GRAS–TYPE CONJECTURES

FOR FUNCTION FIELDS

Cristian D. Popescu

Abstract. Based on results obtained in [15], we construct groups of special S– units for function fields of characteristic p > 0, and show that they satisfy Gras– type Conjectures. We use these results in order to give a new proof of Chinburg’s Ω_{3}–Conjecture on the Galois module structure of the group of S–units, for cyclic extensions of prime degree of function fields.

0. Introduction

Let K/k be a finite, abelian extension of function fields of characteristic p > 0. Let G = G (K/k) and g = |G|. We will denote by F_{q} and F_{q^{ν}} the exact fields of constants of k and K respectively, where q is a power of p and ν is a positive integer. In what follows we will use the same notations as in [15]. For the convenience of the reader, we briefly summarize in this section the main concepts and results of [15] which will be used in our arguments.

For any two finite, nonempty and disjoint sets S and T of primes in k, and any field F, k ⊆ F ⊆ K, U_{F,S} and U_{F,S,T} denote the groups of S–units and respectively (S, T)–units of F; A_{F,S} and A_{F,S,T} are respectively the S–ideal class group and (S, T)–ideal class group of F, as defined in [15, §1.1]. In particular, if F = K, we suppress K from the notation, so U_{K,S} = U_{S}, U_{K,S,T} = U_{S,T} etc.

Let us assume for the moment that for a certain positive integer r, the set of data (K/k, S, T, r) satisfies the following set of hypotheses:

(H)  \{ \begin{matrix}S ≠ ∅, T ≠ ∅, S ∩ T = ∅. \\ S contains all primes which ramify in K/k. \\ S contains at least r primes which split completely in K/k. \\ |S| ≥ r + 1. \end{matrix}

Let (v_{1}, …, v_{r}) be an ordered r–tuple of primes in S which split completely in K/k, and let W = (w_{1}, …, w_{r}), with w_{i} prime in K, w_{i}|v_{i}, for every i = 1, …, r. One can define a regulator map C ∧^{r} U_{S,T} ⟶^{R_{W}} C [G] , by

R_{W} (u_{1} ∧ ⋯ ∧ u_{r}) = det_{i,j} (− ∑_{σ∈G} log |u_{j}^{σ^{−1}}|_{w_{i}} · σ) , ∀u_{1} ∧ ⋯ ∧ u_{r} ∈ ∧^{r} U_{S,T} .

Typeset by AMS-TEX

Research at MSRI is supported in part by NSF grant DMS–9022140

Throughout this paper, exterior powers are viewed over Z [G], |u| = Nw^{−ord_{w}(u)}, for every u ∈ K^{×}, and Nw is the order of the residue field F_{q^{ν}} (w) at w, for every prime w of K. Similarly, if v is a prime in k, Nv denotes the order of its residue field F_{q}(v).

For every irreducible character χ ∈ Ĝ, let L_{S,T} (s, χ) be the associated (S, T)– L–function, as defined in [15, §1.2], and let Θ_{S,T} (s) = ∑_{χ} L_{S,T} (s, χ)e_{χ^{−1}} be the

Stickelberger function, where e_{χ} = ∑_{σ∈G} χ(σ)·σ^{−1} ∈ C [G]. If r_{χ} = ord_{s=0}L_{S,T} (s, χ),

then

r_{χ} = \{ \begin{matrix}card\{v ∈ S : χ|_{G_{v}} = 1_{G_{v}}\}, & if χ ≠ 1_{G} \\ card (S) − 1, & if χ = 1_{G} , \end{matrix}

where G_{v} is the decomposition group of v in K/k. This shows that hypotheses (H) imply that r_{χ} ≥ r, for every χ ∈ Ĝ, and therefore the following definition makes sense in C [G]

Θ_{S,T}^{(r)} (0) =^{def} lim_{s→0} s^{−r}Θ_{S,T} (s) = ∑_{χ∈Ĝ} lim_{s→0} s^{−r}L_{S,T} (s, χ) · e_{χ^{−1}} ∈ C[G].

Definition. For any ring R with 1, and any R [G]–module M, let

M_{r,S} =^{def} {m ∈ M | e_{χ} · m = 0 in M ⊗_{R} C, ∀χ ∈ Ĝ with r_{χ} ≠ r}.

In [15] (see Theorem 3.2.1 and Corollary 3.2.2) we proved the following result:

Theorem 0. If the set of data (K/k, S, T, r) satisfies hypotheses (H) then:

(1) There exists a unique ε_{S,T} ∈ (Z [1/g] ∧^{r} U_{S,T} )_{r,S} such that

R_{W} (ε_{S,T}) = Θ_{S,T}^{(r)} (0).

(2) The element ε_{S,T} satisfies the equality:

Z [1/g] [G] ε_{S,T} = Fitt_{Z[G]} (A_{S,T}) · (Z [1/g] ∧^{r} U_{S,T} )_{r,S} .

Throughout this paper, for any noetherian ring R with 1 and any finitely generated R–module M, Fitt_{R} (M) denotes the Fitting ideal of M. For the definition and properties of the Fitting ideals needed for our purposes, the reader can consult [15, §1.4].

Let us now fix once and for all two finite, nonempty, disjoint sets S and T of primes in k, S containing at least one prime which splits completely in K/k, and T not containing primes which ramify in K/k.

We will use Theorem 0(1) in order to construct two Z [G]–modules of special units E_{S,T} and E_{S}, of finite index in U_{S,T} and U_{S} respectively. Theorem 0(2) will help us prove that statements similar to the ones conjectured by Gras in [9] and proved by Mazur and Wiles in [12] for the classical number field case of cyclotomic units, are satisfied by E_{S,T} and E_{S} (Theorems 1.4, 2.2, 3.10). We use these results in §4 in order to give a new proof to a special case of Chinburg’s Ω_{3}–Conjecture (Theorem 4.2.7).

Before proceeding we would like to make a final useful remark on group–rings and their modules (see also [15, §1.3]). If L is a field of characteristic 0, then Ĝ(L) will denote the set of characters associated to L–irreducible representations of G. If \overline{L} is an algebraic closure of L, then G (\overline{L}/L) acts canonically on Ĝ(\overline{L}) and Ĝ(L) can be viewed as the set of orbits with respect to this action. For χ ∈ Ĝ(\overline{L}) and ψ ∈ Ĝ(L), we write χ|ψ if χ is in the orbit represented by ψ. For a subgroup H ⊆ G (or an element σ ∈ G) we write ψ|_{H} = 1_{H} (or ψ(σ) = 1) if χ|_{H} = 1_{H} (or χ(σ) = 1), for some (i.e. all) χ ∈ Ĝ(\overline{L}), such that χ|ψ.

If R is a Dedekind domain containg Z [1/g], and L is its field of fractions, then one has a canonical decomposition

R [G] = ⨁_{ψ∈Ĝ(L)} D_{ψ} ,

where D_{ψ} = R [G]·e_{ψ} are finite extensions of R, and e_{ψ} = 1/g ∑_{σ∈G} ψ(σ)·σ^{−1} ∈ R [G].

If M is an R [G]–module, then one has a decomposition

M = ⨁_{ψ∈Ĝ(L)} M^{ψ} ,

where M^{ψ} = M ⨂_{R[G]} D_{ψ}, for every ψ ∈ Ĝ(L).

Acknowledgements. It is a pleasure to thank Karl Rubin for his invaluable help and support, and Adebisi Agboola for helpful conversations leading to the result in §4.

1. The Group E_{S,T} of Stark (S, T)–units

Throughout this section the sets of primes S and T as above will be fixed, so that S contains at least one prime which splits completely and T does not contain primes which ramify in K/k. Let F be an intermediate field of K/k and let S′ be a finite set of primes in k containing S. We define

r_{F,S′} = \{ \begin{matrix}card\{v ∈ S′| v splits completely in F/k\}, & for F ≠ k \\ card (S′) − 1, & for F = k. \end{matrix}

In particular, if S′ = S, we will make the notation r_{F,S} = r_{F}.

Let L be any finite field of characteristic 0, and let ψ ∈ Ĝ(L). We define

r_{ψ,S′} = \{ \begin{matrix}card\{v ∈ S′| ψ|_{G_{v}} = 1_{G_{v}}\}, & for ψ ≠ 1_{G} \\ card (S′) − 1, & for ψ = 1_{G}. \end{matrix}

In particular, for S = S′, we make the notation r_{ψ,S} = r_{ψ}.

For ψ ∈ Ĝ(L) as above, let H_{ψ} = ker(ψ) and let K_{ψ} be the maximal subfield of K fixed by H_{ψ}. The injectivity of ψ on G (K_{ψ}/k) obviously implies that r_{K_{ψ},S′} = r_{ψ,S′}, for any S′ as above.

Let S = {v_{0}, …, v_{s}}. For every i = 0, …, s, let us fix a prime w_{i} above v_{i} in K and let w_{i,F} denote the prime in F sitting below w_{i}, for every intermediate field F. Let W_{F} be the ordered r_{F} –tuple of primes in F defined by

W_{F} = \{ \begin{matrix}(w_{i,F} | v_{i} splits completely in F) , & for F ≠ k \\ (v_{0}, …, v_{s−1}) , & for F = k , \end{matrix}

ordered so that w_{i,F} preceeds w_{j,F} iff i < j.

Definition 1.1. A pair (F, S′) consisting of an intermediate field F, F ≠ k, and a finite set of primes S′ in k is called (S, T) –admissible if the following conditions are satisfied:

(1) S ⊆ S′

(2) |S′| ≥ |S| + 2

(3) the set of data (F/k, S′, T, r_{F}) satisfies hypotheses (H) The pair (k, S) is also declared to be (S, T) –admissible.

Let us emphasize that if (F, S′) is an (S, T) –admissible pair, then the set of data (F/k, S′, T, r_{F}) satisfies hypotheses (H). For any such pair (F, S′), let

ε_{F,S′,T} ∈ [Z [1/g] ∧^{r_{F}} U_{F,S′,T} ]_{r_{F} ,S′}

be the unique element associated by Theorem 0 to the set of data (F/k, S′, T, r_{F}) and the regulator R_{W_{F}}.

