## Abstract

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.

Original language | American English |
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Pages (from-to) | 919-945 |

Number of pages | 27 |

Journal | Journal of the Institute of Mathematics of Jussieu |

Volume | 19 |

Issue number | 3 |

DOIs | |

State | Published - 1 May 2020 |

### Bibliographical note

Publisher Copyright:© 2020 Cambridge University Press.

## Keywords

- Hilbert's irreducibility theorem
- inverse Galois problem
- ramification
- specializations