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

  • David P. Roberts
Part of the Fields Institute Communications book series (FIC, volume 77)

Abstract

We use a rigidity argument to prove the existence of two related degree 28 covers of the projective plane with Galois group \(SU_{3}(3).2\mathop{\cong}G_{2}(2)\). 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.

Notes

Acknowledgements

It is a pleasure to thank Zhiwei Yun for a conversation about G2-rigidity from which this paper grew. It is equally a pleasure to thank Michael Dettweiler and Stefan Reiter for helping to make the direct connections to their work [6]. We are also grateful to the Simons Foundation for research support through grant #209472.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Division of Science and MathematicsUniversity of MinnesotaMorrisUSA

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