Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-22T17:45:40.806Z Has data issue: false hasContentIssue false
  • English
  • Français

LIFTING PUZZLES IN DEGREE FOUR

Part of: Numerical analysis Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  06 July 2009

CRIS POOR ,
NATHAN C. RYAN  and
DAVID S. YUEN
CRIS POOR*
Affiliation:
Department of Mathematics, Fordham University, Bronx, NY 10458-5165, USA (email: [email protected])
NATHAN C. RYAN
Affiliation:
Department of Mathematics, Bucknell University, Lewisburg, PA 17837, USA (email: [email protected])
DAVID S. YUEN
Affiliation:
Department of Mathematics and Computer Science, Lake Forest College, 555 North Sheridan Road, Lake Forest, IL 60045, USA (email: [email protected])
*
✉For correspondence; e-mail: [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 identify the majority of Siegel modular eigenforms in degree four and weights up to 16 as being Duke–Imamoḡlu–Ikeda or Miyawaki–Ikeda lifts. We give two examples of eigenforms that are probably also lifts but of an undiscovered type.

Keywords

Siegel modular formsSatake parametersMiyawaki–Ikeda lifts

MSC classification

Secondary: 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms 11F60: Hecke-Petersson operators, differential operators (several variables) 65-04: Explicit machine computation and programs (not the theory of computation or programming)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 80 , Issue 1 , August 2009 , pp. 65 - 82
Copyright
Copyright © Australian Mathematical Society 2009

References

[1] Andrianov, A. N., Quadratic Forms and Hecke Operators, Grundlehren der mathematischen Wissenschaften, 286 (Springer, Berlin, 1987).CrossRefGoogle Scholar
[2] Breulmann, S. and Kuss, M., ‘On a conjecture of Duke–Imamoḡlu’, Proc. Amer. Math. Soc. 128 (2000), 1595–1604.CrossRefGoogle Scholar
[3] Conway, J. H. and Sloane, N. J. A., Sphere Packings, Lattices and Groups, 3rd edn, Grundlehren der mathematischen Wissenschaften, 290 (Springer, New York, 1999).CrossRefGoogle Scholar
[4] Duke, W. and Imamoḡlu, Ö., ‘Siegel modular forms of small weight’, Math. Ann. 310 (1998), 73–82.CrossRefGoogle Scholar
[5] Eichler, M. and Zagier, D., The Theory of Jacobi Forms, Progress in Mathematics, 55 (Birkhäuser, Basel, 1985).CrossRefGoogle Scholar
[6] Freitag, E., Siegelsche Modulfunktionen, Grundlehren der mathematischen Wissenschaften, 254 (Springer, Berlin, 1983), p. x+341.CrossRefGoogle Scholar
[7] Hafner, J. L. and Walling, L. H., ‘Explicit action of Hecke operators on Siegel modular forms’, J. Number Theory 93 (2002), 34–57.CrossRefGoogle Scholar
[8] Ikeda, T., ‘Pullback of the lifting of elliptic cusp forms and Miyawaki’s conjecture’, Duke Math J. 131 (2006), 469–497.CrossRefGoogle Scholar
[9] Ikeda, T., ‘On the lifting of elliptic cusp forms to Siegel cusp forms of degree 2n’, Ann. of Math. (2) 154 (2001), 641–681.CrossRefGoogle Scholar
[10] Katsurada, H., ‘An explicit formula for Siegel series’, Amer. J. Math. 121 (1999), 415–452.CrossRefGoogle Scholar
[11] Katsurada, H. and Kawamura, H., ‘On Ikeda’s conjecture on the period of the Ikeda lift and its application’, in: RIMS Kokyuroku Bessatsu (RIMS Lecture Notes, Extra Volume) (Research Institute for Mathematical Sciences, Kyoto, 2008), pp. 1–22.Google Scholar
[12] Katsurada, H. and Kawamura, H., On Ikeda’s conjecture on the period of the Ikeda lift, Proc. RIMS Workshop on Construction of Automorphic Forms and their Periods with Applications to appear.Google Scholar
[13] King, O. D., ‘A mass formula for unimodular lattices with no roots’, Math. Comp. 72 (2003), 839–863.CrossRefGoogle Scholar
[14] Krieg, A., ‘Das vertauschungsgesetz zwischen Hecke-Operatoren und dem Siegelschenϕ-Operator’, Arch. Math. (Basel) 46 (1986), 323–329.CrossRefGoogle Scholar
[15] Kurokawa, N., ‘Examples of eigenvalues of Hecke operators on Siegel cusp forms of degree two’, Invent. Math. 49 (1978), 149–165.CrossRefGoogle Scholar
[16] Miyawaki, I., ‘Numerical examples of Siegel cusp forms of degree 3 and their zeta-functions’, Mem. Fac. Sci. Kyushu Univ. Ser. A 46 (1992), 307–339.Google Scholar
[17] Ozeki, M., ‘On a property of Siegel theta-series’, Math. Ann. 228 (1977), 249–258.CrossRefGoogle Scholar
[18] Poor, C. and Yuen, D. S., ‘Computations of Spaces of Siegel modular cusp forms’, J. Math. Soc. Japan 59 (2007), 185–222.CrossRefGoogle Scholar
[19] Poor, C. and Yuen, D. S., Authors website, http://math.lfc.edu/∼yuen/comp.Google Scholar
[20] Poor, C. and Yuen, D. S., Authors website, http://math.lfc.edu/∼yuen/puzzles.Google Scholar
[21] Ryan, N. C. and Shemanske, T. R., ‘Inverting the satake isomorphism for Sp n with applications to Hecke operators’, Ramanujan J. 17 (2008), 219–244.CrossRefGoogle Scholar
[22] Witt, E., ‘Eine Identität zwischen Modulformen zweiten Grades’, Abh. Math. Sem. Hansischen Univ. 14 (1941), 323–337.CrossRefGoogle Scholar