Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: New models for the action of Hecke operators in spaces of Maass wave
  forms   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 $\mathrm{SL}_2({\mathbb
R})$.

1
The categories of this paper are: cs
36
The categories of this paper are: cmp-lg
28
The categories of this paper are: alg-geom
31
The categories of this paper are: ao-sci
13
The categories of this paper are: quant-ph
24
The categories of this paper are: dg-ga
30
The categories of this paper are: q-alg
10
The categories of this paper are: q-bio
34
The categories of this paper are: chem-ph
21
The categories of this paper are: adap-org
14
The categories of this paper are: math-ph
22
The categories of this paper are: chao-dyn
4
The categories of this paper are: cond-mat
33
The categories of this paper are: plasm-ph
27
The categories of this paper are: comp-gas
8
The categories of this paper are: hep-ex