Division polynomials with galois group su3(3):2 ≌ g2(2)

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We use a rigidity argument to prove the existence of two related degree 28 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.

Original languageEnglish (US)
Title of host publicationFields Institute Communications
PublisherSpringer New York LLC
Number of pages38
StatePublished - 2015

Publication series

NameFields Institute Communications
ISSN (Print)1069-5265

Bibliographical note

Funding Information:
It is a pleasure to thank Zhiwei Yun for a conversation about G-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 []. We are also grateful to the Simons Foundation for research support through grant #209472. 2

Publisher Copyright:
© Springer Science+Business Media New York 2015.


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