Article
Galois action on regular dessins d’enfant with simple group action
Contemporary Mathematics
(2018)
Abstract
A quasi-platonic action of the group G on the Riemann surface S
is a conformal action of G on S such that S/G is a sphere and the projection
π_G : S → S/G is branched over {0, 1,∞}. The action is induced by a triple of
(a, b, c) ∈ G^3, generating G, with abc = 1. The projection π_G is a regular Belyi
function and induces a regular dessin d’enfant on S, and so S is defined over a
number field. The absolute Galois group Gal(Q) acts on regular dessins, hence
quasi-platonic actions, by acting on the coefficients of a defining equation of S.
The action of ψ ∈ Gal(Q) on triples is (a, b, c) → (ua^tu(−1), vb^tv(−1), wc^tw(−1))
for some (u, v,w) ∈ G^3, according to the branch cycle argument. The integer
t is characterized by the action of ψ on cyclotomic subfields of Q, and (u,v,w)
is determined by the action of ψ away from cyclotomic subfields. In this paper
we try to reconstruct the Galois action from the branch cycle description of
the action and the structure of the group G. To this end, we focus on groups
which are simple or covers of simple groups.
Keywords
- Riemann surface,
- regualr dessin d'enfant,
- simple group,
- absolute Galois group
Disciplines
Publication Date
2018
DOI
https://doi.org/10.1090/conm/703
Citation Information
Sean A Broughton. "Galois action on regular dessins d’enfant with simple group action" Contemporary Mathematics Vol. 703 (2018) p. 13 - 32 ISSN: 0271-4132 Available at: http://works.bepress.com/allen_broughton/45/