Division polynomials with Galois group SU3(3).2 = G2(2)

TL;DR

This paper proves the existence of specific degree twenty-eight covers with Galois group G2(2) using a rigidity argument and constructs related polynomials through independent methods involving algebraic geometry and motives.

Contribution

It introduces two novel constructions of degree twenty-eight polynomials with Galois group G2(2), connecting group theory, algebraic geometry, and motives.

Findings

Existence of degree twenty-eight covers with Galois group G2(2) proven.

Two independent methods for constructing related polynomials provided.

Specializations yield interesting three-point covers and number fields.

Abstract

We use a rigidity argument to prove the existence of two related degree twenty-eight covers of the projective plane with Galois group SU3(3).2 = G2(2). Constructing corresponding two-parameter polynomials directly from the defining group-theoretic data seems beyond feasablity. Instead we provide two independent constructions of these polynomials, one from 3-division points on covers of the projective line studied by Deligne and Mostow, and one from 2-division points of genus three curves studied by Shioda. We explain how one of the covers also arises as a 2-division polynomial for a family of G2 motives in the classification of Dettweiler and Reiter. We conclude by specializing our two covers to get interesting three-point covers and number fields which would be hard to construct directly.