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2 Cyclotomic Function Fields and Artin-Schreier Extensions

We study two criterions of cyclicity for divisor class groups of function fields, the first one involves Artin L-functions and the second one involves ”affine” class groups. We show that, in general, these two criterions are not linked. Let P be a prime of Fq[T ] of degree d and let KP be the P th cyclotomic function field. In this paper we study the relation between the p-part of Cl(KP ) and the zeta function of KP , where p is the characteristic of Fq. Let χ be an even character of the Galois group of KP/Fq(T ), χ 6= 1. Let g(X,χ) be the ”congruent to one modulo p” part of the L-function of KP/Fq(T ) associated to the character χ. We have two criterions of cyclicity ([2], chapter 8): if degXg(X, χ) ≤ 1 then Cl(KP )p(χ) is a cyclic Zp[μqd−1]module, and if Cl(OKP )p(χ) = {0} then Cl(KP )p(χ) is a cyclic Zp[μqd−1]module. David Goss has obtained that if Cl(OKP )p(χ) is trivial then g(X,χ) is of degree at most one ([2], Theorem 8.21.2). Unfortunately, there is a gap in the proof of this result. In fact, we show that in general Cl(OKP )p(χ) = {0} does not imply degXg(X, χ) ≤ 1 (Proposition 3.4). We also prove that if i is a q-magic number and if ωP is the Teichmüller character at P, then g(X,ω P ) has simple roots when i ≡ 0 (mod q − 1) (Proposition 5.1). Note that Goss conjectures that if i is a q-magic number then degXg(X, ω i P) ≤ 1. This problem is still open and can be viewed as an analogue of Vandiver’s Conjecture for function fields (see section 5).