Abstracts

Konstantin Ardakov: A study of the second Drinfeld covering

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.

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Gabriel Dospinescu: On some locally analytic representations of $D^*$

I will report on joint work in progress with Benchao Su, whose aim is to recover an irreducible supersingular representation of $\mathrm{GL}_2(\mathbb{Q}_p)$ from the associated representation of $D^*$ by Scholze's functor. I will describe a (conjectural so far) recipe for this, which works for many representations.

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Claudius Heyer: On Second Adjointness for mod $p$ Representations

The parabolic induction functor for smooth representations admits the Jacquet functor as a left adjoint. For complex representations it is a deep result of Bernstein, called Second Adjointness, that the Jacquet functor for the opposite parabolic is (up to a twist) also right adjoint to parabolic induction. A similar result is also known for mod $\ell \neq p$ representations, yet for mod $p$ representations the story is a bit more intricate. But due to recent work of Hoff–Meier–Spieß the (derived) right adjoint of parabolic induction is now fairly well understood. In this talk I will explain Second Adjointness for mod $p$ representations; this is joint work with Manuel Hoff, Sarah Meier and Michael Spieß. If time permits, I will also sketch a different approach to Second Adjointness through Hyperbolic Localization; this is ongoing joint work in progress with Lucas Mann and Konrad Zou.

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Christine Huyghe: Fourier transform for coadmissible $D$-modules

I will construct a Fourier transform for coadmissible $D$-modules over rigid analytic fiber bundles, and explain some properties of this transformation.

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Wiesława Nizioł: Topological Vector Spaces

The category of Banach-Colmez spaces embeds fully into the category of Vector Spaces (pro-étale $\mathbb{Q}_p$-sheaves) as well as into the category of Topological Vector Spaces (enriched topological presheaves). I will discuss this embeddings and the computation of the Ext-groups of Banach-Colmez spaces. This is based on a joint work with Pierre Colmez.

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Sascha Orlik: On some non-principal locally analytic representations

In my talk I will construct some locally analytic representations which do not lie in the principal series. Further I will discuss some basic properties of them.

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Contributed talks

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Abdelmounaim Bouchikhi: On the Independence of the Fourier coefficients of two $p$-adic modular forms

We study rigidity phenomena for $p$-adic modular forms and their families. Extending a result of Coleman, we show that if two overconvergent modular forms have Fourier coefficients with values in a finite ratio set, then they must be scalar multiples of each other. We further prove that this rigidity persists for $p$-adic families varying over weight space.

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Moqing Chen: $p$-adic Monodromy of Supersingular Abelian Surfaces over $\mathbb{Q}_p$

Over $p$-adic fields, the $p$-adic étale cohomology of algebraic varieties with good reduction carries the structure of a crystalline Galois representation. The associated monodromy group, defined as the Zariski closure of the image of such a representation, provides a key invariant over the Hecke orbits on Shimura varieties of Hodge type. In this talk, I will introduce a coarse moduli space parametrizing $p$-adic Galois representations arising from abelian surfaces over $\mathbb Q_p$ with supersingular good reduction. I will describe how the monodromy groups vary across this space, and in particular identify a generic monodromy group.

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Vincenzo Di Bartolo: Local coherence aspects in the Langlands program

Recent formulations of categorical Langlands program see the interplay of completed group algebras of $p$-adic groups and their ($m$-)coherence properties. We are going to explore these relations, stating as well recent developments for $\mathrm{SL}_n(F)$ and $\mathrm{GL}_n(F)$, for $F/\mathbb{Q}_p$ a finite extension.

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Raoul Hallopeau: Characteristic cycle for $D$-cap-modules in dimension one

Let $X$ be a smooth, rigid analytic space. Ardakov-Wadsley have introduced a sheaf $D$-cap of rapidly converging differential operators over $X$, together with a category of coadmissible $D$-cap-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. In particular, a notion of "sub-holonomicity" for coadmissible $D$-cap-modules over a smooth, rigid curve follows. I will present this construction.

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Adam Jones: Exploring torsion theory for pro-$p$ Iwahori-Hecke modules over $\mathrm{SL}_3(K)$

The behaviour of supersingular representations makes understanding the canonical relationship between smooth representations of $G=\mathrm{SL}_n(K)$ and the pro-p Iwahori-Hecke algebra $H$ very difficult in characteristic $p$. I will present results which advance an approach developed by Schneider and Ollivier using torsion theory, and use new techniques involving coefficient systems of higher rank buildings to provide evidence towards the existence of a category of $H$-modules that is equivalent to the corresponding category of $G$-representations, and that this category encompasses all non-supersingular representations.

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Corentin Lambert: Overconvergent lift of the field of norms

Let $K/\mathbb{Q}_p$ be a finite extension. Let $L/K$ be a strictly APF infinite extension that is Galois with $G=\mathrm{Gal}(L/K)$ and $X_K(L)$ the field of norm of this extension. I study lifts of $X_K(L)$ when the Frobenius is an overconvergent Laurent series. In this case, I show that $G$ is virtually abelian.

