Modular forms and elliptic curves over imaginary quadratic fields

The aim of this thesis is to contribute to an ongoing project to understand the correspondence between cusp forms, for imaginary quadratic fields, and elliptic curves. This contribution mainly takes the form of developing explicit constructions and computing particular examples. It is hoped that as well as being of interest in themselves, they will be helpful in guiding future theoretical developments. Cremona [7] began the programme of extending the classical techniques using modular symbols to the case of imaginary quadratic fields. He was followed by two of his students Whitley [25] and Bygott [5]. Together they have covered the cases where the class number of the field is equal to 1 or 2. This thesis extends their work to treat all fields of odd class number. It describes an algorithm, which holds for any such field, for determining the space of cusp forms, and for computing the eigenforms and eigenvalues for the action of the Hecke algebra on this space. The approach, using modular symbols, closely follows the work of the previous authors, but new techniques and theoretical simplifcations are obtained which hold in the case considered. All of the algorithms presented in this thesis have been implemented in a computer algebra package, Magma [3], and the results obtained for the fields Q(sqrt(-23)) and Q(sqrt(-31)) are included.EThOS - Electronic Theses Online ServiceGBUnited Kingdo