New models for the action of Hecke operators in spaces of Maass wave forms

Kiming, Ian. "New models for the action of Hecke operators in spaces of Maass wave forms." Annales de l’institut Fourier 57.6 (2007): 1863-1882. <http://eudml.org/doc/10280>.

@article{Kiming2007,
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 $\mathrm\{SL\}_2(\{\mathbb\{R\}\})$.},
affiliation = {University of Copenhagen Department of Mathematics Universitetsparken 5 2100 Copenhagen Ø (Denmark)},
author = {Kiming, Ian},
journal = {Annales de l’institut Fourier},
keywords = {Maass wave forms; Hecke operators; Hecke eigenvalues; Poisson transform},
language = {eng},
number = {6},
pages = {1863-1882},
publisher = {Association des Annales de l’institut Fourier},
title = {New models for the action of Hecke operators in spaces of Maass wave forms},
url = {http://eudml.org/doc/10280},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Kiming, Ian
TI - New models for the action of Hecke operators in spaces of Maass wave forms
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 6
SP - 1863
EP - 1882
AB - 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 $\mathrm{SL}_2({\mathbb{R}})$.
LA - eng
KW - Maass wave forms; Hecke operators; Hecke eigenvalues; Poisson transform
UR - http://eudml.org/doc/10280
ER -