## “Separated Belyĭ Maps” by Zachary Scherr and Michael E. Zieve

Zachary Scherr and his graduate advisor Michael E. Zieve have a new paper on the ArXiv entitled “Separated Belyĭ Maps”.

Abstract. We construct Belyĭ 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 $f: C \to \mathbb P^1$ 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 $T \subset C(\overline{\mathbb Q}) \backslash S$ of prescribed cardinality.