Mathematical Research Letters

Volume 21 (2014)

Number 6

Separated Belyi maps

Pages: 1389 – 1406



Zachary Scherr (Department of Mathematics, University of Pennsylvania, Philadelphia, Penn., U.S.A.)

Michael E. Zieve (Department of Mathematics, University of Michigan, Ann Arbor, Mich., U.S.A.; and Mathematical Sciences Center, Tsinghua University, Beijing, China)


We construct Belyi maps having specified behavior at finitely many points. Specifically, for any curve $C$ defined over $\overline{\mathbb{Q}}$, and any disjoint finite subsets $S, T \subset C(\overline{\mathbb{Q}})$, we construct a finite morphism $\varphi : C \to \mathbb{P}^1$ such that $\varphi$ ramifies at each point in $S$, the branch locus of $\varphi$ is $\lbrace 0, 1, \infty \rbrace$, and $\varphi (T) \cap \lbrace 0, 1, \infty \rbrace = \emptyset$. 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 $\varphi$ are required to imply the above conclusion for a single $C$ and $S$ and all sets $T \subset C(\overline{\mathbb{Q}}) \setminus S \:$ of prescribed cardinality.

Accepted 14 August 2014

Published 2 April 2015