arXiv subjects

Takanori Ayano

Publications and source records attributed to Takanori Ayano.

13 recordsLinked to original sources

arXiv2025

Sigma function associated with a hyperelliptic curve with two points at infinity

Baker constructed basic meromorphic functions on the Jacobian variety of a hyperelliptic curve with two points at infinity. We call them Baker functions. The construction is based on the Abel-Jacobi map, which allows us to identify the field of meromorphic functions on the Jacobian variety of the curve with the field of meromorphic functions on the symmetric product of the curve. In our previous paper, a solution to the KP equation was constructed in terms of the Baker function. This paper is devoted to the properties of the Baker functions. In this paper, we construct an entire function whose second logarithmic derivatives are the Baker functions. We prove that the power series expansion of the entire function around the origin is determined only by the coefficients of the defining equation of the curve and a branch point of the curve algebraically. We also describe the quasi-periodicity of the entire function and express the entire function in terms of the Riemann theta function.

math.AG↗

arXiv2025

Hyperelliptic sigma functions and the Kadomtsev-Petviashvili equation

In this paper, a theory of hyperelliptic functions based on multidimensional sigma functions is developed and explicit formulas for hyperelliptic solutions to the Kadomtsev-Petviashvili equations KP-I and KP-II are obtained. The long-standing problem of describing the dependence of these solutions on the variation of the coefficients of the defining equation of a hyperelliptic curve, which are integrals of the equations, is solved.

math-ph↗

arXiv2025

Reduction of bielliptic hyperelliptic functions of genus 3

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 this paper, we consider a hyperelliptic curve of genus 3 which admits a morphism of degree 2 to an elliptic curve. We express the hyperelliptic functions associated with the curve of genus 3 in terms of the Weierstrass elliptic functions and hyperelliptic functions of genus 2.

math.AG↗

arXiv2024

Hurwitz integrality of the power series expansion of the sigma function for a telescopic curve

A telescopic curve is a certain algebraic curve defined by $m-1$ equations in the affine space of dimension $m$, which can be a hyperelliptic curve and an $(n,s)$ curve as a special case. The sigma function $σ(u)$ associated with the telescopic curve of genus $g$ is a holomorphic function on $\mathbb{C}^g$. For a subring $R$ of $\mathbb{C}$ and variables $u={}^t(u_1,\dots, u_g)$, let \[R\langle\langle u \rangle\rangle=\left\{\sum_{i_1,\dots,i_g\ge0}κ_{i_1,\dots,i_g}\frac{u_1^{i_1}\cdots u_g^{i_g}}{i_1!\cdots i_g!}\;\middle|\;κ_{i_1,\dots,i_g}\in R\right\}.\] If the power series expansion of a holomorphic function $f(u)$ on $\mathbb{C}^g$ around the origin belongs to $R\langle\langle u \rangle\rangle$, then $f(u)$ is said to be Hurwitz integral over $R$. In this paper, we show that the sigma function $σ(u)$ associated with the telescopic curve is Hurwitz integral over the ring generated by the coefficients of the defining equations of the curve and $\frac{1}{2}$ over $\mathbb{Z}$. Further, we show that $σ(u)^2$ is Hurwitz integral over the ring generated by the coefficients of the defining equations of the curve over $\mathbb{Z}$. Our results are a generalization of the results of Y. Ônishi for $(n,s)$ curves to telescopic curves.

math.AG↗

arXiv2024

Difference of solutions for the inversion problem of ultra-elliptic integrals

Let $V$ be a hyperelliptic curve of genus 2 defined by $Y^2=f(X)$, where $f(X)$ is a polynomial of degree 5. The sigma function associated with $V$ is a holomorphic function on $\mathbb{C}^2$. For a point $P$ on $V$, we consider the problem to express the $X$-coordinate of $P$ in terms of the image of $P$ under the Abel-Jacobi map. Two meromorphic functions $f_2$ and $g_2$ on $\mathbb{C}^2$ which give solutions of this problem are known. Since $f_2$ and $g_2$ coincide on the zero set of the sigma function, it is expected that $f_2-g_2$ can be divided by the sigma function. In this paper, we decompose $f_2-g_2$ into a product of the sigma function and a meromorphic function explicitly.

math.CV↗

arXiv2022

Relationships Between Hyperelliptic Functions of Genus 2 and Elliptic Functions

The article is devoted to the classical problems about the relationships between elliptic functions and hyperelliptic functions of genus 2. It contains new results, as well as a derivation from them of well-known results on these issues. Our research was motivated by applications to the theory of equations and dynamical systems integrable in hyperelliptic functions of genus 2. We consider a hyperelliptic curve $V$ of genus 2 which admits a morphism of degree 2 to an elliptic curve. Then there exist two elliptic curves $E_i$, $i=1,2$, and morphisms of degree 2 from $V$ to $E_i$. We construct hyperelliptic functions associated with $V$ from the Weierstrass elliptic functions associated with $E_i$ and describe them in terms of the fundamental hyperelliptic functions defined by the logarithmic derivatives of the two-dimensional sigma functions. We show that the restrictions of hyperelliptic functions associated with $V$ to the appropriate subspaces in $\mathbb{C}^2$ are elliptic functions and describe them in terms of the Weierstrass elliptic functions associated with $E_i$. Further, we express the hyperelliptic functions associated with $V$ on $\mathbb{C}^2$ in terms of the Weierstrass elliptic functions associated with $E_i$. We derive these results by describing the homomorphisms between the Jacobian varieties of the curves $V$ and $E_i$ induced by the morphisms from $V$ to $E_i$ explicitly.

