Orthogonal Hypergeometric Groups with a Maximally Unipotent Monodromy

TL;DR

This paper classifies fourteen orthogonal hypergeometric groups with maximally unipotent monodromy, identifying two as arithmetic and analyzing their preserved quadratic forms and orthogonal vectors.

Contribution

It demonstrates the arithmetic nature of two specific orthogonal hypergeometric groups and provides detailed descriptions of their preserved quadratic forms and isotropic vectors.

Findings

Two groups are proven to be arithmetic.

A table of quadratic forms preserved by these groups is provided.

The orthogonal groups of these forms have $ ext{Q}$-rank two.

Abstract

Similar to the symplectic cases, there is a family of fourteen orthogonal hypergeometric groups with a maximally unipotent monodromy (cf. Table 1.1). We show that two of the fourteen orthogonal hypergeometric groups associated to the pairs of parameters $(0, 0, 0, 0, 0)$, $(\frac{1}{6}, \frac{1}{6}, \frac{5}{6}, \frac{5}{6}, \frac{1}{2})$; and $(0, 0, 0, 0, 0)$, $(\frac{1}{4}, \frac{1}{4}, \frac{3}{4}, \frac{3}{4}, \frac{1}{2})$ are arithmetic. We also give a table (cf. Table 2.1) which lists the quadratic forms $\mathrm{Q}$ preserved by these fourteen hypergeometric groups, and their two linearly independent $\mathrm{Q}$- orthogonal isotropic vectors in $\mathbb{Q}^5$; it shows in particular that the orthogonal groups of these quadratic forms have $\mathbb{Q}$- rank two.