ON A REFINED STARK CONJECTURE

FOR FUNCTION FIELDS

Cristian D. Popescu

Abstract. We prove that a refinement of Stark’s Conjecture formulated by Rubin in [14] is true up to primes dividing the order of the Galois group, for finite, abelian extensions of function fields over finite fields. We also show that in the case of constant field extensions a statement stronger than Rubin’s holds true.

0. Introduction

Let L_{K/k}(s, χ) be the Artin L–function associated to a finite, Galois extensions of

global fields K/k, and a character χ ∈ { \atop G (K/k)}. In the 1970’s and early 1980’s, Stark [15] developed a conjecture concerning the leading coefficient of the Taylor expansion of L_{K/k}(s, χ), at s = 0. The classical formulae for the case of a Dedekind Zeta–function provide good hints: if the order of vanishing is r, then the coefficient in question should be a rational number multiplied by a regulator, obtained as a determinant of an r×r matrix involving character values and logarithms of absolute values of units in K.

The examples however suggest that at least in the case of abelian L-functions, the denominator of the above mentioned rational number could be specified. In the last paper of [15], Stark proposed a refined conjecture, which makes this denominator specific, in the case of abelian L-functions of order of vanishing r = 1 at s = 0. This is what Tate (see [16]) calls a conjecture “over Z” and he asks for such a refined statement for abelian L–functions of any order of vanishing r. In [14] Rubin formulates a refined version of Stark’s conjecture, for abelian L–functions of number fields, of any order of vanishing r at s = 0. Rubin’s statement can be easily extended to the more general case of abelian extensions of global fields of any characteristic.

In this paper we study Rubin’s conjecture in the case of finite, abelian extensions of global fields of characteristic p > 0 (i.e. function fields). The case r = 1 in this situation was independently solved by Deligne (see [17, Chpt.V]) and Hayes (see [6]), with methods relying on ℓ–adic homology of 1–motives and rank 1 sign–normalized Drinfeld modules respectively. We adopt methods similar to Deligne’s in order to treat the arbitrary order of vanishing case.

In §1 we set the notation, define the objects involved and some of their properties, and state the conjecture. §2 is concerned with the “dynamics” of Rubin’s conjecture. In particular we show that, in certain cases, one can derive the conjecture at any level r from a stronger form of the conjecture at level r = 0 (see Corollary

Typeset by AMS-TEX

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

2.2). This fact turns out to be of crucial importance in §§3 and 4. In §3 we prove Rubin’s conjecture, up to primes dividing the order of the Galois group G(K/k), for any abelian extension K/k of function fields (see Theorems 3.1.1 and 3.2.1). In §4 we prove that a stronger statement than Rubin’s holds true for constant field extensions of function fields (see Theorems 4.2.9 and 4.3.1).

We use these results in [13] in order to formulate and prove Gras–type Conjectures for global fields of characteristic p > 0.

The main technique employed in this is the ℓ–adic homological (or, equivalently, the ℓ-adic étale cohomological) (ℓ ≠ p) interpretation of L–functions, due to Grothendieck. However, in order to deal with the p–part of the conjecture, we had to rely on the p–adic homology groups, which are known not to give a good theory of L–functions. Therefore, we needed a “bridge” between the ℓ–adic étale and the p–adic étale cohomology theories, and that is provided by the crystalline cohomology theory. The fact that the Frobenius actions on the ℓ-adic étale and the crystalline cohomology groups have the same characteristic polynomials has been known for quite some time (see [8]). However, we needed the slightly stronger statement that the characteristic polynomials of the Frobenius actions on the χ– components of the ℓ–adic étale and crystalline cohomology groups are the same, for every character χ of G (K/k). Due to lack of a good reference, we provide a proof of this fact (see Proposition A.1) along with a proof of Theorem 1.7.4.1, in the Appendix.

Acknowledgements. It is a pleasure to thank Karl Rubin for his invaluable help and support. Our approach to this problem has been greatly influenced by his published and unpublished thoughts and by John Tate’s book [17].

1. The objects involved and their properties

1.1. GENERAL NOTATION

Let us fix a function field k of characteristic p > 0 and of transcendental degree 1 over a finite field. Let K be a finite, abelian extension of k, of Galois group G = G(K/k), and let g = |G|. We 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. For primes v in k and w in K, such that w|v, we denote by F_{q}(v) and by F_{q^{ν}} (w) their corresponding residue fields, and by d_{v} and d_{w} their degrees over F_{q} and F_{q^{ν}} respectively (d_{v} = [F_{q}(v) : F_{q}] and d_{w} = [F_{q^{ν}} (w) : F_{q^{ν}}]). Let Nv = |F_{q}(v)|, Nw = |F_{q^{ν}} (w)|, and let G_{v} be the decomposition group associated to v in K/k . The normalized absolute value | · |_{w} : K^{×} ⟼ q^{Z} associated to w is defined by: |α|_{w} = (Nw)^{−ord_{w}(α)}, for every α ∈ K.

Let S and T be two finite, nonempty and disjoint sets of primes in k, S containing all primes which ramify in K/k. Let us define:

• S_{K} = {primes in K lying above primes in S}

• T_{K} = {primes in K lying above primes in T } • O_{S} = {α ∈ K : |α|_{w} ≤ 1, for all w ∉ S_{K}}

• U_{S} = O_{S}^{×} (the group of S-units in K).

• U_{S,T} = {α ∈ U_{S} : α ≡ 1 mod w, for all w ∈ S_{K}} (This is a free Z–module.)

• A_{S} = the ideal class group of O_{S}

• A_{S,T} = \frac{\{Fractional ideals of O_{S}, prime to T_{K}\}}{\{fO_{S} : f ≡ 1 mod w , ∀w ∈ T_{K}\}}. A_{S,T} is a finite ideal class

group, isomorphic via the Artin reciprocity map with the Galois group of the maximal abelian extension of K, of conductor dividing ∏_{w∈T_{K}} w, in which all primes in

S_{K} split completely.

• X_{S} = { ∑_{w∈S_{K}} a_{w} · w : a_{w} ∈ Z, ∑_{w∈S_{K}} a_{w} = 0}

• λ_{S,T} : U_{S,T} ⟶ R ⨂ X_{S}, is the G–morphism defined by

λ_{S,T} (α) = ∑_{w∈S_{K}} − log(|α|_{w}) · w,

for any α ∈ U_{S,T}. This induces an R [G]–isomorphism R ⊗ U_{S,T} ⟶^{∼} R ⊗ X_{S}.

• R_{S,T} is the absolute value of the determinant of λ_{S,T}, with respect to Z–bases of U_{S,T} and X_{S}.

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

0 ⟶ U_{S,T} ⟶ U_{S} ⟶ ⨁_{w∈T_{K}} F_{q^{ν}} (w)^{×} ⟶ A_{S,T} ⟶ A_{S} ⟶ 0,  (1)

which, together with the usual class–number formula, implies that, if

ζ_{S,T} (s) = ∏_{w∉S_{K}}(1 − Nw^{−s})^{−1} · ∏_{w∈T_{K}}(1 − Nw^{1−s})

is the (S, T)–zeta function associated to K, then:

ord_{s=0}ζ_{S,T} (s) = |S_{K}| − 1 , lim_{s→0} s^{1−|S_{K}|}ζ_{S,T} (s) = −|A_{S,T} | · R_{S,T} .  (2)

(see [4] for (1) and (2) above)

1.2. THE L–FUNCTIONS

Let Ĝ be the group of irreducible, complex valued characters of G. For any χ ∈ Ĝ, L(s, χ) will denote the Artin L–function associated to χ, and e_{χ} = 1/g ∑_{σ∈G} χ(σ) ·

σ^{−1} ∈ C[G]. The Stickelberger function is defined by:

Θ(s) = ∑_{χ∈Ĝ} L(s, χ) · e_{χ^{−1}},

and it can be thought of as a complex meromorphic function, with values in C[G].

Rather than working with the L and Stickelberger functions as defined above, Rubin [14] works with the (S, T)–modified objects, defined by:

L_{S,T} (s, χ) = ∏_{v∈T}(1 − χ(σ_{v}) · Nv^{1−s}) · ∏_{v∉S}(1 − χ(σ_{v}) · Nv^{−s})^{−1}

Θ_{S,T} (s) = ∑_{χ∈Ĝ} L_{S,T} (s, χ) · e_{χ^{−1}},

where σ_{v} is the Frobenius morphism associated to (the unramified) v in G. Due to the conditions imposed on S and T, L_{S,T} (s, χ) and Θ_{S,T} (s) are holomorphic on the whole complex plane. For an integer r ≥ 0, such that s^{−r}Θ_{S,T} (s) is holomorphic at s = 0, let

Θ_{S,T}^{(r)} (0) = 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].

For S and T fixed, and χ ∈ Ĝ, let r_{χ} = ord_{s=0}L_{S,T} (s, χ). These numbers do not depend on T, but they depend on S in the following way:

r_{χ} = \{ \begin{matrix}#\{v ∈ S : χ|_{G_{v}} = 1_{G_{v}}\}, & if χ ≠ 1_{G} \\ #S − 1, & if χ = 1_{G}, \end{matrix}  (3)

or equivalently (and more elegantly)

r_{χ} = ⟨χ, χ_{X_{S}}⟩, for all χ ∈ Ĝ,  (4)

where χ_{X_{S}} is the character associated to the Galois representation ρ_{X_{S}} given by the C [G]–module C ⊗ X_{S}, and ⟨ , ⟩ is the usual inner product on the space of characters Ĝ. (see [17] for (3) and (4) above.)

1.3. GROUP RINGS AND G–MODULES

For a subring R of C, and a Z [G]–module M, R [G] denotes the group ring with coefficients in R, and RM denotes the R [G]–module R ⊗_{Z} M. For χ ∈ Ĝ, R[χ] denotes the minimal ring extension of R, containing the values of χ.

If L is a field of characteristic 0, then Ĝ(L) will denote the set of characters associated to L–irreducible representations of G. (In particular, if L = C, then Ĝ(C) = Ĝ.) If L̄ is an algebraic closure of L, then G (L̄/L) acts canonically on Ĝ(L̄) and Ĝ(L) can be viewed as the set of orbits with respect to this action. For ψ ∈ Ĝ(L) and χ ∈ Ĝ(L̄), we write χ|ψ if χ belongs to the orbit represented by ψ. If S and T are fixed, (3) above implies that r_{χ} = r_{χ^{τ}}, for any χ ∈ Ĝ(L̄), and any τ ∈ G(L̄/L). We can therefore define r_{ψ} as r_{ψ} = r_{χ}, for any ψ ∈ Ĝ(L) and any χ ∈ Ĝ(L̄), such that χ|ψ.

If r is a positive integer and S is fixed, let Ĝ(L, r) = {ψ ∈ Ĝ(L) : r_{ψ} = r}. If R is a ring whose field of fractions is L, and M is an R [G]– module, let

M_{r,S} = {m ∈ M : e_{ψ} · m = 0, for all ψ ∈ Ĝ(L) \ Ĝ(L, r)},

where e_{ψ} = 1/g ∑_{σ∈G} ψ(σ) · σ^{−1}. This is obviously an R [G]–submodule of M.

Remark 1. Let R be a Dedekind domain containing Z [1/g], and let L be its field of fractions. Then one has a decomposition

R [G] = ⨁_{ψ∈Ĝ(L)} D_{ψ} ,

where D_{ψ} = R [G]·e_{ψ}. The D_{ψ}’s are finite extensions of R (and therefore Dedekind domains themselves), and D_{ψ} ⟶^{∼} R[χ] via the map R [G] ⟶^{χ} R[χ], defined as χ( ∑_{σ∈G} a_{σ} · σ) = ∑_{σ∈G} a_{σ}χ(σ) , for any χ ∈ Ĝ(L̄), χ|ψ. This implies that, if M is an

R [G]–module, then one can decompose it into its ψ–components

M = ⨁_{ψ∈Ĝ(L)} M^{ψ} ,

where M^{ψ} = M ⨂_{R} D_{ψ} = {m ∈ M : e_{χ} · m = 0 in L̄M, if χ ∤ ψ}. For a finitely generated R [G]–module M, the following are therefore equivalent:

(i) M is a projective R [G]–module.
(ii) M^{ψ} is a projective D_{ψ}–module, for any ψ.
(iii) M^{ψ} has no D_{ψ}–torsion, for any ψ.
(iv) M has no R–torsion.

1.4. FITTING IDEALS

If A is a commutative, Noetherian ring, and M is a finitely generated A–module, then Fitt_{A}(M) denotes its Fitting ideal. If

A^{n} ⟶^{ϕ} A^{m} ⟶ M ⟶ 0

is a finite presentation of M, one can consider the composition of A–morphisms

∧^{m} A^{n} ⟶^{∧^{m} ϕ} ∧^{m} A^{m} ⟶^{∼}^{det} A .

By definition, we have Fitt_{A}(M) = Im(det ∘ ∧^{m} ϕ).

We will use the following properties of Fitting ideals:

(a) If M is a cyclic A–module then Fitt_{A}(M) = Ann_{A}(M).

(b) If A ⟶^{f} B is a morphism of Noetherian rings, and M is a finitely generated A–module, then Fitt_{B}(M ⨂_{A} B) = f(Fitt_{A}(M))B.

(c) If R is a Dedekind domain, M is a finitely generated R [G]–module, and a ∈ R [G], then

a ∈ Fitt_{R[G]}(M) ⟺ a ∈ Fitt_{R_{λ}[G]}(M ⨂_{R} R_{λ}), ∀λ,

where λ runs through the set of prime ideals of R.

(d) If A is a finite, direct sum of Noetherian rings A = ⨁_{i} A_{i}, N = ⨁_{i} N_{i} is an

A–module, with N_{i} ⟶^{∼} A_{i}^{n_{i}}, for some positive integers n_{i}, and

N = ⨁_{i} N_{i} ⟶^{f=⨁_{i} f_{i}} N = ⨁_{i} N_{i} ⟶ M ⟶ 0

is an exact sequence of A–modules, then det(f) =^{def} ∑_{i} det_{A_{i}}(f_{i}) ∈ Fitt_{A}(M). (Here

det_{A_{i}}(f_{i}) is the determinant of f_{i} with respect to an A_{i} basis of N_{i}.)

Let 0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0 be an exact sequence of finitely generated A–modules. Then:

(e) Fitt_{A}(M′) · Fitt_{A}(M′′) ⊆ Fitt_{A}(M) ⊆ Fitt_{A}(M′′).

(f) If A is a Dedekind domain and M, M′ and M′′ are finite, then

Fitt_{A}(M) = Fitt_{A}(M′) · Fitt_{A}(M′′) and [A : Fitt_{A}(M)] = |M| .

In particular, if A = Z, then Fitt_{Z}(M) = |M| · Z.

(g) If A = Z [G], and G is cyclic, then Fitt_{A}(M) ⊆ Fitt_{A}(M′).

Proofs for all the properties above, except (c) and (d), can be found in the Appendix of [9]. (c) is a consequence of (b), and (d) follows easily from the definition of the Fitting ideal.

1.5. THE MODULES X_{S}, U_{S,T} AND Λ_{S,T}

Let K/k, S, T be as above, and let r be a positive integer. From now on we are going to assume that the set of data (K/k, S, T, r) satisfies the following extended 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}

Hypotheses (H) imply that, for any χ ∈ Ĝ, we have r_{χ} ≥ r (see (3) above) and therefore Θ_{S,T}^{(r)} (0) ∈ C [G] makes sense. From the definitions , we have Θ_{S,T}^{(r)} (0) ∈ C [G]_{r,S}.

Let us choose an r-tuple (v_{1}, …, v_{r}) of r distinct primes in S, which split completely in K/k. Let us fix W = (w_{1}, …, w_{r}), where w_{i} is a prime in K lying above v_{i}, for any 1 ≤ i ≤ r, and let w ∈ S_{K}, such that w |̸v_{i}, for any 1 ≤ i ≤ r.

All the exterior powers considered in this paper are over Z [G], unless otherwise specified. We will be interested in the Z [G]–modules ∧^{r} U_{S,T} and ∧^{r} X_{S}. Let x =^{def}

(w_{1} − w) ∧ ⋯ ∧ (w_{r} − w) ∈ ∧^{r} X_{S}.

Lemma 1.5.1. Let R be a discrete valuation ring, or a field, containing Z [1/g], let L be its field of fractions and let R [G] = ⨁_{ψ∈Ĝ(L)} D_{ψ} be the direct sum decomposition

described in Remark 1 §1.3. Then

(1) RU_{S,T} ⟶^{∼} RX_{S} ⟶^{∼} ⨁_{ψ∈Ĝ(L)} D_{ψ}^{r_{ψ}}, as R [G]–modules.

(2) (R ∧^{r} U_{S,T} )_{r,S} ⟶^{∼} (R ∧^{r} X_{S})_{r,S} ⟶^{∼} R [G]_{r,S}, as R [G]–modules.

(3) (R ∧^{r} X_{S})_{r,S} = R [G]_{r,S} · x.

Proof. (1) is a consequence of Remark 1, §1.3 , equality (4) and the fact that RU_{S,T} and RX_{S} are isomorphic as R [G]–modules (see §1.1).

(2) and (3) follow from Remark 1, §1.3 and Lemma 2.6 of [14]. □

For every ϕ_{1}, …, ϕ_{r} ∈ Hom_{Z[G]} (U_{S,T} , Z [G]), one can define a Q [G]–morphism:

Q ∧^{r} U_{S,T} ⟶^{ϕ_{1}∧···∧ϕ_{r}} Q [G]

by letting ϕ_{1} ∧ ⋯ ∧ ϕ_{r}(u_{1} ∧ ⋯ ∧ u_{r}) = det_{i,j}(ϕ_{i}(u_{j})), for every u_{1} ∧ ⋯ ∧ u_{r} ∈ ∧^{r} U_{S,T}.

