二维σ函数的解析性和数论性质
Analytical and number-theoretical properties of the two-dimensional sigma function
AI总结：
本文研究二维σ函数的解析性和数论性质，探讨其与代数曲线及热方程的关系。
摘要：
本文旨在探讨与整个函数σ(u;λ)相关的经典和现代问题，该函数由一族非奇异亏格为2的代数曲线族定义，其中u=(u₁,u₃)且λ=(λ₄,λ₆,λ₈,λ₁₀)。它类似于椭圆曲线族的Weierstrass σ函数σ(u;g₂,g₃)。σ(u;λ)的二阶及更高阶对数导数生成了在固定参数向量λ的曲线雅可比簇上u=(u₁,u₃)的双曲函数域。我们考虑了三个Hurwitz级数σ(u;λ)=∑_{m,n≥0}a_{m,n}(λ)u₁^m u₃^n/(m!n!)，σ(u;λ)=∑_{k≥0}ξ_k(u₁;λ)u₃^k/k!和σ(u;λ)=∑_{k≥0}μ_k(u₃;λ)u₁^k/k!。本文着重于函数a_{m,n}(λ)、ξ_k(u₁;λ)和μ_k(u₃;λ)的数论性质。其中包括了最新结果，其证明利用了函数σ(u;λ)由四组热方程在六维空间非全微分框架中确定的基本事实。
英文摘要：
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.
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