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Belyi maps from zeroes of hypergeometric polynomials

Raimundas Vidunas

TL;DR

This work links zeros of low-degree hypergeometric polynomials to algebraic surfaces in parameter space and to genus 0 Belyi maps, focusing on ${}_2F_1(-N,b;c;z)=0$ with $N=3,4$. By developing cubic and quartic hypergeometric frameworks, it constructs elliptic fibrations and rational surfaces, enabling explicit parametrizations, factorization patterns, and Mordell–Weil analyses that underpin Belyi map constructions. The paper provides both full and partial sets of genus 0 Belyi maps for two main forms, derives rational points and Pell-type degenerations, and demonstrates maps defined over ${Q}$ or quadratic fields, enriching the arithmetic geometry of Belyi maps with constructive methods. These results connect hypergeometric algebra, elliptic surfaces, and number-theoretic techniques to expand the catalog of explicit Belyi maps and their fields of definition, with practical implications for Galois orbits and map classifications.

Abstract

Evaluation of low degree hypergeometric polynomials to zero defines an algebraic hypersurface in the affine space of the free parameters and the argument. This article investigates the algebraic surfaces 2F1(-N,b;c;z)=0 for N=3 and N=4. As a captivating application, these surfaces parametrize certain families of genus 0 Belyi maps.

Belyi maps from zeroes of hypergeometric polynomials

TL;DR

This work links zeros of low-degree hypergeometric polynomials to algebraic surfaces in parameter space and to genus 0 Belyi maps, focusing on with . By developing cubic and quartic hypergeometric frameworks, it constructs elliptic fibrations and rational surfaces, enabling explicit parametrizations, factorization patterns, and Mordell–Weil analyses that underpin Belyi map constructions. The paper provides both full and partial sets of genus 0 Belyi maps for two main forms, derives rational points and Pell-type degenerations, and demonstrates maps defined over or quadratic fields, enriching the arithmetic geometry of Belyi maps with constructive methods. These results connect hypergeometric algebra, elliptic surfaces, and number-theoretic techniques to expand the catalog of explicit Belyi maps and their fields of definition, with practical implications for Galois orbits and map classifications.

Abstract

Evaluation of low degree hypergeometric polynomials to zero defines an algebraic hypersurface in the affine space of the free parameters and the argument. This article investigates the algebraic surfaces 2F1(-N,b;c;z)=0 for N=3 and N=4. As a captivating application, these surfaces parametrize certain families of genus 0 Belyi maps.
Paper Structure (19 sections, 5 theorems, 184 equations, 3 figures)

This paper contains 19 sections, 5 theorems, 184 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Hypergeometric details
  3. Hypergeometric polynomials
  4. Distinctive roots of hypergeometric polynomials
  5. Cubic hypergeometric polynomials
  6. Complete factorization for some $b,c$
  7. Elliptic fibration by $b$ or $c$
  8. Quartic hypergeometric polynomials
  9. Elliptic fibration by $z$
  10. The rational surface
  11. The fibration by $b$
  12. Belyi maps
  13. Easy maps
  14. Belyi maps of the form (\ref{['eq:g11hm']})
  15. Full sets of Belyi maps
  16. ...and 4 more sections

Key Result

Lemma 2.1

Consider the sequence of hypergeometric polynomials with some $b,c\in\mathbb{C}$. Suppose that for a positive integer $k$ we have $(b)_k\neq 0$, $(c)_{k}\neq 0$, and $(b+c)_k\neq 0$. Then:

Figures (3)

  • Figure 1: The signs of $b$ and $c$ depending: (i) on $e,z$ in formula (\ref{['eq:bcst']}); (ii) on $t,y$ in formula (\ref{['eq:bcuv']}). The upper sign in $\pm,\mp,=$ is for $b$, while the lower one is for $c$. There sign + indicates both $b,c$ positive.
  • Figure 2: The number of Belyi maps (\ref{['eq:g11hm']}) for integer values of $p/r$ and $q/r$. The dark region in the middle, and the lines emanating from it, represent the cases with no Belyi maps. The light grey regions represent $m+1$ Belyi maps; the dashed lines --- unique Belyi maps; the dense dotted lines --- pairs of Belyi maps; the sparser dotted lines --- triples of maps; the two kinds of dashed-dotted lines: $(m-1)$ or $m$ maps.
  • Figure 3: The elliptic curves (\ref{['eq:ecm5']}) and (\ref{['eq:ecm6']}), with the numerator line in (\ref{['eq:ecp5']}) or (\ref{['eq:ecp6']}) that together with the horizontal line $v=0$ determines the sign of $p/r$.

Theorems & Definitions (33)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • proof
  • Lemma 5.1
  • Example 5.2
  • Example 5.3
  • Remark 5.4
  • ...and 23 more