On \(l\)-class groups of global function fields (Q2789368)

Let \(K\) be a finite geometric extension of the rational function field \({\mathbb F}_q(T)\) and let \(L/K\) be a finite cyclic extension of prime degree \(l\). The main objective of this paper is to study the \(l\)-class group of \(Cl(L):=Cl({\mathcal O}_L)\) where \({\mathcal O}_L\) is the integral closure of \({\mathbb F}_q[T]\) in \(L\) under the assumption that \(h({\mathcal O}_K):= |Cl(K)|\) is not divisible by \(l\). The main tools used here are genus theory and Conner-Hurrelbrink exact hexagon for function fields.NEWLINENEWLINEThe first main result is a new proof (Theorem 3.5) of Hasse's formula for \(r_l(Cl(L)_l^G)\) [\textit{H. Hasse}, Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. I: Klassenkörpertheorie. Leipzig: B. G. Teubner (1930; JFM 56.0165.01)] using Conner-Hurrelbrink exact hexagon when \(L/K\) is a cyclic extension of degree \(l\) of number fields with \(l\) being an odd prime and such that \(l\nmid h({\mathcal O}_K)\). Here \(r_l\) denotes the \(l\)-rank. The proof is also valid for function fields. Next the authors prove (Proposition 3.8) the function field analogue of \textit{F. Gerth} III's results [J. Number Theory 8, 84--98 (1976; Zbl 0329.12006)] on \(3\)-class groups of cyclic cubic extensions (assuming that \(3\nmid h({\mathcal O}_K)\)).NEWLINENEWLINEConsidering \(Cl(L)_l\) as \({\mathbb Z}_l[G]\)--module, where \({\mathbb Z}_l\) is the ring of \(l\)-adic integers and \(G=\mathrm{Gal}(L/K)\) and assuming that \(l\nmid h({\mathcal O}_K)\), the authors give the structure of \(Cl(L)_l\) as \({\mathbb Z}_l[G]\)-module which was obtained by \textit{F. Gerth} III for number fields [Math. Comput. 29, 1135--1137 (1975; Zbl 0314.12013)].NEWLINENEWLINEIn the last part of the paper, the authors discuss some relations of class numbers between biquadratic function fields and their subfields for \(q\) odd.