Linear characters of SL_2 over Dedekind domains

TL;DR

This paper explicitly describes the group of linear characters of SL_2 over certain Dedekind domains, including those with rings of integers of number fields, focusing on characters with kernels as congruence subgroups.

Contribution

It provides an explicit characterization of linear characters of SL_2 over a broad class of Dedekind domains, extending previous understanding to include those with non-totally complex number fields.

Findings

Explicit description of linear characters for specific Dedekind domains

Determination of characters with kernels as congruence subgroups

Extension of character theory to broader classes of arithmetic domains

Abstract

For an important class of arithmetic Dedekind domains O including the ring of integers of not totally complex number fields, we describe explicitly the group of linear characters of SL_2(O). For this, we determine, for arbitrary Dedekind domains O, the group of linear characters of SL_2(O) whose kernel is a congruence subgroup.