On the size of determinants in the class number formulae of cyclotomic function fields

Abstract

We discuss the size of the determinants, which appear in the determinant formulae of the relative class numbers of cyclotomic function fields. These are the determinants of integer symmetric matrices, whose entries are 0 or 1. We show that, for a smaller characteristic, the determinants are significantly large (in the absolute value) compared to the determinants of randomly generated such matrices, while for a larger characteristic, it is not the case. We explain why this happens by comparing some upper bounds.