Finiteness results for modular curves of genus at least 2

TL;DR

This paper proves finiteness and computability results for new modular curves of genus at least 2 over Q, including explicit classifications for genus 2 and hyperelliptic cases, using an algorithmic approach.

Contribution

It establishes the finiteness and computability of new modular curves of fixed genus over Q, and provides explicit classifications for genus 2 and hyperelliptic cases.

Findings

Finiteness of new modular curves of genus g over Q for each g ≥ 2.

Explicit classification of all genus 2 new modular curves.

Construction of a complete list of new modular hyperelliptic curves.

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.