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 defined over , and any disjoint finite subsets , we construct a finite morphism such that ramifies at each point in , the branch locus of is , and is disjoint from . 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 are required to imply the above conclusion for a single and and all sets of prescribed cardinality.

You can download the paper here.

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