Sieves and the Minimal Ramification Problem

Abstract: 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.