New Structures and Techniques in p-adic Geometry

27 oct. 2025, 09:00 → 31 oct. 2025, 16:30 Europe/Paris

Centre de conférences Marilyn et James Simons (Le Bois-Marie)

New Structures and Techniques in p-adic Geometry   
   
October 27-31, 2025 at IHES - Marilyn and James Simons Conference Center    
How to get to IHES

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In recent years, p-adic geometry has benefited from several new ideas. The goal of this conference is to invite experts to give lectures explaining some of these ideas and their uses.  The main themes are: motivic methods, categorical p-adic Langlands, and Igusa stacks.  Besides these mini-courses, there will also be some individual lectures by selected participants.

 Registration deadline: June 30, 2025

Speakers:

• Johannes Anschütz (Université Paris-Saclay)
• Ana Caraiani (Imperial College London)
• Gabriel Dospinescu (CNRS, Université de Clermont-Auvergne)
• Veronika Ertl (Université de Caen)
• Eugen Hellmann (Universität Münster)
• Kalyani Kansal (Imperial College London)
• Dongryul Kim (Stanford University)
• Arthur-César Le Bras (Université de Strasbourg)
• Shubhodip Mondal (Purdue University)
• Timo Richarz (TU Darmstadt)
• Alberto Vezzani (Università degli Studi di Milano)
• Mingjia Zhang (IAS, Princeton)
  
  
  

Organizing committee:     
Dustin Clausen (IHES), Toby Gee (Imperial College London),
Wiesława Nizioł (IMJ-PRG)

This conference is supported by the Simons Collaboration on Perfection in algebra, geometry, and topology

POSTER_p-adic_Geometry_Workshop.pdf

Titles&Abstracts_NewStructuresConf_27-31Oct2025.pdf

lun. 27 octobre

09:30 Welcome coffee

1 Non-Archimedean Motives I

We will define the categories of (étale, rational) motives over an adic space $S$ and illustrate their most important properties, focusing on relevant applications in the study of $p$-adic cohomology theories. In particular, we will present the six-functor formalism they are equipped with, the continuity/spreading-out property, compact generation, and the identification between an analytic motive over a local field and a monodromy operator acting on its nearby cycle. We will sketch the proofs of these facts, highlighting the role of homotopies at each stage. Several applications will be presented, especially concerning the definition and study of rigid, de Rham, and Hyodo-Kato cohomologies.

Orateur: Timo Richarz (TU Darmstadt)

Vidéo

11:00 Coffee break

2 Geometric Aspects of the $p$-adic Locally Analytic Langlands Correspondence I

The goal of these lectures is to discuss some aspects of the geometrization of the $p$-adic Langlands program (more precisely, the part of it which deals with locally analytic representations on topological $\mathbb{Q}_p$-vector spaces). The lectures by Hellmann (based on joint work with Hernandez and Schraen and with Heuer) will focus on the « Galois » side, introducing moduli stacks of (variants of) $p$-adic Galois representations and on the study of their geometry. The lectures of Le Bras (based on joint work with Anschütz, Rodriguez Camargo, Scholze) will focus on the « automorphic » side, explaining how to realize locally analytic representations as sheaves on a variant of Fargues-Scholze’s stack of $G$-bundles on the Fargue-Fontaine curve.

Orateur: Eugen Hellmann (Universität Münster)

Vidéo

12:15 Buffet-lunch

3 Igusa Stacks I

Igusa stacks are a new tool to study the cohomology of Shimura varieties with both characteristic 0 and torsion coefficients. Let (G,X) be a Shimura datum of Hodge type. The associated Shimura variety with infinite level at p, considered as a diamond over a p-adic field, can be expressed as a fiber product of an Igusa stack with a purely local object, the diamond associated to a flag variety, over the moduli stack Bun_G of G-bundles over the Fargues-Fontaine curve. This Cartesian diagram, called the Igusa stack diagram, allows us to use techniques from the categorical local Langlands program to study the cohomology of Shimura varieties.

