Hilbert properties of varieties

The Hilbert property for varieties was introduced by Colliot-Thélène, Sansuc and Serre in an attempt to salvage the Noether program. Later, Corvaja and Zannier identified an important geometric obstruction to the Hilbert property for varieties over number fields and suggested a weaker property. They speculate that this weak Hilbert property holds for every variety over a number field with a dense set of rational points, which if true would solve the inverse Galois problem.

In this talk I will motivate and introduce these concepts, survey some of the many recent results on the topic, and then focus in particular on the weak Hilbert property for abelian varieties over finitely generated fields (work of Corvaja, Demeio, Javanpeykar, Lombardo and Zannier) and certain infinite extensions (work of Bary-Soroker, Petersen and the speaker).