Advances in the Theory of Numbers: Proceedings of the Thirteenth Conference of the Canadian Number Theory Association
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 feasibility. 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.
© Springer Science+Business Media New York 2015
David P. Roberts. Division polynomials with Galois group SU3(3).2 = G2(2). Advances in the Theory of Numbers: Proceedings of the Thirteenth Conference of the Canadian Number Theory Association 77 (2015), 169-206.