Extending the work of Phùng on Gauß Manin stratifications to the non relatively compact case, I will deduce a Künneth formula for the regular singular stratified fundamental group, which can be seen as a generalization of the tame fundamental group. As an application I will deduce that for homogeneous spaces the tame fundamental group is trivial if and only if the regular singular is, which is a generalization of a conjecture of Gieseker proposed by Esnault.
Introductory talk: Theta-invariants and geometry of numbers in infinite dimension.
Abstract: I will present a generalization of the classical "geometry of numbers" that allows one to handle some infinite dimensional avatars of euclidean lattices.
Regular Talk: Formal-analytic surfaces over Spec(Z)
Abstract: Formal-analytic surfaces play a role, in Arakelov geometry, similar to the one of complex analytic surfaces in complex algebraic geometry. They may be investigated by using the "infinite dimensional geometry of number" introduced in my previous talk, and lead to diverse Diophantine applications, notably to the algebraicity of certain formal series with integral coefficients.
We will discuss the apparition of ind-hermitian vector bundles as spaces of sections of coherent sheaves in Arakelov geometry. We will then explain how these objects provide a natural framework for the study of arithmetic ampleness.
Dans cet exposé, j’expliquerai un analogue de la géométrie d’Arakelov sur un corps muni d’une valeur absolue triviale. Vu comme une courbe arithmétique, le corps trivialement valué est fantastique car la géométrie des nombres correspondante est très simple. Cependant, la géométrie d’Arakelov au-dessus est aussi riche que sa version classique, comme montré par le problème d’effectivité à équivalence rationnelle près des R-diviseurs arithmétiques pseudo-effectifs. Il s’agit d’un travail en collaboration avec Atsushi Moriwaki.
We will describe a certain twisting process for Galois covers and will show how the arithmetic properties of the twisted cover can be used for some lifting and parametrizing problems in Inverse Galois Theory.
We look at the relation between syntomic and p-adic motivic cohomology to study a localisation triangle for syntomic cohomology. This allows us to compare Besser's rigid syntomic and Nekovar-Nizioł's arithmetic syntomic cohomology.
We explain how to use p to l companions in order to prove integrality of cohomologically rigid connections on a projective smooth complex variety. (Simpson’s conjecture)
Classical local-global principles are given over global fields. This talk will discuss work of the speaker and collaborators on such principles over semi-global fields, which are function fields of curves defined over a complete discretely valued field such as the p-adics.
The tempered fundamental group classifies some étale analytic covering of a non-archimedean analytic space. We will discuss how one can recover some Shimura curves up to isomorphism from their tempered fundamental group.
Let X be a projective scheme over an affine scheme. We develop some technics to prove the existence of closed subschemes of X with various nice properties. Concretes applications will be given as the existence of finite quasi-sections. This is a common work with Gabber and Lorenzini.
The theory of linear series on tropical curves allows for a combinatorial systematic treatment of degenerations of classical linear series, leading to a number of beautiful results connecting classical and tropical viewpoints. The moduli space of spin curves over curves of given genus and its compactification by Cornalba are very well studied spaces and, jointly with their generalized version for moduli of higher roots, are important tools in the understanding of the moduli space of curves itself. Building on the beautiful framework developed by Abramovich, Caporaso and Payne on the relation between the analytification and the tropicalization of the moduli space of curves, I will report on ongoing joint work with Lucia Caporaso and Marco Pacini describing the topicalization of Cornalba's space and speculate on a number of properties and applications of this space.
We give a conjectural description of vanishing orders and special values of zeta functions of arithmetic schemes in terms of Weil-Arakelov cohomology. We also try to explain how our description fits with Deninger's conjectured cohomology. This is joint work with Matthias Flach.
I will present joint work with Evgeny Shinder, where we use Denef and Loeser's motivic nearby fiber and a theorem by Larsen and Lunts to prove that stable rationality specializes in families with mild singularities. I will also discuss an improvement of our results by Kontsevich and Tschinkel, who defined a birational version of the motivic nearby fiber to prove specialization of rationality.
We prove a result about the Galois module structure of the homology of the Fermat curve using commutative algebra, number theory, and algebraic topology. Specifically, we extend work of Anderson about the action of the absolute Galois group of a cyclotomic field on a relative homology group of the Fermat curve. By finding explicit formulae for this action, we determine the maps between several Galois cohomology groups which arise in connection with obstructions for rational points on the generalized Jacobian. Ray class fields of the cyclotomic field and Heisenberg extensions play a key role in the result. This is joint work with R. Davis, V. Stojanoska, and K. Wickelgren.
We use Weil restriction in order to describe affine, smooth group schemes over the ring of dual numbers k[ε] in terms of extensions of group schemes over k.
We deduce a Dieudonné classification for smooth, unipotent group schemes over k[ε]. This is joint work with Dajano Tossici.
We shall describe a proof of a local form of the degree one part of the Adams-Riemann-Roch theorem. This local form provides a canonical isomorphism between the determinant of the cohomology of a vector bundle on a variety and the determinant of the cohomology of a different vector bundle, obtained by tensor operations from the first one and from the sheaf of differentials. This kind of theorem can be found in D. Eriksson’s thesis and also (implicitly) in J. Franke’s work on the functorial form of the Riemann-Roch theorem. Our aim here is to provide a proof this theorem, which is as elementary as possible, to make it more accessible.
We show that the category of ordinary K3 surfaces over finite fields is equivalent to a certain category of linear-algebraic data over Z. This is an analogue of Deligne's theorem on ordinary abelian varieties over finite fields, and refines prior work of Nygaard and Yu. An important ingredient are the new results on integral p-adic Hodge theory by Bhatt, Morrow, and Scholze.
We show that the enriques and bielliptic surfaces defined over algebraically closed fields of characteristic bigger than three do not have any non trivial Fourier-Mukai partners. This is joint work with K. Honigs and M. Lieblich.