logo ANNALES DE L'INSTITUT FOURIER

With cedram.org
Table of contents for this issue | Previous article | Next article
Ian Kiming
New models for the action of Hecke operators in spaces of Maass wave forms
(Des modèles nouveaux pour l’action des opérateurs de Hecke dans les espaces des formes d’onde de Maass)
Annales de l'institut Fourier, 57 no. 6 (2007), p. 1863-1882, doi: 10.5802/aif.2316
Article PDF | Reviews MR 2377889 | Zbl pre05225866
Class. Math.: 11F70, 11R39, 22E50
Keywords: Maass wave forms, Hecke operators, Hecke eigenvalues, Poisson transform.

Résumé - Abstract

Utilizing the theory of the Poisson transform, we develop some new concrete models for the Hecke theory in a space $M_{\lambda }(N)$ of Maass forms with eigenvalue $1/4-\lambda ^2$ on a congruence subgroup $\Gamma _1(N)$. We introduce the field $F_{\lambda } = \mathbb{Q} (\lambda ,\sqrt{n} , n^{\lambda /2} \mid ~n\in \mathbb{N} )$ so that $F_{\lambda }$ consists entirely of algebraic numbers if $\lambda = 0$.

The main result of the paper is the following. For a packet $\Phi = (\nu _p \mid p\nmid N)$ of Hecke eigenvalues occurring in $M_{\lambda }(N)$ we then have that either every $\nu _p$ is algebraic over $F_{\lambda }$, or else $\Phi $ will – for some $m\in \mathbb{N}$ – occur in the first cohomology of a certain space $W_{\lambda ,m}$ which is a space of continuous functions on the unit circle with an action of $\mathrm{SL}_2({\mathbb{R}})$ well-known from the theory of (non-unitary) principal representations of $\mathrm{SL}_2({\mathbb{R}})$.

Bibliography

[1] A. Ash & G. Stevens, “Cohomology of arithmetic groups and congruences between systems of Hecke eigenvalues”, J. Reine Angew. Math. 365 (1990), p. 192-220  MR 826158 |  Zbl 0596.10026
[2] D. Blasius, L. Clozel & D. Ramakrishnan, “Algébricité de l’action des opérateurs de Hecke sur certaines formes de Maass”, C. R. Acad. Sci. Paris 305 (1987), p. 705-708  Zbl 0628.10035
[3] D. Blasius, L. Clozel & D. Ramakrishnan, “Opérateurs de Hecke et formes de Maass: application de formule des traces”, C. R. Acad. Sci. Paris 306 (1988), p. 59-62  MR 929109 |  Zbl 0639.10020
[4] U. Bunke & M. Olbrich, “Cohomological properties of the smooth globalization of a Harish-Chandra module”, Preprint 1995, http://xxx.lanl.gov/abs/math.RT/9508203 arXiv
[5] U. Bunke & M. Olbrich, “Fuchsian groups of the second kind and representations carried by the limit set”, Invent. math. 127 (1996), p. 127-154 Article |  MR 1423028 |  Zbl 0880.30035
[6] M. Flensted-Jensen, Analysis on non-Riemannian symmetric spaces, CBMS Regional Conference Series in Mathematics, AMS, 1986  MR 837420 |  Zbl 0589.43008
[7] G. Henniart, “Erratum à l’exposé No. 711”, Astérisque 201–203 (1991), p. 485-486 Numdam |  Zbl 0748.11057
[8] H. Maaß, “Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen”, Math. Ann. 121 (1949), p. 141-183 Article |  MR 31519 |  Zbl 0033.11702
top