An analogue of Girstmair's formula in function fields

Abstract

Suppose that $p$ is an odd prime and $g>1$ is a primitive root modulo $p$. Let $M$ be a number field contained in the $p$-th cyclotomic field. Girstmair found a surprising relation between the relative class number of $M$ and the digits of $1/p$ in base $g$. In this paper, we consider an analogue of Girstmair's formula in function fields. Suppose that $P\in {\mathbb{F}}_q[T]$ is monic irreducible and $G\in {\mathbb{F}}_q[T]$ is a primitive root modulo $P$. Let $L$ be an extension field of ${\mathbb{F}}_q(T)$ contained in the $P$-th cyclotomic function field. The goal of this paper is to give relations between the plus and minus parts of the divisor class number of $L$ and the digits of $1/P$ in base $G$.