Modular symbols over number fields
Aranes, M. (2010) Modular symbols over number fields. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b2490253~S15
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  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  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 .
|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|
|Institution:||University of Warwick|
|Theses Department:||Mathematics Institute|
|Supervisor(s)/Advisor:||Cremona, J. E.|
|Sponsors:||European Commission (EC) (MRTN-CT-2006-035495)|
|Extent:||v, 140 leaves : charts|
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