Abstract
We construct Belyi maps having specified behavior at finitely many points. Specifically, for any curve C defined over Q-bar, and any disjoint finite subsets S, T in C(Q-bar), we construct a finite morphism f: C -> 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 in C(Q-bar) \ S of prescribed cardinality.
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