Definition 1.5.2. Let Λ_{S,T} be the Z [G]–submodule of Q ∧^{r} U_{S,T} defined by:

Λ_{S,T} = \{ε ∈ (Q ∧^{r} U_{S,T} )_{r,S} | \begin{matrix}ϕ_{1} ∧ ⋯ ∧ ϕ_{r}(ε) ∈ Z [G] , \\ ∀ϕ_{1}, …, ϕ_{r} ∈ Hom_{Z[G]} (U_{S,T} , Z [G]) \end{matrix} \} .

Lemma 1.5.3. The lattice Λ_{S,T} defined above satisfies the equalitites:

(1) Z [1/g] Λ_{S,T} = (Z [1/g] ∧^{r} U_{S,T} )_{r,S}.

(2) If r = 1, then Λ_{S,T} = (U_{S,T})_{1,S}.

(3) If r = 0, then Λ_{S,T} = Z [G]_{0,S}.

Proof. See [14], Prop. 1.2. □

1.6. THE CONJECTURES For every component w_{i} of the r–tuple W, chosen in §1.5, one can define a Z [G]– morphism:

X_{S} ⟶^{w_{i}^{∗}} Z [G] ,

by defining it first on ⨁_{w∈S_{K}} Zw as

w_{i}^{∗}(w) = ∑_{\begin{matrix}σ∈G \\ w^{σ}=w_{i} \end{matrix}} σ, ∀w ∈ S_{K},

and by taking the restricion to X_{S}. One obtains this way a C [G]–morphism

C ∧^{r} X_{S} ⟶^{w_{1}^{∗}∧···∧w_{r}^{∗}} C [G] ,

defined by w_{1}^{∗} ∧ ⋯ ∧ w_{r}^{∗}(x_{1} ∧ ⋯ ∧ x_{r}) = det_{i,j}(w_{i}^{∗}(x_{j})), ∀x_{1} ∧ ⋯ ∧ x_{r} ∈ ∧^{r} X_{S}.

Remark 1. If x = (w_{1} − w) ∧ ⋯ ∧ (w_{r} − w) is the element of ∧^{r} X_{S}, defined in §1.4, then w_{1}^{∗} ∧ ⋯ ∧ w_{r}^{∗}(x) = 1, as one can easily see.

Let

C ∧^{r} U_{S,T} ⟶^{∼}^{λ_{S,T}^{(r)}} C ∧^{r} X_{S}

be the C [G]–isomorphism defined by λ_{S,T}^{(r)} (u_{1} ∧⋯∧u_{r}) = λ_{S,T} (u_{1})∧⋯∧λ_{S,T} (u_{r}),

for every u_{1} ∧ ⋯ ∧ u_{r} ∈ ∧^{r} U_{S,T}.

Definition 1.6.1.

(1) The regulator map associated to the r–tuple W is defined to be the C [G]–

morphism:

C ∧^{r} U_{S,T} ⟶^{R_{W}} C [G] ,

given by R_{W} = (w_{1}^{∗} ∧ ⋯ ∧ w_{r}^{∗}) ∘ λ_{S,T}^{(r)} .

(2) For every χ ∈ Ĝ, we define

C ∧^{r} U_{S,T} ⟶^{R_{W,χ}} C

by R_{W,χ} = χ ∘ R_{W}.

Remark 2. Let us observe that:

R_{W} |_{(C ∧^{r} U_{S,T} )}_{r,S} : (C ∧^{r} U_{S,T} )_{r,S} ⟶ C [G]_{r,S}

is an isomorphism of C [G]–modules. Indeed, since λ_{S,T}^{(r)} is an isomorphism (Lemma 1.5.1), all we have to check is that

(w_{1}^{∗} ∧ ⋯ ∧ w_{r}^{∗})|_{(C ∧^{r} X_{S})}_{r,S} : (C ∧^{r} X_{S})_{r,S} ⟶ C [G]_{r,S}

is an isomorphism. Acording to Lemma 1.5.1 (3), (C ∧^{r} X_{S})_{r,S} = C [G]_{r,S} · x, and

therefore our statement follows from Remark 1 above.

Conjecture A (“over Q”)(Stark, Tate). If the set of data (K/k, S, T, r) satisfies hypotheses (H), then there exists a unique ε_{S,T} ∈ (Q ∧^{r} U_{S,T} )_{r,S}, such that

R_{W} (ε_{S,T}) = Θ_{S,T}^{(r)} (0).

Remark 3. As Rubin shows (see [14], Prop. 2.3), Conjecture A above is equivalent to Stark’s original conjecture “over Q”, for the L–functions associated to characters χ ∈ Ĝ(C, r). As Tate points out (see [17], Chpt.V), Stark’s Conjecture “over Q” is always true in the function field case. Therefore Conjecture A is a theorem in the setting we are considering in this paper. We will refer to Conjecture A as Q · St (K/k, S, T, r) in the sequel.

The following is the refinement Rubin [14] proposes for Conjecture A:

Conjecture B (“over Z”) (Rubin). If the set of data (K/k, S, T, r) satisfies hypotheses (H), then there exists a unique ε_{S,T} ∈ Λ_{S,T}, such that R_{W} (ε_{S,T}) = Θ_{S,T}^{(r)} (0).

Remark 4. If r = 0, this statement is equivalent to Θ_{S,T} (0) ∈ Z [G]_{0,S} (see Lemma 1.5.3(3)). This was proved by Deligne in the function field case ( see [17], Chpt.V).

If r = 1, as Rubin points out (see [14], Prop. 2.5), Conjecture B above, for a fixed S and for all T, is equivalent to Stark’s own refined conjecture for all L–functions associated to characters χ ∈ Ĝ(C, r). In the function field case, this was proved independently and with very different methods by Deligne (see [17], Chpt.V) and Hayes [6]. (Also see [13 §3] for more details on the equivalence between Rubin’s Conjecture for r = 1 and the classical Brumer–Stark Conjecture.)

The main goal of this paper is the study of Conjecture B above (which will be refered to as St (K/k, S, T, r) ), for any value of r.

Remark 5. Remark 2 above implies that the uniqueness of ε_{S,T} in both Conjecture A and B is automatic, once one proves its existence in the appropriate vector space or lattice.

1.7. GEOMETRIC BACKGROUND (see [17])

1.7.1 The corresponding schemes. Let X_{1} ⟶ Y_{1} be a finite morphism of projective, irreducible, smooth curves, defined over F_{q}, corresponding to the inclusion k ↪ K. Let F be an algebraic closure of F_{q} and let X = X_{1} ×_{Spec(F_{q})} Spec(F),

and Y = Y_{1} ×_{Spec(F_{q})} Spec(F) be the smooth, projective curves associated to X_{1}

and Y_{1} respectively, by extending scalars from F_{q} to F. Since F_{q} is algebraically closed in k, k ⨂_{F_{q}} F is a field and, correspondingly, Y is an irreducible, smooth,

projective curve, defined over F. Since the algebraic closure of F_{q} in K is F_{q^{ν}}, K ⨂_{F_{q}} F ⟶^{∼} ⨁_{0≤i≤ν−1} K^{(i)}, with K^{(i)} mutually isomorphic fields. Correspondingly, X = ∐_{0≤i≤ν−1} X^{(i)}, where the X^{(i)}’s are irreducible, smooth, projective curves, de-

fined over F, whose fields of rational functions are the K^{(i)}’s respectively. The X^{(i)}’s are mutually isomorphic over F, and therefore their genera g_{X^{(i)}} are equal. We will denote the common value of these numbers by g_{X}. We will also denote by [X^{(i)}] the generic point of the scheme X^{(i)}, for every 0 ≤ i ≤ ν − 1.

Every σ ∈ G gives isomorphisms

σ^{−1} : K ⟶^{∼} K , σ^{−1} ⊗ 1_{F} : K ⊗ F ⟶^{∼} K ⊗ F ,

which induce the isomorphisms

σ_{X_{1}} : X_{1} ⟶^{∼} X_{1} , σ_{X} : X ⟶^{∼} X

of F_{q}–schemes and F–schemes respectively. This way, G acts on the sets of points of X and it permutes the generic points [X^{(i)}] transitively. We emphasize the fact that σ_{X_{1}}and σ_{X} are associated to σ^{−1} ∈ G. This way, if w is a prime in K, and [w] is the associated closed point on the scheme X_{1}, then σ_{X_{1}}([w]) = [w^{σ}].

1.7.2 The geometric Frobenius. Let F_{X_{1}} : X_{1} ⟶ X_{1} be the geometric Frobenius endomorphism of X_{1}, relative to F_{q}. If U is an affine open subset of X_{1}, then F_{X_{1}}|_{U} corresponds to the q–power map on the F_{q}–algebra Γ (U, O_{X_{1}}), of U-sections of the structural sheaf O_{X_{1}}.

Definition 1.7.2. Let

F = F_{X_{1}} ×_{Spec(F_{q})} 1_{Spec(F)} : X ⟶ X

be the Spec(F)–scheme morphism obtained from F_{X_{1}} by extending scalars to F. F is called the geometric Frobenius endomorphism of X relative to F_{q}.

Remark 1. F permutes transitively the generic points [X^{(i)}] of X, and that F^{ν} fixes each of these. We can therefore suppose from now on that [X^{(i)}] = F^{i} ([X^{(0)}]), for every 0 ≤ i ≤ ν − 1.

Remark 2. The simple fact that σ(x^{q}) = σ(x)^{q} , ∀x ∈ K , ∀σ ∈ G, implies that F_{X_{1}} ∘σ_{X_{1}} = σ_{X_{1}} ∘F_{X_{1}} and therefore, by extending scalars, F ∘σ_{X} = σ_{X} ∘F, ∀σ ∈ G.

1.7.3 The ℓ–adic homology groups of X. We first give explicit descriptions of the homology groups H_{i} (X, Z) of X, with coefficients in Z, and of the actions of G and of the geometric Frobenius map F on each of these. For any i ≥ 0 and any σ ∈ G, we will denote by σ_{∗,i} and F_{∗,i} the maps induced on H_{i} (X, Z) by σ and F respectively.

• H_{i} (X, Z) = 0 for i ≥ 3.

• H_{0} (X, Z) = the free abelian group generated by [X^{(i)}] , 0 ≤ i ≤ ν − 1. The maps σ_{∗,0} and F_{∗,0} act on H_{0} (X, Z) by simply permuting the [X^{(i)}]’s as described in §1.7.2. Remark 1 in §1.7.2 also implies that F_{∗,0}^{ν} = 1_{H_{0}(X,Z)}.

• H_{1} (X, Z) = Pic^{0}(X), where Pic^{0}(X) is the Jacobian variety associated to X. We have Pic^{0}(X) = ∐_{i} Pic^{0}(X^{(i)}), and each Pic^{0}(X^{(i)}) is an irreducible abelian

variety, of dimension g_{X}, whose underlying group is isomorphic to the quotient of the group of divisors of degree zero on X^{(i)}, by the subgroup of principal divisors:

Pic^{0}(X^{(i)}) = \frac{Div^{0} (X^{(i)})}{\{div(f) : f ∈ K^{(i)}^{×}\}}, Pic^{0}(X) = \frac{Div^{0} (X)}{\{div(f) : f ∈ ⨁_{i} K^{(i)}^{×}\}}.

Here Div^{0} (X) is the group of divizors of multidegree zero on X, i.e. of degree zero if restricted to each component X^{(i)}.

The maps σ_{∗,1} and F_{∗,1} are naturally induced by the actions of σ and F on the set of closed points of the scheme X.

• H_{2} (X, Z) = H_{0} (X, Z) ⨂ F^{×}, with σ_{∗,2} = σ_{∗,0} ⊗ 1_{F}^{×}, and F_{∗,2} = F_{∗,0} ⊗ σ_{q}, where σ_{q} : F^{×} ⟶ F^{×} is the q–power map σ_{q}(x) = x^{q}.

If M is an abelian group and ℓ is a prime number, we give the following ad–hoc definition of the ℓ–adic Tate module of M:

T_{ℓ} (M) = \{ \begin{matrix}M ⨂ Z_{ℓ} & , if M has no Z–torsion \\ \underleftarrow{lim}_{n} M[ℓ^{n}] & , otherwise, \end{matrix}

where M[ℓ^{n}] is the group of ℓ^{n}–torsion points of M, and the projective limit is taken with respect to the multiplication by ℓ maps M[ℓ^{n+1}] ⟶^{ℓ} M[ℓ^{n}].

Definition 1.7.3.1. Let ℓ be any prime number. The ℓ–adic homology groups H_{i} (X, Z_{ℓ}) of X are defined by:

H_{i} (X, Z_{ℓ}) = T_{ℓ} (H_{i} (X, Z)) , ∀i ≥ 0.

The Z–linear actions of G and F on H_{i} (X, Z) induce in a natural way Z_{ℓ}–linear actions on H_{i} (X, Z_{ℓ}). According to the definition above, we have:

• H_{i} (X, Z_{ℓ}) = 0, ∀i ≥ 3.

• H_{0} (X, Z_{ℓ}) = H_{0} (X, Z) ⨂ Z_{ℓ} is the free Z_{ℓ}–module generated by [X^{(i)}], for all 0 ≤ i ≤ ν − 1.

• H_{1} (X, Z_{ℓ}) = T_{ℓ} (Pic^{0}(X)) = ⨁_{i} T_{ℓ} (Pic^{0}(X^{(i)})) ⟶^{∼} Z_{ℓ}^{νn}, where n = 2g_{X}, if

ℓ ≠ p, and 0 ≤ n < 2g_{X}, if ℓ = p.

• H_{2} (X, Z_{ℓ}) ⟶^{∼} H_{0} (X, Z_{ℓ}) ⨂ T_{ℓ} (F^{×}) . Since T_{ℓ} (F^{×}) = \underleftarrow{lim} μ_{ℓ^{n}}, where μ_{ℓ^{n}} = {ζ ∈ F^{×} : ζ^{ℓ^{n}} = 1}, we have T_{ℓ} (F^{×}) ⟶^{∼} Z_{ℓ} if ℓ ≠ p, and T_{p} (F^{×}) = {1}. We therefore have H_{2} (X, Z_{ℓ}) ⟶^{∼} H_{0} (X, Z_{ℓ}), if ℓ ≠ p, and H_{2} (X, Z_{p}) = 0. It is worth noticing that the isomorphism H_{2} (X, Z_{ℓ}) ⟶^{∼} H_{0} (X, Z_{ℓ}) is G–equivariant, but not F_{∗}–equivariant. In fact, the action of F_{∗,2} is taken into the action of q · F_{∗,0} by this isomorphism.

If ℓ is a prime number, and R is any ring containing Z_{ℓ}, then, by definition, H_{i} (X, R) = H_{i} (X, Z_{ℓ}) ⨂_{Z_{ℓ}} R. In what follows, σ_{∗,i} and F_{∗,i} will also denote the

R–linear actions induced by σ and F on H_{i} (X, R), for any i ≥ 0, and any R as above.

Remark 1. The H_{i} (X, Z_{ℓ})’s as defined above are in fact the functorial duals of the étale cohomology groups H_{ét}^{i} (X, Z_{ℓ}) of X (see Appendix).

1.7.4 Homological interpretation of Θ_{S,T} (s). Let R be a commutative ring, and V a finitely generated projective R [G]–module. Let W be a finitely generated R [G]–module so that V ⊕ W ⟶^{∼} R [G]^{n}, for some integer n ≥ 0. Let f ∈ End_{R[G]}(V). If u is a variable, one can define the polynomial in u, with coefficients in R [G]:

det_{R[G]} (1 − f · u| V) =^{def} det_{R[G]} (1 − (f ⊕ 0_{W}) · u| V ⊕ W) ,

where the determinant on the right is taken with respect to a basis of the free R [G]–module V ⊕ W. Schanuel’s Lemma easily implies that this definition does not depend on W.

In particular, if R and L are as in Remark 1, §1.3, then one can consider the decompositions R [G] = ⨁_{ψ∈Ĝ(L)} D_{ψ}, and V = ⨁_{ψ∈Ĝ(L)} V^{ψ}, with V^{ψ} free

D_{ψ}–modules of finite rank. It is easy to see that, in this case,

det_{R[G]} (1 − f · u| V) = ∑_{ψ∈Ĝ(L)} det_{D_{ψ}} (1 − f · u| V^{ψ}) ,

in R [G] [u] = ⨁_{ψ∈Ĝ(L)} D_{ψ}[u].

Remark 1. If R′ is an R–algebra, V is a projective R–module, and f ∈ End_{R[G]}(V), then V′ = V ⨂_{R} R′ is a projective R′ [G]–module , f′ = f ⊗ 1_{R′} ∈ End_{R′[G]}(V′), and det_{R′[G]}(1 − f′ · u| V′) = det_{R[G]}(1 − f · u| V).

For every prime number ℓ, and every i = 0 , 1 , 2, let

P_{i,ℓ}(u) =^{def} det_{Q_{ℓ}[G]} (1 − F_{∗,i} · u| H_{i} (X, Q_{ℓ})) .

The following theorem will play an important role in our future arguments.

Theorem 1.7.4.1. For any i = 0, 1, 2 :

(1) P_{i,p}(u) ∈ Z_{p} [1/g] [G] [u], and if ℓ ≠ p, then P_{i,ℓ} ∈ Z [1/g] [G] [u].

(2) If ℓ ≠ p, then P_{i,ℓ}(u) does not depend on ℓ.

Let P_{i}(u) =^{def} P_{i,ℓ}(u), for any ℓ ≠ p. Then:

(3) There exist polynomials Q_{i}(u) ∈ Z_{p} [1/g] [G] [u], such that

P_{i}(u) = P_{i,p}(u) · Q_{i}(u) .

(4) If K/k is a constant field extension, then Q_{i}(u) ∈ Z_{p} [G] [u].

Proof. See Appendix. □

The link between the polynomials P_{i}(u) defined above and the Stickelberger function Θ(s) is given by the following:

Theorem 1.7.4.2. There is an equality of complex, meromorphic functions

Θ(s) = ∏_{0≤i≤2} P_{i}(q^{−s})^{(−1)^{i+1}}.

