Related papers: Dessins on Modular Curves
In this paper we compute the degree of a curve which is the image of a mapping $z\longmapsto (f(z): g(z): h(z))$ constructed out of three linearly independent modular forms of the same even weight $\ge 4$ into $\mathbb P^2$. We prove that…
Consider a Kleinian singularity $ \mathbb{C}^2/\Gamma $, where $ \Gamma $ is a finite subgroup of $ SL_2(\mathbb{C}) $. In this paper, we construct moduli spaces of framed sheaves on a projective Deligne-Mumford stack compactifying the…
The study of bipartite maps (or Grothendieck's dessins d'enfants) is closely connected with geometry, mathematical physics and free probability. Here we study these objects from their permutation factorization formulation using a novel…
Special values of Siegel modular functions for $\operatorname{Sp} (\mathbb{Z})$ generate class fields of CM fields. They also yield abelian varieties with a known endomorphism ring. Smaller alternative values of modular functions that lie…
Discrete analogs of the index transforms with squares of Bessel functions of the first and second kind $J_\nu(z),\ Y_\nu(z)$ are introduced and investigated. The corresponding inversion theorems for suitable classes of functions and…
Moduli spaces of real bundles over a real curve arise naturally as Lagrangian submanifolds of the moduli space of semi-stable bundles over a complex curve. In this paper, we adapt the methods of Atiyah-Bott's "Yang-Mills over a Riemann…
We present two new problems on lower bounds for resolution Betti numbers of monomial ideals generated in a fixed degree. The first concerns any such ideal and bounds the total Betti numbers, while the second concerns ideals that are…
In general, algorithms for computing the Selmer group of the Jacobian of a curve have relied on either homogeneous spaces or functions on the curve. We present a theoretical analysis of algorithms which use functions on the curve, and show…
We develop the theory of Abelian functions associated with cyclic trigonal curves by considering two new cases. We investigate curves of genus six and seven and consider whether it is the trigonal nature or the genus which dictates certain…
Main purpose of this paper is to reconstruct generating function of the Bernstein type polynomials. Some properties this generating functions are given. By applying this generating function, not only derivative of these polynomials but also…
Modular equations occur in number theory, but it is less known that such equations also occur in the study of deformation properties of quasiconformal mappings. The authors study two important plane quasiconformal distortion functions,…
In this paper, we associate a finite dimensional algebra, called a Brauer graph algebra, to every clean dessin d'enfant by constructing a quiver based on the monodromy of the dessin. We show that Galois conjugate dessins d'enfants give rise…
We present a database of several hundred modular forms up to and including weight six on noncongruence subgroups of index $\leq 17$. In addition, our database contains expressions for the Belyi map for genus zero subgroups and equations of…
Recently, it has become evident that submodularity naturally captures widely occurring concepts in machine learning, signal processing and computer vision. Consequently, there is need for efficient optimization procedures for submodular…
In this note we study $SL(2,\mathbb{Z})$-invariant functions such as modular graph functions or coefficient functions of higher derivative corrections in type IIB string theory. The functions solve inhomogeneous Laplace equations and we…
In this paper, we present formulas for the edge zeta function and the second weighted zeta function with respect to the group matrix of a finite abelian group $\Gamma $. Furthermore, we give another proof of Dedekind Theorem for the group…
In 2003, Kedlaya gave an algorithm to compute the zeta function associated to a hyperelliptic curve over a finite field, by computing the rigid cohomology of the curve. Edixhoven remarked that it is actually possible to compute the…
We use sign-reversing involutions to solve two computational problems that arise naturally in the geometry of moduli spaces of curves. In particular, we give an explicit combinatorial formula for arbitrary $\psi$ class intersection products…
We study the Drinfeld modular curves arising from the Hecke congruence subgroups of $\mathrm{SL}_2(\mathbb{F}_q[T])$. Using a combinatorial method of Gekeler and Nonnengardt, we obtain a genus formula for these curves. In cases when the…
We study special values of a modular function $\Lambda$ which is one of generalized $\lambda$ functions. We show special values of $\Lambda$ at imaginary quadratic points are algebraic integers. Further we prove that $\Lambda$ and the…