Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Arithmetic representations of fundamental groups II: finiteness   Let $X$ be a smooth curve over a finitely generated field $k$, and let $\ell$
be a prime different from the characteristic of $k$. We analyze the dynamics of
the Galois action on the deformation rings of mod $\ell$ representations of the
geometric fundamental group of $X$. Using this analysis, we prove analogues of
the Shafarevich and Fontaine-Mazur finiteness conjectures for function fields
over algebraically closed fields in arbitrary characteristic, and a weak
variant of the Frey-Mazur conjecture for function fields in characteristic
zero.
  For example, we show that if $X$ is a normal, connected variety over
$\mathbb{C}$, the (typically infinite) set of representations of
$\pi_1(X^{\text{an}})$ into $GL_n(\overline{\mathbb{Q}_\ell})$, which come from
geometry, has no limit points. As a corollary, we deduce that if $L$ is a
finite extension of $\mathbb{Q}_\ell$, then the set of representations of
$\pi_1(X^{\text{an}})$ into $GL_n(L)$, which arise from geometry, is finite.

37
The categories of this paper are: supr-con
28
The categories of this paper are: alg-geom
13
The categories of this paper are: quant-ph
1
The categories of this paper are: cs
9
The categories of this paper are: nlin
30
The categories of this paper are: q-alg
3
The categories of this paper are: physics
34
The categories of this paper are: chem-ph
32
The categories of this paper are: atom-ph
27
The categories of this paper are: comp-gas
6
The categories of this paper are: astro-ph
15
The categories of this paper are: nucl-ex
10
The categories of this paper are: q-bio
12
The categories of this paper are: nucl-th
21
The categories of this paper are: adap-org
0
The categories of this paper are: hep-ph