Proof. See [17], Chpt.V. □

The natural question which arises is to give a similar homological interpretation for Θ_{S,T} (s). This was done by Deligne (see [17], Chpt.V). We are going to briefly describe his constructions below, as they will be needed in §§3 and 4.

Let S_{X} be the finite set of closed points on X, lying above points corresponding to primes in S, and let Div (S_{X}) be the Z [G]–module of divisors on X, supported on S_{X}. There is an exact sequence of Z [G]–modules:

0 ⟶ ker α_{S} ⟶ Div (S_{X}) ⟶^{α_{S}} H_{0} (X, Z) ⟶ 0,  (5)

where α_{S} is the multidegree map given by:

α_{S} (∑_{j=1}^{m} n_{j}P_{j}) = ∑_{0≤i≤ν−1} ( ∑_{P_{j}∈X^{(i)}} n_{j}) [X^{(i)}] ,

for every ∑_{j=1}^{m} n_{j}P_{j} ∈ Div (S_{X}).

Remark 2. Obviously ker α_{S} is a free Z–module of finite rank and, by definition, T_{ℓ}(ker α_{S}) = ker α_{S}⊗Z_{ℓ}, for any prime number ℓ. α_{S} is F_{∗,0}–invariant and therefore F acts on both ker α_{S} and T_{ℓ}(ker α_{S}). We denote the induced endomorphisms by F_{∗,0}.

Let T_{X} be the finite set of closed points on X, lying above points associated to primes in T, and let H_{0} (T_{X}, Z) be the free abelian group generated by T_{X}. There is an exact sequence of Z [G]–modules

0 ⟶ H_{0} (X, Z) ⊗ F^{×} ⟶^{β_{T}} H_{0} (T_{X}, Z) ⊗ F^{×} ⟶ coker(β_{T}) ⟶ 0,  (6)

where

β_{T} ( ∑_{0≤i≤ν−1} [X^{(i)}] ⊗ f_{i}) = ∑_{0≤i≤ν−1} ( ∑_{P ∈T_{X}∩X^{(i)}} P ) ⊗ f_{i},

for any ∑_{0≤i≤ν−1} [X^{(i)}] ⊗ f_{i} ∈ H_{0} (X, Z) ⊗ F^{×}.

Remark 3. From the definition of β_{T}, one can easily see that, as a group, coker(β_{T}) is a torus, i.e. coker(β_{T}) ⟶^{∼} F^{×}^{m}, for some positive integer m. Therefore we have T_{ℓ}(coker(β_{T})) ⟶^{∼} Z_{ℓ}^{m}, if ℓ ≠ p, and T_{p}(coker(β_{T})) = {1}.

Since F induces an isomorphism F_{∗,2} = F_{∗,0} ⊗ σ_{q} on both H_{0} (X, Z) ⊗ F^{×} and H_{0} (T_{X}, Z) ⊗ F^{×}, and β_{T} is F_{∗,2}–invariant, F induces a Z–linear isomorphism on coker(β_{T}), and a Z_{ℓ}–linear isomorphism on T_{ℓ}(coker(β_{T})). We will denote both these isomorphisms by F_{∗,2}.

Theorem 1.7.4.3. If S and T are two finite, nonempty, disjoint sets of primes in k, S containing all primes which ramify in K/k, and u = q^{−s}, then

(1) Θ_{S,T} (s) ∈ Z [G] [u].

(2) For every prime number ℓ, such that ℓ ≠ p, we have

Θ_{S,T} (s) = det_{Q_{ℓ}[G]} (1 − F_{∗,0} · u| T_{ℓ}(ker α_{S}) ⊗ Q_{ℓ}) ·
· det_{Q_{ℓ}[G]} (1 − F_{∗,1} · u| H_{1} (X, Q_{ℓ})) ·
· det_{Q_{ℓ}[G]} (1 − F_{∗,2} · u| T_{ℓ}(coker(β_{T})) ⊗ Q_{ℓ}) .

Proof. See [17], Chpt.V. □

Remark 4. The theorem above implies in particular that, if the set of data (K/k, S, T, 0) satisfies hypotheses (H), then Θ_{S,T} (0) ∈ Z [G], and therefore (by definition) Θ_{S,T} (0) ∈ Z [G]_{0,S}. This is the proof of St (K/k, S, T, r), for r = 0.

Lemma 1.7.4.4. Let ℓ be a prime number, such that gcd(ℓ, g) = 1. Then (1) If ℓ ≠ p,

Θ_{S,T} (s) = det_{Z_{ℓ}[G]} (1 − F_{∗,0} · u| T_{ℓ}(ker α_{S})) · det_{Z_{ℓ}[G]} (1 − F_{∗,1} · u| H_{1} (X, Z_{ℓ}))

· det_{Z_{ℓ}[G]} (1 − F_{∗,2} · u| T_{ℓ}(coker(β_{T}))) .

(2) If gcd(p, g) = 1, there exists a polynomial Q(u) ∈ Z_{p} [G] [u] such that

Θ_{S,T} (s) = Q(u) · det_{Z_{p}[G]} (1 − F_{∗,1} · u| H_{1} (X, Z_{p})) ·
· det_{Z_{p}[G]} (1 − F_{∗,0} · u| T_{p}(ker α_{S})) .

Proof. Let us first notice that, for ℓ satisfying gcd(ℓ, g) = 1, H_{1} (X, Z_{ℓ}), T_{ℓ}(ker α_{S}) and T_{ℓ}(coker(β_{T})) are Z_{ℓ}–free and therefore Z_{ℓ} [G]–projective modules (see Remark 1, §1.3). This shows that the determinants on the right–hand side of (1) make sense. Statement (1) in the lemma is now a direct consequence of Remark 1 above and Theorem 1.7.4.3(2).

The fact that T_{ℓ}(ker α_{S}) = (ker α_{S} ⊗ Z [1/g]) ⨂_{Z[1/g]} Z_{ℓ} and that ker α_{S} ⊗ Z [1/g]

is Z [G]–projective imply that

det_{Z[1/g][G]}(1 − F_{∗,0} · u| ker α_{S} ⊗ Z [1/g]) = det_{Z_{ℓ}[G]} (1 − F_{∗,0} · u| T_{ℓ}(ker α_{S})) , (7)

for all prime numbers ℓ, such that gcd(ℓ, g) = 1, in particular for ℓ = p. Equality (7) shows that the polynomial on the right is independent of ℓ, as long as gcd(ℓ, g) = 1.

As Deligne shows (see [17], Chpt.V), the exact sequence (6) implies that, if ℓ ≠ p, gcd(ℓ, g) = 1, we have an equality

det_{Z_{ℓ}[G]} (1 − F_{∗,2} · u| T_{ℓ}(coker(β_{T}))) = \frac{∏_{v∈T}(1 − σ_{v}^{−1}(qu)^{d_{v}})}{det_{Z_{ℓ}[G]} (1 − qF_{∗,0} · u| H_{0} (X, Z_{ℓ}))}. (8)

On the other hand, H_{0} (X, Z_{ℓ}) = H_{0} (X, Z [1/g]) ⊗ Z_{ℓ}, and H_{0} (X, Z [1/g]) is a projective Z [1/g] [G]–module. Remark 1 therefore implies that

det_{Z_{ℓ}[G]} (1 − qF_{∗,0} · u| H_{0} (X, Z_{ℓ})) = det_{Z[1/g][G]} (1 − qF_{∗,0} · u| H_{0} (X, Z [1/g])) .

This implies that the right hand–side of equality (8) is an element in the power series ring Z [1/g] [G] [[u]]. On the other hand, the left–hand side of the same equality belongs to Z_{ℓ} [G] [u]. We therefore have

det_{Z_{ℓ}[G]} (1 − F_{∗,2} · u| T_{ℓ}(coker(β_{T}))) ∈ Z [1/g] [G] [u] ,

for all ℓ, such that gcd(ℓ, g) = 1.

Let Q(u) =^{def} Q_{1}(u) · det_{Z_{ℓ}[G]} (1 − F_{∗,2} · u| T_{ℓ}(coker(β_{T}))), where Q_{1}(u) is the polynomial defined in Theorem 1.7.4.1 (3), and ℓ is a prime number, such that ℓ ≠ p and gcd(ℓ, g) = 1. According to the arguments above, Q(u) ∈ Z_{p} [G] [u]. Part (1) of the lemma, in addition to Theorem 1.7.4.1(3) and equality (7) imply that

Θ_{S,T} (s) =Q(u) · det_{Z_{p}[G]} (1 − F_{∗,1} · u| H_{1} (X, Z_{p})) ·
· det_{Z_{p}[G]} (1 − F_{∗,0} · u| T_{p}(ker α_{S})) ,

which concludes the proof of (2). □

2. Dynamics (Changing S and r)

Let us suppose that the set of data (K/k, S, T, r) satisfies hypotheses (H). Let r′ > r, and let v_{r+1}, …, v_{r′} be r′ − r distinct primes in k, not belonging to S or T, and splitting completely in K/k. If S′ = S ∪ {v_{r+1}, …, v_{r′}}, then the set of data (K/k, S′, T, r′) satisfies hypotheses (H) as well. Since T remains fixed throughout this section, we will denote by ε_{S} and ε_{S′} the elements ε_{S,T} and ε_{S′,T} whose existence is predicted by Q · St (K/k, S, T, r) and Q · St (K/k, S′, T, r′) respectively. We will also make the notations A_{S} = A_{S,T}, U_{S} = U_{S,T} etc. A_{S,S′} will denote the subgroup of A_{S} generated by primes in K, lying above v_{i}, for all r + 1 ≤ i ≤ r′. All the Fitting ideals and exterior powers involved from now on in this paper are considered over Z [G], unless otherwise specified, and therefore we will suppress the group ring Z [G] from the notation.

In this section we will list some relations between statements St (K/k, S, T, r) and St (K/k, S′, T, r′), which will be needed in §§3 and 4. For most proofs we refer the reader to [14], §5.

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

0 ⟶ A_{S,S′} ⟶ A_{S} ⟶ A_{S′} ⟶ 0.  (9)

Proposition 2.1. If ε_{S} and ε_{S′} are the elements defined above, then (1) ε_{S} ∈ Z [1/g] Fitt (A_{S}) Λ_{S} ⟺ ε_{S′} ∈ Z [1/g] Fitt (A_{S′}) Λ_{S′}.

(2) If G is cyclic, then ε_{S} ∈ Fitt (A_{S}) Λ_{S} ⟹ ε_{S′} ∈ Λ_{S′}.

Proof. (see also Theorem 5.3 in [14]) According to [14], §5, there exist Q [G]– morphisms:

Q ∧^{r′} U_{S′} ⟶^{Φ} Q ∧^{r} U_{S}, Q ∧^{r′−r} U_{S′} ⟶^{Φ′} Q [G] ,

satisfying the following properties:

(a) Φ is injective if restricted to QΛ_{S′}.

(b) Φ (Z [1/g] Λ_{S′}) = Z [1/g] Fitt (A_{S,S′}) Λ_{S}.

(c) Fitt (A_{S,S′}) Λ_{S} ⊆ Φ (Λ_{S′}).

(d) Φ(ε_{S′}) = ε_{S}.

(e) Φ′ ( ∧^{r′−r} U_{S′}) = Fitt (A_{S,S′}).

(f) Φ(u_{1} ∧ u_{2}) = Φ′(u_{1}) ∧ u_{2}, for every u_{1} ∈ Q ∧^{r′−r} U_{S′} and u_{2} ∈ Q ∧^{r} U_{S}.

(1) The exact sequence (9) and the fact that Z [1/g] [G] is a finite, direct sum of Dedekind domains (see §1.3) imply that there is an equality of Z [1/g] [G]–ideals (see §1.4(f)):

Z [1/g] Fitt (A_{S}) = (Z [1/g] Fitt (A_{S,S′})) · (Z [1/g] Fitt (A_{S′})) .  (10)

Let us suppose that ε_{S} ∈ Z [1/g] Fitt (A_{S}) Λ_{S}. (10) implies that ε_{S} can be written as a finite sum ε_{S} = ∑_{i} γ_{i}δ_{i}λ_{i}, with γ_{i} ∈ Z [1/g] Fitt (A_{S′}), δ_{i} ∈ Fitt (A_{S,S′}) and

λ_{i} ∈ Λ_{S}. According to (c) above, this implies that ε_{S} ∈ Φ (Z [1/g] Fitt (A_{S′}) Λ_{S′}). Thus (a) and (d) imply that ε_{S′} ∈ Z [1/g] Fitt (A_{S′}) Λ_{S′}.

Let us suppose now that ε_{S′} ∈ Z [1/g] Fitt (A_{S′}) Λ_{S′}. This implies that ε_{S} = Φ(ε_{S′}) ∈ Fitt (A_{S′}) Φ (Z [1/g] Λ_{S′}). On the other hand, according to (b) and (10) above, we have the equalities:

Fitt (A_{S′}) Φ (Z [1/g] Λ_{S′}) = Z [1/g] Fitt (A_{S,S′}) Fitt (A_{S′}) Λ_{S} =
= Z [1/g] Fitt (A_{S}) Λ_{S} .

This concludes the proof of (1).

(2) If G is cyclic , §1.4 (g) and the exact sequence (9) imply that Fitt (A_{S}) ⊆ Fitt (A_{S,S′}), and therefore (a) and (c) imply (2). □

Corollary 2.2. Let us suppose that, in the context above, r = 0. Then: (1) Θ_{S,T} (0) ∈ Z [1/g] Fitt (A_{S}) ⟺ ε_{S′} ∈ Z [1/g] Fitt (A_{S′}) Λ_{S′}.

(2) If G is cyclic, Θ_{S,T} (0) ∈ Z [G]_{0,S}Fitt (A_{S}) ⟹ ε_{S′} ∈ Z [G]_{r′,S′} · ∧^{r′} U_{S′}.

Proof. (1) In order to prove (1), one has to rely on the fact that, in the case r = 0, Λ_{S} = Z [G]_{0,S} (see Lemma 1.5.3(3)), and that

Z [1/g] Fitt (A_{S}) · Z [1/g] [G]_{0,S} = Z [1/g] Fitt (A_{S}) ∩ Z [1/g] [G]_{0,S}.

This last equality follows from the fact that Z [1/g] [G] = ⨁_{ψ∈Ĝ(Q)} D_{ψ} and that, with

respect to this decomposition, Z [1/g] [G]_{0,S} = ⨁_{ψ∈Ĝ(Q,0)} D_{ψ} (see §1.3).

Proposition 2.1 (1), in addition to the fact that Θ_{S,T} (0) ∈ C [G]_{0,S}, implies therefore that

Θ_{S,T} (0) ∈ Z [1/g] Fitt (A_{S})⟺ Θ_{S,T} (0) ∈ Z [1/g] Fitt (A_{S}) · Z [1/g] [G]_{0,S}
⟺ ε_{S′} ∈ Z [1/g] Fitt (A_{S′}) Λ_{S′}.

(2) Let us suppose that Θ_{S,T} (0) ∈ Z [G]_{0,S} · Fitt (A_{S}). One can therefore write Θ_{S,T} (0) as a finite sum Θ_{S,T} (0) = ∑_{i} α_{i}·f_{i}, where α_{i} ∈ Z [G]_{0,S}, and f_{i} ∈ Fitt (A_{S}).

Property (e) above together with §1.4(g) imply that there exist elements u_{i} ∈ ∧^{r′} U_{S′} so that f_{i} = Φ′(u_{i}), for every index i, and therefore (f) implies that ε_{S} = Θ_{S,T} (0) = Φ (∑ α_{i} · u_{i}). The obvious fact that, in our context, Z [G]_{0,S} = Z [G]_{r′,S′} therefore implies that:

∑_{i} α_{i} · u_{i} ∈ Z [G]_{r′,S′} · ∧^{r′} U_{S′} ⊆ Λ_{S′}.

Properties (a) and (d) imply then that ε_{S′} = ∑ α_{i} · u_{i} ∈ Z [G]_{r′,S′} · ∧^{r′} U_{S′}. □

We conclude this section with the following statement, whose proof can be found in [14], §5 (see Corollary 5.4). For a more detailed proof, see [12], Chapter II.

Proposition 2.3. Let us suppose that the set of data (K/k, S, T, r) satisfies hypotheses (H), and that ε_{S,T} is the element satisfying Q · St (K/k, S, T, r). If ε_{S,T} ∈ Z [1/g] Λ_{S,T}, then the following statements are equivalent:

(1) ε_{S,T} ∈ Z [1/g] Fitt (A_{S,T}) Λ_{S,T} .

(2) Z [1/g] [G] ε_{S,T} = Z [1/g] Fitt (A_{S,T}) Λ_{S,T} .

We will actually prove that statement (1) in Proposition 2.3 is always satisfied in the function field case (see Theorem 3.2.1(2)), and therefore (2) is always satisfied (see Corollary 3.2.2). This fact turns out to be of crucial importance in proving that Gras–type Conjectures hold true in the function field setting (see [13]).

3. Conjecture B up to primes dividing |G (K/k) |

3.1. THE r = 0 CASE In the context described in §1.7, let us suppose that the set of data (K/k, S, T, 0) satisfies hypotheses (H). The goal of this section is the proof of the following:

Theorem 3.1.1. If the set of data (K/k, S, T, 0) satisfies hypotheses (H), then

Θ_{S,T} (0) ∈ Z [1/g] Fitt_{Z[1/g][G]}(A_{S,T}).

Remark 1. Before starting to prove the theorem above, let us first notice that, according to §1.4 (c), its statement is equivalent to

Θ_{S,T} (0) ∈ Fitt_{Z_{ℓ}[G]} (A_{S,T} ⊗ Z_{ℓ}) ,

for all primes ℓ, such that gcd(ℓ, g) = 1. We will therefore prove the Theorem “prime by prime”.