Let U_{F,S′,T}^{∗} = Hom_{Z[G]} (U_{F,S′,T} , Z [G]) be the Z [G]–dual of U_{F,S′,T}. Then for

every Φ = ϕ_{1} ∧ ⋯ ∧ ϕ_{r_{F} −1} ∈ Z [1/g] ∧^{r_{F} −1} U_{F,S′,T}^{∗} we get a Z [1/g] [G]–morphism

\xymatrix{Z [1/g] ∧^{r_{F}} U_{F,S′,T} \ar[r]^{Φ} & Z [1/g] U_{F,S′,T} ,}

given by

Φ (u_{1} ∧ ⋯ ∧ u_{r_{F}}) = ∑_{1≤i≤r_{F}} (−1)^{i} det_{\begin{matrix}1≤k,j≤r_{F} \\ j≠i \end{matrix}} (ϕ_{k}(u_{j})) · u_{i} ,

for all u_{1} ∧ ⋯ ∧ u_{r_{F}} ∈ ∧^{r_{F}} U_{F,S′,T}.

For every intermediate field F, let μ_{F} be the group of roots of unity in F, and let e_{F} = |μ_{F} |. In particular, if F = K, we have μ_{K} = F_{q^{ν}}^{×} and therefore e_{K} = q^{ν} − 1.

Definition 1.2. Let E^{∼}_{S,T} be the Z [1/g] [G]–submodule of Z [1/g] ⨂_{Z} (K^{×}/μ_{K}) gen-

erated by the set

{Φ (ε_{F,S′,T}) | (F, S′) is (S, T) –admissible , Φ ∈ Z [1/g] ∧^{r_{F} −1} U_{F,S′,T}^{∗} } .

Definition 1.3. The group E_{S,T} of Stark (S, T)–units is defined by

E_{S,T} = E^{∼}_{S,T} ⋂ U_{S,T} ,

where the intersection is viewed inside Z [1/g] ⨂_{Z} (K^{×}/μ_{K}).

The main goal of this section is the proof of the following Gras–type Conjecture for the group E_{S,T} of special (S, T)–units:

Theorem 1.4. Let K/k, S and T be as above. Then

(1) The index [U_{S,T} : E_{S,T}] is finite.

(2) For any prime number ℓ, such that gcd(ℓ, g) = 1, and any ψ ∈ Ĝ(Q_{ℓ})

| (U_{S,T} /E_{S,T} ⊗ Z_{ℓ})^{ψ} | = | (A_{S,T} ⊗ Z_{ℓ})^{ψ} |^{r_{ψ}} .

Before proceeding to the proof of the theorem above, we need a few more lemmas and definitions.

Definition 1.5. Let L be a field of characteristic 0, and let ψ ∈ Ĝ(L). A pair (F, S′) is called (S, T, ψ) –admissible if the following conditions are satisfied

(1) (F, S′) is (S, T) –admissible

(2) K_{ψ} ⊆ F

(3) r_{ψ,S′} = r_{F}.

Let us notice that (k, S) is the only (S, T, 1_{G})–admissible pair, while for ψ ≠ 1_{G} there exist infinitely many (S, T, ψ) –admissible pairs.

Lemma 1.6.

(1) Let ψ ∈ Ĝ(L) and let (F, S′) be an (S, T, ψ) –admissible pair. Then r_{K_{ψ}} = r_{ψ,S′} = r_{F} and (K_{ψ}, S′) is also (S, T, ψ) –admissible.

(2) If (F, S′) and (F, S′′) are (S, T, ψ) –admissible pairs, then (F, S′ ⋃ S′′) is (S, T, ψ) –admissible as well.

Proof. Both statements are clear if ψ = 1_{G}. Let ψ ≠ 1_{G}.

(1) From the definitions above we have the following sequence of inequalities

r_{K_{ψ}} ≥ r_{F} = r_{ψ,S′} ≥ r_{ψ} = r_{K_{ψ}} ,

which shows that r_{K_{ψ}} = r_{ψ,S′} = r_{F}. This obviously implies that (K_{ψ}, S′) is (S, T, ψ) –admissible.

(2) Since r_{F} = r_{ψ,S′} = r_{ψ,S′′}, we have ψ|_{G_{v}} ≠ 1_{G_{v}}, for every v ∈ S′ \ S and every v ∈ S′′ \ S. This shows that r_{F} = r_{ψ,S′∪S′′}, which obviously implies that (F, S′ ⋃ S′′) is (S, T, ψ) –admissible. □

Lemma 1.7. Let ψ ∈ Ĝ(Q) and let (F, S′) be an (S, T, ψ) –admissible pair. Then the natural Z [G]–morphisms involved in the following commutative diagrams:

\xymatrix{U_{S,T} \ar[r] & U_{S′,T} & A_{S,T} \ar[r] & A_{S′,T} \\ U_{F,S,T} \ar[u] \ar[r] & U_{F,S′,T} \ar[u] & A_{F,S,T} \ar[u] \ar[r] & A_{F,S′,T} , \ar[u]}

become Z [1/g] [G]^{ψ}–isomorphisms, when tensored with Z [1/g] [G]^{ψ} over Z [G].

Proof. The vertical morphisms in both diagrams obviously become isomorphisms when tensored with Z [1/g] [G]^{ψ}. (Their inverses are the Z [1/g] [G]^{ψ}–linear extensions of the norm map from K to F at the level of units and ideal class–groups respectively.) This observation settles the lemma above for ψ = 1_{G} and it shows that it is enough to prove that the lower horizontal morphisms in both diagrams become isomorphisms when tensored with Z [1/g] [G]^{ψ}, for ψ ≠ 1_{G}.

Let ψ ≠ 1_{G} and let S be the Z [G]–submodule of A_{F,S,T} generated by the ideal classes

{w| w prime in F, w|v for some v ∈ S′ \ S} .

Let us fix a prime w(v) in F for every prime v ∈ S′\S. We have two exact sequences of Z [G]–modules (see [15, §2 (9)] and [17, §5.1]):

\xymatrix{0 \ar[r] & S \ar[r] & A_{F,S,T} \ar[r] & A_{F,S′,T} \ar[r] & 0}

\xymatrix{0 \ar[r] & U_{F,S,T} \ar[r] & U_{F,S′,T} \ar[r] & ⨁_{v∈S′\S} Z [G] w(v) \ar[r] & S \ar[r] & 0.}

Let us fix v ∈ S′ \ S. Since ψ ≠ 1_{G} and (F, S′) is (S, T, ψ) –admissible, we have ψ|_{G_{v}} ≠ 1_{G_{v}}. This shows that there exists σ ∈ G such that ^{σ}w(v) = w(v) and ψ(σ) ≠ 1. We therefore have an equality

e_{ψ}(σ − 1) · w(v) = 0  (1)

in Z [1/g] [G]^{ψ}, where e_{ψ} is the idempotent associated to ψ in Z [1/g] [G]. The fact that ψ(σ) ≠ 1 easily implies that e_{ψ}(σ−1) is invertible in Z [1/g] [G]^{ψ} and therefore relation (1) above shows that

⨁_{v∈S′\S} Z [1/g] [G]^{ψ} · w(v) = 0.

The last equality proves that, when tensored with Z [1/g] [G]^{ψ} (which is a flat Z [G]–algebra), the exact sequences above yield isomorphisms

(Z [1/g] A_{F,S,T})^{ψ} ⟶^{∼} (Z [1/g] A_{F,S′,T})^{ψ} , (Z [1/g] U_{F,S,T})^{ψ} ⟶^{∼} (Z [1/g] U_{F,S′,T})^{ψ} ,

which concludes the proof of Lemma 1.7. □

Lemma 1.8.

(1) Let ψ ∈ Ĝ(Q). Then E^{∼}_{S,T}^{ψ} is generated as a Z [1/g] [G]^{ψ}–module by the set

{e_{ψ} · Φ (ε_{K_{ψ},S′,T} ) | Φ ∈ Z [1/g] ∧^{r_{ψ}−1} U_{K_{ψ},S′,T}^{∗} } ,

for any S′ such that (K_{ψ}, S′) is (S, T, ψ) –admissible.

(2) E^{∼}_{S,T}^{ψ} ⊆ Z [1/g] U_{S,T} and Z [1/g] E_{S,T} = E^{∼}_{S,T}.

Proof. (1) Let (F, S′) be an (S, T) –admissible pair. If the pair (F, S′) is not (S, T, ψ) –admissible, then either K_{ψ} ⊈ F, or K_{ψ} ⊆ F and r_{ψ,S′} ≠ r_{F}. In both

circumstances ε_{F,S′,T}^{ψ} = 0 in (Z [1/g] ∧^{r_{F}} U_{F,S′,T} )^{ψ}. We thus have

Φ (ε_{F,S′,T}^{ψ} ) = 0, for all Φ ∈ Z [1/g] ∧^{r_{F} −1} U_{F,S′,T}^{∗} .

It is therefore enough to restrict ourselves to (S, T, ψ) –admissible pairs in the defi-

nition of E^{∼}_{S,T}^{ψ}. Let (F, S′) be such a pair. Then, according to Lemma 1.6, (K_{ψ}, S′) is also (S, T, ψ) –admissible and r_{F} = r_{K_{ψ}} = r_{ψ}. The definition and uniqueness of ε_{K_{ψ},S′,T} and ε_{F,S′,T} show that

ε_{F,S′,T} = N_{F/K_{ψ}}^{(r_{ψ})} (ε_{K_{ψ},S′,T} ) ,  (2)

where N_{F/K_{ψ}}^{(r_{ψ})} is the Z [1/g]–linear extension to Z [1/g] ∧^{r_{ψ}} U_{F,S′,T} of the r_{ψ}–exterior power of the norm map

\xymatrix{U_{F,S′,T} \ar[r]^{^{N^{F/K_{ψ}}}} & U_{K_{ψ},S′,T}} .

