Number Theory

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


Galois coverings of the arithmetic line

  • David HarbaterAffiliated withUniversity of Pennsylvania


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 . 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.