Related papers: Elliptic function of level 4
We study elliptic vortices on $\mathbb{C}\times T^2$ by considering the 2d quiver gauge theory describing their moduli spaces. The elliptic genus of these moduli spaces is the elliptic version of vortex partition function of the 4d theory.…
We prove in this paper the global Lorentz estimate in term of fractional-maximal function for gradient of weak solutions to a class of p-Laplace elliptic equations containing a non-negative Schr\"odinger potential which belongs to reverse…
We derive new integral representations for objects arising in the classical theory of elliptic functions: the Eisenstein series $E_s$, and Weierstrass' $\wp$ and $\zeta$ functions. The derivations proceed from the Laplace-Mellin…
In this paper, we prove two structural theorems on the general Berndt-type integrals with the denominator having arbitrary positive degrees by contour integrations involving hyperbolic and trigonometric functions, and hyperbolic sums…
In the paper, properties of orbit functions are reviewed and further developed. Orbit functions on the Euclidean space $E_n$ are symmetrized exponential functions. The symmetrization is fulfilled by a Weyl group corresponding to a…
We give a brief review of the main results of the theory of elliptic hypergeometric functions -- a new class of special functions of mathematical physics. We prove the most general univariate exact integration formula generalizing Euler's…
In this work we describe a construction that leads to an explicit solution of the problem of differentiation of hyperelliptic functions. A classical genus $g=1$ example of such a solution is a result of F.G.Frobenius and L.Stickelberger.…
For a finite field $\mathbb{F}_q$ of characteristic $p\geq 5$ and $K=\mathbb{F}_q(t)$, we consider the family of elliptic curves $E_d$ over $K$ given by $y^2+xy - t^dy=x^3$ for all integers $d$ coprime to $q$. We provide an explicit…
This letter continues the program aimed at analysis of the scalar product of states in the Chern-Simons theory. It treats the elliptic case with group SU(2). The formal scalar product is expressed as a multiple finite dimensional integral…
In this paper we establish the existence and multiplicity of nontrivial solutions to the following problem \begin{align*} \begin{split} (-\Delta)^{\frac{1}{2}}u+u+(\ln|\cdot|*|u|^2)&=f(u)+\mu|u|^{-\gamma-1}u,~\text{in}~\mathbb{R},…
For almost all tuples $(x_1,\dots,x_n)$ of complex numbers, a strong version of Schanuel's Conjecture is true: the $2n$ numbers $x_1,\dots,x_n, {\mathrm e}^{x_1},\dots, {\mathrm e}^{x_n}$ are algebraically independent. Similar statements…
Hinted by the elliptic parameterization of the Ising model, the addition formula of the elliptic function forms to give the integrable SU(2) group relation in the previous paper. We then expect that the addition formula of the Abelian…
We develop a formal group--theoretic framework for the Riemann zeta function by treating its Euler product as an element of the multiplicative formal group $\widehat{\mathbb{G}}_m$ and its logarithm as the associated formal group logarithm.…
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…
Jacobi elliptic functions and complete elliptic integrals are generalized using three parameters. These generalized functions and integrals are closely related to ordinary differential equations involving $p$-Laplacian. In this paper,…
The general analytic solution to the functional equation $$ \phi_1(x+y)= { { \biggl|\matrix{\phi_2(x)&\phi_2(y)\cr\phi_3(x)&\phi_3(y)\cr}\biggr|} \over { \biggl|\matrix{\phi_4(x)&\phi_4(y)\cr\phi_5(x)&\phi_5(y)\cr}\biggr|} } $$ is…
By using the degenerate Whittaker functions, we study the Fourier expansion of the Gan-Gurevich lifts which are Hecke eigen quaternionic cusp forms of weight $k$ ($k\geq 2$, even) on the split exceptional group $G_2$ over $\mathbb{Q}$ which…
The elliptic genus for arbitrary two dimensional $N=2$ Landau-Ginzburg orbifolds is computed. This is used to search for possible mirror pairs of such models. An important aspect of this work is that there is no restriction to theories for…
This is an English translation and digitisation of Frobenius' and Stickelberger's "On the theory of elliptic functions" first published in Journal fur die reine und angewandte Mathematik (Crelle's journal), 83, 175-179 (1877) with the title…
We construct real Jacobi forms with matrix index using path integrals. The path integral expressions represent elliptic genera of two-dimensional N=(2,2) supersymmetric theories. They arise in a family labeled by two integers N and k which…