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Ultraelliptic Integrals and Two-Dimensional Sigma Functions
Functional Analysis and Its Applications ( IF 0.6 ) Pub Date : 2019-10-15 , DOI: 10.1134/s0016266319030018
T. Ayano , V. M. Buchstaber

This paper is devoted to the classical problem of the inversion of ultraelliptic integrals given by basic holomorphic differentials on a curve of genus 2. Basic solutions F and G of this problem are obtained from a single-valued 4-periodic meromorphic function on the Abelian covering W of the universal hyperelliptic curve of genus 2. Here W is the nonsingular analytic curve W = {u =(u1, u3) ∈ ℂ2: σ(u) = 0}, where σ(u) is the two-dimensional sigma function. We show that G(z) = F(ξ(z)), where z is a local coordinate in a neighborhood of a point of the smooth curve W and ξ(z) is the smooth function in this neighborhood given by the equation σ(u1, ξ(u1)) = 0. We obtain differential equations for the functions F(z), G(z), and ξ(z), recurrent formulas for the coefficients of the series expansions of these functions, and a transformation of the function G(z) into the Weierstrass elliptic function ℘ under a deformation of a curve of genus 2 into an elliptic curve.

中文翻译:

超椭圆积分和二维Sigma函数

本文致力于对类2上的曲线由基本全纯微分给出的超椭圆形积分反演的经典问题。此问题的基本解FG是从Abelian覆盖上的单值4周期亚纯函数获得的W¯¯属的通用超椭圆曲线的2。这里w ^是奇异解析曲线w ^ = { ü =(Û 1ü 3)∈ℂ 2σÛ)= 0},其中σÛ)是双维度sigma函数。我们证明Gz)= Fξz)),其中z是平滑曲线W的一个点附近的局部坐标,而ξz)是该附近由方程σu 1)给出的平滑函数,ξu 1))=0。我们获得函数Fz),Gz)和ξz),这些函数的级数展开系数的递推公式以及在类2的曲线变形为椭圆曲线时将函数Gz)转换为Weierstrass椭圆函数transformation 。
更新日期:2019-10-15
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