2020/03/19 by Takanori Ayano, Victor Matveevich Buchstaber, Ayano, Takanori +1
Computer Science · Mathematics · #14H40 #14H42 #14K25 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Analytic Number Theory Research #Coding theory and cryptography #Complex Variables (math.CV) #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2003.08565
openalex publication_date 2020/03/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
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 = (u1,u3) and λ= (λ4, λ6,λ8,λ10). It is an analogue of the Weierstrass sigma function σ(u;g2,g3) 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 = (u1,u3) on the Jacobians of curves with a fixed parameter vector λ. We consider three Hurwitz series σ(\bf u;λ)=∑m,n≥ 0am,n(λ)(u1mu3n)/(m!n!), σ(\bf u;λ) = ∑k≥ 0ξk(u1;λ)(u3k)/(k!) and σ(\bf u;λ) = ∑k≥ 0μk(u3;λ)(u1k)/(k!). The survey is devoted to the number-theoretic properties of the functions am,n(λ), ξk(u1;λ) and μk(u3;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.