Related papers: On Atkin-Swinnerton-Dyer congruence relations

We investigate various aspects of higher-spin anti-de Sitter supergravity in three dimensions as described by Chern-Simons theory based on the finite-dimensional superalgebra sl(N |N - 1), with the particular case of N = 3 as our prime…

We considerably improve Ono's and Ahlgren-Ono's work on the frequent occurrence of Ramanujan-type congruences for the partition function, and demonstrate that Ramanujan-type congruences occur in families that are governed by square-classes.…

Each quiver corresponds to a path semigroup, and such a path semigroup also corresponds to an associative K-algebra over an algebraically closed field K. Let Q be a quiver and S_Q, KQ be its path semigroup, path algebra, respectively. In…

Massless higher-spin fields show no preference for any value of the cosmological constant in $3d$. All matter-free higher-spin gravities in 3d are equivalent to Chern-Simons theories with an appropriate choice of gauge algebra. For various…

The Chern-Simons formulation of $AdS_3$ supergravity is considered. Asymptotic conditions on the Rarita-Schwinger fields are given. Together with the known boundary conditions on the bosonic fields, these ensure that the asymptotic algebra…

The generalized connections of the (anti)-de Sitter space symmetry algebra, which are differential forms of arbitrary degree with values in any irreducible (spin)-tensor representation of the (anti)-de Sitter algebra, are studied. It is…

In 2003, Rodriguez-Villegas conjectured four supercongruences on the truncated ${}_3F_2$ hypergeometric series for certain modular K3 surfaces, which were gradually proved by several authors. Motivated by some supercongruences on…

We study the Aharony duality for three dimensional $\mathcal N=2$ supersymmetric gauge theories for orthogonal gauge groups with matters in vector representation. We provide the evidence for the duality by working out the partition function…

Using the relationship between Siegel cusp forms of degree $2$ and cuspidal automorphic representations of $\mathrm{GSp}(4,\mathbb{A}_{\mathbb{Q}})$, we derive some congruences involving dimensions of spaces of Siegel cusp forms of degree…

We use a (pre)-Kuznetsov type formula to prove a density result for the Borel-type congruence subgroup of GLn. This has some arithmetic applications to optimal lifting and counting considered earlier by A. Kamber and H. Lavner for $GL_3$.

We present a new family of dualities for three-dimensional gauge theories, motivated by the brane realization of the reduction of four-dimensional dualities on a circle. This family can be understood as a generalization of Aharony duality…

The abelian and monoidal structure of the category of smooth weight modules over a non-integrable affine vertex algebra of rank greater than one is an interesting, difficult and essentially wide open problem. Even conjectures are lacking.…

We study Baxter's T-Q equation of XXX spin-chain models under the semiclassical limit where an intriguing SU(N)/SU(2)^{N-3} correspondence emerges. That is, two kinds of 4D \mathcal{N}=2 superconformal field theories having the above…

Ng and Schauenburg proved that the kernel of a $(2+1)$-dimensional topological quantum field theory representation of $\mathrm{SL}(2, \mathbb{Z})$ is a congruence subgroup. Motivated by their result, we explore when the kernel of an…

We prove a version of Blattner's conjecture, for irreducible subquotients of principal series representations with integral infinitesimal character of a real reductive Lie group whose Beilinson-Bernstein D-module is supported on a K-orbit…

We prove the conjectures on the ($L^{\infty}$)-sizes of the spaces of Siegel cusp forms of degree $n$, weight $k$, for any congruence subgroup in the weight aspect as well as for all principal congruence subgroups in the level aspect, in…

In this paper we partially settle Fock-Goncharov's duality conjecture for cluster varieties associated to their moduli spaces of ${\rm G}$-local systems on a punctured surface $\frak{S}$ with boundary data, when ${\rm G}$ is a group of type…

We explicitly determine the locations of G orbifold conformal field theories, G=Z_M, M=2,3,4,6, G=\hat D_n, n=4,5, or G the binary tetrahedral group \hat T, within the moduli space M^{K3} of N=(4,4) superconformal field theories associated…

In this paper we address the issue of existence of cusp forms for almost simple Lie groups using the approach of the second author combined with local information on supercuspidal representations for $p$-adic groups known by the first…

We prove two results on some special generators of finite simple groups and use them to prove that every non-abelian finite simple group $S$ admits a non-congruence presentation (as conjectured in [CLT24]), and that if $S$ has a non-trivial…