(see [17, §6] for (2)). The Z–freeness of U_{F,S′,T} gives an isomorphism of abelian groups

\xymatrix{U_{F,S′,T}^{∗} \ar[r]_{∼}^{ϕ→ϕ_{0}} & _{Hom_{Z} (U_{F,S′,T} , Z)}} ,  (3)

defined by ϕ(u) = ∑_{σ∈G(F/k)} ϕ_{0} (σ^{−1}u)·σ, for every u ∈ U_{F,S′,T}. Obviously the same

type of isomorphism holds true for U_{K_{ψ},S′,T}^{∗} as well. The inclusion U_{K_{ψ},S′,T} ⊆ U_{F,S′,T} induces therefore a surjective morphism

[figure: Z [1/g] U_{F,S′,T}^{∗}; ϕ→ϕ̃; Z [1/g] U_{K_{ψ},S′,T}^{∗} ,]

satisfying the relation ϕ̃ (N_{F/K_{ψ}}u) = πϕ(u), for every u ∈ U_{K_{ψ},S′,T}, where π is the natural projection

\xymatrix{Z [G (F/k)] \ar[r]^{π} & Z [G (K_{ψ}/k)] .}

We therefore obtain a surjective Z [1/g]–morphism

[figure: Z [1/g] ∧^{r_{ψ}−1} U_{F,S′,T}^{∗}; Z [1/g] ∧^{r_{ψ}−1} U_{K_{ψ},S′,T}^{∗}]
\xymatrix{Φ = ϕ_{1} ∧ ⋯ ∧ ϕ_{r_{ψ}−1} \ar[r] & Φ̃ = ϕ̃_{1} ∧ ⋯ ∧ ϕ̃_{r_{ψ}−1}}

satisfying the relation

Φ̃ (N_{F/K_{ψ}}^{(r_{ψ})}(u)) = N_{F/K_{ψ}} (Φ(u)) ,

for every u ∈ Z [1/g] ∧^{r_{ψ}} U_{F,S′,T}. The equality above obviously implies that, for any

(S, T, ψ) –admissible pair (F, S′) and any Φ ∈ Z [1/g] ∧^{r_{ψ}−1} U_{F,S′,T}^{∗}, the following

relation holds true in E^{∼}_{S,T}^{ψ}:

[F : K_{ψ}] e_{ψ}Φ (ε_{F,S′,T}) = e_{ψ}Φ̃ (ε_{K_{ψ},S′,T} ) .  (4)

Let now S′ and S′′ be two sets of primes such that the pairs (K_{ψ}, S′) and (K_{ψ}, S′′) are both (S, T, ψ) –admissible. According to Lemma 1.6(2), the pair (K_{ψ}, S′ ∪ S′′) is (S, T, ψ) –admissible as well. The definition and uniqueness of the elements ε give the following relations (see [17, §6]):

ε_{K_{ψ},S′∪S′′,T} = [ ∏_{v∈S′′\S′} (1 − σ_{v}^{−1})] ε_{K_{ψ},S′,T} = [ ∏_{v∈S′\S′′} (1 − σ_{v}^{−1})] ε_{K_{ψ},S′′,T} ,

where σ_{v} is the Frobenius morphism associated to v in G (K_{ψ}/k). This implies that

[ ∏_{v∈S′′\S′} e_{ψ} (1 − σ_{v}^{−1})] ε_{K_{ψ},S′,T} = [ ∏_{v∈S′\S′′} e_{ψ} (1 − σ_{v}^{−1})] ε_{K_{ψ},S′′,T} . (5)

Relations (4) and (5), the fact that [F : K_{ψ}] and e_{ψ} (σ_{v} − 1) are invertible in Z [1/g] [G]^{ψ}, for v ∈ (S′ \ S′′) ∪ (S′′ \ S′) (see the proof of Lemma 1.7), and Lemma 1.7 show that (1) holds true.

(2) Lemma 1.7 implies that for any S′ such that (K_{ψ}, S′) is (S, T, ψ) –admissible, we have Z [1/g] [G]^{ψ} U_{K_{ψ},S′,T} = Z [1/g] [G]^{ψ} U_{S,T}. We thus have

e_{ψ}Φ (ε_{K_{ψ},S′,T} ) ∈ Z [1/g] [G]^{ψ} U_{S,T} ,

for any Φ ∈ ∧^{r_{ψ}−1} U_{K_{ψ},S′,T}^{∗}. According to this relation, the fact that E_{S,T}^{∼} ⊆ Z [1/g] U_{S,T} follows from statement (1) in our lemma. The equality Z [1/g] E_{S,T} =

E^{∼}_{S,T} is now an obvious consequence of the definition of E_{S,T}. □

Proof of Theorem 1.4. Since U_{S,T} is a finitely generated Z–module, it is obviously enough to prove the second statement of the theorem. Let ℓ be a prime number such that gcd (ℓ, g) = 1, and let

Z_{ℓ} [G] = ⨁_{ψ∈Ĝ(Q_{ℓ})} D_{ψ}

be the decomposition of Z_{ℓ} [G] into a direct sum of discrete valuation rings, as described in §0. Let us fix ψ ∈ Ĝ(Q_{ℓ}) and let S′ be a set of primes such that (K_{ψ}, S′) is (S, T, ψ) –admissible. Since r_{ψ,S′} = r_{ψ}, Lemma 1.5.1(1) of [15] gives us the following isomorphisms of D_{ψ}–modules

(Z_{ℓ}U_{K_{ψ},S′,T} )^{ψ} ⟶^{∼} D_{ψ}^{r_{ψ}} , Hom_{D_{ψ}} ((Z_{ℓ}U_{K_{ψ},S′,T} )^{ψ} , D_{ψ}) ⟶^{∼} D_{ψ}^{r_{ψ}} .

Let {e_{1}, …, e_{r_{ψ}}} be a D_{ψ} basis of (Z_{ℓ}U_{K_{ψ},S′,T} )^{ψ} and let {e_{1}^{∗}, …, e_{r_{ψ}}^{∗}} be its D_{ψ}– dual. We have the following equalities of D_{ψ}–modules

D_{ψ} ⨂_{Z[1/g][G]} (Z [1/g] [G] ∧^{r_{ψ}} U_{K_{ψ},S′,T} )_{r_{ψ},S′} = (Z_{ℓ} ∧^{r_{ψ}} U_{K_{ψ},S′,T} )^{ψ}
= D_{ψ} (e_{1} ∧ ⋯ ∧ e_{r_{ψ}}) .

Let us now combine Theorem 0(2), applied to the set of data (K_{ψ}/k, S′, T, r_{ψ}), with the equalities above. After tensoring with D_{ψ} over Z [1/g] [G], we obtain

D_{ψ}ε_{K_{ψ},S′,T} = a_{ψ} (e_{1} ∧ ⋯ ∧ e_{r_{ψ}}) ,  (6)

where

a_{ψ} = Fitt_{Z[G]} (A_{K_{ψ},S′,T} ) ⨂_{Z[G]} D_{ψ} = Fitt_{D_{ψ}} (A_{K_{ψ},S′,T} ⨂_{Z[G]} D_{ψ}) .  (7)

(For the last equality above see [15, §1.4(b)].) We are now prepared to compute the D_{ψ}–module (Z_{ℓ}E_{S,T})^{ψ} in terms of the chosen basis {e_{1}, …, e_{r_{ψ}}}. Let us first notice that (3) shows that we have the following relations:

(Z_{ℓ}U_{K_{ψ},S′,T}^{∗} ) ⟶^{∼} Hom_{Z_{ℓ}[G]} (Z_{ℓ}U_{K_{ψ},S′,T} , Z_{ℓ} [G]) =

= ⨁_{χ∈Ĝ(Q_{ℓ})} Hom_{D_{χ}} ((Z_{ℓ}U_{K_{ψ},S′,T} )^{χ} , D_{χ}) .

This direct sum decomposition together with the definition of E^{∼}_{S,T}, Lemma 1.8 and equality (6) imply that

\begin{aligned}(Z_{ℓ}E_{S,T})^{ψ} & = (Z_{ℓ}E^{∼}_{S,T} )^{ψ} = \\ & = ∑_{1≤i≤r_{ψ}} (e_{1}^{∗} ∧ ⋯ ∧ ê_{i}^{∗} ∧ ⋯ ∧ e_{r_{ψ}}^{∗}) (D_{ψ}ε_{K_{ψ},S′,T} ) = \\ & = ∑_{1≤i≤r_{ψ}} (e_{1}^{∗} ∧ ⋯ ∧ ê_{i}^{∗} ∧ ⋯ ∧ e_{r_{ψ}}^{∗}) (a_{ψ} · e_{1} ∧ ⋯ ∧ e_{r_{ψ}}) = \\ & = a_{ψ}e_{1} ⊕ ⋯ ⊕ a_{ψ}e_{r_{ψ}} , \end{aligned} (8)

equality viewed inside (Z_{ℓ}U_{K_{ψ},S′,T} )^{ψ} = (Z_{ℓ}U_{S,T})^{ψ} = D_{ψ}^{r_{ψ}} (see Lemma 1.7 above and [15], Lemma 1.5.1(1)). Relation (8), together with (7) and [15, §1.4(f)], obviously give us the following equality

| (U_{S,T} /E_{S,T} ⊗ Z_{ℓ})^{ψ} | = [D_{ψ} : a_{ψ}]^{r_{ψ}} = | (A_{K_{ψ},S′,T} ⊗ Z_{ℓ})^{ψ} |^{r_{ψ}}.

On the other hand, after tensoring with D_{ψ}, Lemma 1.7 shows that

(A_{K_{ψ},S′,T} ⊗ Z_{ℓ})^{ψ} ⟶^{∼} (A_{S,T} ⊗ Z_{ℓ})^{ψ} ,

as D_{ψ}–modules, which, combined with the previous equality, settles statement (2) in our theorem. □

2. The Group E_{S} of Stark S–units With the same assumptions as in the previous section, let us keep S fixed and let T vary, so that S ∩ T = ∅, and T does not contain any prime which ramifies in K/k. We will denote by T the set of all possible choices of sets T. For each T ∈ T we have just defined a Z [G]–submodule E_{S,T} of U_{S,T}.

Definition 2.1. The group of Stark S–units E_{S} is the Z [G]–submodule of U_{S}, generated by

μ_{K} ⋃ ( ⋃_{T ∈T} E_{S,T} ) .

The main goal of this section is the proof of the following Gras–type Conjecture satisfied by the group E_{S} of special S–units:

Theorem 2.2. Let S be a finite set of primes in k, containing at least one prime which splits completely in K/k. Then:

(1) The index [U_{S} : E_{S}] is finite.

(2) For any prime number ℓ, such that gcd(ℓ, g · e_{K}) = 1, and any character ψ ∈ Ĝ(Q_{ℓ}),

| (U_{S}/E_{S} ⊗ Z_{ℓ})^{ψ} | = | (A_{S} ⊗ Z_{ℓ})^{ψ} |^{r_{ψ}}.

