Weierstrass Sigma Function Research Articles

Casting more light in the shadows: dual Somos-5 sequences

Abstract Motivated by the search for an appropriate notion of a cluster superalgebra, incorporating Grassmann variables, Ovsienko and Tabachnikov considered the extension of various recurrence relations with the Laurent phenomenon to the ring of dual numbers. 
Furthermore, by iterating recurrences with specific numerical values, some particular well-known integer sequences, such as the Fibonacci sequence, Markoff numbers, and Somos sequences, were shown to produce associated ``shadow'' sequences when they were extended to the dual numbers. Here we consider the most general version of the Somos-5 recurrence defined over the ring of dual numbers $\mathbb{D}$ with complex coefficients, that is the ring $\mathbb{C}[\varepsilon]$ modulo the relation $\varepsilon^2=0$. We present three different ways to present the general solution of the initial value problem for Somos-5 and its shadow part: in analytic form, using the Weierstrass sigma function with arguments in $\mathbb{D}$; in terms of the solution of a linear difference equation; and using Hankel determinants constructed from $\mathbb{D}$-valued moments, via a connection with a QRT map over the dual numbers.

Semiabelian varieties and transcendence on Weierstrass sigma functions

Abstract We establish new results on complex and $p$ -adic linear independence on a class of semiabelian varieties. As applications, we obtain transcendence results concerning complex and $p$ -adic Weierstrass sigma functions associated with elliptic curves.

System of Partial Differential Equations and Analytical Representations of the Weierstrass Sigma Function

System of Partial Differential Equations and Analytical Representations of the Weierstrass Sigma Function

Solving the Quispel–Roberts–Thompson maps using Kajiwara–Noumi–Yamada’s representation of elliptic curves

It is well known that the dynamical system determined by a Quispel–Roberts–Thompson map (a QRT map) preserves a pencil of biquadratic polynomial curves on . In most cases this pencil is elliptic, i.e. its generic member is a smooth algebraic curve of genus one, and the system can be solved as a translation on the elliptic fiber to which the initial point belongs. However, this procedure is rather complicated to handle, especially in the normalization process. In this paper, for a given initial point on an invariant elliptic curve, we present a method to construct the solution directly in terms of the Weierstrass sigma function, using Kajiwara–Noumi–Yamada’s parametric representation of elliptic curves.

On the Asymptotics and Distribution of Values of the Jacobi Theta Functions and the Estimate of the Type of the Weierstrass Sigma Functions

A refined asymptotics of the Jacobi theta functions and their logarithmic derivatives have been received. The asymptotics of the Nevanlinna characteristics of the indicated functions and the arbitrary elliptic function have been found. The estimation of the type of the Weierstrass sigma functions has been given.

A universal identity for theta functions of degree eight and applications

Previously, we proved an identity for theta functions of degree eight, and several applications of it were also discussed. This identity is a natural extension of the addition formula for the Weierstrass sigma-function. In this paper we will use this identity to reexamine our work in theta function identities in the past two decades. Hundreds of results about elliptic modular functions, both classical and new, are derived from this identity with ease. Essentially, this general theta function identity is a theta identities generating machine. Our investigation shows that many well-known results about elliptic modular functions with different appearances due to Jacobi, Kiepert, Ramanujan and Weierstrass among others, actually share a common source. This paper can also be seen as a summary of my past work on theta function identities. A conjecture is also proposed.

Tropical sequences associated with Somos sequences

Since the seminal note published by M. Somos in 1989, a great deal of attention of specialists in number theory and adjacent areas are attracted by nonlinear sequences that satisfy a quadratic recurrence relation. At the same time, special attention is paid to the construction of Somos integer sequences and their Laurent property with respect to initial values and coefficients of a recurrence. In the fundamental works of Robinson, Fomin and Zelevinsky the Laurent property of the Somos-k sequence for k = 4, 5, 6, 7 was proved. In the works of Hone, representations for Somos-4 and 5 sequences were found via the Weierstrass sigma function on elliptic curves, and for k = 6 via the Klein sigma function on hyperelliptic curve of genus It should also be noted that the Somos sequences naturally arise in the construction of cryptosystems on elliptic and hyperelliptic curves over a finite field. This is explained by the reason that addition theorems hold for the sequences mentioned above, and they naturally arise when calculating multiple points on elliptic and hyperelliptic curves. For k = 4, 5, 6, 7, the Somos sequences are Laurent polynomials of k initial variables and ordinary polynomials in the coefficients of the recurrence relation. Therefore, these Laurent polynomials can be written as an irreducible fraction with an ordinary polynomial in the numerator with initial values and coefficients as variables. In this case, the denominator can be written as a monomial of the initial variables. Using tropical functions, we prove that the degrees of the variables of the above monomial can be represented as quadratic polynomials in the order index of the element of the Somos sequence, whose free terms are periodic sequences of rational numbers. Moreover, in each case these polynomials and the periods of their free terms are written explicitly

