2005/11/30 by Robert M. Guralnick, R. M. Guralnick, Thomas J. Tucker +3
Mathematics · #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #Limits and Structures in Graph Theory #math.AG #math.NT #msc:11G20 #msc:14G15
paper · pdf · doi:10.1093/imrn/rnm004
published as Internat. Math. Res. Notices 2007; Vol. 2007: article ID rnm004 · 19 pages; various minor changes to previous version. To appear in International Mathematics Research Notices
arxiv created 2007/05/15 · openalex publication_date 2008/10/22 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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.