Division polynomials with galois group su(3):2 ≌ g(2)

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Fields Institute Communications, ISSN: 1069-5265, Vol: 77, Page: 169-206

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Roberts, David P.
Springer Nature
Mathematics; Polynomials; Galois group
book chapter description
We use a rigidity argument to prove the existence of two related degree 28 covers of the projective plane with Galois group SU(3):2 ≌ G(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 G 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.