Let S_{X_{1}} and S_{X} be the finite sets of closed points on X_{1} and X respectively, lying above points in Y_{1} associated to primes in S. T_{X_{1}} and T_{X} have similar meanings. If Z is a subset of closed points of X_{1} or X, then Div (Z) and Div^{0} (Z) will denote the groups of divisors and respectively divisors of degree 0, supported on Z. (If Z ⊆ X, degree 0 means multidegree 0, i.e. degree 0 on each connected component X^{(i)}.) Let

Pic^{0} (X_{1}) = \frac{Div^{0} (X_{1})}{\{div(f) : f ∈ K^{×}\}} , Pic^{0}(X) = \frac{Div^{0} (X)}{\{div(f) : f ∈ ⨁_{0≤i≤ν−1} K^{(i)}^{×}\}}

be the Picard groups associated to X_{1} and X respectively. As explained in §1.7.3, Pic^{0}(X) is the underlying group of an abelian variety of dimension g_{X}, on which the geometric Frobenius map induces a bijective endomorphism F_{∗,1}. Pic^{0} (X_{1}) can be naturaly viewed as a finite subgroup of Pic^{0}(X), and it is precisely the part of Pic^{0}(X) fixed by the action of F_{∗,1}.

We will also consider the following groups:

Pic^{0} (X_{1})_{T} = \frac{Div^{0} (X_{1} \ T_{X_{1}})}{\{div(f) : f ∈ K^{×}, f ≡ 1 mod w, ∀w ∈ T_{X_{1}}\}} ,

Pic^{0}(X)_{T} = \frac{Div^{0} (X \ T_{X})}{\{div(f) : f ∈ ⨁_{i} K^{(i)}^{×}, f ≡ 1 mod w, ∀w ∈ T_{X}\}}.

It can be shown that Pic^{0}(X)_{T} is the underlying group of an algebraic group as well. We will not use this fact in what follows. Pic^{0} (X_{1})_{T} sits inside the part of Pic^{0}(X)_{T} fixed by the action F_{∗,1} of the geometric Frobenius morphism and, as Lemma 3.1.4 below shows, it is in fact equal to this. There obviously are surjective group homomorphisms

π_{X_{1}} : Pic^{0} (X_{1})_{T} ⟶ Pic^{0} (X_{1}) , π_{X} : Pic^{0}(X)_{T} ⟶ Pic^{0}(X) .

There is a commutative diagram of Z [G]–modules, with exact rows

0 ⟶ K^{×}/F_{q^{ν}}^{×} ⟶^{div} Div^{0} (X_{1}) ⟶ Pic^{0} (X_{1}) ⟶ 0
π_{K} ↓  ^{ϕ_{S}} ↓  ^{ψ_{S}} ↓
0 ⟶ K^{×}/U_{S} ⟶^{div_{S}} Div (X_{1} \ S_{X_{1}}) ⟶ A_{S} ⟶ 0,

(11)

where:

• π_{K} is the usual projection (F_{q^{ν}}^{×} ⊆ U_{S}).

• ϕ_{S} is the “forgetful” map defined by ϕ ( ∑_{w_{i}∈X_{1}} n_{i}w_{i}) = ∑_{w_{i}∈X_{1}\S_{X_{1}}} n_{i}w_{i}.

• ψ_{S} is the map induced by ϕ_{S}.

• div is the usual divisor map.

• div_{S} is the “forgetful” divisor map, defined by div_{S}(f) = ∑_{w∈X_{1}\S_{X_{1}}} ord_{w}(f)·w,

for any f ∈ K.

The snake lemma applied to (11) and the surjectivity of π_{K} give an isomorphism of Z [G]–modules coker(ϕ_{S}) ⟶^{∼} coker(ψ_{S}). Let dZ = Im (Div (S_{X_{1}}) ⟶^{deg} Z), where

“deg” is the divisor degree map on X_{1}. It is an easy observation that coker(ϕ_{S}) ⟶^{∼} Z/dZ as Z [G]–modules, with G acting trivially on Z/dZ. Therefore we obtain the following isomorphism of Z [G]–modules

coker(ψ_{S}) ⟶^{∼} Z/dZ,  (12)

with G acting trivially on Z/dZ.

There is a commutative diagram of Z [G]–modules with exact rows

0 ⟶ ( ⨁_{w∈T_{X_{1}}} F_{q^{ν}} (w)^{×})/F_{q^{ν}}^{×} ⟶^{i_{T}} Pic^{0} (X_{1})_{T} ⟶^{π_{X_{1}}} Pic^{0} (X_{1}) ⟶ 0
π_{T} ↓  ^{ψ_{S,T}} ↓  ^{ψ_{S}} ↓
0 ⟶ ( ⨁_{w∈T_{X_{1}}} F_{q^{ν}} (w)^{×})/U_{S} ⟶^{i_{S,T}} A_{S,T} ⟶ A_{S} ⟶ 0,

(13)

where:

• The lower row comes from the exact sequence (1).

• i_{T} ({ \atop (a_{w})_{w∈T_{X_{1}}}} ) = { \atop div(f)}, for every (a_{w})_{w∈T_{X_{1}}} ∈ ⨁_{w∈T_{X_{1}}} F_{q^{ν}} (w)^{×} and f ∈ K,

satisfying f ≡ a_{w} mod w, ∀w ∈ T_{X_{1}}.

• ψ_{S,T} ({ \atop ∑_{i} n_{i} · w_{i}}) = the ideal class corresponding to ∏_{i} w_{i}^{n_{i}} in A_{S,T}.

The snake lemma applied to (13) and the surjectivity of π_{T} imply that coker(ψ_{S,T}) ⟶^{∼} coker(ψ_{S}). (12) therefore gives an isomorphism of Z [G]–modules

coker(ψ_{S,T}) ⟶^{∼} Z/dZ,  (14)

with G acting trivially on Z/dZ.

Proposition 3.1.2. If ℓ is a prime number, such that gcd(ℓ, g) = 1, then

det_{Z_{ℓ}[G]} (1 − F_{∗,0}|T_{ℓ}(ker α_{S})) ∈ Fitt_{Z_{ℓ}[G]}(coker(ψ_{S,T}) ⊗ Z_{ℓ}).

Before starting to prove the proposition, let us observe that the determinant in the statement makes sense due to the Z_{ℓ} [G]–projectivity of T_{ℓ}(ker α_{S}) (see Remark 2, §1.7.4 and Remark 1, §1.3). In what follows, if M is an abelian group and f ∈ End(M), then M^{f} denotes the subgroup of M fixed by f.

Proof. Since α_{S} is F_{∗,0}–equivariant, we have a commutative diagram of Z [G]– modules, with exact rows

0 ⟶ ker α_{S} ⟶ Div (S_{X}) ⟶^{α_{S}} H_{0} (X, Z) ⟶ 0
(1−F_{∗,0})_{1}↓  ^{(1−F_{∗,0})_{2}}↓  ^{(1−F_{∗,0})_{3}}↓
0 ⟶ ker α_{S} ⟶ Div (S_{X}) ⟶ H_{0} (X, Z) ⟶ 0.

Obviously, we have the following equalities:

• ker(1 − F_{∗,0})_{2} = Div (S_{X})^{F_{∗,0}} = Div (S_{X_{1}}),

• ker(1 − F_{∗,0})_{3} = the rank 1, free Z–module generated by A = ∑_{0≤i≤ν−1} [X^{(i)}],

with trivial G–action.

The snake lemma gives therefore a long exact sequence of Z [G]–modules

0 ⟶ ker(1 − F_{∗,0})_{1} ⟶ Div (S_{X_{1}}) ⟶^{A·deg} Z · A ⟶

⟶ coker(1 − F_{∗,0})_{1} ⟶ coker(1 − F_{∗,0})_{2} ⟶ coker(1 − F_{∗,0})_{3} ⟶ 0

Since Im(A · deg) = dZ · A, and G fixes A, we get an injective morphism of Z [G]–modules

Z/dZ ↪ coker(1 − F_{∗,0})_{1},  (15)

with G acting trivially on Z/dZ. Since for ℓ ∤ g, Z_{ℓ} [G] is a direct sum of Dedekind domains, (14), (15) and (f), §1.4 imply that

Fitt_{Z_{ℓ}[G]} (coker(1 − F_{∗,0})_{1} ⊗ Z_{ℓ}) ⊆ Fitt_{Z_{ℓ}[G]} (coker(ψ_{S,T}) ⊗ Z_{ℓ}) , ∀ℓ , ℓ ∤ g . (16)

If one considers the decomposition Z_{ℓ} [G] = ⨁_{ψ∈Ĝ(Q_{ℓ})} D_{ψ}, as in §1.3, then the fact

that T_{ℓ}(ker α_{S}) is a projective Z_{ℓ} [G]–module implies that one has an isomorphism T_{ℓ}(ker α_{S}) ⟶^{∼} ⊕_{ψ}D_{ψ}^{n_{ψ}}, for some positive integers n_{ψ} (see Remark 1, §1.3). The exact sequence

\xymatrix{T_{ℓ}(ker α_{S}) \ar[r]^{(1−F_{∗,0})_{1}} & T_{ℓ}(ker α_{S}) \ar[r] & coker(1 − F_{∗,0})_{1} ⊗ Z_{ℓ} \ar[r] & 0}

gives therefore a situation similar to the one described in §1.4 (d), with R = Z_{ℓ} [G] and N = T_{ℓ}(ker α_{S}). We therefore have

det_{Z_{ℓ}[G]}(1 − F_{∗,0}|T_{ℓ}(ker α_{S})) ∈ Fitt_{Z_{ℓ}[G]} (coker(1 − F_{∗,0})_{1} ⊗ Z_{ℓ}) .  (17)

Relations (16) and (17) imply the statement in Proposition 3.1.2. □

Proposition 3.1.3. If ℓ is a prime number, such that gcd(ℓ, g) = 1, then

det_{Z_{ℓ}[G]} (1 − F_{∗,2}| T_{ℓ}(coker(β_{T}))) · det_{Z_{ℓ}[G]} (1 − F_{∗,1}| H_{1} (X, Z_{ℓ})) ∈
∈ Fitt_{Z_{ℓ}[G]} (Pic^{0} (X_{1})_{T} ⊗ Z_{ℓ}) .

Proof. Let us recall that we have an exact sequence of Z [G]–modules

0 ⟶ H_{0} (X, Z) ⊗ F^{×} ⟶^{β_{T}} H_{0} (T_{X}, Z) ⊗ F^{×} ⟶ coker(β_{T}) ⟶ 0 .  (18)

The q–power geometric Frobenius morphism induces isomorphisms F_{∗,2} = F_{∗,0} ⊗ σ_{q} on each term of (18), where σ_{q} : F^{×} ⟶ F^{×} is the q–power map, and β_{T} is F_{∗,2}–invariant (see §1.7.4).

Let Σ = G (F/F_{q}) (topologically generated by σ_{q}), let Σ_{ν} be the closed subgroup of Σ (topologically) generated by σ_{q}^{ν}, and let Γ_{0} be the finite subgroup of Aut_{Z} (H_{0} (T_{X}, Z)) generated by F_{∗,0}. Then obviously the profinite group Γ_{0} × Σ acts continously on each term of (18), considered with the discrete topology.

Let H be the closed subgroup of Γ_{0} × Σ, (topologically) generated by F_{∗,2} = F_{∗,0}⊗σ_{q}. Let H_{ν} be the closed subgroup of H generated by F_{∗,2}^{ν} = F_{∗,0}^{ν}⊗σ_{q}^{ν}. Since the actions of H and G on H_{0} (T_{X}, Z)⊗F^{×} commute, (18) is an exact sequence of H × G–modules.

We also have an exact sequence of G–modules

0 ⟶ coker(β_{T}) ⟶ Pic^{0}(X)_{T} ⟶ Pic^{0}(X) ⟶ 0 ,  (19)

which preserves the Frobenius action on each of its terms (see [17], Chpt.V).

Let ℓ be a prime number. Remark 3, §1.7.4 shows that the multiplication by ℓ maps

coker(β_{T}) [ℓ^{n+1}] ⟶^{ℓ} coker(β_{T}) [ℓ^{n}]

are surjective for every n ≥ 0. The definition of Pic^{0} (X_{1})_{T}, combined with the weak approximation theorem for valuations on K^{(i)}, for all 0 ≤ i ≤ ν − 1, easily implies that the natural maps

Pic^{0} (X_{1})_{T} [ℓ^{n}] ⟶ Pic^{0} (X_{1}) [ℓ^{n}]

are surjective as well, for every n ≥ 0. These two facts show that the short exact sequence (19) gives a short exact sequence of G–modules

0 ⟶ T_{ℓ}(coker(β_{T})) ⟶ T_{ℓ} (Pic^{0}(X)_{T} ) ⟶ T_{ℓ} (Pic^{0}(X)) ⟶ 0 ,  (20)

at the level of ℓ–adic Tate modules, preserving the Frobenius action on its terms. We will need two lemmas:

Lemma 3.1.4. We have the following:

(1) T_{ℓ} (Pic^{0}(X)_{T} ) is a free Z_{ℓ}–module of finite rank, for every prime ℓ.

(2) Pic^{0}(X)_{T}^{F_{∗,1}} ⟶^{∼} Pic^{0} (X_{1})_{T} as Z [G]–modules, where Pic^{0}(X)_{T}^{F_{∗,1}} is the part of Pic^{0}(X)_{T} fixed by F_{∗,1}.

Proof. Statement (1) follows directly from the exact sequence (20) and from the fact that both T_{ℓ}(coker(β_{T})) and T_{ℓ} (Pic^{0}(X)) are free Z_{ℓ}–modules of finite rank (see §1.7.3 and Remark3, §1.7.4).

(2) The fact that F_{∗,0}^{ν} = 1_{H_{0}(X,Z)} implies that there is an H–isomorphism

H_{0} (X, Z) ⊗ F^{×} ⟶^{∼} (Z · [X^{(0)}] ⊗ F^{×}) ⨂_{Z[H_{ν}]} Z [H] .  (21)

Shapiro’s Lemma and Hilbert’s Theorem 90 imply therefore that, at the level of Galois cohomology groups, we have

H^{i} (H, H_{0} (X, Z) ⊗ F^{×}) ⟶^{∼} H^{i} (H_{ν}, Z · [X^{(0)}] ⊗ F^{×}) = H^{i} (Σ_{ν}, F^{×}) = 0, (22)

for i = 1, 2.

The fact that H/H_{ν} is finite cyclic and (21) above imply that there are isomorphisms of G–modules

(H_{0} (X, Z) ⊗ F^{×})^{H} ⟶^{∼} (Z · [X^{(0)}] ⊗ F^{×})^{H_{ν}} ⟶^{∼} F^{×}^{Σ_{ν}} = F_{q^{ν}}^{×} .  (23)

In a similar fashion one can prove that

H^{1} (H, H_{0} (T_{X}, Z) ⊗ F^{×}) = 0  (24)

and that there is an isomorphism of G–modules

(H_{0} (T_{X}, Z) ⊗ F^{×})^{H} ⟶^{∼} ⨁_{w∈T_{X}} F_{q^{ν}} (w)^{×}.  (25)

As a consequence of (22–25), the long exact sequence of H–cohomology groups associated to the short exact sequence (18) starts as

0 ⟶ F_{q^{ν}}^{×} ⟶ ⨁_{w∈T_{X_{1}}} F_{q^{ν}} (w)^{×} ⟶ (coker(β_{T}))^{H} ⟶

⟶ 0 ⟶ 0 ⟶ H^{1} (H, coker(β_{T})) ⟶ 0 ⟶ · · · .

This implies that we have the following G–isomorphisms:

H^{1} (H, coker(β_{T})) = 0 and (coker(β_{T}))^{H} ⟶^{∼} ( ⨁_{w∈T_{X_{1}}} F_{q^{ν}} (w)^{×})/F_{q^{ν}}^{×} . (26)

Let H be the subgroup of Aut_{Z} (Pic^{0} (X_{1})_{T} ) generated by F_{∗,1}. As we mentioned above, the restriction of F_{∗,1} to coker (β_{T}) via the injective morphism coker (β_{T}) ⟶ Pic^{0} (X_{1})_{T} coincides to F_{∗,2}. This observation together with (26) show that the short exact sequence (19) gives a long exact sequence of H–cohomology groups which starts as

0 ⟶ (coker(β_{T}))^{F_{∗,2}} ⟶ Pic^{0}(X)_{T}^{F_{∗,1}} ⟶ Pic^{0}(X)^{F_{∗,1}} ⟶ 0.

We therefore have a commutative diagram with exact rows

\xymatrix{0 \ar[r] & ( ⨁_{w∈T_{X_{1}}} F_{q^{ν}} (w)^{×})/F_{q^{ν}}^{×} \ar[r]^{i_{T}} \ar[d]_{i_{1}} & Pic^{0} (X_{1})_{T} \ar[r]^{π_{X_{1}}} \ar[d]_{i_{2}} & Pic^{0} (X_{1}) \ar[r] \ar[d]_{i_{3}} & 0 \\ 0 \ar[r] & (coker(β_{T}))^{F_{∗,2}} \ar[r] & _{Pic^{0}(X)_{T}_{F_{∗,1}}} \ar[r] & Pic^{0}(X)^{F_{∗,1}} \ar[r] & 0 ,}

(27)

where i_{2} and i_{3} are the natural inclusions and i_{1} is the isomorphism in (26). We already know that i_{3} and i_{1} are isomorphisms (see comments at the beginning of this section and (26) above, respectively). The snake lemma therefore implies that i_{2} is an isomorphism as well, and this concludes the proof of (2). □

Lemma 3.1.5. Let M be a Z [G]–module, which has nontrivial Z–torsion, let f ∈ End_{Z[G]}(M), and ℓ a prime number, such that

(1) T_{ℓ}(M) is a free Z_{ℓ}–module of finite rank.

(2) M^{f} =^{def} {m ∈ M : f(m) = m} is finite. Then there is an isomorphism of Z_{ℓ} [G]–modules

T_{ℓ}(M)/(1 − f)T_{ℓ}(M) ⟶^{∼} Z_{ℓ} ⊗ M^{f}.

