Exceptional covers and bijections on rational points

Robert M. Guralnick, Thomas J. Tucker, Michael E. Zieve

Published 2005-11-10, updated 2007-05-15Version 2

We show that if f: X --> Y is a finite, separable morphism of smooth curves defined over a finite field F_q, where q is larger than an explicit constant depending only on the degree of f and the genus of X, then f maps X(F_q) surjectively onto Y(F_q) if and only if f maps X(F_q) injectively into Y(F_q). Surprisingly, the bounds on q for these two implications have different orders of magnitude. The main tools used in our proof are the Chebotarev density theorem for covers of curves over finite fields, the Castelnuovo genus inequality, and ideas from Galois theory.