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Symplectic Hypergeometric Groups of Degree Six

Jitendra Bajpai, Daniele Dona, Sandip Singh, Shashank Vikram Singh

Abstract

Our computations show that there is a total of $40$ pairs of degree six coprime polynomials $f,g$ where $f(x)=(x-1)^6$, $g$ is a product of cyclotomic polynomials, $g(0)=1$ and $f,g$ form a primitive pair. The aim of this article is to determine whether the corresponding $40$ symplectic hypergeometric groups with a maximally unipotent monodromy follow the same dichotomy between arithmeticity and thinness that holds for the $14$ symplectic hypergeometric groups corresponding to the pairs of degree four polynomials $f,g$ where $f(x)=(x-1)^4$ and $g$ is as described above. As a result we prove that at least $18$ of these $40$ groups are arithmetic in $\mathrm{Sp}(6)$. In addition, we extend our search to all degree six symplectic hypergeometric groups. We find that there is a total of $458$ pairs of polynomials (up to scalar shifts) corresponding to such groups. For $211$ of them, the absolute values of the leading coefficients of the difference polynomials $f-g$ are at most $2$ and the arithmeticity of the corresponding groups follows from Singh and Venkataramana, while the arithmeticity of one more hypergeometric group follows from Detinko, Flannery and Hulpke. In this article, we show the arithmeticity of $160$ of the remaining $246$ hypergeometric groups.

Symplectic Hypergeometric Groups of Degree Six

Abstract

Our computations show that there is a total of pairs of degree six coprime polynomials where , is a product of cyclotomic polynomials, and form a primitive pair. The aim of this article is to determine whether the corresponding symplectic hypergeometric groups with a maximally unipotent monodromy follow the same dichotomy between arithmeticity and thinness that holds for the symplectic hypergeometric groups corresponding to the pairs of degree four polynomials where and is as described above. As a result we prove that at least of these groups are arithmetic in . In addition, we extend our search to all degree six symplectic hypergeometric groups. We find that there is a total of pairs of polynomials (up to scalar shifts) corresponding to such groups. For of them, the absolute values of the leading coefficients of the difference polynomials are at most and the arithmeticity of the corresponding groups follows from Singh and Venkataramana, while the arithmeticity of one more hypergeometric group follows from Detinko, Flannery and Hulpke. In this article, we show the arithmeticity of of the remaining hypergeometric groups.

Paper Structure

This paper contains 7 sections, 3 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Proof of Proposition \ref{['Proposition']}
  3. Sage Code
  4. Table A: hypergeometric groups in $\mathrm{Sp}(6)$ with $\alpha=(0,0,0,0,0,0)$
  5. Table B: More examples of arithmetic hypergeometric groups in $\mathrm{Sp}(6)$
  6. Table-C: Examples of hypergeometric groups in $\mathrm{Sp}(6)$ for which arithmeticity follows from Singh and Venkataramana SV
  7. Table-D: Other Open cases

Key Result

Proposition 1

Let $f,g$ be a pair of degree $n$ polynomials which are products of cyclotomic polynomials, do not have any common roots, form a primitive pair and have the constant terms equal to $1$ (these conditions ensure that $n$ must be even). Let the leading coefficient of the difference polynomial $f-g$ has

Theorems & Definitions (8)

  • Proposition 1
  • Remark 1
  • Example 1
  • Example 2
  • Theorem 1
  • Theorem 2
  • Remark 2
  • Remark 3