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 $N\mathrm{/}\mathbb{Q}$ 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 $N\mathrm{/}\mathbb{Q}$ obtained as a specialization of a branched covering $\phi\colon C\to $\mathbb{P}$^1_{$\mathbb{Q}$}$\mathrm{.}Thisleadstonovelupperboundson$m(G)$, for finite groups $G$ that are realizable as the Galois group of a branched covering. Some instances of our general results are: $$1\le m(S_m)\le 4andn\le m(S_m^n)\le n+4,$$ 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 $n \to \infty$foracertainclassofgroups$G$. 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.