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Georg Linden: Classification of equivariant line bundles on the Drinfeld upper half plane

We explicitly determine the group of isomorphism classes of equivariant line bundles on the Drinfeld upper half plane associated to a non-archimedean local field $F$ for $\mathrm{GL}_2(F)$, for its subgroup of matrices whose determinant has trivial valuation, and for $\mathrm{GL}_2(\mathcal{O}_F)$. Our results extend a recent classification of torsion equivariant line bundles with connection due to Ardakov and Wadsley, but we use a different approach. A crucial ingredient is a construction due to Van der Put which relates invertible analytic functions on the Drinfeld upper half plane to currents on the Bruhat–Tits tree. Another tool we use is condensed group cohomology.

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Nataniel Marquis: Enhanced structures on the $p$-adic cohomologies of Weil restrictions

Let $K$ be a $p$-adic local field and $G_{K,\text{plec}}=\mathrm{Aut}_K(K\otimes \overline{\mathbb{Q}_p})$ be the plectic Galois group. Plectic representations appear implicitely in recent work towards mod $p$ local Langlands correspondance for $\mathrm{GL}_2(K)$. We will briefly give some examples of such representations and explain how they fit into the work of Breuil-Herzig-Hu-Morra-Schraen. As a local mirror of the global plectic conjecture by Nekovar-Scholl, one hopes to put plectic structures on various cohomologies of the Weil restriction to $\mathbb{Q}_p$ of $K$-rigid analytic varieties. Such plectic structure on étale/ pro-étale cohomologies would construct plectic Galois representations. As first steps in this direction, we present the construction of such structures for de Rham and logcrystalline cohomologies, even in situations when Künneth formula doesn't hold.

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Guillaume Pignon-Ywanne: Poincaré duality for $\mathbb{F}_p$-étale cohomology of some non-proper rigid analytic varieties

Étale cohomology of algebraic or rigid analytic varieties over $\mathbb{Q_p}$ form an important source of $\operatorname{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$-representations, that are "of geometric origin". From this point of view, the study of Poincaré Duality asks whether or not the dual of such a representation is still of geometric origin. I this talk, I will explain some results on Poincaré Duality for étale cohomology with $\mathbb{F}_p$-coefficients of rigid analytic varieties, relying on the six functor formalism for solid $\mathcal{O}^+/p$-coefficients developed by Lucas Mann. In particular, I will focus on spaces resembling Drinfeld's upper half plane, for which cohomology groups also admits a natural action of $\operatorname{GL}_n(\mathbb{Q}_p)$, related to the Steinberg representation.

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Gal Porat: Solid locally analytic representations in mixed characteristic

Locally analytic representations of $p$-adic Lie groups with $\mathbb{Q}_p$ coefficients play an important role in $p$-adic Hodge theory and in the $p$-adic Langlands program. This perspective reveals important differential structures, such as the Sen and Casimir operators. A few years ago, Rodrigues Jacinto and Rodriguez Camargo developed a "solid" version of this theory using the language of condensed mathematics, which provides more robust homological tools (comparison theorems, spectral sequences...) for studying these representations. This talk will present work that extends this solid theory to a much broader class of mixed characteristic coefficients, such as $\mathbb{F}_p((X))$ or $\mathbb{Z}_p[[X]]\langle p/X \rangle$, as well as semilinear representations. I plan to explain how these ideas relate to mixed characteristic phenomena in $p$-adic Hodge theory and the Langlands program.

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Julian Reichardt: On $D$-cap-modules of finite length

The notion of holonomicity plays a central role in the classical theory of algebraic $D$-modules, in particular in its applications to geometric representation theory. An important property of holonomic $D$-modules is that they are of finite length. In the $p$-adic setting, one has the analytically enriched notion of analytic $D$-cap-modules due to Ardakov–Wadsley. However, the development of an analogous framework of holonomic $D$-cap-modules is more subtle, and it is not clear that holonomic $D$-cap-modules are of finite length. We show that for quasi-compact rigid analytic spaces, the extension functor sends holonomic $D$-modules to $D$-cap-modules of finite length. Using this, we strengthen results of Bitoun–Bode and Ardakov–Bode–Wadsley on the coadmissibility of $D$-cap-modules, showing finite length for quasi-compact rigid analytic spaces. In particular, this shows that a various natural examples of holonomic $D$-cap-modules are of finite length. This is part of the speaker's PhD thesis.

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James Timmins: Augmented Iwasawa algebras are Gorenstein

Every smooth natural characteristic representation of a $p$-adic Lie group is a module for its augmented Iwasawa algebra, a ring introduced by Kohlhaase. I will present a result that bounds the cohomology of augmented Iwasawa algebras. Ongoing work with Vincenzo di Bartolo.

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Finn Wiersig: Towards a Kashiwara-style constructibility theorem for $p$-adic $D$-modules

Kashiwara proved that the solution complexes of holonomic $D$-modules on complex manifolds have constructible cohomology. Is the same true for $D$-modules on rigid-analytic spaces? In this talk, I will present joint work in progress with Konstantin Ardakov on this question.

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Xinyu Zhou: Dualities between integral local Shimura varieties

Using the theory of $p$-adic shtukas, Scholze and Weinstein show in the "Berkeley lecture notes" that there is a "duality" between local Shimura varieties associated to group $G$ and those associated to an inner twist $G'$, by proving that they become isomorphic at the infinite level. In this talk, we show how to generalize this duality result further to the integral models of local Shimura varieties. As a consequence of our proof, we show there is a close connection between the Newton stratification of the special fiber of an integral local Shimura variety and absolute Banach-Colmez spaces. We also discuss an application to cohomology of $p$-adic groups with some natural mod-$p$ coefficients.