Program
Andrea Pulita
Title: Exponents for p-adic differential equations
Abstract: The index theory for the de Rham cohomology of differential equations on Berkovich curves relies on some local invariants, called the exponents. Roughly speaking, the exponents form a multiset of elements of the base field attached locally to the restriction of the differential module to an annulus of the curve. The index theorem relates the finite dimensionality of de Rham cohomology to the radii of convergence of solutions and to the exponents appearing on certain annuli at the boundary of the curve.
Despite their importance, the theory of exponents remains poorly understood. They are expected to play a role analogous to that of formal exponents, but the existing approaches reveal several discrepancies between the two theories.
In this talk I will describe three different approaches to defining exponents. The first is of Tannakian nature and leads to an abstract definition that also applies, in some cases, to irregular differential modules. The second approach uses the reduction modulo p of the connection. The third corresponds to the classical notion introduced by Christol–Mebkhout and later developed by Dwork, Kedlaya, and Shiho. The first two approaches are new and I have obtained them in collaboration with Christopher Lazda.
These three notions turn out to be largely incompatible with one another. I will illustrate this phenomenon with some relatively explicit examples. Finally, I will present a conjecture which, if true, would greatly simplify the theory of exponents and its role in index theorems.
The talk will begin with a brief overview of the index theory for differential equations on Berkovich curves, in order to explain the motivation for introducing exponents.
Daniel Caro
Title: On the injectivity in crystalline comparisons
Abstract: Let $k$ be a perfect field of positive characteristic. For smooth affine $k$-varieties, we identify a slope obstruction to the injectivity of the comparison morphism from rigid cohomology to rationalised crystalline cohomology. This yields a negative answer to a question of Esnault–Kisin–Petrov concerning the injectivity of the de Rham-to-crystalline comparison map for smooth affine schemes over the Witt vectors that admit good compactifications. In contrast, we establish injectivity for certain subspaces defined by slope conditions as well as in cohomological degree one. For the latter case, we will focus on the result with coefficients in $F$-able overholonomic $D$-modules leveraging a generalisation of Kedlaya’s full faithfulness theorem. This is a common work with Marco D’Addezio.
Konstantin Ardakov
Title: A study of the second Drinfeld covering
Abstract: I will discuss joint work, very much in progress, with Simon Wadsley. The Drinfeld tower provides very interesting examples of equivariant $\mathcal{D}$-modules on the $p$-adic upper half plane. We attempt to generalise Teitelbaum’s work on the first Drinfeld covering to the case of the second covering. This involves computing some invariant and equivariant Picard groups of well-chosen affinoids in the first Drinfeld covering.
Finn Wiersig
Title: A reconstruction theorem for coadmissible D-cap modules Abstract: Let X be a smooth rigid-analytic variety. Ardakov and Wadsley introduced the sheaf D-cap of infinite order differential operators on X, along with the category of coadmissible D-cap modules. In this talk, I present a Riemann-Hilbert correspondence for these coadmissible D-cap modules. Specifically, we interpret a coadmissible D-cap module as a p-adic differential equation, explain what it means to solve such an equation, and describe how to reconstruct the module from its solutions.
Christine Huyghe
Title: Fourier Transform for coadmissible D-modules
Abstract:Fourier transform is a classical tool in D-module theory over the complex analytic varieties introduced by Malgrange, and studied by Sabbah among many other people. In the context of p-adic analytic geometry, such a Fourier transform exists for l-adic constructible sheaves since the work of Ramero. I will explain how the theory of coadmissible D-modules can provide a Fourier transform for p-adic rigid analytic varieties. Note that, in this case, coefficients are p-adic.
Raoul Hallopeau
Title: Characteristic variety for Dcap-modules in dimension one
Abstract: Let X be a smooth rigid analytic space. Ardakov-Wadsley have introduced a sheaf Dcap of rapidly converging differential operators over X, together with a category of coadmissible Dcap-modules that play the role of coherent objects. I have defined a characteristic variety and a characteristic cycle for these modules in the one-dimensional case, using microlocalization sheaves of Dcap. In particular, a notion of “sub-holonomicity” for coadmissible Dcap-modules over a smooth rigid curve follows. I will present the construction of this characteristic variety.
Ruochuan Liu
Title: A p-adic Riemann-Hilbert correspondence over Robba rings
Abstract: We construct overconvergent relative period sheaves for Robba rings and overconvergent de Rham period sheaves on the v-site of a liftable semi-stable formal scheme. Using these constructions, we establish a p-adic Riemann–Hilbert correspondence relating small local systems with Robba ring coefficients on the generic fiber to small flat connections on the associated period ringed spaces. We further analyze the local Frobenius action and show that this correspondence is compatible with the Frobenius structure. As an application, we prove a comparison theorem between the cohomology of these period sheaves and the rigid cohomology of the special fiber, refining a previous result by Le Bras.
Tobias Schmidt
Title: Irreducibility results for equivariant D-modules on rigid spaces
Abstract: Let G be a p-adic Lie group. In this talk, I will first explain some background on coadmissible G-equivariant D-modules on rigid analytic spaces and give some examples. I will then focus on a general irreducibility result for certain geometrically induced modules and discuss some applications. If time permits, I will give some ideas of the proof. This is joint work with Konstantin Ardakov.