Modular curves over Q(i)

Abstract

In his paper, Rational Points On Modular Curves, Barry Mazur stated the following program, which is now known as Mazur’s Program B: “Given a number field K and a subgroup H of GL_2(\hat{Z})=\prod_{p}GL_2(Z_p), classify all elliptic curves E/K whose associated Galois representation on torsion points maps Gal(\bar{K}/K) into H \subset GL_2(\hat{Z}). “ One way to do this is to count the K-rational points on the modular curve associated to H. Over the years, lots of progress has been made on Mazur’s Program B concerning elliptic curves defined over the rational field. However, less is known about modular curves defined over a general number field. The first step in advancing Program B for elliptic curves defined over number fields is enumerating all arithmetically maximal subgroups H \subset GL_2(\hat{Z}). This thesis aims to enumerate all such groups associated to modular curves defined over the number field Q(i).