Arithmeticity of Some Hypergeometric Monodromy Groups in Sp(4)

TL;DR

This paper investigates the arithmetic nature of certain hypergeometric monodromy groups in Sp(4), expanding the classification of which groups are arithmetic among those associated with specific polynomial pairs.

Contribution

It demonstrates that 15 additional hypergeometric monodromy groups are arithmetic, complementing previous classifications and providing new insights into their structure.

Findings

12 of 51 groups are arithmetic

13 groups are thin

15 of the remaining 26 groups are arithmetic

Abstract

The article [14] gives a list of 51 symplectic hypergeometric monodromy groups corresponding to primitive pairs of degree four polynomials, which are products of cyclotomic polynomials, and for which, the absolute value of the leading coefficient of the difference polynomial is greater than 2. It follows from [12] and [14] that 12 of the 51 monodromy groups are arithmetic (cf. Table 1); and the thinness of 13 of the remaining 39 monodromy groups follows from [3] (cf. Table 2). In this article, we show that 15 of the remaining 26 monodromy groups are arithmetic (cf. Table 3).