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 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.
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Acknowledgements
It is a pleasure to thank Zhiwei Yun for a conversation about G 2-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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Roberts, D.P. (2015). Division Polynomials with Galois Group \(SU_{3}(3).2\mathop{\cong}G_{2}(2)\) . In: Alaca, A., Alaca, Åž., Williams, K. (eds) Advances in the Theory of Numbers. Fields Institute Communications, vol 77. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3201-6_8
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DOI: https://doi.org/10.1007/978-1-4939-3201-6_8
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