In the first lecture, we will discuss the Igusa stack diagram, constructed in various levels of generality by Zhang, Daniels-van Hoften-Kim-Zhang, and Kim. In the next two lectures, we will discuss applications of the Igusa stack diagram to the cohomology of Shimura varieties, specifically to results such as torsion-vanishing, Eichler-Shimura relations, and Ihara's lemma. These are due to a number of researchers, including Koshikawa, Hamann-Lee, Daniels-van-Hoften-Kim-Zhang, Yang-Zhu and Yang. In the last lecture, we will discuss the relative intersection cohomology of the Igusa stack, which is joint work in progress of the two mini-course speakers with Linus Hamann.

Orateur: Mingjia Zhang (IAS, Princeton)

Vidéo

15:00 Coffee break

4 Non-Archimedean Motives II

Motivic methods in $p$-adic arithmetic geometry

We will define the categories of (étale, rational) motives over an adic space S and illustrate their most important properties, focusing on relevant applications in the study of p-adic cohomology theories. In particular, we will present the six-functor formalism they are equipped with, the continuity/spreading-out property, compact generation, and the identification between an analytic motive over a local field and a monodromy operator acting on its nearby cycle. We will sketch the proofs of these facts, highlighting the role of homotopies at each stage. Several applications will be presented, especially concerning the definition and study of rigid, de Rham, and Hyodo-Kato cohomologies.

Orateur: Alberto Vezzani (Università degli Studi di Milano)

Vidéo

mar. 28 octobre

09:30 Welcome coffee

5 Higher Pushforwards in Rigid Cohomology via Motives

Berthelot's conjecture states that the higher push-forwards in rigid cohomology of the structure sheaf along a smooth and proper morphism are canonically overconvergent $F$-isocrystals. I will explain how motivic non-archimedean homotopy theory can be used to define solid relative rigid cohomology and prove a version of Berthelot's conjecture.
(Joint work with Alberto Vezzani.)

Orateur: Veronika Ertl (Université de Caen)

Vidéo

11:00 Coffee break

6 Geometric Aspects of the $p$-adic Locally Analytic Langlands Correspondence II

The goal of these lectures is to discuss some aspects of the geometrization of the $p$-adic Langlands program (more precisely, the part of it which deals with locally analytic representations on topological $\mathbb{Q}_p$-vector spaces). The lectures by Hellmann (based on joint work with Hernandez and Schraen and with Heuer) will focus on the « Galois » side, introducing moduli stacks of (variants of) $p$-adic Galois representations and on the study of their geometry. The lectures of Le Bras (based on joint work with Anschütz, Rodriguez Camargo, Scholze) will focus on the « automorphic » side, explaining how to realize locally analytic representations as sheaves on a variant of Fargues-Scholze’s stack of $G$-bundles on the Fargue-Fontaine curve.

Orateur: Arthur-César Le Bras (Université de Strasbourg)

Vidéo

12:15 Buffet-lunch

7 Igusa Stacks II

Igusa stacks are a new tool to study the cohomology of Shimura varieties with both characteristic 0 and torsion coefficients. Let (G,X) be a Shimura datum of Hodge type. The associated Shimura variety with infinite level at p, considered as a diamond over a p-adic field, can be expressed as a fiber product of an Igusa stack with a purely local object, the diamond associated to a flag variety, over the moduli stack Bun_G of G-bundles over the Fargues-Fontaine curve. This Cartesian diagram, called the Igusa stack diagram, allows us to use techniques from the categorical local Langlands program to study the cohomology of Shimura varieties.

In the first lecture, we will discuss the Igusa stack diagram, constructed in various levels of generality by Zhang, Daniels-van Hoften-Kim-Zhang, and Kim. In the next two lectures, we will discuss applications of the Igusa stack diagram to the cohomology of Shimura varieties, specifically to results such as torsion-vanishing, Eichler-Shimura relations, and Ihara's lemma. These are due to a number of researchers, including Koshikawa, Hamann-Lee, Daniels-van-Hoften-Kim-Zhang, Yang-Zhu and Yang. In the last lecture, we will discuss the relative intersection cohomology of the Igusa stack, which is joint work in progress of the two mini-course speakers with Linus Hamann.

