Related papers: The elliptic function ${\rm dn}_3$ of Shen
Classically, Euler developed the theory of the Riemann zeta - function using as his starting point the exponential and partial fraction forms of cot(z) . In this paper we wish to develop the theory of $L$-functions of elliptic curves…
Recently, a class of partition functions associated with higher rank rational and trigonometric integrable models were introduced by Foda and Manabe. We use the dynamical $R$-matrix of the elliptic quantum group $E_{\tau,\eta}(gl_3)$ to…
On page 206 in his lost notebook, Ramanujan recorded the following enigmatic identity for his theta function $\varphi(q)$: \begin{equation*} \varphi(e^{-7\pi\sqrt{7}}) = 7^{-3/4}\varphi(e^{-\pi\sqrt{7}})\big\{1 + (\quad)^{2/7} +…
M5-branes on an associative three-cycle $M_3$ in a $G_2$-holonomy manifold give rise to a 3d $\mathcal{N}=1$ supersymmetric gauge theory, $T_{\mathcal{N}=1} [M_3]$. We propose an $\mathcal{N}=1$ 3d-3d correspondence, based on two…
We study fully nonlinear elliptic equations on Hermitian manifolds through blow-up argument and partial uniform ellipticity. We apply our results to draw geometric conclusions on finding conformal Hermitian metrics with prescribed…
We define a general class of (multiple) integrals of hypergeometric type associated with the Jacobi theta functions. These integrals are related to theta hypergeometric series through the residue calculus. In the one variable case, we get…
We prove a pair of transformation formulas for multivariate elliptic hypergeometric sum/integrals associated to the $A_n$ and $BC_n$ root systems, generalising the formulas previously obtained by Rains. The sum/integrals are expressed in…
The recently proposed expression for the general three point function of exponential fields in quantum Liouville theory on the sphere is considered. By exploiting locality or crossing symmetry in the case of those four-point functions,…
In this paper, we extend the nontangential maximal function estimate obtained by C. Kenig, F. Lin and Z. Shen in \cite{KFS1} to the nonhomogeneous elliptic operators with rapidly oscillating periodic coefficients. The result relies on the…
We study the three-dimensional lens partition function for $\mathcal N=2$ supersymmetric gauge dual theories on $S^3/\mathbb{Z}_r$ by using the gauge/YBE correspondence. This correspondence relates supersymmetric gauge theories to exactly…
We study the root of unity limit of the lens elliptic gamma function solution of the star-triangle relation, for an integrable model with continuous and discrete spin variables. This limit involves taking an elliptic nome to a primitive…
We consider two sequences $a(n)$ and $b(n)$, $1\leq n<\infty$, generated by Dirichlet series of the forms $$\sum_{n=1}^{\infty}\frac{a(n)}{\lambda_n^{s}}\qquad\text{and}\qquad \sum_{n=1}^{\infty}\frac{b(n)}{\mu_n^{s}},$$ satisfying a…
A generalization of Jacobi's elliptic functions is introduced as inversions of hyperelliptic integrals. We discuss the special properties of these functions, present addition theorems and give a list of indefinite integrals. As a physical…
The present paper is devoted to the problem about the reduction of hyperelliptic functions of genus 3. Our research was motivated by applications to the theory of equations and dynamical systems integrable in hyperelliptic functions. In…
Based on the theory of rigid cohomology, we provide an explicit formula of zeta functions of certain K3 families, which we call the hypergeometric type. The central point of our argument is the comparison between the 2nd rigid cohomology of…
An exact formula for partition functions in 3d field theories was recently suggested by Jafferis, and Hama, Hosomichi, and Lee. These functions are expressed in terms of specific $q$-hypergeometric integrals whose key building block is the…
For primes p congruent to 1 mod 12, we present an explicit relation between the traces of Frobenius on a family of elliptic curves with j-invariant 1728/t and values of a particular 2F1-hypergeometric function over F_p. Additionally, we…
We provide elliptic extensions of elementary identities such as the sum of the first $n$ odd or even numbers, the geometric sum and the sum of the first $n$ cubes. Many such identities, and their $q$-analogues, are indefinite sums, and can…
In this paper we give a survey of elliptic theory for operators associated with diffeomorphisms of smooth manifolds. Such operators appear naturally in analysis, geometry and mathematical physics. We survey classical results as well as…
Orbifold elliptic genus and elliptic genus of singular varieties are introduced and relation between them is studied. Elliptic genus of singular varieties is given in terms of a resolution of singularities and extends the elliptic genus of…