In order to prove the theorem above, we need the following

Lemma 2.3. For any T ∈ T , |E_{S}/E_{S,T} | is divisible only by primes dividing

g · ∏_{w∈T_{K}}(Nw − 1).

Proof. Let T_{1}, T_{2} ∈ T and let ψ ∈ Ĝ(Q). Let S′ be a finite set of primes in k such that the pair (K_{ψ}, S′) is (S, T_{1} ∪ T_{2}, ψ)–admissible. Then (K_{ψ}, S′) is obviously (S, T_{1}, ψ)–admissible and (S, T_{2}, ψ)–admissible as well. The definition and uniqueness of the elements ε associated by Theorem 0 to the sets of data (K_{ψ}, S′, T_{i}, r_{ψ}), for i = 1, 2, and (K_{ψ}, S′, T_{1} ∪ T_{2}, r_{ψ}), give the following relations:

\begin{aligned}ε_{K_{ψ},S′,T_{1}∪T_{2}} & = [ ∏_{v∈T_{2}\T_{1}}(1 − σ_{v}^{−1} · Nv)] · ε_{K_{ψ},S′,T_{1}} = \\ & = [ ∏_{v∈T_{1}\T_{2}}(1 − σ_{v}^{−1} · Nv)] · ε_{K_{ψ},S′,T_{2}} \end{aligned}  (9)

(see [17, §6] for (9)). Let us also notice (see Lemma 3.1 below as well) that for any prime v in k, which is unramified in K/k, we have

(1 − σ_{v}^{−1} · Nv) μ_{K} = {1} ,  (10)

where σ_{v} is the Frobenius morphism associated to v in G (K/k). In light of Lemma 1.8(1), equalities (9) and (10) imply that, for any set T ∈ T ,

∏_{v∈T}(1 − σ_{v}^{−1} · Nv) · Z [1/g] E_{S} ⊆ Z [1/g] E_{S,T} ,

and therefore

∏_{w∈T_{K}}(1 − Nw) · Z [1/g] E_{S} ⊆ Z [1/g] E_{S,T} .  (11)

(We are using the fact that (1 − σ_{v}^{−1} · Nv) divides (1−Nw) in the group ring Z [G], for any v ∈ T, and any w in K, such that w|v.) The statement in the lemma follows directly from (11). □

Proof of Theorem 2.2. Since U_{S} is a finitely generated Z–module, it is obviously enough to prove (2) in the statement of the Theorem above.

Due to the fact that K has a divisor of degree 1 over F_{q^{ν}}, supported outside any finite set of primes (see [13]), we have

gcd{(1 − Nw); w ∈ T_{K}, T ∈ T } = e_{K}.

Let us fix a prime number ℓ, such that gcd(ℓ, ge_{K}) = 1, and let us consider T ∈ T , T = {v}, such that gcd(1 − Nw, ℓ) = 1, for every prime w ∈ T_{K}. We have the following exact sequences of Z [G]–modules (see [15, §1.1, (1)]):

0 ⟶ U_{S}/U_{S,T} ⟶ ⨁_{w∈T_{K}} F_{q^{ν}} (w)^{×} ⟶ A_{S,T} ⟶ A_{S} ⟶ 0

0 ⟶ (E_{S} ∩ U_{S,T})/E_{S,T} ⟶ U_{S,T} /E_{S,T} ⟶ U_{S}/E_{S} ⟶ U_{S}/ (U_{S,T} · E_{S}) ⟶ 0.

Lemma 2.3 implies that, when tensored with Z_{ℓ}, the exact sequences above give the following isomorphisms of Z_{ℓ} [G]–modules

A_{S,T} ⊗ Z_{ℓ} ⟶^{∼} A_{S} ⊗ Z_{ℓ}, U_{S,T} /E_{S,T} ⊗ Z_{ℓ} ⟶^{∼} U_{S}/E_{S} ⊗ Z_{ℓ}.

These isomorphisms, combined with Theorem 1.4(2), imply that

\begin{matrix}|(U_{S}/E_{S} ⊗ Z_{ℓ})^{ψ}|^{r_{ψ}} = |(U_{S,T} /E_{S,T} ⊗ Z_{ℓ})^{ψ}|^{r_{ψ}} = \\ = |(A_{S,T} ⊗ Z_{ℓ})^{ψ}|^{r_{ψ}} = |(A_{S} ⊗ Z_{ℓ})^{ψ}|^{r_{ψ}}, \end{matrix}

for every character ψ ∈ Ĝ(Q_{ℓ}). This concludes the proof of Theorem 2.2. □

3. The case |S| = 1 We keep the same notations as in the previous sections, and we assume in addition that S consists of a single prime v_{0} which splits completely in K/k. (If one thinks of v_{0} as the prime at infinity, then this case is the function field analogue of the totally real number field case.) The goal of this section is to remove condition gcd (ℓ, e_{K}) = 1 in Theorem 2.2(2), under the present hypotheses. This result will prove to be of crucial importance in our approach of a particular case of Chinburg’s Ω_{3}–Conjecture (see §4).

Let us first remark that, under the present hypotheses,

r_{F} = \{ \begin{matrix}1, & for F ≠ k \\ 0, & for F = k \end{matrix} , r_{ψ} = \{ \begin{matrix}1, & for ψ ≠ 1_{G} \\ 0, & for ψ = 1_{G}, \end{matrix}  (12)

for any F and ψ as in §1. For any intermediate field F, let U_{F,S} be the group of S–units of F (U_{S} = U_{K,S}), let μ_{F} be the group of roots of unity in F^{×}, let \overline{U}_{F,S} =^{def} U_{F,S}/μ_{K} and \overline{E}_{S} =^{def} E_{S}/μ_{K}, and let a_{F} = Ann_{Z[G(F/k)]} (μ_{F}). As a consequence of |S| = 1, we obviously have the equalities U_{k,S} = μ_{k} and \overline{U}_{k,S} = \{1\}. We also have the following characterization of a_{F} (see [21], Chpt.IV, §1, Lemma 1):

Lemma 3.1 (Tate). The ideal a_{F} is generated over Z [G (F/k)] by the set

{1 − σ_{v}^{−1} · Nv| v prime in k, v ∉ S′} ,

for any finite set of primes S′ of k, containing all the primes which ramify in F/k.

An immediate consequence of Lemma 3.1 is that, if F is any intermediate field, and if π_{F} : Z [G] ⟶ Z [G (F/k)] is the natural projection given by π_{F} (σ) = σ|_{F}, for all σ ∈ G, then

π_{F} (a_{K}) = a_{F} .  (13)

We call a pair (F, S′) as in Definition 1.1 S–admissible if F ≠ k, and for every finite set T of primes in k, such that S′ ∩ T = ∅, the pair (F, S′) is (S, T) –admissible, in the sense of Definition 1.1. For any field L of characteristic 0, and any ψ ∈ Ĝ(L), ψ ≠ 1_{G}, a pair (F, S′) is called (S, ψ) –admissible if, for every T as above, the pair (F, S′) is (S, T, ψ) –admissible, in the sense of Definition 1.5. An obviously equivalent and certainly easier to handle definition of these concepts is the following:

Definition 3.2. A pair (F, S′) consisting of an intermediate field F, F ≠ k, and a finite set S′ of primes in k, is called S–admissible if it satisfies the following conditions:

(1) v_{0} ∈ S′

(2) |S′| ≥ 3

(3) S′ contains all the primes which ramify in F/k.

If ψ is as above and if, in addition to (1)—(3), (F, S′) satisfies

(4) r_{ψ,S′} = 1, then (F, S′) is called (S, ψ) –admissible.

For an S–admissible pair (F, S′), let

Θ_{F,S′}(s) = ∑_{χ∈{ \atop G(F/k)}} L_{S′} (s, χ) · e_{χ^{−1}}

be the S′–Stickelberger function associated to F/k, where L_{S′} (s, χ) is the Artin L–function associated to χ, with Euler factors at primes in S′ removed.

Let us fix a prime w_{0}|v_{0} in K. In what follows we will denote by w_{0} the prime lying below this fixed prime in any intermediate field F as well. For an S–admissible pair (F, S′) we will denote by R_{w_{0}} the regulator map

CU_{F,S′,T} = CU_{F,S′} ⟶^{R_{W}} C [G (F/k)]

associated as in §0 to W = {w_{0}} and r = 1. The following is a reformulation in this context of Brumer–Stark’s Conjecture proved independently by Deligne (see [21], Chpt.V) and Hayes (see [10]):

Theorem 3.3 (Deligne, Hayes). For any S–admissible pair (F, S′) there exists a unique element η_{F,S′} ∈ (\overline{U}_{F,S})_{1,S′} satisfying the following properties:

(1) \frac{1}{e_{F}} R_{w_{0}} (η_{F,S′}) = Θ′_{F,S′}(0)

(2) [ ∏_{v∈T} (1 − σ_{v}^{−1} · Nv)] · (\frac{1}{e_{F}} η_{F,S′}) ∈ (U_{F,S,T})_{1,S′} in QU_{F,S,T}, for every fi-

nite, nonempty set T of primes in k, such that T ∩ S′ = ∅.

Let (F, S′) and T be as in the Theorem above. Rubin’s Conjecture (Conjecture B in [15, §1.6]) applied to the set of data (F, S′, T, 1) predicts the existence of a unique element ε_{F,S′,T} ∈ (U_{F,S′,T})_{1,S′} (see Lemma 1.5.3(2) in [15] as well) satisfying the regulator condition

R_{w_{0}} (ε_{F,S′,T}) = Θ′_{F,S′,T} (0) .

Due to the obvious relation

Θ′_{F,S′,T} (0) = [ ∏_{v∈T} (1 − σ_{v}^{−1} · Nv)] · Θ′_{F,S′}(0)

and to the uniqueness of η_{F,S′} and ε_{F,S′,T}, the link between Theorem 3.3 above and Rubin’s Conjecture in this context is given by

ε_{F,S′,T} = [ ∏_{v∈T} (1 − σ_{v}^{−1} · Nv)] · (\frac{1}{e_{F}} η_{F,S′}) .  (14)

This relation shows that Theorem 3.3 implies Rubin’s Conjecture for r = 1. One can actually show that they are equivalent (see [17]).

