Modular Curves Over Q

Abstract

Among all curves, there is a particularly interesting family consisting of the modular curves, which we shall describe in §1. These parametrize elliptic curves with points of order N, or cyclic subgroups of order N. They form the prototype of higher dimensional versions, modular varieties, which parametrize abelian varieties with other structures involving points of finite order. We have already seen the use of such varieties in Faltings’ proof of the Mordell conjecture, and more specifically of the Shafarevich conjecture, in Chapter IV, §5.