论文标题
二维Sigma函数的分析和数理论特性
Analytical and number-theoretical properties of the two-dimensional sigma function
论文作者
论文摘要
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 $λ= (λ_4, λ_6，λ_8，λ_{10}）$。它是Weierstrass Sigma函数的类似物$σ（u; g_2，g_3）$的椭圆曲线家族。函数2和更高函数的对数衍生物$ {σ（{\ bf u};λ）} $生成curves jacobians $ {\ bf u} =（u_1，u_3）$的高ellip函函数的字段。我们考虑三个hurwz系列$σ（{\ bf u};λ）= \ sum_ {m，n \ ge 0} a_ {m，n}（λ）\ frac {u_1^mu_3^n} 0}ξ_k（u_1;λ）\ frac {u_3^k} {k！} $和$σ（{\ bf u};λ）= \ sum_ {k \ ge 0}μ_k（u_3;λ）该调查专门用于函数的数字理论属性$ a_ {m，n}（λ）$，$ξ_k（u_1;λ）$和$μ_k（u_3;λ）$。它包括最新结果，证明使用了基本事实，即功能$ {σ（{\ bf u};λ）} $由六维空间的非实体框架中的四个加热方程的系统确定。
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.
