2002/08/08 by Noam D. Elkies, Everett W. Howe, Andrew Kresch +3
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Coding theory and cryptography #Commutative Algebra and Its Applications #math.AG #math.NT #msc:11G20 #msc:14G05 #msc:14G15
paper · pdf · doi:10.1215/s0012-7094-04-12224-9
published as Duke Math. J. 122, no. 2 (2004), 399--422 · LaTeX, 18 pages
arxiv created 2002/08/08 · openalex publication_date 2004/04/01 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We resolve a 1983 question of Serre by constructing curves with many points of every genus over every finite field. More precisely, we show that for every prime power q there is a positive constant cq with the following property: for every integer g≥0, there is a genus-g curve over Fq with at least cqg rational points over Fq. Moreover, we show that there exists a positive constant d such that for every q we can choose cq=d log q. We show also that there is a constant c>0 such that for every q and every n>0, and for every sufficiently large g there is a genus-g curve over Fq that has at least cg/n rational points and whose Jacobian contains a subgroup of rational points isomorphic to (ℤ/nℤ)r for some r>cg/n.