In a joint research project with Itay Ben Yaacov, we study a class of fields enriched with a global structure tying together their various valuations by a product formula. This is an elementary class in the sense of continuous logic

A famous Theorem by Artin and Schreier characterizes the real closed fields as being those fields which have a finite non-trivial absolute Galois group. Instances of p-adic analogs of this Theorem are known (Neukirch, Pop, Koenigsmann, Efrat), but there is much more to this story. Namely I will give a 'minimalistic' p-adic analog, which as in the Artin-Schreier Theorem, invoves only finite groups. This aspect of the story relates to the birational p-adic section conjecture, etc.

In this talk, I will explain how to relate the two counting problems in the title by generalizing the McKay correspondence to number-theoretic base fields, that is, local fields and number fields. Over local fields, generalizing the McKay correspondence by Batyrev and Denef-Loeser, one can relate stringy invariants of quotient varieties to mass formulas of extensions of local fields. Over number fields, using the local result and a heuristic argument, one can (less tightly than in the local case) relate Manin's conjecture on rational points of Fano varieties to Malle's […]