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.
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