Related papers: Modular Cauchy kernel corresponding to the Hecke c…

In this paper, we proved an arithmetic Siegel-Weil formula and the modularity of some arithmetic theta function on the modular curve $X_0(N)$ when $N$ is square free. In the process, we also construct some generalized Delta function for…

We give an explicit construction of the quantum-group generators ---local, semi-local, and global --- in terms of the group-valued quantum fields $\tilde g$ and $\tilde g^{-1}$ in the Wess-Zumino-Novikov-Witten (WZNW) theory. The algebras…

This paper consists in a very brief English summary of the results appearing in French in two previous articles. We omit the proofs and focus on explaining our approach and theorems. This paper is not intended to be published. Questions are…

We consider correlation functions for a vertex operator algebra on a genus two Riemann surface formed by sewing two tori together. We describe a generalisation of genus one Zhu recursion where we express an arbitrary genus two $n$--point…

Let $\mathcal{L}(\mathscr{H})$ denote the $C^*$-algebra of adjointable operators on a Hilbert $C^*$-module $\mathscr{H}$. We introduce the generalized Cauchy-Schwarz inequality for operators in $\mathcal{L}(\mathscr{H})$ and investigate…

We begin the systematic study of cohomological Hecke operators of modifications of coherent sheaves on a smooth surface $X$, along a fixed proper curve $Z \subset X$. We develop the necessary geometric foundations in order to define the…

The study of the Dirac system and second-order elliptic equations with complex-valued coefficients on the plane leads to bicomplex Vekua equations. To the difference of complex pseudoanalytic (generalized analytic) functions the theory of…

Let S_{w+2}(\Gamma_0(N)) be the vector space of cusp forms of weight w+2 on the congruence subgroup \Gamma_0(N). We first determine explicit formulas for period polynomials of elements in S_{w+2}(\Gamma_0(N)) by means of Bernoulli…

In the context of arithmetic surfaces, Bost defined a generalized Arithmetic Chow Group (ACG) using the Sobolev space L^2_1. We study the behavior of these groups under pull-back and push-forward and we prove a projection formula. We use…

In this paper, we consider a two-dimensional operator with an antisymmetric integral kernel, recently introduced by Z. Avetisyan and A. Karapetyants in connection to the study of general homogeneous operators. This is the unique…

A commuting $n$-tuple $(T_1, \ldots, T_n)$ of bounded linear operators on a Hilbert space $\clh$ associate a Hilbert module $\mathcal{H}$ over $\mathbb{C}[z_1, \ldots, z_n]$ in the following sense: \[\mathbb{C}[z_1, \ldots, z_n] \times…

The Green functions of the partial differential operators of even order acting on smooth sections of a vector bundle over a Riemannian manifold are investigated via the heat kernel methods. We study the resolvent of a special class of…

Serre proved that any holomorphic cusp form of weight one for $\Gamma_1(N)$ is lacunary while a holomorphic modular form for $\Gamma_1(N)$ of higher integer weight is lacunary if and only if it is a linear combination of cusp forms of…

We prove the following two results 1. For a proper holomorphic function $ f : X \to D$ of a complex manifold $X$ on a disc such that $\{df = 0 \} \subset f^{-1}(0)$, we construct, in a functorial way, for each integer $p$, a geometric…

In this paper, we prove the category of finite length modules for the $\mathbb{Z}_2$-orbifold $M(1)^+$ of the Heisenberg vertex operator algebra whose simple composition factors are $M(1)^\pm$ or $M(1,\lambda)$ for $\lambda \in…

Let $\Gamma$ be a bounded Jordan curve and $\Omega_i,\Omega_e$ its two complementary components. For $s\in(0,1)$ we define $\mathcal{H}^s(\Gamma)$ as the set of functions $f:\Gamma\to \mathbb C$ having harmonic extension $u$ in…

A connection between the zeta functions of zeros and poles of a meromorphic function has been established, and using it, a criterion for the absence of zeros has been derived. Sufficient conditions for the existence of zeros of sums of…

We prove a general theorem for constructing integral quantum cluster algebras over ${\mathbb{Z}}[q^{\pm 1/2}]$, namely that under mild conditions the integral forms of quantum nilpotent algebras always possess integral quantum cluster…

We develop a noncommutative invariant theory for ordinary linear differential operators on Riemann surfaces. For a monic binomially normalized operator $L=\sum_{k=0}^n {n\choose k}a_kD^{\,n-k}$, $a_0=1$, with coefficients in an associative…

It is shown that Ramanujan's cubic transformation of the Gauss hypergeometric function ${}_2F_1$ arises from a relation between modular curves, namely the covering of $X_0(3)$ by $X_0(9)$. In general, when $2\le N\le 7$ the N-fold cover of…