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

2014/11/25 by David P. Roberts, Roberts, David P.
Mathematics · #11R21 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11R21

paper · pdf · doi:10.48550/arxiv.1411.7015

31 pages, 10 figures

arxiv created 2014/11/25 · arxiv updated 2014/11/26

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.

Related

• Plane curves possessing two outer Galois points
• Correspondances compatibles avec une relation binaire, relevement d'extensions de groupe de Galois L3(2) et probleme de Noether pour L3(2)
• Examples of plane rational curves with two Galois points in positive characteristic, II
• Explicit computation of Galois representations occurring in families of curves
• Number fields with discriminant +-2^a 3^b: Examples from three point covers