Lily Khadjavi

Covering maps and a theorem of Belyi's

Abstract: A result of Belyi's, which has been used in work on the Inverse Galois problem, states that an algebraic curve is defined over ${\bf \overline Q}$ if and only if there is a covering map of the curve to the projective line ramified over at most three points. To this, in his {\sl Esquisse d'un Programme}, Grothendieck enthusiastically wrote, {\sl ``...jamais sans doute un r\'esultat profond et d\'eroutant ne fut d\'emontr\'e en si peu de lignes!''}\ (``...never, without a doubt, was such a deep and disconcerting result proved in so few lines!'').

In fact, following Belyi's algorithm, one can construct such a map explicitly. Hence this result has also been used by Elkies to show that an effective $ABC$ conjecture implies an effective Mordell's Theorem. We discuss this theorem and its applications, including bounds on the degree and coefficients of such a covering map.