Symplectic hypergeometric groups of degree six

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 Sp6.In addition, we extend our search to all degree six symplectic hypergeometric groups and 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 163 of the remaining 246 hypergeometric groups.