Аналитические и теоретико-числовые свойства двумерных сигма-функций

Translated title of the contribution: Analytical and number-theoretical properties of the two-dimensional sigma function

T. Ayano, V. M. Buchstaber

Research output: Contribution to journalReview articlepeer-review

1 Citation (Scopus)

Abstract

This survey is devoted to the classical and modern problems related to the entire function σ(u; λ), defined by a family of nonsingular algebraic curves of genus 2, where 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(u; λ) generate fields of hyperelliptic functions of u = (u1, u3) on the Jacobians of curves with a fixed parameter vector λ. We consider three Hurwitz series σ(u; λ) = m,n>0 am,n(λ) umm1!unn3! σ(u; λ) = k>0 ξk(u1; λ) ukk3! and σ(u; λ) = k>0 μk(u3; λ) ukk1! . 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(u; λ) is determined by the system of four heat equations in a nonholonomic frame of six-dimensional space.

Translated title of the contributionAnalytical and number-theoretical properties of the two-dimensional sigma function
Original languageRussian
Pages (from-to)9-50
Number of pages42
JournalChebyshevskii Sbornik
Volume21
Issue number1
DOIs
Publication statusPublished - 2020
Externally publishedYes

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