Fitting ideals of class groups in Carlitz–Hayes cyclotomic extensions

Abstract

We generalize some results of Greither and Popescu to a geometric Galois cover X → Y which appears naturally for example in extensions generated by p n -torsion points of a rank 1 normalized Drinfeld module (i.e. in subextensions of Carlitz-Hayes cyclotomic extensions of global fields of positive characteristic). We obtain a description of the Fitting ideal of class groups (or of their dual) via a formula involving Stickelberger elements and providing a link (similar to the one in [1]) with Goss ζ-function.

Stickelberger series and Main Conjecture for function fields

“…We mention that the interpretation of Ψ y as a C × ∞ -character on S-idéles is the approach suggested in [2, Theorem 3.8 and Remark 3.9]: we shall see a more explicit relation between Ψ y and rec S in the special case presented in the next section. The extension Ψ y ∶ Z[G S ] X → C ∞ X gives an interpolation formula for the Goss Zetafunction (the case of the Carlitz cyclotomic extension is presented in [4,Theorem 4.2]). Theorem 3.8.…”

“…Remark 1.1. For the characters of type 3 we are only able to compute the Fitting ideal of a dual of C n and it is often non principal: we have no arithmetic interpretation (from the point of view of Iwasawa theory) for this situation yet so we decided to present it in a different paper (see [4,Section 3]).…”