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In the arithmetic theory of function fields, there are two distinct analogs of classical Bernoulli numbers. It has been known for a while that one analog was intimately connected with class number formulae. Due to recent progress, one can now begin to understand the arithmetic meaning of the other. Let k = Fr(T), r = p m, and let A = Fr(T). Let P e Spec(A) be associated to a prime of degree d. As in (2) (the notation of (2) will be followed through­ out), one can associate to P the abelian extension k(P) of k with Galois group isomorphic to A/P*. This extension is very closely analogous to Q{çp)/Q', Çp a primitive pth root of unity. We will be interested here in the "totally real" subfield k(P)+ of k(P) obtained as the fixed field of F* Ç A/P*. In particular, our results are concerned with the p-primary class group and with the class group Cl(P)+ of the ring of ^-integers 0(P)+ of k(P)+. One knows that the natural projection of C1(P)+ to C1(P)+ is surjective

Units and class-groups in the arithmetic theory of function fields