Finiteness results for modular curves of genus at least 2
| dc.creator | Baker, Matthew | |
| dc.creator | Gonzalez-Jimenez, Enrique | |
| dc.creator | Gonzalez, Josep | |
| dc.creator | Poonen, Bjorn | |
| dc.date | 2002-11-26 | |
| dc.date | 2003-12-22 | |
| dc.date.accessioned | 2026-07-07T04:53:16Z | |
| dc.date.available | 2026-07-07T04:53:16Z | |
| dc.description | A curve X over the field Q of rational numbers is modular if it is dominated by X_1(N) for some N; if in addition the image of its jacobian in J_1(N) is contained in the new subvariety of J_1(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 J_0(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. | |
| dc.description | 53 pages; minor revisions made | |
| dc.identifier | https://arxiv.org/abs/math/0211394 | |
| dc.identifier | http://arxiv.org/abs/math/0211394 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/65781 | |
| dc.subject | Number Theory | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 11G18; 14G35 | |
| dc.title | Finiteness results for modular curves of genus at least 2 | |
| dc.type | text |