Lifts of projective congruence groups

TL;DR

This paper investigates the prevalence of noncongruence subgroups of SL_2(Z) that are projectively equivalent to congruence subgroups, exploring their existence, properties, and implications for modular forms and elliptic surfaces.

Contribution

It demonstrates the widespread existence of such subgroups for principal and some other congruence groups and develops algorithms to identify and classify them.

Findings

Noncongruence subgroups projectively equivalent to principal congruence groups exist for levels N>2.

Many cases of such subgroups are found for Gamma_0(N).

Algorithms are provided to construct and determine the nature of these subgroups.

Abstract

We show that noncongruence subgroups of SL_2(Z) projectively equivalent to congruence subgroups are ubiquitous. More precisely, they always exist if the congruence subgroup in question is a principal congruence subgroup Gamma(N) of level N>2, and they exist in many cases also for Gamma_0(N). The motivation for asking this question is related to modular forms: projectively equivalent groups have the same spaces of cusp forms for all even weights whereas the spaces of cusp forms of odd weights are distinct in general. We make some initial observations on this phenomenon for weight 3 via geometric considerations of the attached elliptic modular surfaces. We also develop algorithms that construct all subgroups projectively equivalent to a given congruence subgroup and decides which of them are congruence. A crucial tool in this is the generalized level concept of Wohlfahrt.