Workshop on recent trends in p-adic cohomology

25th-27th March, 2015
Imperial College London

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All talks and coffee breaks will be at 170 Queen's Gate. The conference dinner will be held at Ognisko restaurant.

Speaker: Tomoyuki Abe

Title: Arithmetic \(\mathcal{D}\)-modules and the existence of crystalline companions

Abstract: We will show the existence of the crystalline companion of a smooth \(\ell\)-adic sheaf on a curve. This is shown by using the method of Drinfel'd and Lafforgue, after establishing a \(p\)-adic Langlands correspondence. The theory of arithmetic \(\mathcal{D}\)-modules plays a central role in the construction of a six functor formalism for certain algebraic stacks.

Speaker: Jennifer Balakrishnan

Title: Variations on quadratic Chabauty

Abstract: Let \(X\) be a curve defined over a number field \(F\). Following Coleman and Gross, we view a global \(p\)-adic height pairing on the Jacobian of \(X\) as a sum of local height pairings corresponding to the places of \(F\). The contribution above \(p\) is computed using \(p\)-adic cohomology and can be interpreted as sum of double Coleman integrals. We describe how to use these Coleman integrals to recover integral and rational points on certain elliptic and hyperelliptic curves. In particular, we discuss how to apply the method to find integral points when \(F\) is quadratic (joint work with Amnon Besser and Steffen Mueller) and how to use it to find rational points on certain bielliptic genus 2 curves (joint work with Netan Dogra).

Speaker: Daniel Caro

Title: Betti number estimates in \(p\)-adic cohomology

Abstract: Let \(k\) be a perfect field of characteristic \(p\) and \(\ell\) be a prime number different to \(p\). When \(k\) is algebraically closed, within the framework of Grothendieck's \(\ell\)-adic etale cohomology of \(k\)-varieties, Bernstein, Beilinson and Deligne in their famous paper on perverse sheaves established some Betti number estimates. The goal of this talk is to get the same estimates in the context of Berthelot's arithmetic \(\mathcal{D}\)-modules. One emerging problem when we follow the original \(\ell\)-adic proof is that we still do not have vanishing cycles theory. Instead, we use some Fourier transform and Abe-Marmora formula relating the irregularity of an isocrystal with the rank of its Fourier transform.

Speaker: Bruno Chiarellotto

Title: Good reduction criteria

Abstract: We will deal with the problem of detecting good special fiber reduction for a family, in both the geometric and arithmetic cases. In the geometric case we will sketch the work done with Di Proietto and Shiho on a criterium of good reduction for a family of curves in characteristic 0 without using trascendental methods. In the arithmetic case we will consider the case of K3 and Enriques surfaces.

Speaker: Christopher Davis

Title: Canonical lifts of norm fields

Abstract: Witt vectors provide a canonical and functorial way to lift perfect fields of characteristic \(p\) to \(p\)-adically complete DVRs of characteristic zero. In the theory of \((\phi,\Gamma)\)-modules, it is desirable to be able to similarly lift certain imperfect characteristic p fields called norm fields. In this talk I will describe why such a lift is desirable, and how we can construct such a lift in many cases using new constructions involving Witt vectors. This is joint work with Bryden Cais.

Speaker: Frédéric Déglise

Title: Étale motives and syntomic modules

Abstract: The theory of triangulated motives, initiated by Beilinson and Voevodsky, has in recent years reached maturity with the establishment of a complete 6 functors formalism theory on the model of SGA4 and SGA5. These developments end up in an integral étale theory, which encompass both the theory of SGA4 and the theory of rational motives, as developed independently by Ayoub and Cisinski-Déglise. I will present this theory and the notion of homotopy completion we introduced with Cisinski to give a new interpretation of the \(\ell\)-adic realization functor of triangulated mixed motives. In the \(p\)-adic local case, the description of the essential image of this realization functor is a very mysterious question. I will give the first results of a collaboration with Niziol which try to understand this realization functor through syntomic cohomology and the recent work of Beilinson on the \(p\)-adic comparison theorems. We introduce the syntomic dg-algebra which represents syntomic cohomology and begin the study of modules over it, in the sense of motivic homotopy theory.

Speaker: Kiran Kedlaya

Title: Connections on nonarchimedean curves: behavior at type 4 points

Abstract: Much of the progress made in the last 50 years on nonarchimedean differential equations, and especially the convergence of local solutions, can be neatly packaged in the language of Berkovich spaces. Specifically, given a connection on a nonarchimedean analytic curve, one records the convergence of local horizontal sections in a Newton polygon which varies in a reasonable way (in particular continuously for Berkovich's topology); precise statements of this form have been articulated by Poineau-Pulita and Baldassarri-Kedlaya. In this talk, we will focus on a key property which lies somewhat deeper: the local constancy of the convergence polygon around a type 4 point. This ends up being closely related to the proof of semistable reduction for overconvergent \(F\)-isocrystals.

