Gras{type Conjectures for Function Fields

Based on results obtained in [15], we construct groups of special S– units for function fields of characteristic p > 0, and show that they satisfy Gras– type Conjectures. We use these results in order to give a new proof of Chinburg’s Ω3–Conjecture on the Galois module structure of the group of S–units, for cyclic extensions of prime degree of function fields. 0. Introduction Let K/k be a finite, abelian extension of function fields of characteristic p > 0. Let G = G (K/k) and g = |G|. We will denote by Fq and Fqν the exact fields of constants of k andK respectively, where q is a power of p and ν is a positive integer. In what follows we will use the same notations as in [15]. For the convenience of the reader, we briefly summarize in this section the main concepts and results of [15] which will be used in our arguments. For any two finite, nonempty and disjoint sets S and T of primes in k, and any field F , k ⊆ F ⊆ K, UF,S and UF,S,T denote the groups of S–units and respectively (S, T )–units of F ; AF,S and AF,S,T are respectively the S–ideal class group and (S, T )–ideal class group of F , as defined in [15, §1.1]. In particular, if F = K, we suppress K from the notation, so UK,S = US, UK,S,T = US,T etc. All the exterior powers considered in this paper are taken over the group ring Z [G], unless stated otherwise. Let us assume for the moment that for a certain positive integer r, the set of data (K/k, S, T, r) satisfies the following set of hypotheses: (H)    S 6= ∅, T 6= ∅, 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. Let (v1, . . . , vr) be an ordered r–tuple of primes in S which split completely in K/k, and let W = (w1, . . . , wr), with wi prime in K, wi|vi, for every i = 1, . . . , r. One can define a regulator map C r ∧US,T RW −−→ C [G] , by RW (u1 ∧ · · · ∧ ur) = det i,j ( − ∑ σ∈G log |u −1 j |wi · σ ) , ∀u1 ∧ · · · ∧ ur ∈ r ∧US,T .