Related papers: Modularity from $q$-series
Two $q$-supercongruences of truncated basic hypergeometric series containing two free parameters are established by employing specific identities for basic hypergeometric series. The results partly extend two $q$-supercongruences that were…
Let $h(x,y)$ be a non-degenerate binary cubic form with integral coefficients, and let $S$ be an arbitrary finite set of prime numbers. By a classical theorem of Mahler, there are only finitely many pairs of relatively prime integers $x,y$…
Using the method of the $q$-exponential differential operator, we give an extension of the Sears $_4\phi_3$ transformation formula. Based on this extended formula and a $q$-series expansion formula for an analytic function around the…
We introduce the combinatorial notion of a $q$-fatorization graph intended as a tool to study and express results related to the classification of prime simple modules for quantum affine algebras. These are directed graphs equipped with…
A sequence of rational functions in a variable $q$ is $q$-holonomic if it satisfies a linear recursion with coefficients polynomials in $q$ and $q^n$. We prove that the degree of a $q$-holonomic sequence is eventually a quadratic…
Consider the polynomial ring in countably infinitely many variables over a field of characteristic zero, together with its natural action of the infinite general linear group G. We study the algebraic and homological properties of finitely…
We study finite-dimensional representations of quantum affine algebras using q-characters. We prove the conjectures from math.QA/9810055 and derive some of their corollaries. In particular, we prove that the tensor product of fundamental…
In this paper, we construct a new class of modules over the Block algebra $\BB(q)$, where $q$ is a nonzero complex number. We determined the irreducibilities of these modules and the isomorphisms among them.
Let $q=e^{2\pi i\tau}$, $\Im\tau>0$, $x=e^{2\pi i\xi}\in\CC$ and $(x;q)_\infty=\prod_{n\ge 0}(1-xq^n)$. Let $(q,x)\mapsto(q^*,\iota_q x)$ be the classical modular substitution given by $q^*=e^{-2\pi i/\tau}$ and $\iota_q x=e^{2\pi…
Let $\lambda(n)$ denote the Fourier coefficients of a fixed modular form and $h(n)$ a Steinhaus or Rademacher random multiplicative function. In this paper, we determine the order of magnitude of \[ \E|\sum_{n \leq x} h(n)\lambda(n)|^{2q}…
The basic properties of the Temperley-Lieb algebra $TL_n$ with parameter $\beta = q + q^{-1}$, for $q$ any non-zero complex number, are reviewed in a pedagogical way. The link and standard (cell) modules that appear in numerous physical…
In 1975, Goldfeld gave an effective solution to Gauss's conjecture on the class numbers of imaginary quadratic fields. In this paper, we generalize Goldfeld's theorem to the setting of totally real number fields.
The concept of $q$-deformation, or ``$q$-analogue'' arises in many areas of mathematics. In algebra and representation theory, it is the origin of quantum groups; $q$-deformations are important for knot invariants, combinatorial…
Sander Zwegers showed that Ramanujan's mock theta functions are $q$-hypergeometric series, whose $q$-expansion coefficients are half of the Fourier coefficients of a non-holomorphic modular form. George Andrews, Henri Cohen, Freeman Dyson,…
Recently, Alanazi, Munagi, and Saikia employed the theory of modular forms to investigate the arithmetic properties of the function $\overline{R_{\ell,\mu}}(n)$, which enumerates the overpartitions of $n$ where no part is divisible by…
We consider a $q$-analogue of abstract simplicial complexes, called $q$-complexes, and discuss the notion of shellability for such complexes. It is shown that $q$-complexes formed by independent subspaces of a $q$-matroid are shellable.…
The coset $G$-space of a finite group and a subgroup is a fundamental module of study of Schur and others around 1930; for example, its endomorphism algebra is a Hecke algebra of double cosets. We study and review its generalization $Q$ to…
This paper provides some statistics for the coefficients of Russell- Type modular equations for the modular function, {\lambda}({\tau}). The results hold uniformly for all odd primes. They do not rely on any numerical evaluations of…
Theta series for indefinite quadratic lattices were introduced by Zwegers, for signature (m-1,1), Alexandrov, Banerjee, Manschot and Pioline, for signature (m-2,2), and Nazaroglu, for signature (m-q,q). These series are modular modular…
Using the $L^2$ norm of the Higgs field as a Morse function, we study the moduli spaces of $U(p,q)$-Higgs bundles over a Riemann surface. We require that the genus of the surface be at least two, but place no constraints on $(p,q)$. A key…