Proof. Let us consider the exact sequence of Z_{ℓ}–modules

0 ⟶ Z_{ℓ} ⟶ Q_{ℓ} ⟶ Q_{ℓ}/Z_{ℓ} ⟶ 0.

In virtue of hypothesis (1) above,

0 ⟶ T_{ℓ}(M) ⟶ T_{ℓ}(M) ⨂_{Z_{ℓ}} Q_{ℓ} ⟶ T_{ℓ}(M) ⨂_{Z_{ℓ}} Q_{ℓ}/Z_{ℓ} ⟶ 0

is still exact. We therefore get a commutative diagram with exact rows

0 ⟶ T_{ℓ}(M) ⟶ T_{ℓ}(M) ⨂_{Z_{ℓ}} Q_{ℓ} ⟶ T_{ℓ}(M) ⨂_{Z_{ℓ}} Q_{ℓ}/Z_{ℓ} ⟶ 0
(1−f)_{1}↓  ^{(1−f)_{2}}↓  ^{(1−f)_{3}}↓
0 ⟶ T_{ℓ}(M) ⟶ T_{ℓ}(M) ⨂_{Z_{ℓ}} Q_{ℓ} ⟶ T_{ℓ}(M) ⨂_{Z_{ℓ}} Q_{ℓ}/Z_{ℓ} ⟶ 0,

where (1 − f)_{1} = (1 − f) at the level of Tate modules, (1 − f)_{2} = (1 − f)_{1} ⊗ 1_{Q_{ℓ}}, and (1 − f)_{3} = (1 − f)_{1} ⊗ 1_{Q_{ℓ}/Z_{ℓ}}. Hypothesis (2) above implies that (1 − f)_{1} is injective and therefore (1) implies that (1 − f)_{2} is an isomorphism of Q_{ℓ}–vector spaces. Snake lemma applied to the diagram above gives therefore an isomorphism of Z_{ℓ} [G]–modules

ker(1 − f)_{3} ⟶^{∼} T_{ℓ}(M)/(1 − f)T_{ℓ}(M).  (28)

On the other hand

T_{ℓ}(M) ⨂_{Z_{ℓ}} Q_{ℓ}/Z_{ℓ} ⟶^{∼} ⋃_{i≥0} M[ℓ^{i}].

Let M[ℓ^{∞}] = ⋃_{i≥0} M[ℓ^{i}]. Then the commutative diagram

T_{ℓ}(M) ⨂_{Z_{ℓ}} Q_{ℓ}/Z_{ℓ} ⟶^{∼} M[ℓ^{∞}]
(1−f)_{3}↓  ^{(1−f)}↓
T_{ℓ}(M) ⨂_{Z_{ℓ}} Q_{ℓ}/Z_{ℓ} ⟶^{∼} M[ℓ^{∞}],

in addition to (28) and hypothesis (2) give the isomorphisms of Z_{ℓ} [G]–modules

T_{ℓ}(M)/(1 − f)T_{ℓ}(M) ⟶^{∼} M[ℓ^{∞}]^{f} ⟶^{∼} M^{f} ⊗ Z_{ℓ}.

This concludes the proof of Lemma 3.1.5. □

We now return to the proof of Proposition 3.1.3. Lemma 3.1.4 and the finiteness of Pic^{0} (X_{1})_{T} (see the upper row of commutative diagram (27) and keep in mind that Pic^{0} (X_{1}) is finite) imply that Pic^{0} (X_{1})_{T} together with its endomorphism F_{∗,1}, and any prime number ℓ, satisfy the hypotheses in Lemma 3.1.5. We therefore have an isomorphism of Z_{ℓ} [G]–modules

T_{ℓ} (Pic^{0}(X)_{T} ) /(1 − F_{∗,1})T_{ℓ} (Pic^{0}(X)_{T} ) ⟶^{∼} Pic^{0} (X_{1})_{T} ⊗ Z_{ℓ},  (29)

for any prime number ℓ. The exact sequence (20) gives a commutative diagram of Z_{ℓ} [G]–modules with exact rows

0 ⟶ T_{ℓ}(coker(β_{T})) ⟶ T_{ℓ} (Pic^{0}(X)_{T} ) ⟶ T_{ℓ} (Pic^{0}(X)) ⟶ 0
(1−F_{∗,2})↓  ^{(1−F_{∗,1})_{1}}↓  ^{(1−F_{∗,1})_{2}}↓
0 ⟶ T_{ℓ}(coker(β_{T})) ⟶ T_{ℓ} (Pic^{0}(X)_{T} ) ⟶ T_{ℓ} (Pic^{0}(X)) ⟶ 0

The finiteness of Pic^{0}(X)_{T}^{F_{∗,1}} = Pic^{0} (X_{1})_{T} and Pic^{0}(X)^{F_{∗,1}} = Pic^{0} (X_{1}) respectively implies that the maps (1 − F_{∗,1})_{1} and (1 − F_{∗,1})_{2} are both injective. Snake lemma applied to the diagram above, and (29), give an exact sequence of Z_{ℓ} [G]– modules

0 ⟶ coker(1 − F_{∗,2}) ⟶ Pic^{0} (X_{1})_{T} ⊗ Z_{ℓ} ⟶ coker(1 − F_{∗,1})_{2} ⟶ 0

Property (e) §1.4 implies that

Fitt_{Z_{ℓ}[G]}(coker(1 − F_{∗,1})_{2}) · Fitt_{Z_{ℓ}[G]}(coker(1 − F_{∗,2})) ⊆
∈ Fitt_{Z_{ℓ}[G]} (Pic^{0} (X_{1})_{T} ⊗ Z_{ℓ}) .

(30)

On the other hand, both T_{ℓ}(coker(β_{T})) and T_{ℓ} (Pic^{0}(X)) are free Z_{ℓ}–modules, and therefore projective Z_{ℓ} [G]–modules (see Remark 1 §1.3). An argument similar to the one used in proving (17) shows that

det_{Z_{ℓ}[G]}(1 − F_{∗,2}|T_{ℓ}(coker(β_{T}))) ∈ Fitt_{Z_{ℓ}[G]}(coker(1 − F_{∗,2})) and
det_{Z_{ℓ}[G]}((1 − F_{∗,1})_{1}|T_{ℓ}(Pic^{0}(X))) ∈ Fitt_{Z_{ℓ}[G]}(coker(1 − F_{∗,1})_{2}).

(31)

According to (30) and (31) we can finally conclude that Proposition 3.1.3 is true. □

Proof of Theorem 3.1.1. Let us consider a prime ℓ, such that gcd(ℓ, g) = 1, and the exact sequence of Z_{ℓ} [G]–modules

\xymatrix{Pic^{0} (X_{1})_{T} ⊗ Z_{ℓ} \ar[r]^{ψ_{S,T} ⊗1_{Z_{ℓ}}} \ar[r] & A_{S,T} ⊗ Z_{ℓ} ⟶ coker(ψ_{S,T}) ⊗ Z_{ℓ} \ar[r] & 0}

(see diagram (21) for the definition of ψ_{S,T}). Property (e) §1.4 implies that

Fitt_{Z_{ℓ}[G]}(coker (ψ_{S,T}) ⊗ Z_{ℓ}) · Fitt_{Z_{ℓ}[G]}(Pic^{0} (X_{1})_{T} ⊗ Z_{ℓ}) ⊆

⊆ Fitt_{Z_{ℓ}[G]}(A_{S,T} ⊗ Z_{ℓ}). (32)

Step 1. Let us first suppose that ℓ ≠ p. Then Propositions 3.1.2 and 3.1.3, in addition to (32) above, and Lemma 1.7.4.4 (1) for s = 0 (and therefore u = 1), imply that

Θ_{S,T} (0) =det_{Z_{ℓ}[G]} (1 − F_{∗,0}|T_{ℓ}(ker α_{S})) · det_{Z_{ℓ}[G]}(1 − F_{∗,1}|H_{1} (X, Z_{ℓ}))·
·det_{Z_{ℓ}[G]}(1 − F_{∗,2}|T_{ℓ}(coker(β_{T}))) ∈ Fitt_{Z_{ℓ}[G]}(A_{S,T} ⊗ Z_{ℓ}).

Step 2. Let us suppose now that ℓ = p, and gcd(p, g) = 1. The first observation we have to make is that T_{p}(coker(β_{T})) = 1 and therefore Proposition 3.1.3 implies that

det_{Z_{p}[G]} (1 − F_{∗,1}| H_{1} (X, Z_{p})) ∈ Fitt_{Z_{p}[G]} (Pic^{0} (X_{1})_{T} ⊗ Z_{p}) .  (33)

Proposition 3.1.2 , (32) and (33) above, in addition to Lemma 1.7.4.4 (2) imply therefore that

Θ_{S,T} (0) =Q(1) · det_{Z_{p}[G]} (1 − F_{∗,0}|T_{p}(ker α_{S})) ·
·det_{Z_{p}[G]}((1 − F_{∗,1})_{1}|H_{1} (X, Z_{p})) ∈ Fitt_{Z_{p}[G]}(A_{S,T} ⊗ Z_{p}).

According to Remark 1, Step 1 and Step 2 above conclude the proof of Theorem 3.1.1. □

3.2. ARBITRARY r

The following theorem shows that, for general abelian extensios of function fields, Conjecture B is true, up to primes dividing the order of the Galois group.

Theorem 3.2.1. If the set of data (K/k, S, T, r) satisfies hypotheses (H), then (1) There exists a unique element ε_{S,T} ∈ Z [1/g] Λ_{S,T} satisfying

Θ_{S,T}^{(r)} (0) = R_{W} (ε_{S,T}).

(2) The element ε_{S,T} above satisfies

ε_{S,T} ∈ Z [1/g] Fitt (A_{S,T}) Λ_{S,T} .

Proof. Let {v_{1}, …, v_{r}} be the distinguished set of r distinct primes in S, which split completely in K/k, and let S_{0} = S \ {v_{1}, ⋯, v_{r}}. Then obviously the set of data (K/k, S_{0}, T, 0) satisfies hypotheses (H). Theorem 3.1.1 implies that

Θ_{S_{0},T} (0) ∈ Z [1/g] Fitt (A_{S_{0},T}) .

Corollary 2.2 (1), with S′ = S and S = S_{0}, implies therefore that the element ε_{S,T} satisfying Q · St (K/k, S, T, r), also satisfies ε_{S,T} ∈ Z [1/g] Fitt (A_{S,T}) Λ_{S,T}. This settles the proof of both (1) and (2) above, as the uniqueness of ε_{S,T} is a consequence of Remark 5, §1.6. □

Corollary 3.2.2 ( Raw form of Gras’ Conjecture ). The element ε_{S,T}, whose existence is proved in the Theorem 3.2.1, satisfies

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

Proof. This is a direct consequence of Theorem 3.2.1 (2), Proposition 2.3 and Lemma 1.5.3(1). □

In [13] it will become apparent that Corollary 3.2.2 implies Gras-type conjectures for function fields.

4. Proof of a stronger form of Conjecture B for constant field extensions

Througout this section K/k is a constant field extension of function fields of characteristic p.

4.1. PRELIMINARY CONSIDERATIONS Let F_{q} and F_{q^{ν}} be the exact fields of constants of k and K respectively. Then clearly G(K/k) ⟶^{∼} G(F_{q^{ν}} /F_{q}) is cyclic of order ν, with a distinguished generator σ, satisfying σ(ζ) = ζ^{q}, for every ζ in F_{q^{ν}}, and K/k is everywhere unramified. We remind the reader that for primes v in k, and w in K, such that w|v, σ_{v} denotes the Frobenius morphism of v relative to K/k, d_{v} = [F_{q}(v) : F_{q}] and d_{w} = [F_{q^{ν}} (w) : F_{q^{ν}}]. Simple arithmetic considerations show that σ_{v} = σ^{d_{v}}, that there are precisely r_{v} = gcd(d_{v}, ν) primes in K lying above v, and that d_{w} = d_{v}/r_{v}. (This implies in particular that v splits completely in K/k if and only if ν|d_{v}.)

Lemma 4.1.1. For any prime number ℓ, there are Z_{ℓ} [G]–isomorphisms

(1) H_{0} (X, Z_{ℓ}) ⟶^{∼} Z_{ℓ} [G], which sends the action of F_{∗,0} on H_{0} (X, Z_{ℓ}) into multiplication by σ^{−1} on Z_{ℓ} [G].

(2) H_{2} (X, Z_{ℓ}) ⟶^{∼} \{ \begin{matrix}Z_{ℓ} [G] , & if ℓ ≠ p \\ 0, & if ℓ = p, \end{matrix}

which sends the action of F_{∗,2} on H_{2} (X, Z_{ℓ}) into multiplication by qσ^{−1} on Z_{ℓ} [G].

(3) H_{1} (X, Z_{ℓ}) ⟶^{∼} \{ \begin{matrix}Z_{ℓ} [G]^{2g_{X}} , & if ℓ ≠ p, \\ Z_{p} [G]^{n} , & if ℓ = p , \end{matrix}

for some integer n, 0 ≤ n < 2g_{X}.

Proof. Let us first notice that

F_{∗,0} ([X^{(i)}]) = σ_{∗,0}^{−1} ([X^{(i)}]) , for every 0 ≤ i ≤ ν − 1.  (34)

(1) We know that H_{0} (X, Z) is Z–free of basis {[X^{(i)}] : 0 ≤ i ≤ ν − 1}, that G is cyclic of order ν and it permutes the elements [X^{(i)}] transitively. We have also chosen the indices i so that [X^{(i)}] = F_{∗,0}^{i} ([X^{(0)}]). (34) implies therefore that we can define an isomorphism of Z [G]–modules

H_{0} (X, Z) ⟶^{∼}^{ρ} Z [G] ,  (35)

given by ρ ([X^{(i)}]) = σ^{−i}, ∀0 ≤ i ≤ ν − 1. Obviously, under this isomorphism, the action of F_{∗,0} on the left corresponds to multiplication by σ^{−1} on the right. Statement (1) in the lemma can now be obtained by tensoring (35) with Z_{ℓ}.

(2) We already know (see §1.7.3) that there is an isomorphism

H_{2} (X, Z_{ℓ}) ⟶^{∼}^{f} H_{0} (X, Z_{ℓ})

which takes the action of F_{∗,2} on the right hand–side into the action of qF_{∗,0} on the left hand–side. The map (ρ ⊗ 1_{Z_{ℓ}}) ∘ f gives therefore a Z_{ℓ} [G]–isomorphism satisfying the requirements in (2).

(3) The fact that |G| = ν and that G permutes transitively and isomorphically the ν irreducible components X^{(i)} of X, and therefore their corresponding Picard groups Pic^{0}(X^{(i)}), implies that there is an isomorphism of Z [G]–modules

H_{1} (X, Z) = ⨁_{0≤i≤ν−1} Pic^{0}(X^{(i)}) ⟶^{∼} Z [G] ⨂_{Z} Pic^{0}(X^{(0)}) ,

with G acting trivially on Pic^{0}(X^{(0)}). Passing to ℓ-adic Tate modules, one obtains the desired Z_{ℓ} [G]–isomorphisms. □

4.2. THE r = 0 CASE

Throughout this section we will assume that the set of data (K/k, S, T, 0) satisfies hypotheses (H).

Lemma 4.2.1. If u = q^{−s}, we have the following:

(1) For any prime number ℓ ≠ p

Θ_{S,T} (u) = \frac{∏_{v∈S}(1 − (σ^{−1}u)^{d_{v}})}{(1 − σ^{−1}u)} · \frac{∏_{v∈T}(1 − (qσ^{−1}u)^{d_{v}})}{(1 − (qσ^{−1}u))} ·det_{Z_{ℓ}[G]}(1−F_{∗,1}·u| H_{1} (X, Z_{ℓ})).

(2) There exists a polynomial Q(u) ∈ Z_{p} [G] [u], such that

\begin{matrix}Θ_{S,T} (u) = & Q(u) · ^{\frac{∏_{v∈S}(1−(σ^{−1}u)^{dv})}{(1−σ^{−1}u)}} · ^{\frac{∏_{v∈T}(1−(qσ^{−1}u)^{dv})}{(1−(qσ^{−1}u))}}· \\ ·det_{Z_{p}[G]}(1 − F_{∗,1} · u| H_{1} (X, Z_{p})). \end{matrix}

Proof. (1) Let ℓ ≠ p be a prime number. The freeness of the H_{i} (X, Z_{ℓ})’s as Z_{ℓ} [G]–modules implied by Lemma 4.1.1, and Theorem 1.7.4.2, show that

Θ(s) = ∏_{0≤i≤2} det_{Z_{ℓ}[G]}(1 − F_{∗,i} · u| H_{i} (X, Z_{ℓ}))^{(−1)^{i+1}},  (36)

for any prime number ℓ ≠ p. Lemma 4.1.1 (1) and (2) and equality (36) give

Θ(s) = \frac{det_{Z_{ℓ}[G]}(1 − F_{∗,1} · u| H_{1} (X, Z_{ℓ}))}{(1 − σ^{−1}u) · (1 − qσ^{−1}u)}.  (37)

On the other hand, since K/k is unramified everywhere, we have

Θ_{S,T} (s) = ∏_{v∈S}(1 − σ_{v}^{−1}Nv^{−s}) · ∏_{v∈T}(1 − σ_{v}^{−1}Nv^{1−s}) · Θ(s).

This equality and (37) give (1) in Lemma 4.2.1.

(2) This follows directly from (1) and Theorem 1.7.4.1 (3), with i = 1 and Q(u) = Q_{1}(u). □

Proposition 4.2.2. Under the assumptions above, we have the following: (1) Θ_{S,T} (0) ∈ Fitt_{Z[G]}(A_{S,T}) ∩ Z [G]_{0,S}.

(2) If ℓ is a prime number, such that gcd(ℓ, g) = 1, or gcd(ℓ, d_{w}) = 1, for some w ∈ S_{K}, then Θ_{S,T} (0) ∈ Fitt_{Z_{ℓ}[G]}(A_{S,T} ⊗ Z_{ℓ}) · Z_{ℓ} [G]_{0,S}.

