Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-24T00:19:03.905Z Has data issue: false hasContentIssue false
  • English
  • Français

EFFICIENTLY GENERATED SPACES OF CLASSICAL SIEGEL MODULAR FORMS AND THE BÖCHERER CONJECTURE

Part of: Computational number theory Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  01 April 2011

MARTIN RAUM
MARTIN RAUM*
Affiliation:
MPI für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We state and verify up to weight 172 a conjecture on the existence of a certain generating set for spaces of classical Siegel modular forms. This conjecture is particularly useful for calculations involving Fourier expansions. Using this generating set, we verify the Böcherer conjecture for nonrational eigenforms and discriminants with class number greater than one. As a further application we verify another conjecture for weights up to 150 and investigate an analog of the Victor–Miller basis. Additionally, we describe some arithmetic properties of the basis we found.

Keywords

Siegel modular formsBöcherer conjectureHecke actioncomputation

MSC classification

Secondary: 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms 11Y99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 89 , Issue 3 , December 2010 , pp. 393 - 405
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Bergström, J., Faber, C. and van der Geer, G., ‘Siegel modular forms of genus 2 and level 2: cohomological computations and conjectures’, Preprint, 2008.CrossRefGoogle Scholar
[2]Böcherer, S., ‘Bemerkungen über die Dirichletreihen von Koecher und Maaß (remarks on the Dirichlet series of Koecher and Maaß)’, Math. Gottingensis 68 (1986), 1–36.Google Scholar
[3]Böcherer, S. and Schulze-Pillot, R., ‘The Dirichlet series of Koecher and Maaß and modular forms of weight 3/2’, Math. Z. 209 (1992), 273–287.CrossRefGoogle Scholar
[4]Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system. I. The user language’, J. Symbolic Comput. 24 (1997), 235–265.CrossRefGoogle Scholar
[5]Brumer, A. and Kramer, K., ‘Abelian surfaces over ℚ’, Preprint, 2009.Google Scholar
[6]Deligne, P., ‘Valeurs de fonctions L et périodes d’intégrales’, in: Automorphic Forms, Representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proceedings of Symposia in Pure Mathematics, XXXIII (American Mathematical Society, Providence, RI, 1979), pp. 313–346.Google Scholar
[7]Eichler, M. and Zagier, D., The Theory of Jacobi Forms (Birkhäuser, Boston, MA, 1985).CrossRefGoogle Scholar
[8]Gritsenko, V. A., ‘Modulformen zur Paramodulgruppe und Modulräume der abelschen Varietäten’, Math. Gottingensis 12 (1995), 1–89.Google Scholar
[9]Igusa, J., ‘On Siegel modular forms of genus two’, Amer. J. Math. 84 (1962), 175–200.CrossRefGoogle Scholar
[10]Kohnen, W. and Kuß, M., ‘Some numerical computations concerning spinor zeta functions in genus 2 at the central point’, Math. Comp. 71(240) (2002), 1597–1607.CrossRefGoogle Scholar
[11]Kohnen, W. and Zagier, D., Modular forms with rational periods’, Modular forms, Durham Symposium, England, 1983, 197–249, 1984.Google Scholar
[12]Krieg, A. and Raum, M., ‘The functional equation for the twisted spinor-L-function of genus 2’, arXiv:0907.2767 [math.NT].Google Scholar
[13]Poor, C. and Yuen, D., ‘Paramodular cups forms’, 2009, arXiv:0912.0049v1 [math.NT].Google Scholar
[14]Raum, M., ‘A computational approach to the Böcherer conjecture’, talk at the AKLS meeting Cologne, May 5th 2010.Google Scholar
[15]Raum, M., Ryan, N., Skoruppa, N. P. and Tornaría, G., ‘Theoretical and algorithmic aspects of an implementation of siegel modular forms’, Preprint, 2010.Google Scholar
[16]Ryan, N. C. and Tornaría, G., ‘A Böcherer-type conjecture for paramodular forms’, 2010, http://arxiv.org/abs/1006.1582v1 [math.NT].Google Scholar
[17]Skoruppa, N. P., ‘Computations of Siegel modular forms of genus two’, Math. Comp. 58(197) (1992), 381–398.CrossRefGoogle Scholar
[18]Stein, W. A.et al., ‘Sage mathematics software (version 4.1.2)’, 2009, http://www.sagemath.org.Google Scholar