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 $\sigma (\mathbf{u};\lambda )$, 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 $\sigma (\mathbf{u};\lambda )={\sum }_{m,n\ge 0}{a}_{m,n}(\lambda )\frac{u_1^m{u}_3^n}{m!n!}$, $\sigma (\mathbf{u};\lambda )={\sum }_{k\ge 0}{\xi }_k(u_1;\lambda )\frac{u_3^k}{k!}$ 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.

43 pages

References in corpus (2)

• Multi-Dimensional Sigma-Functions
• Universal elliptic functions

Cited by in corpus (1)

• Relationships Between Hyperelliptic Functions of Genus 2 and Elliptic Functions