DS10 - Défi de tous les savoirs

Definability in non-archimedean geometry – Défigéo

Submission summary

The use of model theoretic tools in non-archimedean geometry with applications to number theory can be traced back to the work of Ax-Kochen-Ersov on the Artin conjecture in the sixties. Another spectacular application was provided by Denef in the eighties, with his proof of rationality of the Poincaré series associated with the p-adic points on a variety.

More recently, these tools played a central role in the development of motivic integration, initiated by Kontsevich, which led to important applications in fields as diverse as algebraic geometry, number theory or the Langlands program.They also lay at the heart of the recent progress in the study of the topology of Berkovich spaces. The concept of definability has been ubiquitous in all of these achievements. For the convenience of the presentation, we chose to subdivide our directions of research into four main themes, each one dedicated to some aspect of definability.

1) Berkovich geometry
Model-theoretic methods have been quite successful in the study of topological properties of Berkovich spaces. It is now time to go one step further and to start investigating properties of a more algebraic nature. This includes the study of polyhedral structures within Berkovich spaces as well as the étale and motivic homotopy type of Berkovich spaces. Our motivation is to obtain an equivalence of categories between suitably localized homotopy categories of varieties or definable sets over a valued field, resp. over the sort RV in the Hrushovski-Kazhdan sense. This should ultimately provide the right framework for categorification of motivic integration. On the more analytical side, we intend to develop Kähler geometry on Berkovich spaces.

2) Motivic Milnor fibre and the monodromy conjecture
The monodromy conjecture relates poles of non-archimedean integrals to eigenvalues of the local monodromy. As of today, it remains an important open problem. Recently the Hrushovski-Kazhdan approach of motivic integration allowed to obtain a better understanding of the trace of iterates of the monodromy in terms of arcs. This gives some hope this approach could help to make progress on the conjecture. These questions are closely related to the study of the motivic Milnor fibre of Denef-Loeser and to the issue of categorification of motivic integration raised in 1). We also intend to explore the connections of the motivic Milnor fibre with rigid motives in the sense of Ayoub. Another question we would like to address is the construction of the right avatar of the motivic Milnor fibre over the reals.

3) Applications to diophantine geometry
Pila's spectacular unconditional proof of the André-Oort conjecture for the j-line has attracted recently considerable interest. It is based on a counting theorem by Pila and Wilkie on the number of rational points in o-minimal structures. Recently, members of the project have proved a p-adic version of the Pila-Wilkie theorem. In the perspective of potential applications to diophantine problems, we intend to study the definability properties of various objects being subject to $p$-adic uniformization.

4) Applications of motivic integration to non-archimedean harmonic analysis and representation theory
The definability of various properties and objects arising in representation theory of p-adic groups have strong consequences in harmonic analysis (transfer principle for the Fundamental Lemma and its many variants, integrability of characters). Although a lot of progress has been made, we are still suffering some limitations which one should remedy. In particular, the lack of a satisfactory theory of motivic integration allowing multiplicative is a problem which we plan to address. This would allow to develop a motivic version of the Mellin transform and to extend the range of validity of the motivic Poisson formula. In another direction it seems important to develop a genuine theory of motivic wave front sets.

Project coordination

François Loeser (Institut de Mathématiques de Jussieu-Paris Rive Gauche)

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.

Partner

IMJ-PRG Institut de Mathématiques de Jussieu-Paris Rive Gauche
IRMAR - UR1 Institut de Recherche Mathématiques de Rennes
LAMA Laboratoire de Mathématiques

Help of the ANR 268,580 euros
Beginning and duration of the scientific project: September 2015 - 48 Months

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