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International Mathematics Research Notices (2007) Vol. 2007 : article ID rnm004, 19 pages, doi:10.1093/imrn/rnm004 published on May 24, 2007
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Copyright © The Author 2007. Published by Oxford University Press.

Exceptional Covers and Bijections on Rational Points

Robert M. Guralnick1, Thomas J. Tucker2 and Michael E. Zieve3,

1 Department of Mathematics, Univrsity of Souhern California, Los Angeles, CA 90089–2532, USA
2 Department of Mathematics, Hylan Building, University of Rochester, Rochester, NY 14627, USA
3 Center for Communications Research, 805 Bunn Drive, Princeton, NJ 08540, USA

Correspondence: Correspondence to be sent to: Michael E. Zieve, Center for Communications Research, 805 Bunn Drive, Princeton, NJ 08540, USA. e-mail: zieve{at}idaccr.org

We show that if f: X -> Y is a finite, separable morphism of smooth curves defined over a finite field Fq, where q is larger than an explicit constant depending only on the degree of f and the genus of X, then f maps X(Fq) surjectively onto Y(Fq) if and only if f maps X(Fq) injectively into Y(Fq). 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.

Communicated by Bjorn Poonen



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This Article
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