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

This survey is devoted to the classical and modern problems related to the entire function ${\sigma({\mathbf{u}};\lambda)}$, defined by a family of nonsingular algebraic curves of genus $2$, where ${\mathbf{u}} = (u_1,u_3)$ and $\lambda = (\lambda_4, \lambda_6,\lambda_8,\lambda_{10})$. It is an analogue of the Weierstrass sigma function $\sigma({{u}};g_2,g_3)$ of a family of elliptic curves. Logarithmic derivatives of order $2$ and higher of the function ${\sigma({\mathbf{u}};\lambda)}$ generate fields of hyperelliptic functions of ${\mathbf{u}} = (u_1,u_3)$ on the Jacobians of curves with a fixed parameter vector $\lambda$. We consider three Hurwitz series $\sigma({\mathbf{u}};\lambda)=\sum_{m,n\ge0}a_{m,n}(\lambda)\frac{u_1^mu_3^n}{m!n!}$, $\sigma({\mathbf{u}};\lambda) = \sum_{k\ge 0}\xi_k(u_1;\lambda)\frac{u_3^k}{k!}$ and $\sigma({\mathbf{u}};\lambda) = \sum_{k\ge 0}\mu_k(u_3;\lambda)\frac{u_1^k}{k!}$. The survey is devoted to the number-theoretic properties of the functions $a_{m,n}(\lambda)$, $\xi_k(u_1;\lambda)$ and $\mu_k(u_3;\lambda)$. It includes the latest results, which proofs use the fundamental fact that the function ${\sigma ({\mathbf{u}};\lambda)}$ is determined by the system of four heat equations in a nonholonomic frame of six-dimensional space.