The cusp amplitudes and quasi-level of a congruence subgroup of SL2 over any Dedekind domain

TL;DR

This paper generalizes properties of classical modular groups to groups over Dedekind domains, focusing on cusp amplitudes, levels, and introducing quasi-levels to better understand congruence subgroups.

Contribution

It extends algebraic properties of SL_2(Z) to SL_2(D) over Dedekind domains, introduces quasi-amplitude and quasi-level concepts, and provides new conditions for congruence subgroups.

Findings

Extended properties of cusp amplitudes and levels to Dedekind domains

Introduced quasi-amplitude and quasi-level notions

Derived new necessary conditions for congruence subgroups

Abstract

We extend some algebraic properties of the classical modular group SL_2(Z) to equivalent groups in the theory of Drinfeld modules, in particular properties which are important in the theory of modular curves. We study cusp amplitudes and the level of a (congruence) subgroup of SL_2(D) for any Dedekind domain D, as ideals of D. In particular, we extend a remarkable result of Larcher. We introduce finer notions of quasi-amplitude and quasi-level, which are not required to be ideals and encode more information about the subgroup. Our results also provide several new necessary conditions for a subgroup of SL_2(D) to be a congruence subgroup.