Maass forms and spectral theory for noncongruence subgroups
Heilbronn Number Theory Seminar
31st January 2018, 4:00 pm – 5:00 pm
Howard House, 4th Floor Seminar Room
The spectral theory for the modular group is well-understood and we know, for instance, that there is an infinite number of cusp forms and we also know that are no small eigenvalues (neither exceptional, nor residual).
That there exists an infinite number of cusp forms can easily be shown for congruence subgroups using, e.g. the Selberg Trace Formula and explicit expressions for the scattering determinant.
It was conjectured by Selberg that congruence subgroups should have no small "exceptional" (i.e. cuspidal) eigenvalues and this has been proven in many cases (the most extensive proofs are obtained by Booker and Strömbergsson using the trace formula).
When it comes to non-congruence subgroups, however, the situation is more complicated (interesting) and even the existence of cusp forms is not known in general (except for those coming from overgroups, of course).
For exceptional eigenvalues, It is known, however, that any subgroup of the modular group which satisfies a certain inequality between index and genus must have at least one small exceptional eigenvalue although no explicit example has been given.
In this talk I will present some interesting conjectures about the spectral theory for (general) subgroups of the modular group together with algorithms and numerical results, including numerical examples of exceptional and residual eigenvalues.
One of the main obstacles to using the same theoretical tools which are available for congruence subgroups to the setting of non-congruence subgroups is that the scattering determinant does not have a simple expression in terms of quotients of L-functions and it turns out, for instance, that the scattering determinant has poles very close to the half-line. Because of this it is not possible to use the standard Weyl’s law to show that there is an infinite number of cusp forms and indeed we will present a "numerical" version of an average Wey’s law to give evidence that the computations are complete and consistent.