Derived geometry and arithmetic

• Applicant: Dr. Federico Bambozzi, Ph.D.
• Subject Area: Mathematics
• Term: from 2018 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 414260436

One of the recent trends in the research on the p-adic Langlands program is the entering of geometric considerations, concepts and intuitions on topics that were classically considered to belong to algebra. Probably the more interesting and influential of such developments is the theory of the Fargues-Fontaine curve, which permits to interpret local Galois representations (objects of algebraic nature) as some specific vector bundle on a geometric object (technically a scheme) which has many formal properties which remind the ones of a projective curve of genus 0 (although this scheme is not of finite type over a field).My project aims to study question of related nature from a global point of view, i.e. not just focusing on the local phenomena related to p-adic Langlands. Recent progress I have made with my collaborators suggests that the local theory of the Fargues-Fontaine curve is a local shadow of some unified theory which is not only relevant in p-adic geometry but which also relates to classical topics like Dirichlet series of complex geometry. Building on our previous work, the project is developed around three main goals:1) to complete the foundations of bornological homotopy theory, in particular by developing the theory of bornological spectra and the relative spectral geometry as a homotopical generalization of classical analytic geometry;2) to continue to investigate how our ideas can be helpful for understanding Hodge Theory (both complex and p-adic) - in particular to understand the relations between our archimedean Fargues-Fontaine curve and the Twistor curve defined by Carlos Simpson;3) to study the Topological Hochschild Homology (THH) functor in this context.In particular, for reaching the second and the third goal we would like to use the notion of global (φ, Γ)-module, which is a global version of the (φ, Γ)-modules studied in p-adic Hodge Theory. We expect that the cohomology groups of arithmetic varieties have a canonical structure of global (φ, Γ)-module in analogy with the fact that varieties over ℚ_p have a structure of classical (φ, Γ)-module. In this way, we expect, for example, to be able to recover the categories of representations of the local Galois groups from purely global data, by specializations at primes. Moreover, we expect that the THH functor, via the Tate spectrum construction, will naturally take value in the derived category of global (φ, Γ)-module because the Tate cohomology of THH is endowed with an action of all Frobenii φ_p, for all primes p. These methods can give contributions to understanding the cohomology of arithmetic varieties and some related open problems like the Mazur-Fontaine conjecture. Applications to the Tate conjecture, the motivic Galois group and towards understanding of motives over ℤ are possible.

• DFG Programme: Research Fellowships
• International Connection: United Kingdom
• Host: Professor Dr. Yakov Kremnizer