Finiteness results for modular curves of genus at least 2

A curve X over [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] 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 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 J₀(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.