Related papers: A three dimensional ball quotient
A triangular form is defined to be an integer-valued quadratic polynomial of the form $a_1P_3(x_1)+a_2P_3(x_2)+\cdots+a_kP_3(x_k)$ where $a_i's$ are positive integers and $P_3(x)=x(x+1)/2$. A triangular form is called regular if it…
We construct examples of modular rigid Calabi--Yau threefolds, which give a realization of some new weight 4 cusp forms.
The purpose of this paper is to construct a crepant resolution of quotient singularities by finite subgroups of SL(3,C) of monomial type, and prove that the Euler number of the resolution is equal to the number of conjugacy classes. This…
We study a class of triangulated categories obtained as Verdier quotients of 3-Calabi-Yau categories combinatorially described by quivers with potential from (decorated) marked surfaces. We study their bounded t-structures and consider in…
In this paper, we find formulas for the number of representations of certain diagonal octonary quadratic forms with coefficients $1,2,3,4$ and $6$. We obtain these formulas by constructing explicit bases of the space of modular forms of…
We shall construct a 5-dimensional linear system of holomorphic automorphic forms on a 3-dimensional complex ball by applying Borcherds theory of automorphic forms. We shall show that this linear system gives the dual map from the Segre…
We undertake a systematic scan of vector bundles over spaces from the largest database of known Calabi-Yau three-folds, in the context of heterotic string compactification. Specifically, we construct positive rank five monad bundles over…
We study algebras of meromorphic modular forms whose poles lie on Heegner divisors for orthogonal and unitary groups associated to root lattices. We give a uniform construction of $147$ hyperplane arrangements on type IV symmetric domains…
Recently, Allen, Grove, Long, and Tu proposed an explicit Hypergeometric-Modularity method which gives a concrete link between certain hypergeometric objects and modular forms. The theory is exemplified by a collection of 199 weight 3…
Using the theory of polarised flag type quotients, we determine mass formulae for all principally polarised supersingular abelian threefolds defined over an algebraically closed field $k$ of characteristic $p$. We combine these results with…
Every $\mathbb{A}^{1}-$bundle over the complex affine plane punctured at the origin, is trivial in the differentiable category but there are infinitely many distinct isomorphy classes of algebraic bundles. Isomorphy types of total spaces of…
In analogy with the classical theory of Eichler integrals for integral weight modular forms, Lawrence and Zagier considered examples of Eichler integrals of certain half-integral weight modular forms. These served as early prototypes of a…
The classical modular equations involve bivariate polynomials that can be seen to be univariate with coefficients in the modular invariant $j$. Kiepert found modular equations relating some $\eta$-quotients and the Weber functions…
Given a holomorphic newform $f$ of weight $k$ and with rational coefficients, a question of Mazur and van Straten asks if there is an associated Calabi-Yau variety $X$ over ${\mathbb Q}$ of dimension $k-1$ such that the $\ell$-adic Galois…
In this letter we give an overview on recent developments in representation theory of star product algebras. In particular, we relate the *-representation theory of *-algebras over rings C = R(i) with an ordered ring R and i^2 = -1 to the…
In the 90s, based on presentations of 3-manifolds by Heegaard diagrams, Kuperberg associated a scalar invariant of 3-manifolds to each finite dimensional involutory Hopf algebra over a field. We generalize this construction to the case of…
In this paper, we give a method to construct a classical modular form from a Hilbert modular form. By applying this method, we can get linear formulas which relate the Fourier coefficients of the Hilbert and classical modular forms. The…
In this paper, we study Fontaine-Laffaille, self-dual deformations of a mod p non-semisimple Galois representation of dimension n with its Jordan-Holder factors being three mutually non-isomorphic absolutely irreducible representations. We…
We find and propose an explanation for a large variety of modularity-related symmetries in problems of 3-manifold topology and physics of 3d $\mathcal{N}=2$ theories where such structures a priori are not manifest. These modular structures…
We introduce one of the most beautiful algebraic varieties known, a quintic hypersurface in projective five-space, which is invariant under the action of the Weyl group of $E_6$. This variety is intricately related with many other moduli…