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F和G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 ₁，ü ₃）∈ℂ ²：σ（Û）= 0}，其中σ（Û）是双维度sigma函数。我们证明G（z）= F（ξ（z）），其中z是平滑曲线W的一个点附近的局部坐标，而ξ（z）是该附近由方程σ（u _1）给出的平滑函数，ξ（u ₁））=0。我们获得函数F（z），G（z）和ξ（z），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G（z）转换为Weierstrass椭圆函数transformation 。