Proof. Let us fix a prime v_{0} ∈ S and let S_{0} = {v_{0}}. Obviously the set of data (K/k, S, T, 0) still satisfies hypotheses (H). (Recall that K/k is everywhere unramified.) There is a surjective morphism of Z [G]–modules A_{S_{0},T} ⟶ A_{S,T} (see (9)), which gives a Z_{ℓ} [G]–surjection A_{S_{0},T} ⊗ Z_{ℓ} ⟶ A_{S,T} ⊗ Z_{ℓ}, for every ℓ as above. These imply that

Fitt_{Z[G]}(A_{S_{0},T}) ⊆ Fitt_{Z[G]}(A_{S,T}) , Fitt_{Z_{ℓ}[G]} (A_{S_{0},T} ⊗ Z_{ℓ}) ⊆ Fitt_{Z_{ℓ}[G]} (A_{S,T} ⊗ Z_{ℓ})

(see §1.4(f)). These relations together with

Θ_{S,T} (0) = ∏_{v∈S\{v_{0}}}(1−σ_{v}^{−1})·Θ_{S_{0},T} (0), and ∏_{v∈S\{v_{0}}}(1−σ_{v}^{−1})·Z [G]_{0,S_{0}} ⊆ Z [G]_{0,S} ,

imply that it is enough to prove the proposition above in the case |S| = 1. From now on we will assume that S = {v_{0}}, and we will fix a prime w_{0} ∈ S_{K}. We will need three lemmas.

Lemma 4.2.3. Let us consider the Z [G]–module Z/d_{w_{0}}Z with trivial G–action. Then

\frac{1 − σ^{−d_{w_{0}}}}{1 − σ^{−1}} ∈ Fitt_{Z[G]} (Z/d_{w_{0}}Z) .

Proof. Since G acts trivially on Z/d_{w_{0}}Z, \frac{1−σ^{−dw_{0}}}{1−σ^{−1}} ∈ Ann_{Z[G]}(Z/d_{w_{0}}Z). On the other hand, Z/d_{w_{0}}Z is a cyclic Z [G]–module, and therefore §1.4(a) shows that

Fitt_{Z[G]} (Z/d_{w_{0}}Z) = Ann_{Z[G]}(Z/d_{w_{0}}Z).

□

Lemma 4.2.4. For any v ∈ T, the following hold true:

(1) (1 − (qσ^{−1})^{d_{v}}) ∈ Fitt_{Z[G]}(⨁_{w|v} F_{q^{ν}} (w)^{×}).

(2) \frac{(1−(qσ^{−1})^{dv})}{1−qσ^{−1}} ∈ Fitt_{Z[G]} ((⨁_{w|v} F_{q^{ν}} (w)^{×})/F_{q^{ν}}^{×} ) ,

where F_{q^{ν}}^{×} is embeded in ⨁_{w|v} F_{q^{ν}} (w)^{×} diagonally ( x ⟶ ⨁_{w|v}(x mod w) ).

Proof. Let us fix a prime w_{0} in K, such that w_{0}|v. There is an isomorphism of Z [G]–modules

⨁_{w|v} F_{q^{ν}} (w)^{×} ⟶^{∼} F_{q^{ν}} (w_{0})^{×} ⨂_{Z[G_{v}]} Z [G] ,

where G_{v} is the decomposition group of v relative to K/k. Since F_{q^{ν}} (w_{0})^{×} is a cyclic Z[G_{v}]–module, both ⨁_{w|v} F_{q^{ν}} (w)^{×} and (⨁_{w|v} F_{q^{ν}} (w)^{×})/F_{q^{ν}}^{×} are Z [G]–cyclic and

therefore their Fitting ideals over Z [G] are equal to their Z [G]–annihilators (see §1.4(a)).

(1) The equality σ_{v} = σ^{d_{v}} implies that

1 − (qσ^{−1})^{d_{v}} ∈ Ann_{Z[G_{v}]}(F_{q^{ν}} (w)^{×}) ⊆ Ann_{Z[G]}(⨁_{w|v} F_{q^{ν}} (w)^{×}).

(2) Let I = {ν, ν − 1, …, ν − r_{v} + 1}. Then {σ^{i}| i ∈ I} is a set of coset representatives for the quotient G/G_{v} and {w_{0}^{σ^{i}}| i ∈ I} = {w| w in K, w|v}. Let

j_{w} : F_{q^{ν}}^{×} ⟶ F_{q^{ν}} (w)^{×},

defined by j_{w}(x) = x mod w, for every w|v.

One can easily show that

j_{w_{0}^{σ^{i}}} (x) = σ^{i}[j_{w_{0}}(x^{q^{ν−i}})], ∀ i ∈ I, ∀x ∈ F_{q^{ν}}^{×} .  (38)

We will identify F_{q^{ν}}^{×} with a subgroup of F_{q^{ν}} (w_{0})^{×}, via j_{w_{0}}. If ζ is a generator of F_{q^{ν}} (w_{0})^{×} then, under this identification,

F_{q^{ν}}^{×} = ⟨ζ^{α}⟩, where α = \frac{q^{νd_{v}/r_{v}} − 1}{q^{ν} − 1}.  (39)

Relations (38) and (39) imply that the image of the composition of maps

J : F_{q^{ν}}^{×} ⟶^{⨁_{w|v} j_{w}} ⨁_{w|v} F_{q^{ν}} (w)^{×} ⟶^{∼} F_{q^{ν}} (w_{0})^{×} ⨂_{Z[G_{v}]} Z [G] ⟶^{∼} ⨁_{i∈I} σ^{i} · F_{q^{ν}} (w_{0})^{×}

satisfies the equality

Im(J) = {∑_{i∈I} σ^{i} · ζ^{α·q^{ν−i}·n}| n ∈ Z/(q^{ν} − 1)Z}.  (40)

Let us also observe that ⨁_{i∈I} σ^{i} · F_{q^{ν}} (w_{0})^{×} is generated over Z [G] by the element \overline{ζ}

having ζ = σ^{ν}ζ in the component corresponding to i = ν, and 0 in all the other components.

Let us consider the “q^{d_{v}}–evaluation map”:

ϕ_{q} : Z[G_{v}] ⟶ Z/(q^{d_{v}ν/r_{v}} − 1)Z,

defined by ϕ_{q}( ∑_{j≥0} a_{j}σ_{v}^{j}) = ∑_{j≥0} a_{j}(q^{d_{v}})^{j}. The equality σ^{d_{v}}(ζ) = ζ^{q^{dv}} shows that if

\overline{α} = ∑_{i∈I} σ^{i} · α_{i} ∈ Z [G], with α_{i} ∈ Z [G_{v}], ∀i ∈ I, then \overline{α} · \overline{ζ} = ∑_{i∈I} σ^{i} · ζ^{ϕ_{q}(α_{i})} . In

light of these observations, equality (40) shows that:

Ann_{Z[G]} ((⨁_{w|v} F_{q^{ν}} (w)^{×})/F_{q^{ν}}^{×} ) = Ann_{Z[G]}(\overline{ζ}) =

= {∑_{i∈I} σ^{i} · α_{i}| α_{i} ∈ Z[G_{v}], ∃n ∈ Z, ϕ_{q}(α_{i}) ≡ n · αq^{ν−i} mod (q^{νd_{v}/r_{v}} − 1), ∀ i ∈ I}.

(41)

One can write

\frac{1 − (qσ^{−1})^{d_{v}}}{1 − qσ^{−1}} = ∑_{i∈I} σ^{i} · [q^{ν−i} · A],

where A = \frac{1−(qσ^{−1})^{dv}}{1−(qσ^{−1})^{rv}}. According to (41), in order to prove statement (2) in our lemma, we would simply have to show that α|ϕ_{q}(A) in Z/(q^{d_{v}ν/r_{v}} − 1)Z. This is an easy consequence of the fact that gcd(d_{v}/r_{v}, ν/r_{v}) = 1 (see comments at the beginning of §4.1). □

Lemma 4.2.5. Let ℓ be any prime number. Then

det_{Z_{ℓ}[G]}(1 − F_{∗,1}| H_{1} (X, Z_{ℓ})) ∈ Fitt_{Z_{ℓ}[G]} (Pic^{0} (X_{1}) ⊗ Z_{ℓ}) .

Proof. Lemma 3.1.5 applied to M = Pic^{0}(X) and f = F_{∗,1} gives an exact sequence of Z_{ℓ} [G]–modules

0 ⟶ T_{ℓ}(Pic^{0}(X)) ⟶^{1−F_{∗,1}} T_{ℓ}(Pic^{0}(X)) ⟶ Pic^{0} (X_{1}) ⊗ Z_{ℓ} ⟶ 0.  (42)

(take into account that Pic^{0}(X)^{F_{∗,1}} = Pic^{0} (X_{1})). Lemma 4.1.1 (3) shows that T_{ℓ}(Pic^{0}(X)) = H_{1} (X, Z_{ℓ}) is a free Z_{ℓ} [G]–module, for every prime number ℓ. The exact sequence (42) gives therefore a free Z_{ℓ} [G]–resolution of Pic^{0} (X_{1}) ⊗ Z_{ℓ}. Lemma 4.2.5 is now a direct consequence of the definition of the Fitting ideal. □

We are now ready to conclude the proof of Proposition 4.2.2.

(1) By definition, Θ_{S,T} (0) ∈ Z [G]_{0,S}, and therefore all we have to prove is that Θ_{S,T} (0) ∈ Fitt_{Z[G]}(A_{S,T}) which, according to §1.4(c), is equivalent to

Θ_{S,T} (0) ∈ Fitt_{Z_{ℓ}[G]}(A_{S,T} ⊗ Z_{ℓ}), ∀ℓ prime .

Let us fix a prime number ℓ. We will refer again to commutative diagram (11) . With the same notations, we have an exact sequence of Z [G]–modules

Pic^{0} (X_{1}) ⟶^{ψ_{S}} A_{S} ⟶ coker(ψ_{S}) ⟶ 0,  (43)

and coker(ϕ_{S}) ⟶^{∼} coker(ψ_{S}) as Z [G]–modules. The only difference is that, in our situation, due to the fact that S = {v_{0}}, d_{w} = d_{w_{0}}, for any w ∈ S_{K}. We therefore have a Z [G]–isomorphism coker(ϕ_{S}) ⟶^{∼} Z/d_{w_{0}}Z, which gives a Z [G]–isomorphism

coker(ψ_{S}) ⟶^{∼} Z/d_{w_{0}}Z,  (44)

with G acting trivially on Z/d_{w_{0}}Z. Lemmas 4.2.3 and 4.2.5, in addition to exact sequence (43) and property (e), §1.4, imply that

\frac{1 − σ^{−d_{w_{0}}}}{1 − σ^{−1}} · det_{Z_{ℓ}[G]}(1 − F_{∗,1}| H_{1} (X, Z_{ℓ})) ∈ Fitt_{Z_{ℓ}[G]}(A_{S} ⊗ Z_{ℓ}).  (45)

Let us fix a prime v_{T} ∈ T and let T_{0} = {v_{T} }. We have the following exact sequences of Z [G]–modules:

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

( ⨁_{w|v_{T}} F_{q^{ν}} (w)^{×})/F_{q^{ν}}^{×} ⟶ ( ⨁_{w|v_{T}} F_{q^{ν}} (w)^{×})/(U_{S}/U_{S,T_{0}}) ⟶ 0.

(The first exact sequence above is another way of writing (1), with T = T_{0}, and the second one is a consequence of the obvious injection F_{q^{ν}}^{×} ↪ U_{S}/U_{S,T_{0}}.) These exact sequences combined with (45), Lemma 4.2.4 (2) and §1.4(e), imply that

\frac{1 − σ^{−d_{w_{0}}}}{1 − σ^{−1}} · \frac{1 − (qσ^{−1})^{d_{v_{T}}}}{1 − qσ^{−1}} · det_{Z_{ℓ}[G]}(1 − F_{∗,1}| H_{1} (X, Z_{ℓ})) ∈ Fitt_{Z_{ℓ}[G]}(A_{S,T_{0}} ⊗ Z_{ℓ}).

(46)

And finally, in order to pass from T_{0} to T, let us consider the following exact sequences of Z [G]–modules

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

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

These exact sequences, combined with (46), Lemma 4.2.4 (1) and §1.4(e), imply

\frac{1 − σ^{−d_{w_{0}}}}{1 − σ^{−1}} · \frac{∏_{v∈T} 1 − (qσ^{−1})^{d_{v}}}{1 − qσ^{−1}} · det_{Z_{ℓ}[G]}(1 − F_{∗,1}| H_{1} (X, Z_{ℓ})) ∈ Fitt_{Z_{ℓ}[G]}(A_{S,T} ⊗ Z_{ℓ}).

Let us denote by L the left hand–side of (47). Lemma 4.2.1, with s = 0 (and therefore u = 1), implies that

Θ_{S,T} (0) = \{ \begin{matrix}\frac{1−σ^{−dv}}{1−σ^{−dw_{0}}} · L, & if ℓ ≠ p, \\ Q(1) · \frac{1−σ^{−dv}}{1−σ^{−dw_{0}}} · L, & if ℓ = p. \end{matrix}  (48)

Since d_{w_{0}} = d_{v}/r_{v} and Q(1) ∈ Z_{p} [G], (47) together with (48) show that

Θ_{S,T} (0) ∈ Fitt_{Z_{ℓ}[G]}(A_{S,T} ⊗ Z_{ℓ}),

for every prime number ℓ. This, as we remarked earlier, concludes the proof of Proposition 4.2.2 (1).

(2) Let ℓ be a prime number as in Proposition 4.2.2 (2). Let us first notice that, if gcd(ℓ, g) = 1, then (2) is a direct consequence of (1). Indeed, in this situation

Z_{ℓ} [G] = ⨁_{ψ∈Ĝ(Q_{ℓ})} D_{ψ}, Z_{ℓ} [G]_{0,S} = ⨁_{ψ∈Ĝ(Q_{ℓ},0)} D_{ψ},

which, together with Fitt_{Z_{ℓ}[G]}(A_{S,T} ⊗ Z_{ℓ}) = ⊕_{ψ}Fitt_{D_{ψ}}(A_{S,T} ⊗ D_{ψ}), imply that

Fitt_{Z_{ℓ}[G]}(A_{S,T} ⊗ Z_{ℓ}) · Z_{ℓ} [G]_{0,S} = Fitt_{Z_{ℓ}[G]}(A_{S,T} ⊗ Z_{ℓ}) ∩ Z_{ℓ} [G]_{0,S} .

Let us then suppose that ℓ|g and gcd(ℓ, d_{w_{0}}) = 1. With notations as in the proof of (1), this implies that

coker(ψ_{S}) ⊗ Z_{ℓ} = 0.  (49)

The exact sequence (43) and Lemma 4.2.5 imply that (45) can be rewritten as

det_{Z_{ℓ}[G]}(1 − F_{∗,1}| H_{1} (X, Z_{ℓ})) ∈ Fitt_{Z_{ℓ}[G]}(A_{S} ⊗ Z_{ℓ}),

which shows that in (45–47) above, the factor \frac{1−σ^{−dw_{0}}}{1−σ^{−1}} is not needed anymore. (47) and (48) therefore imply that

Θ_{S,T} (0) ∈ \frac{1 − σ^{−d_{v}}}{1 − σ^{−1}} · Fitt_{Z_{ℓ}[G]}(A_{S,T} ⊗ Z_{ℓ}).  (50)

On the other hand, \frac{1−σ^{−dv}}{1−σ^{−1}} ∈ Z [G]_{0,S}. Indeed, if χ ∈ Ĝ(C) \ Ĝ(C, 0), then χ ≠ 1_{G} and χ(σ_{v}) = 1 (see (3) and take into account that |S| = 1.) This implies that χ(\frac{1−σ^{−dv}}{1−σ^{−1}} ) = 0, which proves our assertion. This fact together with (50) imply Proposition 4.2.2 (2). □

Our next goal is the proof of the following:

Proposition 4.2.6. Let ℓ be a prime number, such that ℓ | g, and ℓ | d_{w}, for every w ∈ S_{K}. Then

Θ_{S,T} (0) ∈ Fitt_{Z_{ℓ}[G]}(A_{S,T} ⊗ Z_{ℓ}) · Z_{ℓ} [G]_{0,S} .

An argument similar to the one given at the begining of the proof of Proposition 4.2.2 shows that it is enough to prove this statement in the case |S| = 1. We therefore assume that S = {v_{0}}, and fix w_{0} ∈ S_{K}. In what follows, C_{ℓ} denotes a completion of the algebraic closure of Q_{ℓ} with respect to the ℓ–adic valuation. W denotes a finite ring-extension of Z_{ℓ}, inside C_{ℓ}, containing the values of all χ ∈ Ĝ(C_{ℓ}). We will extend scalars to W and prove Proposition 4.2.6 inside W [G].

Let ℓ^{m} be the exact power of ℓ dividing g = |G|. Let L be the ℓ–Sylow subgroup of G. There is a decomposition G = Δ × L, where Δ is the maximal subgroup of G, satisfying gcd(ℓ, |Δ|) = 1. Let K_{L} and K_{Δ} be the maximal subfields of K fixed by L and Δ respectively. Let \overline{L} = G(K_{Δ}/k) and \overline{Δ} = G(K_{L}/k). Restriction maps give isomorphisms Δ ⟶^{∼} \overline{Δ} and L ⟶^{∼} \overline{L}.

Remark 1. Due to d_{w_{0}} = d_{v}/gcd(d_{v}, g), the fact that ℓ | d_{w_{0}} implies that ℓ^{m} | d_{v}, and therefore v splits completely in K_{Δ} (see §4.1). This implies that σ_{v} ∈ Δ. Let us consider the decomposition Ĝ(C_{ℓ}) = Δ̂(C_{ℓ}) × L̂(C_{ℓ}) and let χ, δ and λ denote generic elements of Ĝ(C_{ℓ}), Δ̂(C_{ℓ}) and L̂(C_{ℓ}) respectively. We can therefore conclude that a character χ = (δ, λ) belongs to Ĝ(C_{ℓ}, 0) if and only if δ(σ_{v}) ≠ 1, or χ = 1_{G} (i.e. δ = 1_{Δ} and λ = 1_{L}) (see (3)).