Speaker: Andreas Langer

Title: Grothendieck-Messing deformation theory for varieties of K3-type

Abstract: For an artinian local ring \(R\) with perfect residue field we define higher displays over the small Witt ring. The second crystalline cohomology of a variety of K3-type \(X\) (for example the Hilbert schemes of zero-dimensional subschemes of a K3-surface) is equipped with the additional structure of a 2-display. We extend the Grothendieck-Messing lifting theory from p-divisible groups to such varieties. The deformations of \(X\) over a nilpotent pd-thickening correspond uniquely to selfdual liftings of the associated Hodge filtration. For the proof we give an algebraic definition of the Beauville-Bogomolov form on the second de Rham cohomology of \(X\) and show that for ordinary varieties the deformations of \(X\) correspond uniquely to selfdual deformations of the 2-display endowed with its Beauville-Bogomolov form. This is joint work with Thomas Zink.

Speaker: Christopher Lazda

Title: Rigid cohomology over Laurent series fields

Abstract: If \(F\) is a characterstic \(p\) local field (that is, a Laurent series field), then every \(\ell\)-adic representation (\(\ell\neq p\)) of its absolute Galois group is potentially semistable: this is Grothendieck's \(\ell\)-adic local monodromy theorem, and it can be viewed as a cohomological incarnation of potentially semistable reduction. If we consider the the \(p\)-adic setting, however, then the objects produced by rigid cohomology (i.e. \((\varphi,\nabla)\)-modules over the Amice ring) are not the objects to which the correpsonding local monodromy theorem applies (that is i.e. \((\varphi,\nabla)\)-modules over the Robba ring). I will explain some work in constructing a refinement of rigid cohomology that bridges the gap between these two kinds of objects, and which therefore 'reconnects' the monodromy theorem to the geometry of varieties over \(F\). I will also discuss how a construction of Marmora leads to a formulation versions of weight monodromy and \(\ell\)-independence which include the case \(\ell=p\), and hopefully explain why everything works in the case of smooth curves. This is joint work with Ambrus Pál.

Speaker: Bernard Le Stum

Title: On constructible isocrystals

Abstract: One can replace the coherence condition on an overconvergent isocrystal by the weaker condition of being constructible. I conjecture that there exists a 6 operations formalism on the (derived) category of constructible \(F\)-isocrystals. We will show that, in the case of a smooth curve over a prefect field, the category of constructible isocrystals is actually equivalent to a perverse category of holonomic arithmetic \(\mathcal{D}\)-modules. As a consequence, the conjecture can be shown to hold for smooth curves.

Speaker: Ambrus Pál

Title: Crystalline Chebotarev density theorems

Abstract: I will formulate a conjectural analogue of Chebotarev's density theorem for convergent \(F\)-isocrystals over a smooth geometrically irreducible curve defined over a finite field using the Tannakian formalism. I will talk about the proof of this analogue in several special classes, including all semi-simple convergent \(F\)-isocrystals which have a filtration by isoclinic \(F\)-isocrystals of pair-wise different slopes whose monodromy groups are reductive and abelian over a non-empty open subcurve. The methods used include the theory of reductive groups and \(p\)-adic analysis, but at some point also a little bit of Diophantine geometry, too. This is joint work with Urs Hartl.

Speaker: Atsushi Shiho

Title: Convergent isocrystals on simply connected varieties

Abstract: It is conjectured by de Jong that, if \(X\) is a connected smooth projective variety over an algebraically closed field \(k\) of characteristic \(p>0\) with trivial étale fundamental group, then any isocrystal on \(X\) is trivial. We prove this conjecture under two additional assumptions. This is joint work with Hélène Esnault.

Speaker: Fabien Trihan

Title: On the Geometric Iwasawa and ETN Conjectures for abelian varieties

Abstract: We will talk about our recent progress toward the formulation of the Equivariant Tamagawa Number conjecture and non-commutative Iwasawa Main conjecture for abelian varieties over function fields of characteristic \(p\).

Speaker: Nobuo Tsuzuki

Title: On Lefschetz type problem and variation of slopes for \(F\)-isocrystals

Abstract: I will prove that irreducibility of \(F\)-isocrystals on a projective smooth variety of dimension \(>1\) is preserved under taking a certain hyperplane section. I also discuss the difficulty of a Lefschetz type problem which comes from the non-constancy of the Newton polygon of Frobenius structures of \(F\)-isocrystals.