(Both the published version and the arXiv version are available online.)
Let h : C→D 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.
Michael Zieve: home page publication list