Exceptional covers and bijections on rational points

Details Exceptional covers and bijections on rational points Exceptional covers and bijections on rational points

2005-11-10

arxiv.org/abs/math/0511276v2

Internat. Math. Res. Notices 2007; Vol. 2007: article ID rnm004

Mathematics

Number Theory

19 pages; various minor changes to previous version. To appear in International Mathematics Research Notices

Scientific paper

10.1093/imrn/rnm004

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.

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