Aranes, M. (2010) Modular symbols over number fields. PhD thesis, University of Warwick.

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Abstract

Let K be a number field, R its ring of integers. For some classes of fields, spaces of cusp forms of weight 2 for GL(2;K) have been computed using methods based on modular symbols. J.E. Cremona [9] began the programme of extending the classical methods over Q to the case of imaginary quadratic fields. This work was continued by some of his Ph.D. students [35, 6, 22], and results have been obtained for some imaginary quadratic fields with small class number. More recently, P. Gunnells and D. Yasaki [18] have developed related algorithms for real quadratic fields.

The aim of this thesis is to contribute to the extension of the modular symbols method, when possible developing algorithms and implementations for effective computations. Some parts of the theory are purely algebraic and can be extended to all number fields. We generalise the theory for cusps and Manin symbols; we also describe a generalisation of Atkin-Lehner involutions and study other normaliser elements. On the other hand, all previous explicit computations for the imaginary quadratic field case were done only for specific fields. In the last part of this thesis we begin work towards a general implementation of the techniques used in this case. In particular, we are able to compute a fundamental domain of the hyperbolic 3-space for any imaginary quadratic field.

Implementations of the algorithms described in this thesis have been written by the author in the open-source mathematics software Sage [31].

Item Type:	Thesis or Dissertation (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Algebraic fields, Modules (Algebra)
Official Date:	December 2010
Dates:	Date Event December 2010 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Cremona, J. E.
Sponsors:	European Commission (EC) (MRTN-CT-2006-035495)
Extent:	v, 140 leaves : charts
Language:	eng
URI:	http://wrap.warwick.ac.uk/id/eprint/35128

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