Related papers: Modular forms on noncongruence subgroups and Atkin…

We point out the existence of nonlinear $\sigma$-models on group manifolds which are left symmetric and right Poisson-Lie symmetric. We discuss the corresponding rich T-duality story with particular emphasis on two examples: the anisotropic…

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…

We study orbifolds of ${\cal N} = 4$ U(n) super-Yang-Mills theory given by discrete subgroups of SU(2) and SU(3). We have reached many interesting observations that have graph-theoretic interpretations. For the subgroups of SU(2), we have…

We classify the irreducible representations of smooth, connected affine algebraic groups over a field, by tackling the case of pseudo-reductive groups. We reduce the problem of calculating the dimension for pseudo-split pseudo-reductive…

We prove the modularity of a positive proportion of abelian surfaces over $\mathbf{Q}$. More precisely, we prove the modularity of abelian surfaces which are ordinary at $3$ and are $3$-distinguished, subject to some assumptions on the…

In this paper we express certain multiplicities in modular representation-theoretic categories of type A in terms of affine p-Kazhdan-Lusztig polynomials. The representation-theoretic categories we deal with include the categories of…

We study inert, and ambiguous conjugacy classes in the modular group $\mathrm{PSL}(2,\mathbb{Z})$ from a purely combinatorial perspective. Using word length in the free product representation $\mathbb{Z}_2 * \mathbb{Z}_3$ of the modular…

We define Modular Linear Differential Equations (MLDE) for the level-two congruence subgroups $\Gamma_\vartheta$, $\Gamma^0(2)$ and $\Gamma_0(2)$ of $\text{SL}_2(\mathbb Z)$. Each subgroup corresponds to one of the spin structures on the…

Spinor structure is understood as a totality of tensor products of biquaternion algebras, and the each tensor product is associated with an irreducible representation of the Lorentz group. A so-defined algebraic structure allows one to…

In this paper the authors consider four questions of primary interest for the representation theory of reductive algebraic groups: (i) Donkin's Tilting Module Conjecture, (ii) the Humphreys-Verma Question, (iii) whether $\operatorname{St}_r…

The purpose of this paper is to show how a congruence between (the Fourier coefficients of) a Hilbert cusp form and a Hilbert Eisenstein series of parallel weight $2$ gives rise to congruences between algebraic parts of critical values of…

A pseudomodular group is a discrete subgroup $\Gamma \leq PGL(2,\mathbb{Q})$ which is not commensurable with $PSL(2,\mathbb{Z})$ and has cusp set precisely $\mathbb{Q}\cup\{\infty\}$. The existence of such groups was proved by Long and…

In this note, I find a new property of the congruence lattice, Con$L$, of an SPS lattice $L$ (slim, planar, semimodular, where "slim" is the absence of~$\mathsf M_3$ sublattices) with more than $2$ elements: \emph{there are at least two…

The coset Sp(2,R)/U(1) is parametrized by two real scalar fields. We generalize the formalism of auxiliary tensor (bispinor) fields in U(1) self-dual nonlinear models of abelian gauge fields to the case of Sp(2,R) self-duality. In this new…

We study the congruence problem for subgroups of the modular group that appear as Veech groups of square-tiled surfaces in the minimal stratum of abelian differentials of genus two.

We study congruences modulo powers of a prime $p$ between pairs of $p$-new modular Hecke eigenforms of level $\Gamma_0(p)$ and same weight $k$. Based on explicit computations, we conjecture that every such eigenform $f$ admits a twin to…

We develop the theory of spinorial polyforms associated with bundles of irreducible Clifford modules of non-simple real type, obtaining a precise characterization of the square of an irreducible real spinor in signature $(p-q) =…

We continue our study of Yoshida's lifting, which associates to a pair of automorphic forms on the adelic multiplicative group of a quaternion algebra a Siegel modular form of degree 2. We consider here the case that the automorphic forms…

We consider the quantum-mechanical algebra of observables generated by canonical quantization of $SL(2,R)$ Chern-Simons theory with rational charge on a space manifold with torus topology. We produce modular representations generalizing the…

We extend our previous work arXiv:1012.5721 [hep-th] on the non-compact N=2 SCFT_2 defined as the supersymmetric SL(2,R)/U(1)-gauged WZW model. Starting from path-integral calculations of torus partition functions of both the axial-type…