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

David P. Roberts

doi:10.1007/978-1-4939-3201-6_8

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

David P. Roberts

Mathematics (Morris)

Research output: Chapter in Book/Report/Conference proceeding › Chapter

1 Scopus citations

Abstract

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 language	English (US)
Title of host publication	Fields Institute Communications
Publisher	Springer New York LLC
Pages	169-206
Number of pages	38
DOIs	https://doi.org/10.1007/978-1-4939-3201-6_8
State	Published - 2015

Publication series

Name	Fields Institute Communications
Volume	77
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.

Publisher link

10.1007/978-1-4939-3201-6_8

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abstract = "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.",

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