Noncongruence subgroups and Maass waveforms

Abstract

The main topic of the paper is spectral theory for noncongruence subgroups of the modular group. We have studied a selection of the main conjectures in the area: the Roelcke-Selberg and Phillips-Sarnak conjectures and Selberg's conjecture on exceptional eigenvalues. The first two concern the existence and nonexistence of an infinite discrete spectrum for certain types of Fuchsian groups and last states that there are no exceptional eigenvalues for congruence subgroups, or in other words, there is a specific spectral gap in the cuspidal spectrum.Our main theoretical result states that if the corresponding surface has a reflectional symmetry which preserves the cusps then the Laplacian on this surface has an infinite number of "new" discrete eigenvalues. We define old and new spaces of Maass cusp forms for noncongruence subgroups in a way that provides a natural generalization of the usual definition from congruence subgroups and give a method for determining the structure of the old space algorithmically.In addition to the theoretical result we also present computational data, including a table of subgroups of the modular group and eigenvalues of Maass forms for noncongruence subgroups. We also present, for the first time, numerical examples of both exceptional and (non-trivial) residual eigenvalues. To be able to (even heuristically) cerify lists of computed eigenvalues we also proved an explicit average Weyl's law in this setting.

Seiberg-Witten geometry, modular rational elliptic surfaces and BPS quivers

“…(2.37) Doran showed that there are 33 modular RES [87], which can be extended to 47 distinct RES, as we will see in the next subsection. Our approach is different from that of [87] and closer to the construction of Shioda [27]; we will use the classification of subgroups of PSL(2, Z) [88][89][90] and the map (2.37) in order to find all modular rational elliptic surfaces. This will be done explicitly in section 4.…”

“…As shown in [87], this is a set of 33 surfaces, up to quadratic twists, which then extends to 47 different surfaces in Persson's classification [3,4]. The subgroups of PSL(2, Z) associated to each modular rational elliptic surface can be deduced from the list of genus 0 subgroups of PSL(2, Z) of index less or equal to 12 [88][89][90], together with the map (2.37).…”

“…Let us also briefly comment on the notation used for these subgroups. Apart from the well-established notation for Γ 0 (N ), Γ 1 (N ) and the principal congruence subgroup Γ(N ), as well as their conjugates, we follow the notation of [90], denoting conjugacy classes by the index, the genus of associated Riemann surface, the number of cusps and elliptic elements, as well as the level: 8 Γ (n Γ ; g, c, e 2 , e 3 ; N ) . (4.1)…”

“…These configurations have been classified in [87], leading to 33 modular RES, which we extend to 47 different surfaces by considering 'quadratic twists', which, however, do not change the modular properties of the elliptic surface. In this work, we review this classification from a different perspective, namely by using the classification of subgroups of PSL(2, Z) [88][89][90]. These are of two types, called congruence or noncongruence, with the former having the principal congruence subgroup Γ(N ) as a subgroup.…”

Computations of Eigenvalues and Resonances on Perturbed Hyperbolic Surfaces with Cusps

“…Versions of the Weyl law with error estimates are available in the constant curvature case, see e.g. [34]. 7 Examples and numerical studies 7.1 General set-up In this section we will consider several examples of manifolds with cusps.…”