Аналитические и теоретико-числовые свойства двумерных сигма-функций

Обзор посвящен классическим и современным задачам, связанным с целой функцией $\sigma({\bf u};\lambda)$, которая определяется семейством неособых алгебраических кривых рода 2, где ${\bf u}= (u_1,u_3)$, $\lambda=(\lambda_4, \lambda_6, \lambda_8, \lambda_{10})$. Эта функция является аналогом сигма-функции Вейерштрасса $\sigma({{u}};g_2,g_3)$ семейства эллиптических кривых. Логарифмические производные порядка 2 и выше функции ${\sigma({\mathbf{u}};\lambda)}$ порождают поле гиперэллиптических функций от ${\mathbf{u}} = (u_1,u_3)$ на якобианах кривых с фиксированным значением вектора параметров $\lambda$. Мы рассматриваем три ряда Гурвица $\sigma({\mathbf{u}};\lambda)=\sum_{m,n\ge0}a_{m,n}(\lambda)\frac{u_1^mu_3^n}{m!n!}$, $\sigma({\mathbf{u}};\lambda) = \sum_{k\ge 0}\xi_k(u_1;\lambda)\frac{u_3^k}{k!}$ и $\sigma({\mathbf{u}};\lambda) = \sum_{k\ge 0}\mu_k(u_3;\lambda)\frac{u_1^k}{k!}$. Обзор посвящен теоретико-числовым свойствам функций $a_{m,n}(\lambda)$, $\xi_k(u_1;\lambda)$ и $\mu_k(u_3;\lambda)$. Он включает самые последние результаты, доказательства которых использует тот фундаментальный факт, что функция $\sigma ({\mathbf{u}};\lambda)$ определяется системой четырех уравнений теплопроводности в неголономном репере шестимерного пространства.

On Products of Weierstrass Sigma Functions

We prove the following result. Let f : ℂ → ℂ be an even entire function. Assume that there exist 𝛼j, βj : ℂ → ℂ with$$ f\left(x+y\right)f\left(x-y\right)=\sum \limits_{\mathrm{j}=1}^4{\alpha}_j(x){\beta}_j(y),\kern0.5em x,y\in \mathbb{C}. $$ Then f(z) = σL(z) · σΛ(z) · eAz2+C where L and Λ are lattices in ℂ, σL is the Weierstrass sigma function associated with the lattice L, and A,C ∈ ℂ.

A Hirota bilinear equation for Painlevé transcendents PIV, PII and PI

We present some observations on the tau-function for the fourth Painlevé equation. By considering a Hirota bilinear equation of order four for this tau-function, we describe the general form of the Taylor expansion around an arbitrary movable zero. The corresponding Taylor series for the tau-functions of the first and second Painlevé equations, as well as that for the Weierstrass sigma function, arise naturally as special cases, by setting certain parameters to zero.

A modular-invariant modified Weierstrass sigma-function as a building block for lowest-Landau-level wavefunctions on the torus

A “modified” variant of the Weierstrass sigma, zeta, and elliptic functions is proposed whereby the zeta function is redefined by ζ(z) ↦ ζ̃(z) ≡ ζ(z)−γ2z, where γ2 is a lattice invariant related to the almost-holomorphic modular invariant of the quasi-modular-invariant weight-2 Eisenstein series. If ωi is a primitive half-period, ζ̃(ωi) = πωi*/A, where A is the area of the primitive cell of the lattice. The quasiperiodicity of the modified sigma function is much simpler than that of the original, and it becomes the building-block for the modular-invariant formulation of lowest-Landau-level wavefunctions on the torus. It is suggested that the “modified” sigma function is more natural than the original Weierstrass form, which was formulated before quasi-modular forms were understood. For the high-symmetry (square and hexagonal) lattices, the modified and original sigma functions coincide.

Functional equations and Weierstrass sigma-functions

It is proved that if an entire function f: ℂ → ℂ satisfies an equation of the form α 1(x)β 1(y) + α 2(x)β 2(y) + α 3(x)β 3(y), x,y ∈ C, for some α j , β j : ℂ → ℂ and there exist no \({\widetilde \alpha _j}\) and ˜\({\widetilde \beta _j}\) for which \(f\left( {x + y} \right)f\left( {x - y} \right) = {\overline \alpha _1}\left( x \right){\widetilde \beta _1}\left( y \right) + {\overline \alpha _2}\left( x \right){\widetilde \beta _2}\left( y \right)\), then f(z) = exp(Az 2 + Bz + C) ∙ σ Γ(z - z 1) ∙ σ Γ(z - z 2), where Γ is a lattice in ℂ; σ Γ is the Weierstrass sigma-function associated with Γ; A,B,C, z 1, z 2 ∈ ℂ; and \({z_1} - {z_2} \notin \left( {\frac{1}{2}\Gamma } \right)\backslash \Gamma \).

