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 […]
In recent work Harpaz and Wittenberg established a general fibration theorem for the existence of rational points, conditional on a conjecture on locally split values of polynomials. In this talk we report on joint work with Tim Browning, which establishes a special case of their conjecture. We achieve this in proving strong approximation off a non-empty finite set of places for some varieties which are defined using norm forms.
Les conjectures de Manin et Peyre décrivent la répartition des points rationnels de hauteur bornée sur une variété de Fano en terme d'invariants géométriques de la variété. Suivant l'approche développée par La Bretèche, Browning et Peyre, on présentera au cours de cet exposé une preuve de la conjecture de Manin pour une surfaces de Châtelet définie comme modèle minimal propre et lisse d'une variété affine de la forme Y^2+Z^2=F(X,1) avec F polynôme à coefficients entiers de degré 4 sans racine multiple de la forme F=L_1L_2Q avec L_1 et L_2 deux […]
