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

Abstract

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.