u ≡ 1 ⟺ ι(u) ≡ 1 ⟺ λ(u) ≡ 0 (mod p^{ν}). Hence t_{d} is the largest ν for which this happens if u = w^{ε_{d}} is a Kummer unit at d (i.e., ε_{d} ∈ ZG lifts the appropriate idempotent e_{d} ∈ F_{p}G). We have already seen that it is 1 in the case p = 37, d = 32.

WRONG: We ain’t seen nothin’ yet. The ε_{d} used here should ideally be the appropriate idempotent from Z_{p}G — or at least a ZG-lift of its truncation to (Z/p^{μ}Z)G, where μ can be taken = 2 if we just want to decide whether or not t_{d} ≤ 1.

∗∗) They can even be reduced to Z/p^{n}Z — by Lemma 2.1 of JPAA1 (cf. also “ Addyt” for a less mysterious proof).