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# Mathematical Research Letters

## Volume 21 (2014)

### Number 6

### Separated Belyi maps

Pages: 1389 – 1406

DOI: http://dx.doi.org/10.4310/MRL.2014.v21.n6.a10

#### Authors

#### Abstract

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.