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
DOI:
https://doi.org/10.1090/S0025-5718-06-01914-4
Published electronically:
December 4, 2006
MathSciNet review:
2291849
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Abstract | References | Similar Articles | Additional Information
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.
- Caterina Consani and Jasper Scholten, Arithmetic on a quintic threefold, Internat. J. Math. 12 (2001), no. 8, 943–972. MR 1863287, DOI https://doi.org/10.1142/S0129167X01001118
- Henri Darmon and Adam Logan, Periods of Hilbert modular forms and rational points on elliptic curves, Int. Math. Res. Not. 40 (2003), 2153–2180. MR 1997296, DOI https://doi.org/10.1155/S1073792803131108
- L. Dembélé, Explicit computations of Hilbert modular forms on $\mathbb {Q}(\sqrt {5})$. Ph.D. Thesis at McGill University, 2002.
- Lassina Dembélé, Explicit computations of Hilbert modular forms on ${\Bbb Q}(\sqrt {5})$, Experiment. Math. 14 (2005), no. 4, 457–466. MR 2193808
- L. Dembélé, F. Diamond and Robert; Numerical evidences of the weight part of the Serre conjecture for Hilbert modular forms. (preprint).
- Stephen S. Gelbart, Automorphic forms on adèle groups, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1975. Annals of Mathematics Studies, No. 83. MR 0379375
- Benedict H. Gross, Algebraic modular forms, Israel J. Math. 113 (1999), 61–93. MR 1729443, DOI https://doi.org/10.1007/BF02780173
- H. Jacquet and R. P. Langlands, Automorphic forms on ${\rm GL}(2)$, Lecture Notes in Mathematics, Vol. 114, Springer-Verlag, Berlin-New York, 1970. MR 0401654
- Magma algebraic computing system. http://magma.maths.usyd.edu.au
- Ju. I. Manin, Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 19–66 (Russian). MR 0314846
- Loïc Merel, Universal Fourier expansions of modular forms, On Artin’s conjecture for odd $2$-dimensional representations, Lecture Notes in Math., vol. 1585, Springer, Berlin, 1994, pp. 59–94. MR 1322319, DOI https://doi.org/10.1007/BFb0074110
- Arnold Pizer, An algorithm for computing modular forms on $\Gamma _{0}(N)$, J. Algebra 64 (1980), no. 2, 340–390. MR 579066, DOI https://doi.org/10.1016/0021-8693%2880%2990151-9
- D. Pollack, Explicit Hecke action on modular forms. Ph.D. thesis at Havard University 1998.
- Goro Shimura, The special values of the zeta functions associated with Hilbert modular forms, Duke Math. J. 45 (1978), no. 3, 637–679. MR 507462
- Jude Socrates and David Whitehouse, Unramified Hilbert modular forms, with examples relating to elliptic curves, Pacific J. Math. 219 (2005), no. 2, 333–364. MR 2175121, DOI https://doi.org/10.2140/pjm.2005.219.333
- W. Stein, Explicit approaches to Abelian varieties. Ph.D. thesis at University of California at Berkeley, 2000.
- Richard Taylor, On Galois representations associated to Hilbert modular forms, Invent. Math. 98 (1989), no. 2, 265–280. MR 1016264, DOI https://doi.org/10.1007/BF01388853
- Marie France Guého, Le théorème d’Eichler sur le nombre de classes d’idéaux d’un corps de quaternions totalement défini et la mesure de Tamagawa, Journées Arithmétiques (Grenoble, 1973) Soc. Math. France, Paris, 1974, pp. 107–114. Bull. Soc. Math. France Mém., No. 37 (French). MR 0376621, DOI https://doi.org/10.24033/msmf.136
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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
Email:
dembele@math.ucalgary.ca
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.