A Class–field Theoretical Calculation

In this paper, we give the complete characterization of the p–torsion subgroups of certain idèle–class groups associated to characteristic p function fields. As an application, we answer a question which arose in the context of Tan’s approach [Ta1] to an important particular case of a generalization of a conjecture of Gross [Gro] on special values of L–functions. 1. Notation and motivation Let p be a prime number. As usual, the term characteristic p function field (equivalently, characteristic p global field) refers to a finite extension of a field Fp(T ), where Fp is the finite field of p elements and T is a variable. Let K/k be a finite abelian extension of characteristic p function fields, of Galois group Γ := Gal(K/k). Let S be a finite, nonempty set of primes in k, containing all the primes which ramify in K/k. We denote by SK the set of primes in K dividing primes in S. Let K S be the maximal pro–p abelian extension of K, unramified outside SK . Since SK is Γ–invariant, K ab,p S /k is a Galois extension. Let G := Gal(K ab,p S /k), and H := Gal(K S /K). As usual, the group Γ acts by lift–and–conjugation on H. More precisely, γ ∗ h := γ̃hγ̃−1, for all h ∈ H and γ ∈ Γ, where γ̃ denotes any lift of γ to G with respect to the usual epimorphism G −→ Γ. This way, since H is an abelian group, H is endowed with a natural Z[Γ]–module structure. In what follows, we denote by IΓ the augmentation ideal in the group ring Z[Γ], i.e. the ideal of Z[Γ] generated by {γ−1 | γ ∈ Γ}. Let [G, G] denote the commutator subgroup of G, generated by the commutators [x, y] = xyx−1y−1 of all the elements x, y ∈ G. Since, by definition (γ − 1) ∗ h = γ̃hγ̃−1h−1 = [γ̃, h], for all γ ∈ Γ and h ∈ H, we have an inclusion IΓ ·H ⊆ [G,G] . In [Ta1], the following question arises in the context of Tan’s approach to an important particular case (the so–called “p–primary part in characteristic p”–case) of a generalization of a conjecture of Gross [Gro]. Question 0. Under what conditions do we have an equality