Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Arithmetic and geometry of the Hecke groups   We study the arithmetic and geometry properties of the Hecke group $G_q$. In
particular, we prove that $G_q$ has a subgroup $X $ of index $d$, genus $g$
with $v_{\infty} $ cusps, and $\tau_2$ (resp. $v_{r_i}$) conjugacy classes of
elements that are conjugates of $S$ (resp. $R^{q/r_i}$) if and only if (i) $
2g-2 + \tau_2/2 +\sum_{i=1}^k v_{r_i}(1-1/r_i) + v_{\infty} = d(1/2-1/q)$, and
(ii) $ m _0= 4g-4 +\tau_2 + 2 v_{\infty} + \sum _{i=1}^k v_{r_i}(2-q/r_i)\ge 0$
is a multiple of $q-2$, (iii) $m \ge 0$. In the case $q$ is odd, (ii) is a
consequence of (i).

11
The categories of this paper are: hep-lat
25
The categories of this paper are: patt-sol
26
The categories of this paper are: bayes-an
20
The categories of this paper are: acc-phys
27
The categories of this paper are: comp-gas
10
The categories of this paper are: q-bio
0
The categories of this paper are: hep-ph
3
The categories of this paper are: physics
8
The categories of this paper are: hep-ex
23
The categories of this paper are: solv-int
30
The categories of this paper are: q-alg
35
The categories of this paper are: mtrl-th
16
The categories of this paper are: stat
28
The categories of this paper are: alg-geom
32
The categories of this paper are: atom-ph
33
The categories of this paper are: plasm-ph