Orateur: Ana Caraiani (Imperial College London)

Vidéo

15:00 Coffee break

8 Igusa Stacks III

Igusa stacks are a new tool to study the cohomology of Shimura varieties with both characteristic 0 and torsion coefficients. Let (G,X) be a Shimura datum of Hodge type. The associated Shimura variety with infinite level at p, considered as a diamond over a p-adic field, can be expressed as a fiber product of an Igusa stack with a purely local object, the diamond associated to a flag variety, over the moduli stack Bun_G of G-bundles over the Fargues-Fontaine curve. This Cartesian diagram, called the Igusa stack diagram, allows us to use techniques from the categorical local Langlands program to study the cohomology of Shimura varieties.

In the first lecture, we will discuss the Igusa stack diagram, constructed in various levels of generality by Zhang, Daniels-van Hoften-Kim-Zhang, and Kim. In the next two lectures, we will discuss applications of the Igusa stack diagram to the cohomology of Shimura varieties, specifically to results such as torsion-vanishing, Eichler-Shimura relations, and Ihara's lemma. These are due to a number of researchers, including Koshikawa, Hamann-Lee, Daniels-van-Hoften-Kim-Zhang, Yang-Zhu and Yang. In the last lecture, we will discuss the relative intersection cohomology of the Igusa stack, which is joint work in progress of the two mini-course speakers with Linus Hamann.

Orateur: Mingjia Zhang (IAS, Princeton)

Vidéo

mer. 29 octobre

09:30 Welcome coffee

9 Divisors on Fargues-Fontaine Curves

We will explain how vector bundles on different moduli spaces of degree 1 divisors on Fargues-Fontaine curves geometrize $(\varphi,N,{\rm Gal}_{Q_p})$-modules and $(\varphi,\Gamma)$-modules, all or with the restriction of being de Rham, and how this leads to a definition of (perfect) analytic prismatization over $Q_p$.

Orateur: Johannes Anschütz (Université Paris-Saclay)

Vidéo

11:00 Coffee break

10 On the mod $p$ and $p$-adic Jacquet-Langlands correspondence for $GL_2(Q_p)$ and $D^*$

I will discuss some consequences of the 6 functor formalism of Mann to the study of the mod $p$ and $p$-adically completed cohomology of the Drinfeld tower. This is joint work with Colmez and Niziol, and independently with Rodriguez Camargo.

Orateur: Gabriel Dospinescu (CNRS, Université de Clermont-Auvergne)

Vidéo

12:15 Buffet-lunch

14:00 Free afternoon

jeu. 30 octobre

09:30 Welcome coffee

11 Non-Archimedean Motives III

We will define the categories of (étale, rational) motives over an adic space $S$ and illustrate their most important properties, focusing on relevant applications in the study of $p$-adic cohomology theories. In particular, we will present the six-functor formalism they are equipped with, the continuity/spreading-out property, compact generation, and the identification between an analytic motive over a local field and a monodromy operator acting on its nearby cycle. We will sketch the proofs of these facts, highlighting the role of homotopies at each stage. Several applications will be presented, especially concerning the definition and study of rigid, de Rham, and Hyodo-Kato cohomologies.

Orateur: Timo Richarz (TU Darmstadt)

Vidéo

11:00 Coffee break

12 Geometric Aspects of the $p$-adic Locally Analytic Langlands Correspondence III

The goal of these lectures is to discuss some aspects of the geometrization of the $p$-adic Langlands program (more precisely, the part of it which deals with locally analytic representations on topological $\mathbb{Q}_p$-vector spaces). The lectures by Hellmann (based on joint work with Hernandez and Schraen and with Heuer) will focus on the « Galois » side, introducing moduli stacks of (variants of) $p$-adic Galois representations and on the study of their geometry. The lectures of Le Bras (based on joint work with Anschütz, Rodriguez Camargo, Scholze) will focus on the « automorphic » side, explaining how to realize locally analytic representations as sheaves on a variant of Fargues-Scholze’s stack of $G$-bundles on the Fargue-Fontaine curve.

