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). We will first talk about the analysis
that leads to these conclusions.
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 have also developed an algorithms that constructs 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. If time permits we will sketch
the principles of this algorithm.
This is joint work with M. Schuett and H. Verrill.