Functional Equations and Weierstrass Sigma-Functions

В работе доказывается, что если целая функция $f\colon\mathbb{C}\to\mathbb{C}$ удовлетворяет уравнению $f(x+y) f(x-y) = \alpha_1(x)\beta_1(y)+ \alpha_2(x)\beta_2(y) + \alpha_3(x)\beta_3(y)$, $x,y\in \mathbb{C}$, с некоторыми $\alpha_j,\beta_j\colon\mathbb{C}\to\mathbb{C}$, причем не существует таких $\tilde \alpha_j$, $\tilde\beta_j$, что $f(x+y) f(x-y) = \tilde\alpha_1(x)\tilde\beta_1(y)+ \tilde\alpha_2(x)\tilde\beta_2(y)$, то $f(z) = \exp(Az^2+ Bz + C) \cdot \sigma_\Gamma (z-z_1)\cdot \sigma_\Gamma (z-z_2)$, где $\Gamma$ - некоторая решетка в $\mathbb{C}$, $\sigma_\Gamma$ - сигма-функция Вейерштрасса, ассоциированная с $\Gamma$, а $A,B,C,z_1,z_2\in\mathbb{C}$, причем $z_1-z_2\notin (\frac{1}{2}\Gamma)\setminus \Gamma$.

General complex envelope solutions of coupled-mode optics with quadratic or cubic nonlinearity

The analytic general solutions for the complex field envelopes are derived using Weierstrass elliptic functions for two and three mode systems of differential equations coupled via quadratic $\chi_2$ type nonlinearity as well as two mode systems coupled via cubic $\chi_3$ type nonlinearity. For the first time, a compact form of the solutions is given involving simple ratios of Weierstrass sigma functions (or equivalently Jacobi theta functions). A Fourier series is also given. All possible launch states are considered. The models describe sum and difference frequency generation, polarization dynamics, parity-time dynamics and optical processing applications.

The periods of the generalized Jacobian of a complex elliptic curve

Abstract We show that the toroidal Lie group G = ℂ2/Λ, where Λ is the lattice generated by (1, 0), (0, 1) and (τ̂, τ͂), with τ̂ ∉ ℝ, is isomorphic to the generalized Jacobian JL of the complex elliptic curve C with modulus τ̂, defined by any divisor class L ≡ (M) + (N) of C fulfilling M − N = [℘ (τ͂) : ℘´(τ͂) : 1] ∈ C. This follows from an apparently new relation between the Weierstrass sigma and elliptic functions.

The Complex Multivariate Gaussian Distribution

Here I introduce cmvnorm, a complex generalization of the mvtnorm package. A complex generalization of the Gaussian process is suggested and numerical results presented using the package. An application in the context of approximating the Weierstrass sigma function using a complex Gaussian process is given.

Series and integral representations of the Taylor coefficients of the Weierstrass sigma-function

We provide two kinds of representations for the Taylor coefficients of the Weierstrass σ-function σ(⋅;Γ) associated to an arbitrary lattice Γ in the complex plane \(\mathbb{C}=\mathbb{R}^{2}\), the first one in terms of the so-called Hermite–Gauss series over Γ and the second one in terms of Hermite–Gauss integrals over \(\mathbb{C}\).

On elliptic solutions of a coupled nonlinear Schrödinger system

An explicit formula is obtained for single-phase bounded elliptic solutions of the Manakov system of integrable coupled nonlinear Schrödinger equations in terms of the Weierstrass sigma function with a real quasiperiod. The parametrization is effective in the sense that the reality conditions are completely characterized for each of the three possible couplings: focusing–focusing, defocusing–defocusing and focusing–defocusing. The Manakov soliton is recovered in the soliton limit and the small-wave-modulation limit is shown to satisfy the linearized dispersion relation of planewave solutions.

On higher genus Weierstrass sigma-function

The goal of this paper is to propose a new way to generalize the Weierstrass sigma-function to higher genus Riemann surfaces. Our definition of the odd higher genus sigma-function is based on a generalization of the classical representation of the elliptic sigma-function via the Jacobi theta-function. Namely, the odd higher genus sigma-function σχ(u) (for u∈Cg) is defined as a product of the theta-function with odd half-integer characteristic βχ, associated with a spin line bundle χ, an exponent of a certain bilinear form, the determinant of a period matrix and a power of the product of all even theta-constants which are non-vanishing on a given Riemann surface.We also define an even sigma-function corresponding to an arbitrary even spin structure. Even sigma-functions are constructed as a straightforward analog of a classical formula relating even and odd sigma-functions. In higher genus the even sigma-functions are well-defined on the moduli space of Riemann surfaces outside of a subspace defined by vanishing of the corresponding even theta-constant.

Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type I. The Eigenfunction Identities

In this series of papers we study Hilbert-Schmidt integral operators acting on the Hilbert spaces associated with elliptic Calogero-Moser type Hamiltonians. As shown in this first part, the integral kernels are joint eigenfunctions of differences of the latter Hamiltonians. On the relativistic (difference operator) level the kernel is built from the elliptic gamma function, whereas the building block in the nonrelativistic (differential operator) limit is basically the Weierstrass sigma-function. For the AN−1 case we consider all of the commuting Hamiltonians at once, the eigenfunction properties reducing to a sequence of elliptic identities. For the BCN case we only treat the defining Hamiltonians. The functional identities encoding the eigenfunction properties have a remarkable corollary in the relativistic BC1 case: They imply that the sum over eight-fold products of the four Jacobi theta functions is invariant under the Weyl group of E8.