Lemma 3.4. Let ℓ be a prime number, such that gcd (ℓ, g) = 1, let ψ ∈ Ĝ(Q_{ℓ}), ψ ≠ 1_{G}, and let α_{ψ} ∈ a_{K} ⊗Z_{ℓ} such that (a_{K} ⊗ Z_{ℓ})^{ψ} = α_{ψ}D_{ψ}. (Such an element always exists because D_{ψ} is a discrete valuation ring.) Then (\overline{E}_{S} ⊗ Z_{ℓ})^{ψ} is generated as a D_{ψ}–module by

α_{ψ} · [\frac{1}{e_{K_{ψ}}} η_{K_{ψ},S′}] ,

for any S′ such that the pair (K_{ψ}, S′) is (S, ψ) –admissible.

Proof. Let us fix a set S′ such that the pair (K_{ψ}, S′) is (S, ψ) –admissible. According to Lemma 1.8(1) (for r_{ψ} = 1) and to the definition of E_{S}, we have the following equality

(\overline{E}_{S} ⊗ Z_{ℓ})^{ψ} = ∑_{T} D_{ψ} · ε_{K_{ψ},S′,T} ,  (15)

where the sum is taken with respect to all finite, nonempty sets T of primes in k, such that T ∩ S′ = ∅. Lemma 3.1 shows that for any such T, ∏_{v∈T} (1 − σ_{v}^{−1} · Nv) ∈

a_{K_{ψ}}, and therefore (14) and (15) show that

(\overline{E}_{S} ⊗ Z_{ℓ})^{ψ} ⊆ D_{ψ} [α_{ψ} · \frac{1}{e_{K_{ψ}}} η_{K_{ψ},S′}] .

On the other hand, Lemma 3.1 also shows that one can write

α_{ψ} = ∑_{v∉S′} a_{v} · (1 − σ_{v}^{−1}Nv) ,

with a_{v} ∈ D_{ψ}, almost all equal to 0. Let T_{v} = {v}, for any v ∉ S′. The equality above, combined with (14) and the definition of E_{S}, gives

α_{ψ} · \frac{1}{e_{K_{ψ}}} η_{K_{ψ},S′} = ∑_{v∉S′} a_{v} · (1 − σ_{v}^{−1}Nv) (\frac{1}{e_{K_{ψ}}} η_{K_{ψ},S′}) =

= ∑_{v∉S′} a_{v} · ε_{K_{ψ},S′,T_{v}} ∈ (\overline{E}_{S} ⊗ Z_{ℓ})^{ψ} .

This concludes the proof of Lemma 3.4. □

Lemma 3.5. For any ℓ, ψ and S′ as in Lemma 3.4, the natural morphisms of Z [G]–modules involved in the following commutative diagrams:

\xymatrix{_{\overline{U}_{S}} \ar[r] & _{\overline{U}_{S′}} & A_{S} \ar[r] & A_{S′} \\ _{\overline{U}_{K_{ψ},S}} \ar[u] \ar[r] & _{\overline{U}_{K_{ψ},S′}} \ar[u] & A_{K_{ψ},S} \ar[u] \ar[r] & A_{K_{ψ},S′} \ar[u]}

become D_{ψ}–isomorphisms, when tensored with D_{ψ} over Z [G].

Proof. Same as the proof of Lemma 1.7. □

Proposition 3.6. Let ℓ, ψ and S′ be as in Lemma 3.4. Then

| (U_{S}/E_{S} ⊗ Z_{ℓ})^{ψ} | ≥ | (A_{S} ⊗ Z_{ℓ})^{ψ} | .

Proof. Let T be a finite, nonempty set of primes in k, such that T ∩ S′ = ∅. After tensoring with D_{ψ} over Z [1/g] [G], Theorem 0 (2) applied to the set of data (K_{ψ}, S′, T, 1) gives us the inclusion

D_{ψ} · ε_{K_{ψ},S′,T} ⊆ Fitt_{D_{ψ}} ((A_{K_{ψ},S′,T} ⊗ Z_{ℓ})^{ψ}) · (U_{K_{ψ},S′,T} ⊗ Z_{ℓ})^{ψ} .  (16)

On the other hand, after tensoring with D_{ψ} over Z [G], the surjective Z [G]– morphism

[figure: A_{K_{ψ},S′,T}; A_{K_{ψ},S′}]

(see [15, (1)]) gives a surjective morphism of D_{ψ}–modules:

[figure: (A_{K_{ψ},S′,T} ⊗ Z_{ℓ})^{ψ}; (A_{K_{ψ},S′} ⊗ Z_{ℓ})^{ψ}]

which, according to [15, §1.4(e)], gives the following inclusion at the level of Fitting ideals

Fitt_{D_{ψ}} (A_{K_{ψ},S′,T} ⊗ Z_{ℓ})^{ψ} ⊆ Fitt_{D_{ψ}} (A_{K_{ψ},S} ⊗ Z_{ℓ})^{ψ} .  (17)

If we now combine inclusions (16) and (17) with the obvious U_{S′,T} ⊆ \overline{U}_{S′} and with Lemma 3.5, we obtain

D_{ψ} · ε_{K_{ψ},S′,T} ⊆ Fitt_{D_{ψ}} (A_{S} ⊗ Z_{ℓ})^{ψ} · (\overline{U}_{S} ⊗ Z_{ℓ})^{ψ} ,

for every T as above. This last relation together with (15) show that

(\overline{E}_{S} ⊗ Z_{ℓ})^{ψ} ⊆ Fitt_{D_{ψ}} (A_{S} ⊗ Z_{ℓ})^{ψ} · (\overline{U}_{S} ⊗ Z_{ℓ})^{ψ} .  (18)

On the other hand, since r_{ψ,S} = 1, we have an isomorphism of D_{ψ}–modules (\overline{U}_{S} ⊗ Z_{ℓ})^{ψ} ⟶^{∼} D_{ψ}. Relation (18), together with [15, §1.4(f)], therefore imply that

| (U_{S}/E_{S} ⊗ Z_{ℓ})^{ψ} | = | (\overline{U}_{S}/\overline{E}_{S} ⊗ Z_{ℓ})^{ψ} | ≥
≥ [D_{ψ} : Fitt_{D_{ψ}} (A_{S} ⊗ Z_{ℓ})^{ψ}] =
= | (A_{S} ⊗ Z_{ℓ})^{ψ} | ,

which concludes the proof of Proposition 3.6. □

Our next goal is to prove the following:

Proposition 3.7. Let ℓ be a prime number such that gcd(ℓ, g) = 1. Then

∏_{ψ≠1_{G}} | (U_{S}/E_{S} ⊗ Z_{ℓ})^{ψ} | = ∏_{ψ≠1_{G}} | (A_{S} ⊗ Z_{ℓ})^{ψ} | ,

where the products above are taken with respect to characters ψ ∈ Ĝ(Q_{ℓ}).

In order to prove the statement above we employ techniques similar to those used by Rubin in [16]. Let us fix a prime number ℓ, gcd(ℓ, g) = 1, a completion C_{ℓ} ofthe algebraic closure of Q_{ℓ}, with respect to the normalized ℓ–adic valuation v, an embedding C_{ℓ} ↪ C, and a finite, unramified extension O of Z_{ℓ}, inside C_{ℓ}, containing the values of all characters χ ∈ Ĝ(C_{ℓ}). In what follows we will denote by ψ a generic element of Ĝ(Q_{ℓ}), and by χ a generic element of Ĝ(C_{ℓ}). The extension of the normalized ℓ–adic valuation v to C_{ℓ} (which will also be denoted by v) gives an equivalence relation ∼ on C_{ℓ}^{∗}, defined by:

a ∼ b if v(ab^{−1}) = 0, ∀a, b ∈ C_{ℓ}^{∗}.

Let X_{S} be the Z [G]–module defined by

X_{S} = {∑_{w|v_{0}} a_{w} · w| a_{w} ∈ Z, ∑_{w|v_{0}} a_{w} = 0}.

As in [15, §1], we have an injective Z [G]–morphism

\xymatrix{_{\overline{U}_{S}} \ar[r]^{λ_{S}} & RX_{S} ,}

given by λ_{S}(ᾱ) = ∑_{w|v_{0}} − log |α|_{w} · w, for all α ∈ U_{S}, which becomes a C [G]–

isomorphism

C\overline{U}_{S} ⟶^{∼^{λ_{S}}} CX_{S} ,

when extended by C–linearity to C\overline{U}_{S}.

Let us notice that since S = {v_{0}}, the definitions of R_{w_{0}} and λ_{S} show that

λ_{S}(u) = R_{w_{0}}(u) · w_{0}, ∀u ∈ \overline{U}_{S}.  (19)

Let Z_{ℓ} [G] = ⨁_{ψ∈Ĝ(Q_{ℓ})} D_{ψ} be the usual decomposition of Z_{ℓ} [G] in a direct sum of

Dedekind domains. An argument similar to the one in [15], Lemma 1.5.1, combined with the second equality in (12) shows that we have isomorphisms of Z_{ℓ} [G]–modules

Z_{ℓ}\overline{U}_{S} ⟶^{∼} Z_{ℓ}X_{S} ⟶^{∼} ⨁_{\begin{matrix}ψ∈Ĝ(Q_{ℓ}) \\ ψ≠1_{G} \end{matrix}} D_{ψ},  (20)

and equalities

(OX_{S})^{χ} = O · (e_{χ}w_{0}), ∀χ ∈ Ĝ(C_{ℓ}), χ ≠ 1_{G}.  (21)

These show in particular that one can find an injective Z [G]–morphism

\xymatrix{X_{S} \ar[r]^{f} & \overline{U}_{S} ,}

which becomes a Z_{ℓ} [G]–isomorphism, when tensored with Z_{ℓ}.

