Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Quaternionic Manin symbols, Brandt matrices, and Hilbert modular forms

Author: Lassina Dembélé
Journal: Math. Comp. 76 (2007), 1039-1057
MSC (2000): Primary 11-xx; Secondary 11Gxx
Published electronically: December 4, 2006
MathSciNet review: 2291849
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Abstract: In this paper, we propose a generalization of the algorithm we developed previously. Along the way, we also develop a theory of quaternionic $ M$-symbols whose definition bears some resemblance to the classical $ M$-symbols, except for their combinatorial nature. The theory gives a more efficient way to compute Hilbert modular forms over totally real number fields, especially quadratic fields, and we have illustrated it with several examples. Namely, we have computed all the newforms of prime levels of norm less than 100 over the quadratic fields $ \mathbb{Q}(\sqrt{29})$ and $ \mathbb{Q}(\sqrt{37})$, and whose Fourier coefficients are rational or are defined over a quadratic field.

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Additional Information

Lassina Dembélé
Affiliation: Department of mathematics and statistics, University of Calgary, 2500 University Drive N.W., Calgary, AB, Canada T2N 1N4

Keywords: Hilbert modular forms, automorphic forms, Brandt matrices.
Received by editor(s): April 8, 2004
Received by editor(s) in revised form: January 18, 2006
Published electronically: December 4, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.