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Finiteness results for modular curves of genus at least 2 | | Matthew Baker
; Enrique Gonzalez-Jimenez
; Josep Gonzalez
; Bjorn Poonen
; | | Date: |
26 Nov 2002 | | Subject: | Number Theory; Algebraic Geometry MSC-class: 11G18; 14G35 | math.NT math.AG | | Abstract: | 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. | | Source: | arXiv, math.NT/0211394 | | Services: | Forum | Review | PDF | Favorites | |
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