math.AG↗

arXiv2020

Analytical and number-theoretical properties of the two-dimensional sigma function

This survey is devoted to the classical and modern problems related to the entire function ${σ({\bf u};λ)}$, defined by a family of nonsingular algebraic curves of genus $2$, where ${\bf u} = (u_1,u_3)$ and $λ= (λ_4, λ_6,λ_8,λ_{10})$. It is an analogue of the Weierstrass sigma function $σ(u;g_2,g_3)$ of a family of elliptic curves. Logarithmic derivatives of order 2 and higher of the function ${σ({\bf u};λ)}$ generate fields of hyperelliptic functions of ${\bf u} = (u_1,u_3)$ on the Jacobians of curves with a fixed parameter vector $λ$. We consider three Hurwitz series $σ({\bf u};λ)=\sum_{m,n\ge 0}a_{m,n}(λ)\frac{u_1^mu_3^n}{m!n!}$, $σ({\bf u};λ) = \sum_{k\ge 0}ξ_k(u_1;λ)\frac{u_3^k}{k!}$ and $σ({\bf u};λ) = \sum_{k\ge 0}μ_k(u_3;λ)\frac{u_1^k}{k!}$. The survey is devoted to the number-theoretic properties of the functions $a_{m,n}(λ)$, $ξ_k(u_1;λ)$ and $μ_k(u_3;λ)$. It includes the latest results, which proofs use the fundamental fact that the function ${σ({\bf u};λ)}$ is determined by the system of four heat equations in a nonholonomic frame of six-dimensional space.

math.AG↗

arXiv2019

Construction of Two Parametric Deformation of KdV-Hierarchy and Solution in Terms of Meromorphic Functions on the Sigma Divisor of a Hyperelliptic Curve of Genus 3

Buchstaber and Mikhailov introduced the polynomial dynamical systems in $\mathbb{C}^4$ with two polynomial integrals on the basis of commuting vector fields on the symmetric square of hyperelliptic curves. In our previous paper, we constructed the field of meromorphic functions on the sigma divisor of hyperelliptic curves of genus 3 and solutions of the systems for $g=3$ by these functions. In this paper, as an application of our previous results, we construct two parametric deformation of the KdV-hierarchy. This new system is integrated in the meromorphic functions on the sigma divisor of hyperelliptic curves of genus 3. In Section 8 of our previous paper [Funct. Anal. Appl. 51 (2017), 162-176], there are miscalculations. In appendix of this paper, we correct the errors.

math.AG↗

arXiv2018

The field of meromorphic functions on a sigma divisor of a hyperelliptic curve of genus 3 and applications

The field of meromorphic functions on a sigma divisor of a hyperelliptic curve of genus $3$ is described in terms of the gradient of it's sigma function. Solutions of corresponding families of polynomial dynamical systems in $\mathbb{C}^4$ with two polynomial integrals are constructed as an application. These systems were introduced in the work of V. M. Buchstaber and A. V. Mikhailov on the basis of commuting vector fields on the symmetric square of algebraic curves.

math.AG↗

arXiv2016

On Jacobi Inversion Formulae for Telescopic Curves

For a hyperelliptic curve of genus $g$, it is well known that the symmetric products of $g$ points on the curve are expressed in terms of their Abel-Jacobi image by the hyperelliptic sigma function (Jacobi inversion formulae). Matsutani and Previato gave a natural generalization of the formulae to the more general algebraic curves defined by $y^r=f(x)$, which are special cases of $(n,s)$ curves, and derived new vanishing properties of the sigma function of the curves $y^r=f(x)$. In this paper we extend the formulae to the telescopic curves proposed by Miura and derive new vanishing properties of the sigma function of telescopic curves. The telescopic curves contain the $(n,s)$ curves as special cases.

math.AG↗

arXiv2013

On Addition Formulae for Sigma Functions of Telescopic Curves

A telescopic curve is a certain algebraic curve defined by $m-1$ equations in the affine space of dimension $m$, which can be a hyperelliptic curve and an $(n,s)$ curve as a special case. We extend the addition formulae for sigma functions of $(n,s)$ curves to those of telescopic curves. The expression of the prime form in terms of the derivative of the sigma function is also given.

math.AG↗

arXiv2013

Sigma Functions for Telescopic Curves

In this paper, we consider the sigma functions for algebraic curves expressed by a canonical form using a finite sequence $(a_1,...,a_t)$ of positive integers whose greatest common divisor is equal to one (Miura [13]). The idea is to express a non-singular algebraic curve by affine equations of $t$ variables whose orders at infinity are $(a_1,...,a_t)$. We construct a symplectic basis of the first cohomology group and the sigma functions for telescopic curves, i.e., the curves such that the number of defining equations is exactly $t-1$ in the Miura canonical form. The largest class of curves for which such construction has been obtained thus far is $(n,s)$-curves ([3][15]), which are telescopic because they are expressed in the Miura canonical form with $t=2$, $a_1=n$, and $a_2=s$, and the number of defining equations is one.

math.AG↗

arXiv2011

Rates of convergence for nearest neighbor estimators with the smoother regression function

In regression analysis one wants to estimate the regression function from a data. In this paper we consider the rate of convergence for the nearest neighbor estimator in case that the regression function is $(p,C)$-smooth. It is an open problem whether the optimal rate can be achieved by some nearest neighbor estimator in case that $p$ is on (1,1.5]. We solve the problem affirmatively. This is the main result of this paper. Throughout this paper, we assume that the data is independent and identically distributed and as an error criterion we use the expected $L_2$ error.

math.ST↗