2002/11/26 by Matthew Baker, Baker, Matthew, Enrique González–Jiménez +7 · 3 citations
Computer Science · Mathematics · #11G18 #14G35 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #math.AG #math.NT #msc:11G18 #msc:14G35
paper · pdf · doi:10.48550/arxiv.math/0211394
53 pages; minor revisions made
openalex publication_date 2002/11/26 · arxiv created 2003/12/22 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A curve X over the field Q of rational numbers is modular if it is dominated by X1(N) for some N; if in addition the image of its jacobian in J1(N) is contained in the new subvariety of J1(N), then X is called a new modular curve. We prove that for each integer g at least 2, the set of new modular curves over Q of genus g is finite and computable. For the computability result, we prove an algorithmic version of the de Franchis-Severi Theorem. Similar finiteness results are proved for new modular curves of bounded gonality, for new modular curves whose jacobian is a quotient of the new part of J0(N) with N divisible by a prescribed prime, and for modular curves (new or not) with levels in a restricted set. We study new modular hyperelliptic curves in detail. In particular, we find all new modular curves of genus 2 explicitly, and construct what might be the complete list of all new modular hyperelliptic curves of all genera. Finally we prove that for each field k of characteristic zero and each integer g at least 2, the set of genus g curves over k dominated by a Fermat curve is finite and computable.