Separated Belyi Maps
Algebraic Geometry
2016-03-04 v1 Number Theory
Abstract
We construct Belyi maps having specified behavior at finitely many points. Specifically, for any curve C defined over Q-bar, and any disjoint finite subsets S, T in C(Q-bar), we construct a finite morphism f: C -> P^1 such that f ramifies at each point in S, the branch locus of f is {0,1, infty}, and f(T) is disjoint from {0,1, infty}. This refines a result of Mochizuki's. We also prove an analogous result over fields of positive characteristic, and in addition we analyze how many different Belyi maps f are required to imply the above conclusion for a single C and S and all sets T in C(Q-bar) \ S of prescribed cardinality.
Keywords
Cite
@article{arxiv.1310.2555,
title = {Separated Belyi Maps},
author = {Zachary Scherr and Michael E. Zieve},
journal= {arXiv preprint arXiv:1310.2555},
year = {2016}
}
Comments
14 pages