Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-29T05:22:58.962Z Has data issue: false hasContentIssue false
  • English
  • Français

Explicit Calculations of Automorphic Forms for Definite Unitary Groups

Published online by Cambridge University Press:  01 February 2010

David Loeffler
David Loeffler
Affiliation:
Department of Mathematics, Imperial College, South Kensington, London SW7 2AZ, United Kingdom, [email protected]

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.

I give an algorithm for computing the full space of automor-phic forms for definite unitary groups over ℚ, and apply this to calculate the automorphic forms of level G(hat{Z}) and various small weights for an example of a rank 3 unitary group. This leads to some examples of various types of endoscopic lifting from automorphic forms for U1 × U1 × U1 and U1 × U2, and to an example of a non-endoscopic form of weight (3, 3) corresponding to a family of 3-dimensional irreducible ℓ-adic Galois representations. I also compute the 2-adic slopes of some automorphic forms with level structure at 2, giving evidence for the local constancy of the slopes.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 11 , 2008 , pp. 326 - 342
Copyright
Copyright © London Mathematical Society 2008

References

1.Blasius, D. and Rogawski, J. D., ‘Tate classes and arithmetic quotients of the two-ball’, The zeta functions of Picard modular surfaces (University of Montréal, 1992) 421–444.Google Scholar
2.Bosma, W., Cannon, J. and Playoust, C., ‘The MAGMA algebra system. I. The user language’, J. Symbolic Comput. 24 (1997) 235–265.CrossRefGoogle Scholar
3.Chenevier, G., ‘Families p-adiques de formes automorphes et applications aux conjectures de Bloch-Kato’, PhD thesis, Université Paris VII (2003).CrossRefGoogle Scholar
4.Dembélé, L., ‘Quaternionic Manin symbols, Brandt matrices, and Hilbert modular forms’, Math. Gomp. 76 (2007) 1039–1057 (electronic).Google Scholar
5.Fulton, W. and Harris, J., Representation theory: a first course, Graduate Texts in Mathematics 129 (Springer, 1991).Google Scholar
6.Gan, W. T., Hanke, J. P. and Yu, J.-K., ‘On an exact mass formula of Shimura’, Duke Math. J. 107 (2001) 103–133.CrossRefGoogle Scholar
7.Gross, B. H., ‘Algebraic modular forms’, Israel J. Math. 113 (1999) 61–93.CrossRefGoogle Scholar
8.Lansky, J. and Pollack, D., ‘Hecke algebras and automorphic forms’, Compositio Math. 130 (2002) 21–48.CrossRefGoogle Scholar
9.Loeffler, D., ‘Adventures with polynomials: a criterion for Weil numbers’, Eureka 59, to appear. Available from http://www.dpmms.cam.ac.uk/~d1267/maths/cubics.pdf.Google Scholar
10.Platonov, V. and Rapinchuk, A., Algebraic groups and number theory, Pure and Applied Mathematics 139 (Academic Press, 1994).Google Scholar
11.Rogawski, J. D., Automorphic representations of unitary groups in three variables, Annals of Mathematics Studies 123 (Princeton University Press, 1990).CrossRefGoogle Scholar
12.Stein, W., Modular forms, a computational approach, Graduate Studies in Mathematics 79 (American Mathematical Society, 2007).CrossRefGoogle Scholar
13.Stein, W., SAGE Mathematics Software, version 2.8.9 (The Sage Group, 2007), http://www.sagemath.org/.Google Scholar
14.Washington, L. C., Introduction to cyclotomic fields, Graduate Texts in Mathematics 83 (Springer, 1982).CrossRefGoogle Scholar
Supplementary material: File

JCM 11 Loeffler Appendix A Part 1

Loeffler Appendix A Part 1

Download JCM 11 Loeffler Appendix A Part 1(File)
File 655.8 KB
Supplementary material: File

JCM 11 Loeffler Appendix A readme

Loeffler Appendix A readme.txt

Download JCM 11 Loeffler Appendix A readme(File)
File 2.1 KB
Supplementary material: File

JCM 11 Loeffler Appendix B Part 1

Loeffler Appendix B Part 1

Download JCM 11 Loeffler Appendix B Part 1(File)
File 4.3 KB
Supplementary material: File

JCM 11 Loeffler Appendix B readme

Appendix B readme.txt

Download JCM 11 Loeffler Appendix B readme(File)
File 384 Bytes