Let R̂_{S}, R_{S,χ} and R_{K,S} be the determinants of (λ_{S} ∘ f), (λ_{S} ∘ f)^{χ} and λ_{S}, respectively, computed with respect to O–bases of OX_{S}, (OX_{S})^{χ}, and Z–bases of X_{S} and \overline{U}_{S} respectively, for every χ ∈ Ĝ(C_{ℓ}). For every ψ ∈ Ĝ(Q_{ℓ}) we set R_{S,ψ} =^{def} ∏_{χ|ψ} R_{S,χ}. As a consequence of gcd (#coker(f), ℓ) = 1, we have the following

ℓ–adic equivalences:

R_{K,S} ∼ R̂_{S} ∼ ∏_{χ∈Ĝ(C_{ℓ})} R_{S,χ} = ∏_{ψ∈Ĝ(Q_{ℓ})} R_{S,ψ} .  (22)

Lemma 3.8. Let ψ ∈ Ĝ(Q_{ℓ}), ψ ≠ 1_{G}, and u ∈ Z_{ℓ}\overline{U}_{S}, such that the index [(Z_{ℓ}\overline{U}_{S})^{ψ} : D_{ψ}u] is finite. Then

[(Z_{ℓ}\overline{U}_{S})^{ψ} : D_{ψ}u] ∼ R_{S,ψ}^{−1} · (∏_{χ|ψ} R_{w_{0},χ}(u)) ,

where R_{w_{0},χ} =^{def} χ ∘ R_{w_{0}}.

Proof. As a direct consequence of (19), (20) and (21), one can easily prove that

[(O\overline{U}_{S})^{χ} : O (e_{χ}u)] = ℓ^{v(R_{S,χ}^{−1}·R_{w_{0},χ}(u))·[O:Z_{ℓ}]}, ∀χ|ψ.

By taking the product of the equalities above, for all χ|ψ, one obtains the statement in the lemma. □

Corollary 3.9. Let ℓ, ψ and α_{ψ} be as in Lemma 5.3.4. Then

| (U_{S}/E_{S} ⊗ Z_{ℓ})^{ψ} | = R_{S,ψ}^{−1} · ∏_{χ|ψ} R_{w_{0},χ} (α_{ψ} · \frac{1}{e_{K_{ψ}}} η_{K_{ψ},S_{ψ}}) .

Proof. Immediate consequence of Theorem 2.2(1), Lemma 3.4 and Lemma 3.8. □

Let χ ∈ Ĝ(C). (Under the chosen embeding C_{ℓ} ↪ C, χ can be viewed as an element of Ĝ(C_{ℓ}) as well.) Let L_{S}(s, χ) be the Artin L–function associated to χ, with the Euler factor at v_{0} removed:

L_{S}(s, χ) = (1 − χ(σ_{v_{0}})Nv_{0}^{−s}) · L(s, χ) ,

where L(s, χ) is the “complete” Artin L–function associated to χ. For every ψ ∈ Ĝ(Q_{ℓ}), ψ ≠ 1_{G}, let S_{ψ} be a fixed finite set of primes in k, such that the pair (K_{ψ}, S_{ψ}) is (S, ψ) –admissible. Then every χ ∈ Ĝ(C_{ℓ}), χ|ψ, gives a relation

L_{S_{ψ}}(s, χ) = ∏_{v∈S_{ψ}\S′_{ψ}} (1 − χ(σ_{v})Nv^{−s}) · L_{S}(s, χ) ,

where S′_{ψ} = {v ∈ S_{ψ}| v ramifies in K_{ψ}/k, or v = v_{0}}. Due to the fact that (K_{ψ}, S_{ψ}) is (S, ψ) –admissible, we have r_{ψ,S_{ψ}} = 1, and therefore χ(σ_{v}) is a root of unity of order g, different from 1, for all v ∈ S_{ψ} \ S′_{ψ} and all χ|ψ. Since gcd(ℓ, g) = 1, this shows that for χ and v as above we have (1 − χ(σ_{v})) ∈ O^{×}. We therefore have the following ℓ–adic equivalences

L′_{S_{ψ}}(0, χ) = ∏_{v∈S_{ψ}\S′_{ψ}} (1 − χ(σ_{v})) · L′_{S}(0, χ) ∼ L′_{S}(0, χ) ,  (23)

for every ψ and χ as above. We are now prepared to prove Proposition 3.7.

Proof of Proposition 3.7. Let ζ_{K,S}(s) and ζ_{k,S}(s) be the S–zeta functions of K and k respectively. Let h_{K,S} = |A_{S}|, h_{k,S} = |A_{k,S}|, and for every ψ ∈ Ĝ(Q_{ℓ}), let h_{K,S}^{ψ} = | (A_{S} ⊗ Z_{ℓ})^{ψ} | and e_{K}^{ψ} = | (μ_{K} ⊗ Z_{ℓ})^{ψ} |. We have the equality

\frac{ζ_{K,S}(s)}{ζ_{k,S}(s)} = ∏_{ψ≠1_{G}} (∏_{χ|ψ} L_{S} (χ, s)) .  (24)

Since S consists of a single prime which splits completely in K/k, we have the following equalities (see [21], Chpt.I):

ord_{s=0}ζ_{k,S}(s) = 0 , ord_{s=0}ζ_{K,S}(s) = [K : k] − 1 , ord_{s=0}L_{S}(χ, s) = 1 , ∀χ ≠ 1_{G}.

The S–class number formula (see [21], Chpt.I) together with equality (24) therefore give us

\frac{h_{K,S}}{h_{k,S}} · \frac{R_{K,S}}{R_{k,S}} · \frac{e_{K}}{e_{k}} = ∏_{ψ≠1_{G}} (∏_{χ|ψ} L′_{S} (χ, 0)) .  (25)

According to (22), (25), and Theorem 3.3(1) applied to the pairs (K_{ψ}, S_{ψ}), for all ψ ∈ Ĝ(Q_{ℓ}), ψ ≠ 1_{G}, we therefore have

∏_{ψ≠1_{G}} h_{K,S}^{ψ} · ∏_{ψ≠1_{G}} R_{S,ψ} · ( ∏_{ψ≠1_{G}} e_{K}^{ψ} )^{−1} ∼ ∏_{ψ≠1_{G}} (∏_{χ|ψ} R_{w_{0},χ} (\frac{1}{e_{K_{ψ}}} η_{K_{ψ},S_{ψ}})) . (26)

Let us choose α_{ψ} ∈ (a_{K} ⊗ Z_{ℓ}) as in Lemma 3.4, for every ψ as above. The flatness of D_{ψ} and the cyclicity of μ_{K} as Z [G]–modules give us the following exact sequence of D_{ψ}–modules

\xymatrix{0 \ar[r] & (a_{K} ⊗ Z_{ℓ})^{ψ} \ar[r] & D_{ψ} \ar[r] & (μ_{K} ⊗ Z_{ℓ})^{ψ} \ar[r] & 0 ,}

for every ψ. This together with (13) give

e_{K_{ψ}} = e_{K}^{ψ} = [D_{ψ} : α_{ψ}D_{ψ}] .  (27)

Let ψ as above, and for a fixed χ_{0}|ψ, let Z_{ℓ} [χ_{0}] be the discrete valuation domain obtained by adjoining the values of χ_{0} to Z_{ℓ}. Then we have the equalities

\begin{aligned}[D_{ψ} : α_{ψ}D_{ψ}] & = [Z_{ℓ} [χ_{0}] : (χ_{0} (α_{ψ}))] = \\ & = ℓ^{v(χ_{0}(α_{ψ}))·[Z_{ℓ}[χ_{0}]:Z_{ℓ}]} = \\ & = ℓ^{v( ∏_{χ|ψ} χ(α_{ψ}))} . \end{aligned}  (28)

Equalities (27) and (28) show that we have an ℓ–adic equivalence

e_{K_{ψ}} ∼ ∏_{χ|ψ} χ (α_{ψ}) ,

for every ψ ≠ 1_{G}. This equivalence together with the G–linearity of R_{w_{0}} show that,

if we multiply (26) by ∏_{ψ≠1_{G}} R_{S,ψ}^{−1} ( ∏_{χ|ψ} χ (α_{ψ})), we obtain

∏_{ψ≠1_{G}} h_{K,S}^{ψ} ∼ ∏_{ψ≠1_{G}} [R_{S,ψ}^{−1} · ∏_{χ|ψ} R_{w_{0},χ} (α_{ψ} · \frac{1}{e_{K_{ψ}}} η_{K_{ψ},S_{ψ}})] .  (29)

If we now combine (29) with Corollary 3.9, we obtain

∏_{ψ≠1_{G}} h_{K,S}^{ψ} ∼ ∏_{ψ≠1_{G}} | (U_{S}/E_{S} ⊗ Z_{ℓ})^{ψ} | ,

which concludes the proof of Proposition 3.7. □

By combining Propositions 3.6 and 3.7 we obtain the desired result Theorem 3.10. Let S be a set consisting of a single prime of k, which splits completely in K/k. Then:

(1) The index [U_{S} : E_{S}] is finite.

(2) For every prime number ℓ such that gcd(ℓ, g) = 1, and every ψ ∈ Ĝ(Q_{ℓ}), ψ ≠ 1_{G},

| (U_{S}/E_{S} ⊗ Z_{ℓ})^{ψ} | = | (A_{S} ⊗ Z_{ℓ})^{ψ} | .

Remark. Equalities similar to the ones in Theorems 1.4(2), 2.2(2) and 3.10(2) were proved in the number field case, for k = Q, as consequences of the Main Conjecture in Iwasawa Theory (see [12]), or by using Kolyvagin–Euler System techniques (see[11], Appendix). It is conceivable, as Rubin shows in [17, §6], that the elements Φ (ε_{F,S′,T}) give Kolyvagin–Euler Systems in the function field case as well. One could therefore hope to be able to prove the theorems above by using Kolyvagin– Euler System techniques. However, there would be a major drawback: these techniques would fail to give information at the prime p = char(k). The methods we are using do not force us to stay away from the prime p . This is mainly due to the fact that we are not avoiding p in Theorem 0. The cause for this lies deeper, in the links between crystalline and p–adic étale cohomology theories, and in the fact that the action of the geometric Frobenius morphism on the crystalline cohomology groups gives the right L–functions (see Appendix of [15] or [14]). The idea of using crystalline cohomology in order to understand the structure of the p–part of class groups of function fields has been used in the past by Goss and Sinnott (see [7]).

4. On Chinburg’s Ω_{3}–Conjecture for function fields

4.1. THE CONJECTURE

Let K/k be any finite, Galois extension of global fields (number fields or function fields of characteristic p > 0.) Let G = G (K/k), and let K_{0} (Z [G]) be the Grothendieck group of isomorphism classes of finitely generated, projective Z [G]–modules. If P is a finitely generated, projective Z [G]–module, then P ⊗_{Z} Q is Q [G]–free (see [18]) and we define

rank (P) =^{def} rank_{Q[G]} (P ⊗_{Z} Q) .

