An analogue of Girstmair's formula in function fields

TL;DR

This paper explores an analogue of Girstmair's relation between class numbers and digit expansions, extending it from number fields to function fields involving cyclotomic extensions over finite fields.

Contribution

It establishes a new relation connecting the divisor class number parts of function field extensions to digit expansions in base G, generalizing Girstmair's formula to function fields.

Findings

Derived relations between class number parts and digit expansions in function fields.

Extended Girstmair's formula from number fields to function fields.

Provided explicit formulas involving cyclotomic function field extensions.

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 $G$.