Event № 272
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/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/Q obtained as a specialization of a branched covering φ: C → P^1(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 ≤ m(S_m) ≤ 4 and n ≤ m(S^n_m) ≤ n + 4, for all n, m > 0. Here S_m denotes the symmetric group on m letters, and S^n_m is the direct product of n copies of S_m. We also get the correct asymptotic of m(G^n), as n → ∞ 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. Joint work with Lior Bary-Soroker.