We obtain this way a surjective group morphism

[figure: K_{0} (Z [G]); rank; Z ,]

defined by rank ([P]) = rank (P), for the class [P] in K_{0} (Z [G]) of any finitely generated, projective Z [G]–module P.

Definition 4.1.1. The projective class–group Cl (Z [G]) of Z [G] is defined by

Cl (Z [G]) = ker ( \xymatrix{K_{0} (Z [G]) \ar[r]^{rank} & Z} ) .

In [6] Fröhlich defined an invariant of analytic nature W_{K/k} ∈ Cl (Z [G]) for finite, Galois extensions K/k of number fields, by means of the Artin root numbers associated to the irreducible, symplectic representations of G. In [5] Chinburg shows that the definition of W_{K/k} can be carried through in the function field case as well. In particular, if the group G has no irreducible, symplectic representations (as it is the case when G is abelian, for example), then W_{K/k} = 0 in Cl (Z [G]).

In [3] and [4] Chinburg defined an invariant of arithmetic nature Ω (K/k, 3) ∈ Cl (Z [G]) for finite, Galois extensions of number fields, measuring the complexity of the Z [G]–module structure of the group U_{S} of S–units in K, for a sufficiently large finite set S of primes in k. In [5] he extends this definition to the function field case.

The link between the invariants described above is believed to be the following (see [3], [4] and [5]):

Conjecture (Chinburg). For any finite, Galois extension of global fields

Ω (K/k, 3) = W_{K/k} .

Chinburg shows in [3, §IX] how, in the case of a cyclic extension of prime degree of the field Q of rational numbers, the conjecture above follows from Gras’ Conjecture satisfied by the group of cyclotomic units in that context. Following Chinburg’s ideas, we are going to show in the next section that the conjecture above holds true for cyclic extensions of prime degree of function fields, as a consequence of Theorem 3.10 (our analogue of Gras’ Conjecture in the function field setting). Chinburg’s Conjecture in this situation follows from work of S. Bae [1] as well. However Bae’s approach is completely different from ours.

4.2. CYCLIC EXTENSIONS OF PRIME DEGREE

Throughout this section we are going to assume K/k to be a cyclic function field extension of prime degree. Since in this case there are no irreducible, symplectic representations of G, we have

W_{K/k} = 0 ,  (29)

and therefore Chinburg’s Conjecture asserts that Ω (3, K/k) = 0 in Cl (Z [G]). Since |G| = 2 implies Cl (Z [G]) = 0 (see [3, §IX]), we can assume from now on that |G| > 2.

Let G_{0} (Z [G]) be the Grothendieck group of isomorphism classes of finitely generated Z [G]–modules, and let

\xymatrix{h : K_{0} (Z [G]) \ar[r] & G_{0} (Z [G])}

be the Cartan morphism, taking the class [P] of a finitely generated, projective Z [G]–module P in K_{0} (Z [G]), into its class (P) in G_{0} (Z [G]).

For a finite set S of primes of k, let S_{K} be the set of primes of K lying above primes in S. Let us consider the following Z [G]–modules:

Y_{S} = { ∑_{w∈S_{K}} a_{w} · w | a_{w} ∈ Z}, X_{S} = { ∑_{w∈S_{K}} a_{w} · w | a_{w} ∈ Z, ∑_{w∈S_{K}} a_{w} = 0}.

The following statements are proved in [5] and [19] respectively:

Proposition 4.2.1 (Chinburg). The image of Ω (3, K/k) via the Cartan map h satisfies

h (Ω (3, K/k)) = (U_{S}) − (X_{S}) − (A_{S}) ,

for any finite, sufficiently large set S of primes in k.

Proposition 4.2.2 (Reiner). If G is a cyclic group of prime order then, the Cartan map

\xymatrix{h : K_{0} (Z [G]) \ar[r] & G_{0} (Z [G])}

is injective.

According to these Propositions, all we have to do in order to prove the Conjecture above under the present assumptions, is show that (U_{S}) − (X_{S}) − (A_{S}) = 0 in G_{0} (Z [G]), for a sufficiently large set of primes S. Although one can make the expression “sufficiently large” very precise (see [5] ), the next result shows that there is no need for that in the present situation.

Lemma 4.2.3. The class

c_{S} =^{def} (U_{S}) − (X_{S}) − (A_{S}) ∈ G_{0} (Z [G])

does not depend on the finite, nonempty set S of primes in k.

Proof. There is an exact sequence of Z [G]–modules

\xymatrix{0 \ar[r] & X_{S} \ar[r] & Y_{S} \ar[r]^{s} & Z \ar[r] & 0 ,}

where s ( ∑_{w∈S_{K}} a_{w} · w) =^{def} ∑_{w∈S_{K}} a_{w}, for all ∑_{w∈S_{K}} a_{w} · w ∈ Y_{S}. This gives the

following relation in G_{0} (Z [G])

(X_{S}) = (Y_{S}) − (Z) .  (30)

In order to prove the desired statement, it is obviously enough to prove that, if S_{1} and S_{2} are two sets of primes as in the statement of the lemma, satisfying S_{1} ⊆ S_{2}, then c_{S_{1}} = c_{S_{2}} in G_{0} (Z [G]).

Let us fix such sets S_{1} and S_{2}. Let A_{S_{1},S_{2}} be the Z [G]–submodule of A_{S_{1}} generated by the classes of the ideals in S_{1,2} =^{def} S_{2,K} \ S_{1,K}, and let Y_{S_{1},S_{2}} be the free abelian group generated by the ideals in S_{1,2}. We have the following exact sequences of finitely generated Z [G]–modules:

\xymatrix{0 \ar[r] & Y_{S_{1}} \ar[r] & Y_{S_{2}} \ar[r]^{α} & Y_{S_{1},S_{2}} \ar[r] & 0}
\xymatrix{0 \ar[r] & U_{S_{1}} \ar[r] & U_{S_{2}} \ar[r]^{β} & Y_{S_{1},S_{2}} \ar[r]^{γ} & A_{S_{1},S_{2}} \ar[r] & 0}
\xymatrix{0 \ar[r] & A_{S_{1},S_{2}} \ar[r] & A_{S_{1}} \ar[r] & A_{S_{2}} \ar[r] & 0 ,}

where α ( ∑_{w∈S_{2,K}} a_{w} · w) = ∑_{w∈S_{1,2}} a_{w} · w, β(u) = ∑_{w∈S_{1,2}} ord_{w}(u) · w, for all u ∈ U_{S_{2}},

and γ(w) = ŵ (the class of w in A_{S_{1},S_{2}}), for all w ∈ S_{1,2}. If we write down the relations given in G_{0} (Z [G]) by the three exact sequences above, and eliminate the classes (Y_{S_{1},S_{2}}) and (A_{S_{1},S_{2}}), we obtain

(Y_{S_{2}}) − (U_{S_{2}}) − (A_{S_{2}}) = (Y_{S_{1}}) − (U_{S_{1}}) − (A_{S_{1}}) .

This relation combined with (30) for S = S_{1} and S = S_{2} respectively, gives the desired c_{S_{1}} = c_{S_{2}} in G_{0} (Z [G]). □

Lemma 4.2.3 shows that, in order to prove Chinburg’s Conjecture under the present assumptions, we have to find a finite, nonempty set S of primes in k, satisfying c_{S} = 0.

Let S = {v_{0}, …, v_{s}} be a fixed set of primes in k satisfying the following properties:

(1) v_{0} splits completely in K/k.

(2) |S| ≥ 3

(3) S contains all the primes which ramify in K/k.

(4) G_{v_{i}} = G, for all i = 1, …, s. We intend to prove that S satisfies c_{S} = 0. In order to do this, we return to some of the notations and techniques employed in §3.

Let S_{0} = {v_{0}}. Properties (1)—(3) satisfied by S show that (K, S) is an S_{0}– admissible pair. Let η_{K,S} ∈ (\overline{U}_{K,S_{0}})_{1,S} be the unique element provided by Theorem 3.3 (with F = K, S = S_{0} and S′ = S). Let \overline{E}_{S} be the Z [G]–submodule of \overline{U}_{S} = U_{S}/μ_{K} defined by

\overline{E}_{S} = (a_{K} · \frac{1}{e_{K}} η_{K,S}) ⨁ (h_{k,S}Zu_{1} ⊕ Zu_{2} ⊕ ⋯ ⊕ Zu_{s}) ,

where \{u_{1}, …, u_{s}\} is a Z–basis of \overline{U}_{k,S}. (Since η_{K,S} ∈ (\overline{U}_{K,S_{0}})_{1,S}, and r_{1_{G},S} ≥ 2 (see property (2) satisfied by S), we have e_{1_{G}} · η_{K,S} = 0 and therefore the relation a_{K} · \frac{1}{e_{K}} η_{K,S} ∩ \overline{U}_{k,S} = \{0\}.)

Let E_{S} ⊆ U_{S} be the preimage of \overline{E}_{S} ⊆ \overline{U_{S}} under the projection

[figure: U_{S}; \overline{U_{S}} = U_{S}/μ_{K}] .

Then we obviously have an exact sequence of finitely generated Z [G]–modules

\xymatrix{0 \ar[r] & μ_{K} \ar[r] & E_{S} \ar[r] & _{\overline{E}_{S}} \ar[r] & 0} ,

which gives the following relation in G_{0} (Z [G])

(E_{S}) = (\overline{E}_{S}) + (μ_{K}) .  (31)

Proposition 4.2.4.

(1) The index [U_{S} : E_{S}] is finite.

(2) For every prime number ℓ such that gcd(ℓ, g) = 1, and every ψ ∈ Ĝ(Q_{ℓ})

| (U_{S}/E_{S} ⊗ Z_{ℓ})^{ψ} | = | (A_{S} ⊗ Z_{ℓ})^{ψ} | .

Proof. As remarked many times before, it is enough to prove (2) above.

