Related papers: Quantum modularity of partial theta series with pe…

The study of arithmetic properties of coefficients of modular forms $f(\tau) = \sum a(n)q^n$ has a rich history, including deep results regarding congruences in arithmetic progressions. Recently, work of C.-S. Radu, S. Ahlgren, B. Kim, N.…

An eta-quotient of level $N$ is a modular form of the shape $f(z) = \prod_{\delta | N} \eta(\delta z)^{r_{\delta}}$. We study the problem of determining levels $N$ for which the graded ring of holomorphic modular forms for $\Gamma_{0}(N)$…

Let $f$ be a newform of weight $k\geq 2$, level $N$ with coefficients in a number field $K$, and $A$ the adjoint motive of the motive $M$ associated to $f$. We carefully discuss the construction of the realisations of $M$ and $A$, as well…

We compute the rational cohomology of the moduli space of non-singular non-hyperelliptic complex projective curves of genus 3 with an odd theta characteristic.

In this paper we study of the BGG-categories $\mathcal O_q$ associated to quantum groups. We prove that many properties of the ordinary BGG-category $\mathcal O$ for a semisimple complex Lie algebra carry over to the quantum case. Of…

We suggest a new construction for the Quantum Groups - Jones polynomials of torus knots in terms of the PBW theorem of DAHA for any root systems and weights (justified for type A). The main focus is on the DAHA super-polynomials, a stable…

In 2015, Lovejoy and Osburn discovered twelve $q$-hypergeometric series and proved that their Fourier coefficients can be understood as counting functions of ideals in certain quadratic fields. In this paper, we study their modular and…

The colored Jones polynomial is a $q$-polynomial invariant of links colored by irreducible representations of a simple Lie algebra. A $q$-series called a tail is obtained as the limit of the $\mathfrak{sl}_2$ colored Jones polynomials…

We examine values of certain Tornheim's type of double series with odd weight. As a result, an affirmative answer to a conjecture about the parity theorem for the zeta function of the root system of the exceptional Lie algebra $G_2$,…

In this work we demonstrate that the q-numbers and their two-parameter generalization, the q,p-numbers, can be used to obtain some polynomial invariants for torus knots and links. First, we show that the q-numbers, which are closely…

We extend the table of Garoufalidis, Le and Zagier concerning conjectural Rogers-Ramanujan type identities for tails of colored Jones polynomials to all alternating knots up to 10 crossings. We then prove these new identities using q-series…

In the course of the proof of the irrationality of zeta(2) R. Apery introduced numbers b_n = \sum_{k=0}^n {n \choose k}^2{n+k \choose k}. Stienstra and Beukers showed that for the prime p > 3 Apery numbers satisfy congruence b((p-1)/2) =…

The paper is concerned with cohomology of the small quantum group at a root of unity, and of its upper triangular subalgebra, with coefficients in a tilting module. We relate it to a certain t-structure on the derived category of…

We determine the structure over $\mathbb{Z}$ of the ring of symmetric Hermitian modular forms with respect to $\mathbb{Q}(\sqrt{-1})$ of degree $2$ (with a character), whose Fourier coefficients are integers. Namely, we give a set of…

We exhibit a quantum algorithm for determining the zeta function of a genus g curve over a finite field F_q, which is polynomial in g and log(q). This amounts to giving an algorithm to produce provably random elements of the class group of…

We prove Zagier's conjecture on the value at s=4 of the Dedekind zeta-function of a number field F. For any field F, we define a map from the appropriate pieces of algebraic K-theory of F to the cohomology of the weight 4 polylogarithmic…

We prove that, for the moduli space of flat SU(2)-connections on the torus, the Weyl quantization and the quantization using the quantum group of SL(2,C) are unitarily equivalent. This is done by comparing the matrices of the operators…

We study families of two dimensional quantum field theories, labeled by a dimensionful parameter $\mu$, that contain a holomorphic conserved $U(1)$ current $J(z)$. We assume that these theories can be consistently defined on a torus, so…

The problem on the construction of antisymmetric paramodular forms of canonical weight 3 was open since 1998. Any cusp form of this type determines a canonical differential form on any smooth compactification of the moduli space of Kummer…

We address two linked problems at the interface of quantum topology and number theory: deriving asymptotic expansions of the Witten--Reshetikhin--Turaev invariants for 3-manifolds and establishing quantum modularity of false theta…