Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sieves and the Minimal Ramification Problem   The minimal ramification problem may be considered as a quantitative version
of the inverse Galois problem. For a nontrivial finite group $G$, let $m(G)$ be
the minimal integer $m$ for which there exists a Galois extension
$N/\mathbb{Q}$ that is ramified at exactly $m$ primes (including the infinite
one). So, the problem is to compute or to bound $m(G)$.
  In this paper, we bound the ramification of extensions $N/\mathbb{Q}$
obtained as a specialization of a branched covering $\phi\colon C\to
\mathbb{P}^1_{\mathbb{Q}}$. This leads to novel upper bounds on $m(G)$, for
finite groups $G$ that are realizable as the Galois group of a branched
covering. Some instances of our general results are: $$ 1\leq m(S_m)\leq 4
\quad \mbox{and} \quad n\leq m(S_m^n) \leq n+4, $$ for all $n,m>0$. Here $S_m$
denotes the symmetric group on $m$ letters, and $S_m^n$ is the direct product
of $n$ copies of $S_m$. We also get the correct asymptotic of $m(G^n)$, as $n
\to \infty$ for a certain class of groups $G$.
  Our methods are based on sieve theory results, in particular on the
Green-Tao-Ziegler theorem on prime values of linear forms in two variables, on
the theory of specialization in arithmetic geometry, and on finite group
theory.

29
The categories of this paper are: funct-an
26
The categories of this paper are: bayes-an
17
The categories of this paper are: q-fin
10
The categories of this paper are: q-bio
13
The categories of this paper are: quant-ph
36
The categories of this paper are: cmp-lg
28
The categories of this paper are: alg-geom
11
The categories of this paper are: hep-lat
6
The categories of this paper are: astro-ph
14
The categories of this paper are: math-ph
5
The categories of this paper are: gr-qc
32
The categories of this paper are: atom-ph
24
The categories of this paper are: dg-ga
16
The categories of this paper are: stat
18
The categories of this paper are: econ
21
The categories of this paper are: adap-org