Finiteness results for modular curves of genus at least 2

By Matthew Baker, Enrique González–Jiménez, Josep González, Bjorn Poonen. 2002

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A curve X over [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] 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 g ≥ 2, the set of new modular curves over [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] 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 J 0 ( N ) new 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 g ≥ 2, the set of genus- g curves over k dominated by a Fermat curve is finite and computable.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/62f27d354210cc7d55125dd7479fb856c4ab645e