(Both the published version and the arXiv version are available online.)

Let X be a smooth, projective, geometrically irreducible algebraic curve defined over C. Belyi's theorem says that X can be defined over a number field if and only if there exists a finite morphism  f : X→P¹ which has exactly three branch points. We study the amount of flexibility there is in the choice of this "Belyi map" f.  We show that, if X is defined over a number field K, and S and T are disjoint finite subsets of X(K), then there is a Belyi map  f  on X which ramifies at every point in S but for which no point in T maps to the branch locus of  f.  We also prove a refinement of this result in which S is fixed but T can vary: for any positive integer n, there are n+1 Belyi maps f₁, f₂, ... , f_n+1 on X such that, for any n-element subset T of X(Q)∖S, at least one of the maps f_i  has the properties required of  f  in the previous result. These results refine previous results of Mochizuki's, which are used in his proof of the abc conjecture. We also prove analogous results over fields of positive characteristic.

Additional comment added January 2016:  The results of this paper were used in Dimitrov's proof that Mochizuki's result implies an effective version of the abc conjecture.

Michael Zieve:  home page   publication list