×

zbMATH — the first resource for mathematics

Quaternionic Manin symbols, Brandt matrices, and Hilbert modular forms. (English) Zbl 1158.11023
Author’s summary: The author proposes a generalization of the algorithm he developed previously [Exp. Math. 14, No. 4, 457–466 (2005; Zbl 1152.11328)]. Along the way, he also develops 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 he illustrates it with several examples. Namely, he computes 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.

MSC:
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11Y16 Number-theoretic algorithms; complexity
Software:
Magma
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Caterina Consani and Jasper Scholten, Arithmetic on a quintic threefold, Internat. J. Math. 12 (2001), no. 8, 943 – 972. · Zbl 1111.11306
[2] Henri Darmon and Adam Logan, Periods of Hilbert modular forms and rational points on elliptic curves, Int. Math. Res. Not. 40 (2003), 2153 – 2180. · Zbl 1038.11035
[3] L. Dembélé, Explicit computations of Hilbert modular forms on \( \mathbb{Q}(\sqrt{5})\). Ph.D. Thesis at McGill University, 2002.
[4] Lassina Dembélé, Explicit computations of Hilbert modular forms on \Bbb Q(\sqrt 5), Experiment. Math. 14 (2005), no. 4, 457 – 466. · Zbl 1152.11328
[5] L. Dembélé, F. Diamond and Robert; Numerical evidences of the weight part of the Serre conjecture for Hilbert modular forms. (preprint).
[6] 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. · Zbl 0329.10018
[7] Benedict H. Gross, Algebraic modular forms, Israel J. Math. 113 (1999), 61 – 93. · Zbl 0965.11020
[8] H. Jacquet and R. P. Langlands, Automorphic forms on \?\?(2), Lecture Notes in Mathematics, Vol. 114, Springer-Verlag, Berlin-New York, 1970. · Zbl 0236.12010
[9] Magma algebraic computing system. http://magma.maths.usyd.edu.au
[10] Ju. I. Manin, Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 19 – 66 (Russian). · Zbl 0243.14008
[11] 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. · Zbl 0844.11033
[12] Arnold Pizer, An algorithm for computing modular forms on \Gamma \(_{0}\)(\?), J. Algebra 64 (1980), no. 2, 340 – 390. · Zbl 0433.10012
[13] D. Pollack, Explicit Hecke action on modular forms. Ph.D. thesis at Havard University 1998.
[14] Goro Shimura, The special values of the zeta functions associated with Hilbert modular forms, Duke Math. J. 45 (1978), no. 3, 637 – 679. · Zbl 0394.10015
[15] Jude Socrates and David Whitehouse, Unramified Hilbert modular forms, with examples relating to elliptic curves, Pacific J. Math. 219 (2005), no. 2, 333 – 364. · Zbl 1109.11029
[16] W. Stein, Explicit approaches to Abelian varieties. Ph.D. thesis at University of California at Berkeley, 2000.
[17] Richard Taylor, On Galois representations associated to Hilbert modular forms, Invent. Math. 98 (1989), no. 2, 265 – 280. · Zbl 0705.11031
[18] 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). · Zbl 0288.12008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.