The regular inverse Galois problem over non-large fields

Abstract

By a celebrated theorem of Harbater and Pop, the regular inverse Galois problem is solvable over any field containing a large field. Using this and the Mordell conjecture for function fields, we construct the first example of a field $K$ over which the regular inverse Galois problem can be shown to be solvable, but such that $K$ does not contain a large field. The paper is complemented by model-theoretic observations on the diophantine nature of the regular inverse Galois problem.