On Hilbert class fields in characteristic $p>0$ and their $L$-functions

Abstract:

Let k be a global field of characteristic $p > 0$ with field of constants ${{\mathbf {F}}_q}$. Let $\bar k$ be an algebraic closure of k. In this note we study the subfields of $\bar k$ which are maximal unramified abelian extensions of k with field of constants ${{\mathbf {F}}_q}$. Each of these fields may be regarded as an analogue of the Hilbert class field of algebraic number theory [1, p. 79]. In §1 we recall the construction of these class fields and in §2 we show that if k has genus one, they are all ${{\mathbf {F}}_q}$-isomorphic. In §3 we show that this is not necessarily the case if the genus of k is greater than one. The argument there is based on an observation about the L-functions of the fields.