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. In 1994, 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 $P\in F_q\left[T\right]$ is monic irreducible and $G\in F_q\left[T\right]$ is a primitive root modulo P. Let L be a field extension of $F_q\left(T\right)$ which is contained in the P-th cyclotomic function field. Our goal 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.