RÉSUMÉ

A curve X over Q 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 > 1, the set of new modular curves over Q of genus g is finite. Similar finiteness results are proved for new modular curves of bounded gonality. 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.

WWW page design par Thomas Papanikolaou & Christine Bachoc