Class number of \((v,n,M)\)-extensions (Q2716957)

Let \({\mathbb F}_q\) be the finite field of \(q\) elements, and let \(k= {\mathbb F}_q (T)\) be the rational function field over \({\mathbb F}_q\). \textit{L. Carlitz} [Duke Math. J. 1, 137-168 (1935; Zbl 0012.04904)] investigated the cyclotomic extensions of \(k\), which are analogous to the classical cyclotomic extensions of the field of rational numbers \({\mathbb Q}\). \textit{D. R. Hayes} [Trans. Am. Math. Soc. 189, 77-91 (1974; Zbl 0292.12018)] constructed the maximal abelian extension of \(k\) and proved the analogue of the classical Kronecker-Weber Theorem. As a consequence of Hayes' result, any finite abelian extension of \(k\) is contained in some field of the form \(N:= k _n \cdot k(\Lambda _M)\cdot L_v\), called a \((v,n,M)\)-extension, where \(k_n\) is a constant extension, \(k(\Lambda _M)\) is the \(M\)-th cyclotomic function field with \(M\in {\mathbb F}_q[T]\) and \(L_v\) is some field constructed by Hayes where the infinite prime of \(k\) is wildly and fully ramified and no other prime divisor is ramified in \(L_v/k\). NEWLINENEWLINENEWLINEThe main result of the paper under review is an analytic class number formula for \(N\) when \(M\) has a nonzero constant term, which generalizes the analytic class number for \(k(\Lambda _M)\) found by \textit{S. Galovich} and \textit{M. Rosen} [J. Number Theory 13, 363-375 (1981; Zbl 0473.12014)] when \(M = P^\alpha\) for some prime polynomial \(P \in {\mathbb F}_q[T]\). This result is stated at the end of Section 2, and its proof is based on the evaluation of character sums and the values of some \(L\)-series at \(s=0\). NEWLINENEWLINENEWLINEIn the last section, the authors give some examples of the formula, and in particular they state a general class number formula of \(k(\Lambda _M)\) for \(k= {\mathbb F}_p(T)\).