Class groups of imaginary function fields: The inert case

Abstract:

Let $\mathbb {F}$ be a finite field and $T$ a transcendental element over $\mathbb {F}$. An imaginary function field is defined to be a function field such that the prime at infinity is inert or totally ramified. For the totally imaginary case, in a recent paper the second author constructed infinitely many function fields of any fixed degree over $\mathbb {F}(T)$ in which the prime at infinity is totally ramified and with ideal class numbers divisible by any given positive integer greater than 1. In this paper, we complete the imaginary case by proving the corresponding result for function fields in which the prime at infinity is inert. Specifically, we show that for relatively prime integers $m$ and $n$, there are infinitely many function fields $K$ of fixed degree $m$ such that the class group of $K$ contains a subgroup isomorphic to $(\mathbb {Z}/n\mathbb {Z})^{m-1}$ and the prime at infinity is inert.