Related papers: On modular forms of rational weight satisfying the…
In this paper, we explore a two-way connection between quasimodular forms of depth $1$ and a class of second-order modular differential equations with regular singularities on the upper half-plane and the cusps. Here we consider the cases…
We consider the Knizhnik-Zamolodchikov system of linear differential equations. The coefficients of this system are rational functions generated by elements of the symmetric group $S_{n}$. We assume that parameter $\rho=\pm{1}$. In previous…
In this paper we show that Atkin and Swinnerton-Dyer type of congruences hold for weakly modular forms (modular forms that are permitted to have poles at cusps). Unlike the case of original congruences for cusp forms, these congruences are…
If $\beta\in\CC^d$ is integral but not a strongly resonant parameter for the homogeneous matrix $A\in\ZZ^{d\times n}$ with $\ZZ A=\ZZ^d$, then the associated GKZ-system carries a naturally defined mixed Hodge module structure. We study here…
The elliptic quantum Knizhnik-Zamolodchikov-Bernard (qKZB) difference equations associated to the elliptic quantum group $E_{\tau,\eta}(sl_2)$ is a system of difference equations with values in a tensor product of representations of the…
We prove that weights of two Siegel modular forms of nonquadratic nebentypus should satisfy some congruence relations if these modular forms are congruent to each other. Applying this result, we prove that there are no mod $p$ singular…
Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra, $U_q(\mathfrak{g})$ its quantum group, and $U_q(\mathfrak{k}) \subset U_q(\mathfrak{g})$ a quantum symmetric pair subalgebra determined by a Lie algebra automorphism $\theta$. We…
We construct special solutions to the rational quantum Knizhnik-Zamolodchikov equation associated with the Lie algebra $gl_N$. The main ingredient is a special class of the shifted non-symmetric Jack polynomials. It may be regarded as a…
We study the weight modules over affine Kac-Moody algebras from the view point of vertex algebras, and determine the abelian category of weight modules for the simple affine vertex algebra $L_k(\mathfrak{sl}_2)$ at any non-integral…
For each prime $\ell$, let $|\cdot|_\ell$ be an extension to $\bar \Q$ of the usual $\ell$-adic absolute value on $\Q$. Suppose $g(z) = \sum_{n=0}^\infty c(n)q^n \in M_{k+\half}(N)$ is an eigenform whose Fourier coefficients are algebraic…
We establish an explicit bijection between the sets of singular solutions of the (super) KZ equations associated to the Lie superalgebra, of infinite rank, of type $\mf{a, b,c,d}$ and to the corresponding Lie algebra. As a consequence, the…
Let k be an algebraically closed field and A be a finitely generated, centrally finite, non- negatively graded (not necessarily commutative) k-algebra. In this note we construct a representation scheme for graded maximal Cohen-Macaulay A…
A simple relation between inhomogeneous transfer matrices and boundary quantum KZ equations is exhibited for quantum integrable systems with reflecting boundary conditions, analogous to an observation by Gaudin for periodic systems. Thus…
We construct integral representations of solutions to the boundary quantum Knizhnik-Zamolodchikov equations. These are difference equations taking values in tensor products of Verma modules of quantum affine $\mathfrak{sl}_2$, with the…
We consider a composite generalized quantum integrable model solvable by the nested algebraic Bethe ansatz. Using explicit formulas of the action of the monodromy matrix elements onto Bethe vectors in the GL(3)-based quantum integrable…
We prove general Dwork-type congruences for Hasse--Witt matrices attached to tuples of Laurent polynomials. We apply this result to establishing arithmetic and $p$-adic analytic properties of functions originating from polynomial solutions…
We give a new, simple proof of the trace formula for Hecke operators on modular forms for finite index subgroups of the modular group. The proof uses algebraic properties of certain universal Hecke operators acting on period polynomials of…
Using the framework relating hypergeometric motives to modular forms, we define an explicit family of weight 2 Hecke eigenforms with complex multiplication. We use the theory of ${}_2F_1(1)$ hypergeometric series and Ramanujan's theory of…
Zhao and the second author (2013) constructed a functor from o(k)-Mod to o(k + 2)-Mod. In this paper, we use the functor successively to obtain an universal first-order differential operator realization for any highest-weight representation…
We prove a higher weight general Gross--Zagier formula for CM cycles on Kuga--Sato varieties over modular curves of arbitrary levels. To formulate and prove this result, we prove several results on the modularity of CM cycles, in the sense…