Related papers: A Database of Modular Forms on Noncongruence Subgr…

We present two approaches that can be used to compute modular forms on noncongruence subgroups. The first approach uses Hejhal's method for which we improve the arbitrary precision solving techniques so that the algorithm becomes about up…

Determining Fourier coefficients of modular forms of a finite index noncongruence subgroups of the modular group is still a non-trivial task. In this brief note we describe a new algorithm to reliably calculate an approximation for a…

We announce a database of rigorously computed Maass forms on congruence subgroups $\Gamma_0(N)$ and briefly describe the methods of computation.

Classically, congruence subgroups of the modular group, which can be described by congruence relations, play important roles in group theory and modular forms. In reality, the majority of finite index subgroups of the modular group are…

In this paper we give an explicit formula for the number of subgroups of the modular group of a given index that are genus zero and torsion-free and a formula for their conjugacy classes. We do so by exhibiting a correspondence between…

We study equivariant primitives of Eisenstein series for principal congruence subgroups and show that they are precisely the corresponding non-holomorphic Eisenstein series. We present closed formulas that naturally generalise existing…

We define the notion of a $G$-structure for elliptic curves, where $G$ is a finite 2-generated group. When $G$ is abelian, a $G$-structure is the same as a classical congruence level structure. There is a natural action of…

In this paper, we consider modular forms for finite index subgroups of the modular group whose Fourier coefficients are algebraic. It is well-known that the Fourier coefficients of any holomorphic modular form for a congruence subgroup…

We study modular forms for the minimal index noncongruence subgroups of the modular group. Our main theorem is a proof of the unbounded denominator conjecture for these groups, and we also provide a study of the Fourier coefficients of…

Deligne proved that the weights of Siegel modular forms on any congruence subgroup of the Siegel modular group of genus g>1 must be integral or half integral. We give a different proof for this. It uses Mennicke's result that subgroups of…

In this paper we will use experimental and computational methods to find modular forms for non-congruence subgroups, and the modular forms for congruence subgroups that they are associated with via the Atkin--Swinnerton-Dyer correspondence.…

We give algebraic and geometric classifications of $6$-dimensional complex nilpotent anticommutative algebras. Specifically, we find that, up to isomorphism, there are $14$ one-parameter families of $6$-dimensional nilpotent anticommutative…

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…

We calculate the hauptmodul and Belyi map of a genus zero subgroup of the modular group defined via a canonical homomorphism by the Conway group group Co3. Our main result is the Belyi map and its field of definition.

Motivated by a demand for explicit genus 1 Belyi maps from theoretical physics, we give an efficient method of explicitly computing genus one Belyi maps by (1) composing covering maps from elliptic curves to the Riemann sphere with simpler…

In this paper, we explicitly compute two kinds of algebraic Belyi functions on Bring's curve. One is related to a congruence subgroup of ${\rm SL}_2(\mathbb{Z})$ and the other is related to a congruence subgroup of the triangle group…

We consider the 33 conjugacy classes of genus zero, torsion-free modular subgroups, computing ramification data and Grothendieck's dessins d'enfants. In the particular case of the index 36 subgroups, the corresponding Calabi-Yau threefolds…

We give new examples of weight three cusp forms on noncongruence subgroups of SL(2, Z) whose Scholl representation is modular and which satisfy three term Atkin-Swinnerton-Dyer relations.

We compute tables of paramodular forms of degree two and cohomological weight via a correspondence with orthogonal modular forms on quinary lattices.

We identify a class of "semi-modular" forms invariant on special subgroups of $GL_2(\mathbb Z)$, which includes classical modular forms together with complementary classes of functions that are also nice in a specific sense. We define an…