Modular Symbols Modulo Eisenstein Ideals for Bianchi Spaces

Abstract

The goal of this thesis is two-fold. First, it gives an efficient method for calculating the action of Hecke operators in terms of "Manin" symbols, otherwise known as "M-symbols," in the first homology group of Bianchi spaces. Second, it presents data that may be used to understand and better state an unpublished conjecture of Fukaya, Kato, and Sharifi concerning the structure of Bianchi Spaces modulo Eisenstein ideals [5]. Swan, Cremona, and others have studied the homology of Bianchi spaces characterized as certain quotients of hyperbolic 3-space [3], [13]. The first homology groups are generated both by modular symbols and a certain subset of them: the Manin symbols. This is completely analogous to the study of the homology of modular curves. For modular curves, Merel developed a technique for calculating the action of Hecke operators completely in terms of "Manin" symbols [10]. For Bianchi spaces, Bygott and Lingham outlined methods for calculating the action of Hecke operators in terms of modular symbols [2], [9]. This thesis generalizes the work of Merel to Bianchi spaces. The relevant Bianchi spaces are characterized by imaginary quadratic fields K. The methods described in this thesis deal primarily with the case that the ring of integers of K is a PID. Let p be an odd prime that is split in K. The calculations give the F_p-dimension of the homology modulo both p and an Eisenstein ideal. Data is given for primes less than 50 and the five Euclidean imaginary quadratic fields Q(√-1), Q(√-2), Q(√-3), Q(√-7), and Q(√-11). All of the data presented in this thesis comes from computations done using the computer algebra package Magma.