Since gcd(ℓ, |Δ|) = 1, we can decompose the group ring W [G] into its δ– components:

W [G] = W [Δ][L] ⟶^{∼}^{⨁_{δ∈_{Δ}(C_{ℓ})} δ} ⨁_{δ∈Δ̂(C_{ℓ})} W_{δ}[L],

where W_{δ} = W and δ( ∑_{σ∈Δ,τ∈L} a_{σ,τ} στ) = ∑_{τ∈L}( ∑_{σ∈Δ} a_{σ,τ} δ(σ))·τ, for every δ ∈ Δ̂(C_{ℓ}).

According to Remark 1 above, this induces an isomorphism

W [G]_{0,S} ⟶^{∼} ( ⨁_{δ(σ_{v})≠1} W_{δ}[L]) ⨁ W_{1_{Δ}} · N_{L},

where N_{L} = ∑_{σ∈L} σ. We thus have a W [G]–isomorphism

W [G]_{0,S} · Fitt_{W [G]}(A_{S,T} ⊗ W) ⟶^{∼}( ⨁_{δ(σ_{v})≠1} Fitt_{W_{δ}[L]}(A_{S,T} ⊗ W)^{δ}) ⨁
⨁ Fitt_{W [L]}(A_{S,T} ⊗ W)^{Δ} · N_{L}.

This implies that, in order to prove

Θ_{S,T} (0) ∈ W [G]_{0,S} · Fitt_{W [G]}(A_{S,T} ⊗ W),

it is necessary and sufficient to show the following:

(i) If δ(σ_{v}) = 1, δ ≠ 1_{Δ}, then δ(Θ_{S,T} (0)) = 0.

(ii) If δ(σ_{v}) ≠ 1, then δ(Θ_{S,T} (0)) ∈ Fitt_{W_{δ}[L]}(A_{S,T} ⊗ W)^{δ}.

(iii) 1_{Δ}(Θ_{S,T} (0)) ∈ Fitt_{W [L]}(A_{S,T} ⊗ W)^{Δ} · N_{L}.

Assertions (i) and (ii) above are direct consequences of Proposition 4.2.2 (1), after extending scalars to W. Therefore, in order to prove Proposition 4.2.6, one has to prove (iii) above.

Proof of Prop. 4.2.6. In the following arguments, if F is an intermediate field of K/k, then A_{F,S,T} denotes its (S, T)–ideal class group, as defined in §1.1.

The fact that Θ_{S,T} (s) ∈ W [G]_{0,S}, combined with the class-number formula (2), implies that

1_{Δ}(Θ_{S,T} (0)) = ζ_{k,S,T} (0) · \frac{N_{L}}{|L|} = \frac{|A_{k,S,T} |}{|L|} · N_{L} .

Via the isomorphism W [L] ⟶^{∼} W [\overline{L}], we can therefore say that, inside W [\overline{L}],

1_{Δ}(Θ_{S,T} (0)) = \frac{|A_{k,S,T} |}{|\overline{L}|} · N\overline{L},  (51)

where N\overline{L} = ∑_{σ∈\overline{L}} σ. The fact that gcd(ℓ, |Δ|) = 1, combined with a standard

class-field theoretical argument based on the class-field interpretation of the groups A_{F,S,T} (see §1.1), implies that we have an isomorphism of W [\overline{L}]–modules

(A_{S,T} ⊗ W)^{Δ} ⟶^{∼} A_{K_{Δ},S,T} ⊗ W.

This obviously implies that

Fitt_{W [\overline{L}]}(A_{S,T} ⊗ W)^{Δ} = Fitt_{W [\overline{L}]}(A_{K_{Δ},S,T} ⊗ W).  (52)

Equalities (51) and (52) above show that (iii) would be a consequence (after extending scalars to W) of the following

Lemma 4.2.7. With notations as above,

\frac{|A_{k,S,T} |}{|\overline{L}|} · N\overline{L} ∈ Fitt_{Z[\overline{L}]}(A_{K_{Δ},S,T}) · N\overline{L}.

Proof of Lemma 4.2.7. Let us consider the exact sequence

0 ⟶ I\overline{L} ⟶ Z[\overline{L}] ⟶^{s} Z ⟶ 0,

where s( ∑_{σ∈\overline{L}} a_{σ} · σ) = ∑_{σ∈\overline{L}} a_{σ}, for every ∑_{σ∈\overline{L}} a_{σ} · σ ∈ Z[\overline{L}], and I\overline{L} = ker(s).

Obviously I\overline{L} = Ann_{Z[\overline{L}]}(N\overline{L}), and therefore the statement in Lemma 4.2.7 is equivalent to

\frac{|A_{k,S,T} |}{|\overline{L}|} ∈ s(Fitt_{Z[\overline{L}]}(A_{K_{Δ},S,T})).

On the other hand, properties (b) and (f), §1.4 imply that

s(Fitt_{Z[\overline{L}]}(A_{K_{Δ},S,T})) = Fitt_{Z}(A_{K_{Δ},S,T} /I\overline{L}A_{K_{Δ},S,T}) = |A_{K_{Δ},S,T} /I\overline{L}A_{K_{Δ},S,T} | · Z.

The statement in Lemma 4.2.7 is therefore equivalent to

|A_{K_{Δ},S,T} /I\overline{L}A_{K_{Δ},S,T} | | \frac{|A_{k,S,T} |}{|\overline{L}|} .

The next lemma shows that something even stronger holds true.

Lemma 4.2.8. With notations as above,

|A_{K_{Δ},S,T} /I\overline{L}A_{K_{Δ},S,T} | = \frac{|A_{k,S,T} |}{|\overline{L}|}.

Proof of Lemma 4.2.8. Let K_{Δ,S,T} and k_{S,T} be the (S, T)–ray class-fields of K_{Δ} and k respectively. We remind the reader that these are the maximal abelian extensions of K_{Δ} and k respectively, which are tamely ramified at primes above T and unramified outside T, and in which all primes in S split completely (see classfield theoretical interpretation of A_{S,T} in §1.1). The corresponding Artin reciprocity maps give the following group isomorphisms:

A_{K_{Δ},S,T} ⟶^{∼} G(K_{Δ,S,T} /K_{Δ})  (53)
A_{k,S,T} ⟶^{∼} G(k_{S,T} /k).  (54)

The maximality of K_{Δ,S,T}, the fact that v_{0} splits completely in K_{Δ} (see Remark 1), and that K_{Δ}/k is everywhere unramified, imply that K_{Δ} ⊆ k_{S,T} ⊆ K_{Δ,S,T} and that K_{Δ,S,T} /k is Galois. Let G = G(K_{Δ,S,T} /k). We have an exact sequence of groups

1 ⟶ G(K_{Δ,S,T} /K_{Δ}) ⟶ G ⟶ \overline{L} ⟶ 1.  (55)

Since G(K_{Δ,S,T} /K_{Δ}) is abelian, \overline{L} acts on it by (lift and) conjugation and this way (53) becomes an isomorphism of Z[\overline{L}]–modules.

Let [G, G] denote the group theoretical commutator of G. The maximality of k_{S,T} implies that

[G, G] = G(K_{Δ,S,T} /k_{S,T}).  (56)

An easy group theoretical argument, based on the exact sequence (55) and on the fact that \overline{L} is cyclic, shows that [G, G] is generated by elements of the form γ^{σ−1}, with γ ∈ G(K_{Δ,S,T} /K_{Δ}) and σ ∈ \overline{L}. On the other hand, the ideal I\overline{L} is generated over Z by elements of the form σ − 1. This observation and (56) imply that, under the Artin isomorphism (53), we have

I\overline{L} · A_{K_{Δ},S,T} ⟶^{∼} G(K_{Δ,S,T} /k_{S,T}).

We therefore have the equalities

|A_{K_{Δ},S,T} /I\overline{L}A_{K_{Δ},S,T} | = \frac{|G(K_{Δ,S,T} /K_{Δ})|}{|G(K_{Δ,S,T} /k_{S,T})|} = \frac{|G(k_{S,T} /k)|}{|\overline{L}|} = \frac{|A_{k,S,T} |}{|\overline{L}|}.

This concludes the proof of Lemma 4.2.8, Lemma 4.2.7 and Proposition 4.2.6. □

Theorem 4.2.9. Let K/k be a finite constant field extension of function fields, and let us suppose that the set of data (K/k, S, T, 0) satisfies hypotheses (H). Then

Θ_{S,T} (0) ∈ Fitt_{Z[G]}(A_{S,T}) · Z [G]_{0,S} .

Proof. This statement is obviously equivalent to

Θ_{S,T} (0) ∈ Fitt_{Z_{ℓ}[G]}(A_{S,T} ⊗ Z_{ℓ}) · Z_{ℓ} [G]_{0,S} , ∀ ℓ prime,

which follows from Propositions 4.2.2(2) and 4.2.6 above. □

4.3. ARBITRARY r

Theorem 4.3.1. Let K/k be a finite constant field extension of function fields, and let us suppose that the set of data (K/k, S, T, r) satisfies hypotheses (H). Then there exists a unique element

ε_{S,T} ∈ Z [G]_{r,S} · (∧^{r} U_{S,T}),

such that Θ_{S,T} (0) = R_{W} (ε_{S,T}).

Proof. Let {v_{1}, ⋯, v_{r}} be the distinguished set of r distinct primes in S, which split completely in K/k. Let S_{0} = S \ {v_{1}, ⋯, v_{r}}. The set of data (K/k, S_{0}, T, 0) satisfies hypotheses (H) as well. Theorem 4.2.9 therefore implies that

Θ_{S_{0},T} (0) ∈ Fitt_{Z[G]}(A_{S_{0},T}) · Z [G]_{0,S_{0}} .

The statement in Theorem 4.3.1 is now a direct consequence of the relation above, and Corollary 2.2 (2), with S′ = S and S = S_{0}. □

Remark. The inclusion

Z [G]_{r,S} · (∧^{r} U_{S,T}) ⊆ Λ_{S,T} ,

shows that Theorem 4.2.9 settles Conjecture B, for arbitrary r, under the assumption that K/k is a constant field extension. The inclusion above is strict in most cases, and therefore Theorem 4.2.9 provides a refinement of Conjecture B, under the present assumptions.

Appendix

The goal of this appendix is the proof of Theorem 1.7.4.1. With notations as in §1.7, let us recall that, for any prime number ℓ, and any i = 0, 1, 2,

P_{i,ℓ}(u) =^{def} det_{Q_{ℓ}[G]} (1 − F_{∗,i} · u| H_{i} (X, Q_{ℓ})) ,

where H_{i} (X, Q_{ℓ}) is the i–th ℓ–adic homology group with coefficients in Q_{ℓ}, defined in §1.7. The statement we would like to prove is the following:

Theorem. For any i = 0, 1, 2 :

(1) P_{i,p}(u) ∈ Z_{p} [1/g] [G] [u], and if ℓ ≠ p, then P_{i,ℓ} ∈ Z [1/g] [G] [u].

(2) If ℓ ≠ p, then P_{i,ℓ}(u) does not depend on ℓ.

Let P_{i}(u) =^{def} P_{i,ℓ}(u), for any ℓ ≠ p. Then:

(3) There exist polynomials Q_{i}(u) ∈ Z_{p} [1/g] [G] [u] such that

P_{i}(u) = P_{i,p}(u) · Q_{i}(u).

(4) If K/k is a constant field extension, then Q_{i}(u) ∈ Z_{p} [G] [u].

Before proceeding to the proof of the Theorem above, we need to summarize a few considerations made in [17], Chpt.V, on the Stickelberger function Θ(s).

Let |Y_{1}| be the set of closed points of the F_{q}–scheme Y_{1}. These are in one–to–one correspondence with the primes v of the base field k. For every v ∈ |Y_{1}|, let I_{v} be its innertia group in G, let \overline{σ_{v}} ∈ G_{v}/I_{v} be its Frobenius class, and let

F_{v} =^{def} \frac{1}{|I_{v}|} ∑_{τ∈\overline{σ_{v}}} τ^{−1} ∈ Z [1/g] [G] .

Then, if u = q^{−s}, we have the following equality

Θ(u) = ∏_{v∈|Y_{1}|} (1 − F_{v} · u^{−degv}) ∈ 1 + uZ [1/g] [G] [[u]] ,  (a.1)

valid for |u| < q^{−1} (which corresponds to Re(s) > 1).

Remark 1. Let us observe that, if K/k is a constant field extension, then |I_{v}| = 1, and therefore F_{v} ∈ Z [G], for every v ∈ |Y_{1}|. Equality (a.1) therefore shows that, under the present hypothesis,

Θ(u) ∈ Z [G] [[u]].  (a.2)

For every v ∈ |Y_{1}|, let X_{v} be the finite set of closed points of the F–scheme X above v, and let H_{0} (X_{v}, Q) be the Q–vector space generated by X_{v}, with the obvious G–action and q–power geometric Frobenius morphism action F_{∗}. Lemma 2.1 in [17], Chpt.V, shows that

1 − F_{v}u^{−degv} = det_{Q[G]} (1 − F_{∗} · u| H_{0} (X_{v}, Q)) , ∀v ∈ |Y_{1}| .

If we combine these equalities with (a.1), we obtain

Θ(u) = ∏_{v∈|Y_{1}|} det_{Q[G]} (1 − F_{∗} · u| H_{0} (X_{v}, Q)) .  (a.3)

We are now prepared to prove (1) and (2) in the Theorem above.

Proof of (1). Since H_{i} (X, Z_{p}) is a free Z_{p}–module,

H_{i} (X, Z_{p} [1/g]) = H_{i} (X, Z_{p}) ⊗ Z_{p} [1/g]

is a projective Z_{p} [1/g] [G]–module (see Remark 1, §1.3), and therefore

P_{i,p}(u) = det_{Z_{p}[1/g][G]} (1 − F_{∗,i}| H_{i} (X, Z_{p} [1/g])) ∈ Z_{p} [1/g] [G] [u] .

Let ℓ ≠ p be a prime number. Since H_{0} (X, Z [1/g]) is Z [1/g] [G]–projective and H_{0} (X, Q_{ℓ}) = H_{0} (X, Z [1/g]) ⨂_{Z[1/g]} Q_{ℓ} , we have the equality

P_{0,ℓ}(u) = det_{Z[1/g][G]} (1 − F_{∗,0} · u| H_{0} (X, Z [1/g])) .  (a.4)

This obviously implies that P_{0,ℓ} ∈ Z [1/g] [G] [u].

The Q_{ℓ} [G]–isomorphism H_{2} (X, Q_{ℓ}) ⟶^{∼} H_{0} (X, Q_{ℓ}), carying the action of F_{∗,2} into the action of qF_{∗,0} (see §1.7.3), gives the equality

P_{2,ℓ}(u) = det_{Q_{ℓ}[G]} (1 − qF_{∗,0} · u| H_{0} (X, Q_{ℓ})) ,  (a.5)

which, according to the remarks above, implies that

P_{2,ℓ}(u) = det_{Z[1/g][G]} (1 − qF_{∗,0} · u| H_{0} (X, Z [1/g])) .  (a.6)

The last equality obviously shows that P_{2,ℓ}(u) ∈ Z [1/g] [G] [u].

Theorem 1.7.4.2 shows that

P_{1,ℓ}(u) = Θ(u) · P_{0,ℓ}(u) · P_{2,ℓ}(u)  (a.7)

and since Θ(u) ∈ Z [1/g] [G] [[u]] (see (a.1)), we obviously have P_{1,ℓ} ∈ Z [1/g] [G] [u] as well. □

Proof of (2). Equalities (a.4) and (a.6) obviously show that P_{0,ℓ}(u) and P_{1,ℓ}(u) do not depend on ℓ, ℓ ≠ p. Equality (a.7) therefore shows that P_{1,ℓ}(u) is independent of ℓ, ℓ ≠ p, as well. □

Before proceeding to the proof of statements (3) and (4) in the Theorem above, we will need to make a few considerations on the p–adic étale and the crystalline cohomology groups of X.

Let H (X) = ⨁_{i=0,1,2} H^{i} (X) be either the ℓ–adic étale cohomology of X with

coefficients in C_{ℓ}, ⨁_{i=0,1,2} H^{i} (X, C_{ℓ}), ℓ ≠ p, or the crystalline cohomology of X

with coefficients in C_{p}, ⨁_{i=0,1,2} H_{cris}^{i} (X, C_{p}). Here C_{ℓ} denotes the completion of

the algebraic closure of Q_{ℓ} with respect to the extension of the normalized ℓ–adic valuation, for any prime number ℓ. The definition of H^{i} (X, C_{ℓ}) is

H^{i} (X, C_{ℓ}) = H^{i} (X, Z_{ℓ}) ⨂_{Z_{ℓ}} C_{ℓ} ,

where H^{i} (X, Z_{ℓ}) = H_{i} (X, Z_{ℓ})^{∗} (functorial dual), for all i = 0, 1, 2, and every ℓ ≠ p (see [10]).

Let W (F) be the Witt vector ring of F. Then

H_{cris}^{i} (X, C_{p}) =^{def} H_{cris}^{i} (X/W (F)) ⨂_{W (F)} C_{p} ,

where H_{cris}^{i} (X/W (F)) are finitely generated W (F)–modules, functorial in the pair X/F. For a precise definition of the modules H_{cris}^{i} (X/W (F)) the reader can consult [1] and [2]. We will not need it for the present considerations.

