Families of $\phi$-congruence subgroups of the modular group

TL;DR

This paper introduces and analyzes new families of finite index subgroups of the modular group, called $\,\phi$-congruence subgroups, which generalize classical congruence subgroups through homomorphisms into algebraic groups.

Contribution

It defines $\,\phi$-congruence subgroups via homomorphisms into algebraic groups and studies their properties, including examples involving quasi-unipotent and symplectic groups.

Findings

Analysis of $\,\phi$-congruence subgroups and their properties.

Connection between quasi-unipotent cases and modular forms.

Description of the tower of curves related to these subgroups.

Abstract

We introduce and study families of finite index subgroups of the modular group that generalize the congruence subgroups. Such groups, termed $\phi$-congruence subgroups, are obtained by reducing homomorphisms $\phi$ from the modular group into a linear algebraic group modulo integers. In particular, we examine two families of examples, arising on the one hand from a map into a quasi-unipotent group, and on the other hand from maps into symplectic groups of degree four. In the quasi-unipotent case we also provide a detailed discussion of the corresponding modular forms, using the fact that the tower of curves in this case contains the tower of isogenies over the elliptic curve $y^2=x^3-1728$ defined by the commutator subgroup of the modular group.