Let ℓ be as above and let us fix ψ ∈ Ĝ(Q_{ℓ}), ψ ≠ 1_{G} for the beginning. The fact that |G| is prime implies that K_{ψ} = K. Properties (1)–(4) satisfied by S imply that the pair (K = K_{ψ}, S) is (S_{0}, ψ)–admissible. Lemma 3.5 therefore shows that

(Z_{ℓ}\overline{U}_{S})^{ψ} = (Z_{ℓ}\overline{U}_{S_{0}})^{ψ} and (Z_{ℓ}A_{S})^{ψ} = (Z_{ℓ}A_{S_{0}})^{ψ} .  (32)

The definition of \overline{E}_{S} and Lemma 5.3.4 show that

(Z_{ℓ}\overline{E}_{S})^{ψ} = (Z_{ℓ}a_{K})^{ψ} (\frac{1}{e_{K}} η_{K,S}) = (Z_{ℓ}E_{S_{0}})^{ψ} .  (33)

If we now combine equalities (32) and (33) with Theorem 3.10 (for S = S_{0}) we obtain

| (U_{S}/E_{S} ⊗ Z_{ℓ})^{ψ} | = | (\overline{U}_{S}/\overline{E}_{S} ⊗ Z_{ℓ})^{ψ} | =
= | (A_{S} ⊗ Z_{ℓ})^{ψ} | ,

which concludes the proof of statement (2), for ψ ≠ 1_{G}.

Let ψ = 1_{G}. We obviously have the following equalities:

(Z_{ℓ}\overline{E}_{S})^{1_{G}} = h_{k,S}Z_{ℓ}u_{1} ⊕ Z_{ℓ}u_{2} ⊕ ⋯ ⊕ Z_{ℓ}u_{s} , (Z_{ℓ}\overline{U}_{S})^{1_{G}} = Z_{ℓ}\overline{U}_{k,S} ,
(Z_{ℓ}A_{S})^{1_{G}} = Z_{ℓ}A_{k,S} .

These imply that

| (U_{S}/E_{S} ⊗ Z_{ℓ})^{1_{G}} | = | (\overline{U}_{S}/\overline{E}_{S} ⊗ Z_{ℓ})^{1_{G}} | =
= | (Z_{ℓ} ⊗ A_{S})^{1_{G}} | ,

which concludes the proof of Proposition 4.2.4 (2), for ψ = 1_{G} as well. □

For a finite Z [G]–module M, we have a Z [G]–module direct sum decomposion

M = ⨁_{ℓ} (M ⊗ Z_{ℓ}) ,

with respect to all prime numbers ℓ. Let S (M ⊗ Z_{ℓ}) denote the Z [G]– (or equivalently, the Z_{ℓ} [G]–) semisimplification of M ⊗ Z_{ℓ}, for any ℓ. We obviously have the following relation in G_{0} (Z [G])

(M) = ∑_{ℓ} (S (M ⊗ Z_{ℓ})) .  (34)

On the other hand, if ℓ = |G| (recall that |G| is a prime number), and N is a simple Z [G]–module of order a power of ℓ, then the maximal G–fixed submodule N^{G} of N is nontrivial, and therefore N^{G} = N. This fact, combined again with the simplicity of N, shows that |N| = ℓ. We therefore have an exact sequence of Z [G]–modules

\xymatrix{0 \ar[r] & Z \ar[r]^{×ℓ} & Z \ar[r] & N \ar[r] & 0 ,}

which shows that (N) = 0 in G_{0} (Z [G]).

This remark and (34) show that, for any finite Z [G]–module M, we have the following relation in G_{0} (Z [G])

(M) = ∑_{ℓ≠|G|} (S (M ⊗ Z_{ℓ})) .  (35)

Lemma 4.2.5. The equality

(U_{S}) − (E_{S}) − (A_{S}) = 0

holds true in G_{0} (Z [G]).

Proof. Greenberg shows in [8, §5] that, if G is a finite abelian group, ℓ is a prime number such that gcd (ℓ, |G|) = 1, and M_{1}, M_{2} are two finite Z [G]–modules, the following statements are equivalent:

(1) S (M_{1} ⊗ Z_{ℓ}) ⟶^{∼} S (M_{2} ⊗ Z_{ℓ}) as G–modules.

(2) | (M_{1} ⊗ Z_{ℓ})^{ψ} | = | (M_{2} ⊗ Z_{ℓ})^{ψ} |, for all ψ ∈ Ĝ(Q_{ℓ}).

The statement in Lemma 4.2.5 follows now from Proposition 4.2.4, relation (35) and Greenberg’s observation, with M_{1} = U_{S}/E_{S} and M_{2} = A_{S}. □

Proposition 4.2.6. The equality (E_{S}) = (X_{S}) holds true in G_{0} (Z [G]).

Proof. Let us fix w_{i} ∈ S_{K}, w_{i}|v_{i}, for every i = 0, …, s. Properties (1) and (4) satisfied by S imply that we have Z [G]–isomorphisms

Z [G] w_{0} ⟶^{∼} Z [G] , Z [G] w_{i} ⟶^{∼} Z , ∀i = 1, …, s ,

with G acting trivially on Z. The definition of Y_{S} therefore shows that

Y_{S} = ⨁_{0≤i≤s} Z [G] w_{i} ⟶^{∼} Z [G] ⊕ Z^{s}

as Z [G]–modules. Relation (30) therefore shows that the following holds true in

G_{0} (Z [G])
(X_{S}) = (Z [G]) + (s − 1) (Z) .  (36)

We are now going to compute the class (E_{S}) in G_{0} (Z [G]). According to (31) and the definition of \overline{E}_{S}, we obviously have

(E_{S}) = (μ_{K}) + (a_{K} · \frac{1}{e_{K}} η_{K,S}) + s (Z) .  (37)

Since μ_{K} is a cyclic Z [G]–module, the definition of a_{K} implies

(a_{K}) = (Z [G]) − (μ_{K}) .  (38)

Proposition 4.2.4 (1) implies that

rank_{Z}\overline{E}_{S} = rank_{Z}\overline{U}_{S} = [K : k] + (s − 1) ,

and therefore, from the definition of \overline{E}_{S}, we have

rank_{Z} (a_{K} · \frac{1}{e_{K}} η_{K,S}) = [K : k] − 1 .  (39)

Let K be the Z [G]–module defined by the exact sequence

\xymatrix{0 \ar[r] & K \ar[r] & a_{K} \ar[r]^{π} & a_{K} · \frac{1}{e_{K}} η_{K,S} \ar[r] & 0 ,}  (40)

where π(α) = α · \frac{1}{e_{K}} η_{K,S}, for all α ∈ a_{K}. Then (39), combined with the obvious rank_{Z} (a_{K}) = [K : k], shows that rank_{Z} (K) = 1. But since |G| > 2, the only ideal of Z [G] of Z–rank equal to 1 is isomorphic to Z with trivial G–action. This shows that (K) = (Z) and therefore the following equality holds true in G_{0} (Z [G])

(a_{K} · \frac{1}{e_{K}} η_{K,S}) = (a_{K}) − (Z) .

Combining this equality with (37) and (38) one obtains

(E_{S}) = (Z [G]) + (s − 1) (Z) ,

which, according to (36), concludes the proof of Proposition 4.2.6. □

We are now prepared to prove the main result of this section:

Theorem 4.2.7. If K/k is a cyclic extension of prime degree of function fields of characteristic p > 0, then Ω (K/k, 3) = W_{K/k}(= 0) in Cl (Z [G]).

Proof. Proposition 4.2.6 together with Lemma 4.2.5 show that, for the set S of primes in k fixed above, we have c_{S} = 0 in G_{0} (Z [G]). Proposition 4.2.1 and Lemma 4.2.3 thus show that h (Ω (K/k, 3)) = 0 in G_{0} (Z [G]). The injectivity of h (see Proposition 4.2.2) together with (29), therefore imply that Ω (K/k, 3) = 0 = W_{K/k} in Cl (Z [G]), which concludes the proof. □

References

1. Bae, S., On the conjecture of Lichtenbaum and of Chinburg over function fields, Math. Ann. 285 (1989), 417–445.

2. Cassou–Noguès, Ph., Chinburg, T., Fröhlich, A., Taylor, M.J., L–functions and Galois modules, L–functions and Arithmetic (Proceedings of the Durham Symposium, July 1989, J. Coates and M.J. Taylor editors), London Math. Soc. Lecture Notes Series, vol. 153.

3. Chinburg, T., On the Galois structure of algebraic integers and S–units, Invent. Math 74 (1983), 321–349.

4. Chinburg, T, Exact sequences and Galois module structure, Ann. of Math. 121 (1985), 351–376.

5. Chinburg, T., Galois structure invariants of global fields, Working paper from a talk at Durham, July 1989.

6. Fröhlich, A., Some problems of Galois module structure for wild extensions, Proc. London Math. Soc. 27 (1978), 193–212.

7. Goss, D. and Sinnott, W., Class-groups of function fields, Duke Math. J. 52 (1985), 507– 516.

8. Greenberg, R., On p–adic L–functions and cyclotomic fields II, Nagoya Math. J. 67 (1977), 139–158.

9. Gras, G., Classes d’ideaux des corps abeliens et nombres de Bernoulli generalises, Ann. Inst. Fourier 27 (1977), 1–66.

10. Hayes, D.R., Stickelberger elements in function fields, Comp. Math. 55 (1985), 209–239.

11. Lang, S., Cyclotomic Fields I and II (Combined Second Edition), Springer Verlag, 1990.

12. Mazur, B. and Wiles, A., Class fields of abelian extensions of Q., Invent. Math. 76 (1984), 179–330.

13. Moreno, C., Algebraic Curves over Finite Fields, Cambridge University Press, 1991.

14. Popescu, C.D., On a Refined Stark Conjecture for Function Fields, Ph.D. Thesis, Ohio State University, 1996.

15. Popescu, C.D., On a refined Stark Conjecture for function fields, to appear.

16. Rubin, K., Stark units and Kolyvagin–Euler systems, J. für die Reine und Angew. Math. 425 (1992), 141–154.

17. Rubin, K., A Stark conjecture “over Z” for abelian L–functions with multiple zeros, Ann. Inst. Fourier 46 (1996).

18. Swan, R.G., Induced representations and projective modules, Ann. of Math. 71, No. 3 (1960), 552–578.

19. Swan, R.G. and Evans, E.G., K–theory of Finite Groups and Orders, Lecture Notes in Math. 149, Springer Verlag, 1970.

20. Tate, J., The cohomology groups of tory in finite Galois extensions of number fields, Nagoya Math. J. 27 (1966), 709–719.

21. Tate, J., Les Conjectures de Stark sur les Fonctions L d’Artin en s = 0, Progr. in Math. 47 (1984), Birkhaüser, Boston Basel Stuttgart.

Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA, 94720–5070, USA

E-mail address: popescu@msri.org