In what follows 𝕂 will denote either C_{p} or C_{ℓ}, (ℓ ≠ p), depending on which cohomology theory is being used. The correspondences X ⟶ H^{i} (X) are contravariant functors from the category of smooth, projective F–schemes, to the category of 𝕂–vector spaces. For every morphism of F–schemes f : X ⟶ X′, we will denote by f^{∗} the 𝕂–linear maps

\xymatrix{H^{i} (X′) \ar[r]^{H^{i}(f)} & H^{i} (X) ,}

induced by f at the level of cohomology. In particular, every σ ∈ G induces an isomorphism of F–schemes σ_{X} : X ⟶ X, and therefore isomorphisms of 𝕂–vector spaces

σ_{X}^{∗} : H^{i} (X) ⟶ H^{i} (X) .

We can define an action of G on H (X) by setting

σ · h =^{def} (σ_{X}^{∗})^{−1} (h) ,

for all σ ∈ G and h ∈ H (X). This way the étale cohomology groups H^{i} (X, C_{ℓ}) are dual as Galois modules to the homology groups H_{i}(X, C_{ℓ}) defined in §1.7.4. In particular, for any character χ ∈ Ĝ(C_{ℓ}) we have C_{ℓ}–linear isomorphisms

H^{i} (X, C_{ℓ})^{χ} ⟶^{∼} H_{i}(X, C_{ℓ})^{χ^{−1}} ,  (a.8)

carrying the action F^{∗} of the q–power geometric Frobenius morphism on the left hand–side into its action F_{∗,i} on the right hand–side.

Let P_{H}^{i}(u) =^{def} det_{𝕂} (1 − F^{∗} · u| H^{i} (X)). Theorem 1 in [8] shows that the polynomials P_{H}^{i}(u) do not depend on the cohomology theory H (X), and therefore we can make the notation P^{i}(u) = P_{H}^{i}(u), for all i = 0, 1, 2. The polynomials P^{i}(u) have coefficients in Z and their reciprocal roots {α_{ij}} are algebraic integers, satisfying the relations

|α_{ij}| = q^{i/2} ,  (a.9)

for all i = 0, 1, 2, and all j = 1, …, dim_{𝕂}H^{i} (X) (see [8]). Relation (a.9) is the dimension 1 case of Riemann’s Hypothesis, proved by Deligne for smooth, projective F_{q}–varieties of any dimension.

Let P_{H}^{i,χ}(u) = det_{𝕂} (1 − F^{∗} · u | H^{i} (X)^{χ}), for all i = 0, 1, 2, and χ ∈ Ĝ(𝕂). We obviously have the relations

P^{i}(u) = ∏_{χ∈Ĝ(𝕂)} P_{H}^{i,χ}(u) ,  (a.10)

for every i = 0, 1, 2, and every cohomology theory H (X) in our list.

Remark 2. Relations (a.9) and (a.10) imply that for a given cohomology theory H (X) and a given χ ∈ Ĝ(𝕂), the polynomials P_{H}^{i,χ}(u) have mutually disjoint sets of roots, for distinct values of i.

Our next goal is to prove the following:

Proposition A.1. The polynomials P_{H}^{i,χ}(u) do not depend on the cohomology theory H (X).

Before proceeding to proving the statement above, we need an elementary lemma and a few remarks.

Lemma A.2. If V is a finitely generated Q [G]–module, and f ∈ End_{Q[G]} (V), then

(1) Tr_{Q[G]} (f ; V) = 1/g ∑_{σ∈G} Tr_{Q} (σ^{−1}f ; V ) · σ.

(2) Tr_{𝕂} (f ; (V ⊗ 𝕂)^{χ}) = 1/g ∑_{σ∈G} Tr_{𝕂} (σ^{−1}f ; V ⊗ 𝕂) · χ(σ), for all χ ∈ Ĝ(𝕂).

Proof. See [17], Chpt. V, Lemme 2.6. □

In what follows we will freely use the mutually inverse isomorphisms

\xymatrix{1 + uQ [G] [[u]] \ar[r]^{log} & uQ [G] [[u]] , \ar[l]^{exp}}

and also the identity

log det_{Q[G]} (1 − f · u| V) = − ∑_{n≥1} Tr_{Q[G]} (f^{n}; V) · u^{n}/n ,  (a.11)

for any V and f as in Lemma A.2 (see [17], Chpt.V).

Any cohomology theory H (X) in the list above satisfies the following form of the Lefschetz fixed point formula :

Theorem A.3. Let f : X ⟶ X be a morphism of F–schemes, with isolated, multiplicity 1 fixed points. Let Λ (f, X) be the number of fixed points of f. Then

Λ (f, X) = ∑_{i=0,1,2} Tr_{𝕂} (f^{∗}; H^{i} (X)) .

For a proof of the theorem above in the case of étale cohomology, the reader can consult [10]. The case of crystalline cohomology is treated in [1] and [2]. We now recall the definition of a fixed point of multiplicity 1 (see [7], Appendix C).

Definition A.4. A closed point x ∈ X, fixed by f, has multiplicity 1 if the map

\xymatrix{Ω_{X,x} \ar[r]^{1−df_{x}} & Ω_{X,x}} ,

induced by f at the level of the x–germ Ω_{X,x} of the sheaf of differentials Ω_{X} of X, is injective.

We are now prepared to prove Proposition A.1.

Proof of Proposition A.1. (Compare [17], Chpt.V, Thm.2.5) According to (a.3) and (a.11), we have

log Θ(u) = ∑_{v∈|Y_{1}|} log det_{Q[G]} (1 − F_{∗} · u| H_{0} (X_{v}, Q)) =

= ∑_{v,n} Tr_{Q[G]} (F_{∗}^{n}; H_{0} (X_{v}, Q)) · u^{n}/n .

Lemma A.2 (1) shows that

Tr_{Q[G]} (F_{∗}^{n}; H_{0} (X_{v}, Q)) = 1/gTr_{Q} (σ_{∗}^{−1}F_{∗}^{n}; H_{0} (X_{v}, Q)) .

But since σ_{∗}^{−1}F_{∗}^{n} acts on H_{0} (X_{v}, Q) by simply permuting the points of X_{v}, we also have the relation

Tr_{Q} (σ_{∗}^{−1}F_{∗}^{n}; H_{0} (X_{v}, Q)) = Λ (σ_{X}^{−1}F^{n}; X_{v}) ,

where the left hand–side in the last equality represents the number of fixed points of σ_{X}^{−1}F^{n}|_{X_{v}}. This implies that

log Θ(u) = 1/g ∑_{v,n,σ} Λ (σ_{X}^{−1}F^{n}; X_{v}) · σ · u^{n}/n ,

and since X = ∐_{v∈|Y_{1}|} X_{v}, we obtain

log Θ(u) = 1/g ∑_{n,σ} Λ (σ_{X}^{−1}F^{n}; X) · σ · u^{n}/n ,

where Λ (σ_{X}^{−1}F^{n}; X) is the number of fixed points of the F–scheme morphism σ_{X}^{−1}F^{n} : X ⟶ X, for all n ≥ 1.

Let x ∈ X be a point fixed by σ_{X}^{−1}F^{n}, for some n ≥ 1. One can easily show that x is isolated (dimX = 1), and that it has multiplicity 1, in the sense of Definition A.4. We are therefore entitled to use Theorem A.3 in order to compute the numbers Λ (σ_{X}^{−1}F^{n}; X), for all n ≥ 1. We thus have

log Θ(u) = 1/g ∑_{n,σ,i}(−1)^{i}Tr_{𝕂} ((σ_{X}^{−1}F^{n})^{∗} ; H^{i} (X)) · σ · u^{n}/n

= 1/g ∑_{n,σ,i}(−1)^{i}Tr_{𝕂} (σF^{∗n}; H^{i} (X)) · σ · u^{n}/n .

This implies that, for any χ ∈ Ĝ(𝕂), we have

log L(u, χ) = log (χ^{−1}Θ(u)) = χ^{−1} (log Θ(u)) =

= 1/g ∑_{n,σ,i}(−1)^{i}Tr_{𝕂} (σ^{−1}F^{∗n}; H^{i} (X)) · χ(σ)u^{n}/n .

Let us now combine the last equality with (a.11) and with Lemma A.2 (2), for V = H^{i} (X) and f = F^{∗n}. We obtain

log L(u, χ) = ∑_{n,i}(−1)^{i}Tr_{𝕂} (F^{∗n}; H^{i} (X)^{χ}) · u^{n}/n =

= ∑_{i}(−1)^{i+1} log det_{𝕂} (1 − F^{∗} · u; H^{i} (X)^{χ}) =

= log [∏_{i} det_{𝕂} (1 − F^{∗} · u; H^{i} (X)^{χ})^{(−1)^{i+1}}] =

= log [∏_{i} P_{H}^{i,χ}(u)^{(−1)^{i+1}}] .

And now, by taking exp of both sides of the last equality, we obtain:

L(u, χ) = ∏_{i} P_{H}^{i,χ}(u)^{(−1)^{i+1}} ,

for all χ and H (X) as above. This shows that the product ∏_{i} P_{H}^{i,χ}(u)^{(−1)^{i+1}} does

not depend on the cohomology theory H (X). Remark 2 therefore shows that the individual polynomials P_{H}^{i,χ}(u) do not depend on H (X), which concludes the proof of Proposition A.1. □

Let us make the notation P^{i,χ}(u) = P_{H}^{i,χ}(u), for all i = 0, 1, 2, χ ∈ Ĝ, and some cohomology theory H (X) in our list. The isomorphism (a.8) shows that, for all prime numbers ℓ ≠ p, we have

P^{i,χ}(u) = det_{C_{ℓ}} (1 − F_{∗,i} · u| H_{i} (X, C_{ℓ})^{χ^{−1}}) .

Thus, for all i and χ as above, we have

P_{i}(u) = ∑_{χ} P^{i,χ}(u) · e_{χ^{−1}} =

= ∑_{χ} det_{C_{p}} (1 − F^{∗} · u| H_{cris}^{i} (X, C_{p})^{χ}) · e_{χ^{−1}}.

(a.12)

Let L be a local field of characteristic 0, let V be a finitely generated L [G]– module and let f ∈ End_{L[G]} (V). We have a (unique up to an L [f]–isomorphism) splitting of V as a finite direct sum of L [G]–submodules

V = ⨁_{j∈J} L[f] · v_{j} ,  (a.13)

where L[f] · v_{j} are cyclic L[f]–modules, isomorphic to L[u]/(q_{j}(u)), with q_{j}(u) powers of irreducible polynomials, satisfying det (u − f| V) = ∏_{j∈J} q_{j}(u) . We will denote by J_{0} the subset of J consisting of those indices j such that the roots of q_{j}(u) in an algebraic closure L̄ of L have valuation 0. Let

V_{0} = ⨁_{j∈J_{0}} L[f] · v_{j} , V_{1} = ⨁_{j∉J_{0}} L[f] · v_{j} .

Since f is G–invariant, we have a direct sum decomposition of L [G]–modules V = V_{0} ⊕ V_{1}, thus the following divisibility holds in L [G] [u]:

det_{L[G]} (1 − f · u| V_{0}) | det_{L[G]} (1 − f · u| V) .  (a.14)

We will apply the general remarks above to a situation involving the étale and crystalline cohomology groups of X, described in [3], §3, and which we briefly now summarize. Let L be the quotient field of the Witt vector ring W (F) and let

H_{cris}^{i} (X, L) =^{def} H_{cris}^{i} (X/W (F)) ⨂_{W (F)} L ,

be the crystalline cohomology group of X with coefficients in L.

Let H^{i} (X, Q_{p}) be the étale cohomology groups of X with coefficients in Q_{p}, defined by H^{i} (X, Q_{p}) = H_{i} (X, Q_{p})^{∗} (functorial dual). Let

H^{i} (X, C_{p}) =^{def} H^{i} (X, Q_{p}) ⨂_{Q_{p}} C_{p} .

As in the case ℓ ≠ p explained in detail above, we have C_{p}–linear isomorphisms

H^{i} (X, C_{p})^{χ} ⟶^{∼} H_{i} (X, C_{p})^{χ^{−1}} , ∀χ ∈ Ĝ ,

preserving the action of the q–power geometric Frobenius morphism, thus

P_{i,p}(u) = ∑_{χ} det_{C_{p}} (1 − F_{∗,i} · u| H_{i} (X, C_{p})^{χ}) · e_{χ}

= ∑_{χ} det_{C_{p}} (1 − F^{∗} · u| H^{i} (X, C_{p})^{χ}) · e_{χ^{−1}} , ∀i = 0, 1, 2.

(a.15)

There is an exact sequence of Q_{p} [G]–modules (see [3], Th. 3.2)

\xymatrix{0 \ar[r] & H^{i} (X, Q_{p}) \ar[r]^{j} & H_{cris}^{i} (X, L) \ar[r]^{1−ϕ} & H_{cris}^{i} (X, L) \ar[r] & 0} ,

where ϕ is the map induced at the cohomology level by the p–power geometric Frobenius morphism of X (viewed as an F_{p}–scheme), and the injective morphism j : H^{i} (X, Q_{p}) ⟶ H_{cris}^{i} (X, L) preserves the action of F^{∗} on both sides (see [3], Lemma 3.3).

Let H_{cris}^{i} (X, L) = H_{cris}^{i} (X, L)_{0} ⨁ H_{cris}^{i} (X, L)_{1} be the decomposition described above for V = H_{cris}^{i} (X, L) and f = F^{∗}. Then the proof of Lemma 3.3 in [3] shows that H_{cris}^{i} (X, L)_{0} and H_{cris}^{i} (X, L)_{1} are ϕ–stable, j (H^{i} (X, Q_{p})) ⊆ H_{cris}^{i} (X, L)_{0} , and via j we have an equality of L [G]–modules

H_{cris}^{i} (X, L)_{0} = H^{i} (X, Q_{p}) ⨂_{Q_{p}} L .

Obviously this last equality can be written as

H_{cris}^{i} (X, L)_{0} = H^{i} (X, Q_{p}) ⨂_{Q_{p}[G]} L [G] .

This shows that

det_{Q_{p}[G]} (1 − F^{∗} · u| H^{i} (X, Q_{p})) = det_{L[G]} (1 − F^{∗} · u| H_{cris}^{i} (X, L)_{0}) ,

which, according to (a.14), implies that, inside C_{p} [G] [u], we have

det_{C_{p}[G]} (1 − F^{∗} · u| H^{i} (X, C_{p})) | det_{C_{p}[G]} (1 − F^{∗} · u| H_{cris}^{i} (X, C_{p})) .

If we break the last divisibility into χ–components, we obtain the following divisibilities in C_{p}[u]:

det_{C_{p}} (1 − F^{∗} · u| H^{i} (X, C_{p})^{χ}) | det_{C_{p}} (1 − F^{∗} · u| H_{cris}^{i} (X, C_{p})^{χ}) , ∀χ ∈ Ĝ .

According to (a.12) and (a.15), these relations imply that there exist polynomials Q_{i}(u) ∈ C_{p} [G] [u] such that

P_{i}(u) = Q_{i}(u) · P_{i,p}(u) , ∀i = 0, 1, 2 .

In order to prove statements (3) and (4) in our theorem, we will just need to show that Q_{i}(u) ∈ Z_{p} [1/g] [G] [u] and that, in the case when K/k is a constant field extension, Q_{i}(u) ∈ Z_{p} [G] [u]. These statements will easily follow from the next elementary lemma.

Lemma A.5. Let S ⊆ R be two commutative rings with 1. Let u be a variable and let f(u) ∈ S[u], g(u) ∈ S[u] and h(u) ∈ R[u], such that g(0) = 1, and f(u) = g(u) · h(u). Then h(u) ∈ S[u] as well.

Proof. Let f(u) = ∑_{i≥0} f_{i}u^{i}, g(u) = ∑_{i≥0} g_{i}u^{i}, and h(u) = ∑_{i≥0} h_{i}u^{i}, with f_{i}, g_{i} ∈ S,

h_{i} ∈ R and g_{0} = 1. The hypotheses of our lemma imply that h_{0} = f_{0} and f_{1} = g_{1}h_{0} + h_{1}. Since f_{0}, f_{1}, g_{1} ∈ S, these equalities imply that h_{0}, h_{1} ∈ S. Let us supose that h_{0}, h_{1} , …, h_{s} ∈ S, for 0 ≤ s ≤ deg (h) − 1. We have a relation

f_{s+1} = h_{s+1} + (h_{s}g_{1} + h_{s−1}g_{2} + …) ,

which shows that

h_{s+1} = f_{s+1} − (h_{s}g_{1} + h_{s−1}g_{2} + …).

The induction hypothesis now shows that h_{s+1} ∈ S. □

We are now ready to conclude the proof of Theorem 1.7.4.1 (3) and (4).

Proof of (3). According to (1) and (2) in our Theorem, the desired statement follows from Lemma A.5 applied to f(u) = P_{i}(u), g(u) = P_{i,p}(u), h(u) = Q_{i}(u), S = Z_{p} [1/g] [G] [u] and R = C_{p} [G] [u]. □

Proof of (4). Let K/k be a finite, constant field extension of function fields. Remark 1 above shows that

Θ(u) ∈ Z [G] [[u]] .  (a.16)

The proof of Lemma 4.2.1 shows that

P_{0}(u) = 1 − σ^{−1}u ∈ Z [G] [u] , P_{2}(u) = 1 − qσ^{−1}u ∈ Z [G] [u] ,

where σ is the distinguished generator of G described in §4.1. According to (a.7) and (a.16), we therefore have

P_{i}(u) ∈ Z [G] [u] , ∀i = 0, 1, 2 .  (a.17)

Since the H_{i} (X, Z_{p})’s are free Z_{p} [G]–modules in the constant field extension case (see Lemma 4.1.1), we also have

P_{i,p}(u) = det_{Z_{p}[G]} (1 − F_{∗} · u| H_{i} (X, Z_{p})) ∈ Z_{p} [G] [u] , ∀i = 0, 1, 2 . (a.18)

According to (a.17) and (a.18), statement (4) in our Theorem follows from Lemma A.5 applied to f(u) = P_{i}(u), g(u) = P_{i,p}(u), h(u) = Q_{i}(u), S = Z_{p} [G] [u] and R = C_{p} [G] [u]. □

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

E-mail address: : popescu@msri.org