Division Polynomials with Galois Group $$SU_{3}(3).2\mathop{\cong}G_{2}(2)$$

Abstract

We use a rigidity argument to prove the existence of two related degree twenty-eight covers of the projective plane with Galois group SU 3 (3).2 ∼ = G 2 (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 G 2 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.