Wojciech J. Gajda

Hilbert’s irreducibility theorem, which is a statement about rational points on finite covers of the projective line P1Q, is one of the fundamental results in diophantine geometry with plenty of applications throughout algebra. It led Colliot-Thelene and Sansuc in the 80s to the definition of varieties with the Hilbert property (HP). It turns out, however, that this is a quite restricted class of varieties: If a variety over a number field satisfies HP, then its geometric fundamental group is trivial according to a recent theorem of Corvaja and Zannier. This led Corvaja and Zannier in 2017 to the definition of the so-called weak Hilbert property (WHP), which suffices for many applications and is enjoyed by a much larger class of varieties. The class of WHP varieties has been studied extensively since then by many people. The aim of my talk is to survey this development including a variety of open problems from the field. In particular we report on recent joint work with Bary-Soroker and Fehm concerning permanence principles for WHP varieties under extension of the ground field, relying in turn on joint work with Gajda on finiteness properties of torsion fields of abelian varieties.