Stickelberger ideals and relative class numbers in function fields
Generalizing an earlier result by K. Iwasawa, \textit{W. Sinnott} [Ann. Math. (2) 108, 107-134 (1978; Zbl 0395.12014)] defined a Stickelberger ideal for every cyclotomic field \({\mathbb Q}(\zeta_m)\) and showed that its index in the minus part of the group ring \({\mathbb Z}[\text{Gal}({\mathbb Q}(\zeta_m)/{\mathbb Q})]\) equals the minus class number \(h^-({\mathbb Q}(\zeta_m))\) times a certain power of \(2\). The present paper treats the analogue of this result over any global function field \(k\). Labeling one place of \(k\) as \(\infty\) gives us a ring of integers \({\mathbb A}\) of \(k\). After fixing a sign function sgn on \(k\), one can define the narrow ray class group \(G_{\mathfrak m}\) for every ideal \({\mathfrak m}\) of \({\mathbb A}\). The corresponding narrow ray class field \(K=K_{\mathfrak m}\) is also called the \({\mathfrak m}\)-th cyclotomic extension of \((k, \infty, \text{sgn})\). Let \(K^+\) be its ``maximal real'' subfield and \(O_K\) and \(O_{K^+}\) the corresponding integral closures of \({\mathbb A}\). The integer \(h(O_K)^-:=h(O_K)/h(O_{K^+})\) is called the relative ideal class number of \(O_K\). The author defines a Stickelberger ideal and shows that its index in a suitable submodule of \({\mathbb Z}[G_{\mathfrak m}]\) is \((q-1)^a h(O_K)^-\), where \({\mathbb F}_q\) is the constant field of \(k\) and \(a\) depends on the number of different prime divisors of \({\mathfrak m}\). An important ingredient of the proof is a recent method by \textit{G. Anderson} by which the \(G_{\mathfrak m}\)-module structure of a certain cohomology group could be determined.
scientific article; zbMATH DE number 1420270