Kummer's criterion over global function fields
Let \(k\) be a global function field (i.e., a finite extension of a rational function field \(\mathbb{F}_ p (T)\) over the finite field with \(p\) elements) and \(\infty\) a fixed place of degree one over the field \(\mathbb{F}_ q\) of constants. Let further \(A\) be the Dedekind ring of elements of \(k\) regular off \(\infty\), and \({\mathfrak p}\) a (non-zero) prime ideal. With \({\mathfrak p}\), one can associate an abelian extension \(k({\mathfrak p})\) of \(k\), which is similar to the field of \(p\)-th roots of unity \(k(p):= \mathbb{Q}(\zeta_ p)\) over \(k:= \mathbb{Q}\). We have \(k\subset H_ \infty \subset k^ + ({\mathfrak p}) \subset k({\mathfrak p})\), where \(H_ \infty\) is the ring class field of \(A\) and \(k^ + ({\mathfrak p})\) the ``maximal real subfield of \(k({\mathfrak p})\)'', i.e., the maximal abelian extension of \(k\) unramified off \({\mathfrak p}\) and completely split at \(\infty\). The Galois groups are as follows (where \(C\ell\) means ``class group''): \[ C\ell(A) {\overset \cong \longrightarrow} \text{Gal} (H_ \infty/k), \qquad (A/{\mathfrak p})^* {\overset \cong \longrightarrow} \text{Gal} (k({\mathfrak p})/k), \qquad \mathbb{F}_ q^* {\overset \cong \longrightarrow} \text{Gal} (k({\mathfrak p}) /k^ +({\mathfrak p})). \] In this situation, there are several natural Galois modules: divisor class groups of \(k({\mathfrak p})\), \(k^ +({\mathfrak p})\), \(H_ \infty\), of their respective rings of \(A\)-integers, groups of units, of cyclotomic units etc. The connection between these objects has been studied by Galovich-Rosen, Goss, Hayes, Okada, Oukhaba, and others. In particular, in the case where \(k= \mathbb{F}_ q(T)\) and \(\infty\) is the usual infinite place, D. Goss proved a useful criterion for \(C\ell(O^ +({\mathfrak p}))\) to have a non-trivial \(p\)-part [cf. \textit{D. Goss}, The arithmetic of function fields; de Gruyter, 313-402 (1992; Zbl 0806.11028); \(O^ + ({\mathfrak p})= A\)-integers in \(k^ +({\mathfrak p})\)]. It is analogous with the classical Kummer criterion and involves the divisibility by \({\mathfrak p}\) of the so-called ``Bernoulli-Carlitz numbers'' \(B_ i\in k= \mathbb{F}_ q (T)\), which may be defined and calculated from the exponential function associated with the Carlitz module. In the present article, the author extends Goss' Kummer-like criterion to non-rational function fields (but \(\infty\) still of degree one). The presence of non-principal ideals creates considerable technical difficulties, in particular, the ``numbers'' \(B_ i\) have to be replaced by ideals. The precise result is given as ``Main Theorem 2'' (in fact, it encompasses the author's ``Main Theorem 1''): If a certain isotypic component \(C\ell^{(p)} (O^ + ({\mathfrak p})) (\omega^ i)\) of the \(p\)- part of \(C\ell(O^ + ({\mathfrak p}))\) is non-zero, the prime \({\mathfrak p}\) must divide the ``\(i\)-th Bernoulli ideal'' \(B_ i\) of \(A\). The line of proofs is as follows. As in the classical case, the relevant class numbers may be expressed as unit indices (Theorem 4.2). Then, following a construction due to \textit{S. Okada} [J. Number Theory, 38, No. 2, 212-215 (1991; Zbl 0728.11031)], a ``Kummer homomorphism'' from a local unit group of \(k({\mathfrak p})\) to \(O(H_ \infty)/ {\mathfrak p} O(H_ \infty)\) is defined (section 3), whose effect may be described through the ideals \(B_ i\) (Proposition 4.7). This gives the wanted relationship. In an appendix, the Bernoulli ideals \(B_ i\) introduced by the author in section 2 are identified as those defined earlier by Goss.
scientific article; zbMATH DE number 710375