A determinant formula for relative congruence zeta functions for cyclotomic function fields

TL;DR

This paper extends Rosen's determinant formula from relative class numbers to relative congruence zeta functions for cyclotomic function fields, providing a broader mathematical framework.

Contribution

It introduces a new determinant formula for relative congruence zeta functions, generalizing previous results on relative class numbers in cyclotomic function fields.

Findings

Derived a determinant formula for relative congruence zeta functions

Generalized Rosen's formula for class numbers to zeta functions

Enhanced understanding of cyclotomic function field invariants

Abstract

Rosen M. gave a determinant formula for relative class numbers for the P-th cyclotomic function fields in the case of the monic irreducible polynomial P, which is regarded as an analogue of the classical Maillet determinant. In this paper, we will give a determinant formula for the relative congruence zeta functions for cyclotomic function fields. Our formula is regarded as a generalization of the determinant formula for the relative class number.