TY - JOUR

T1 - Sieves and the Minimal Ramification Problem

AU - Bary-Soroker, Lior

AU - Schlank, Tomer M.

N1 - Publisher Copyright:
© 2020 Cambridge University Press.

PY - 2020/5/1

Y1 - 2020/5/1

N2 - The minimal ramification problem may be considered as a quantitative version of the inverse Galois problem. For a nontrivial finite group , let be the minimal integer for which there exists a-Galois extension that is ramified at exactly primes (including the infinite one). So, the problem is to compute or to bound. In this paper, we bound the ramification of extensions obtained as a specialization of a branched covering. This leads to novel upper bounds on , for finite groups that are realizable as the Galois group of a branched covering. Some instances of our general results are: For all 0$]]>. Here denotes the symmetric group on letters, and is the direct product of copies of. We also get the correct asymptotic of , as for a certain class of groups. 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.

AB - The minimal ramification problem may be considered as a quantitative version of the inverse Galois problem. For a nontrivial finite group , let be the minimal integer for which there exists a-Galois extension that is ramified at exactly primes (including the infinite one). So, the problem is to compute or to bound. In this paper, we bound the ramification of extensions obtained as a specialization of a branched covering. This leads to novel upper bounds on , for finite groups that are realizable as the Galois group of a branched covering. Some instances of our general results are: For all 0$]]>. Here denotes the symmetric group on letters, and is the direct product of copies of. We also get the correct asymptotic of , as for a certain class of groups. 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.

KW - Hilbert's irreducibility theorem

KW - inverse Galois problem

KW - ramification

KW - specializations

UR - http://www.scopus.com/inward/record.url?scp=85048783735&partnerID=8YFLogxK

U2 - 10.1017/S1474748018000257

DO - 10.1017/S1474748018000257

M3 - מאמר

AN - SCOPUS:85048783735

VL - 19

SP - 919

EP - 945

JO - Journal of the Institute of Mathematics of Jussieu

JF - Journal of the Institute of Mathematics of Jussieu

SN - 1474-7480

IS - 3

ER -