Recent Developments around the Hilbert Property (I & II)

The Hilbert property, which provides a bridge between Algebraic Geometry and Number Theory, is one of the few general and powerful tool in Arithmetic Geometry. The gist of it is that it makes it possible, given an algebraic situation described by \(n\geq 1\) variables and depending on \(r\geq 1\) parameters, to specialize the parameters and preserve the structure of the situation.
The flagship application, which goes back to Hilbert, is to reduce the number-theoretic Inverse Galois Problem to the search of geometric Galois covers of the line defined over the rationals with a given automorphism group. This classical context will be reviewed as an introductory part of the talk. The Hilbert property has recently been used to address questions of various origins, leading to progress on different topics. The rest of the talk will be a discussion of these developments with an attempt to put some perspective on them. It will still be intended for a wide audience. The discussion will be divided into three main parts, corresponding to whether the situation is algebraic or not, i.e., \(n=1\) or \(n>1\), and within the algebraic situation, to whether the parameters are independent or not.
Topics include the Malle conjecture and other counting problems in Galois Theory, the field theoretic Noether problem and the related notion of parametric extensions, a polynomial version of the number-theoretic Schinzel Hypothesis, an arithmetic Bertini theorem, etc.

Introductory references
• M. Fried and M. Jarden, Field Arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics and Related Areas, Springer-Verlag, Berlin, 3rd edition (2008).
• B. Collas, P. Dèbes and M. D. Fried, Arithmetic geometry and symmetries around Galois and fundamental groups. MFO mini-workshop held April 15-21, 2018, Oberwolfach Rep. 15 no. 2, (2019), 60 pages.

Recent references
• A. Bodin, P. Dèbes and S. Najib, The Schinzel hypothesis for polynomials. Transactions A.M.S., 373/12 (2020), 8339 - 8364.
• P. Dèbes, Groups with no parametric Galois realizations. Annales Sci. E.N.S., vol. 51, (2018), 143-179.
• P. Dèbes, On the Malle conjecture and the self-twisted cover. Israel J. Math., vol. 218 (2017), 101-131.