Chapter

Number Theory

Volume 1240 of the series Lecture Notes in Mathematics pp 165-195

Date:

Galois coverings of the arithmetic line

  • David HarbaterAffiliated withUniversity of Pennsylvania

Abstract

This paper concerns Galois branched coverings of the line, first over the complex numbers and then over the p-adics. We construct such covers with arbitrary Galois group, and then descend these to covers defined over number fields. In particular, every finite group is shown to occur as a Galois group over https://static-content.springer.com/image/chp%3A10.1007%2FBFb0072980/978-3-540-47756-3_9_IEq1_HTML.gif . This is a consequence of a more general result that also implies that complete local domains other than fields are never Hilbertian — thus answering a question of Lang.