Robert M. Guralnick, Thomas J. Tucker, and Michael E. Zieve:
Exceptional covers and bijections on rational points,
Internat. Math. Res. Notices 2007: article ID rnm004. MR 2008e:14029

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

Let  h : CD  be a finite, separable morphism of smooth curves defined over a finite field k. We study the injectivity and surjectivity of the map  hk : C(k)→D(k). Let g be the genus of C, let n be the degree of h, and let q be the cardinality of k. We show:
(1) If hk is injective and  q½ > 2n2+4ng,  then hk is surjective.
(2) If hk is surjective and  q½ > n! (3g+3n),  then hk is injective.

Our proof of (2) involves an analysis of the decomposition and inertia groups of places of the Galois closure of the function field extension  k(C) / k(D), using an analogue of Chebotarev's density theorem to translate injectivity, surjectivity, and exceptionality into group-theoretic properties which are shown to be equivalent via purely group-theoretic arguments. In our proof of (1) we study the geometrically irreducible components of the fibred product C xD C:  by combining Weil's bound (on the number of k-rational points on a curve) with Castelnuovo's bound on the arithmetic genus of curves in C x C, we show that the hypotheses of (1) imply that the diagonal is the only geometrically irreducible component of C xD C which is defined over k. But Lenstra has shown that this last condition implies bijectivity of hk, so (1) follows.

We also discuss potential higher-dimensional analogues of our results.

Additional comment added July 2007:  Our results have been generalized to surfaces by Achter.


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