Orateur: Eugen Hellmann (Universität Münster)

Vidéo

12:15 Buffet-lunch

13 Igusa Stacks IV

Igusa stacks are a new tool to study the cohomology of Shimura varieties with both characteristic 0 and torsion coefficients. Let (G,X) be a Shimura datum of Hodge type. The associated Shimura variety with infinite level at p, considered as a diamond over a p-adic field, can be expressed as a fiber product of an Igusa stack with a purely local object, the diamond associated to a flag variety, over the moduli stack Bun_G of G-bundles over the Fargues-Fontaine curve. This Cartesian diagram, called the Igusa stack diagram, allows us to use techniques from the categorical local Langlands program to study the cohomology of Shimura varieties.

In the first lecture, we will discuss the Igusa stack diagram, constructed in various levels of generality by Zhang, Daniels-van Hoften-Kim-Zhang, and Kim. In the next two lectures, we will discuss applications of the Igusa stack diagram to the cohomology of Shimura varieties, specifically to results such as torsion-vanishing, Eichler-Shimura relations, and Ihara's lemma. These are due to a number of researchers, including Koshikawa, Hamann-Lee, Daniels-van-Hoften-Kim-Zhang, Yang-Zhu and Yang. In the last lecture, we will discuss the relative intersection cohomology of the Igusa stack, which is joint work in progress of the two mini-course speakers with Linus Hamann.

Orateur: Ana Caraiani (Imperial College London)

Vidéo

15:00 Coffee break

14 Towards mod $p$ Local Global Compatibility for Partial Weight one Hilbert Modular Forms

Let $p > 5$ be a prime, and let $F$ be a totally real field in which $p$ is unramified. We study mod $p$ Hilbert modular forms for $F$ of level prime to $p$ and weight $(k, l)$, where $k$ and $l$ are tuples of integers. To a mod $p$ Hilbert modular Hecke eigenform of weight $(k, l)$, Diamond and Sasaki associate a two-dimensional mod $p$ Galois representation of ${\rm Gal}(Fp/F)$. The local–global compatibility (LGC) conjecture predicts that, at each place above $p$, the restriction of this representation admits crystalline lifts with Hodge–Tate weights determined explicitly by $(k, l)$. In this talk, we will discuss a proof showing that LGC for regular $p$-bounded weights (each entry of $k$ between 2 and $p+1$) implies LGC in the partial weight one $p$-bounded case (each entry of $k$ between 1 and $p+1$). Our approach combines computations of scheme-theoretic intersections on the Emerton–Gee stack with weight-changing arguments on quaternionic Shimura varieties, using restriction to Goren–Oort strata. This is joint work in progress with Brandon Levin and David Savitt.

Orateur: Kalyani Kansal (Imperial College London)

Vidéo

ven. 31 octobre

09:30 Welcome coffee

15 Non-Archimedean Motives IV

Motivic methods in $p$-adic arithmetic geometry

We will define the categories of (étale, rational) motives over an adic space $S$ and illustrate their most important properties, focusing on relevant applications in the study of $p$-adic cohomology theories. In particular, we will present the six-functor formalism they are equipped with, the continuity/spreading-out property, compact generation, and the identification between an analytic motive over a local field and a monodromy operator acting on its nearby cycle. We will sketch the proofs of these facts, highlighting the role of homotopies at each stage. Several applications will be presented, especially concerning the definition and study of rigid, de Rham, and Hyodo-Kato cohomologies.

Orateur: Alberto Vezzani (Università degli Studi di Milano)

Vidéo

11:00 Coffee break

16 Geometric Aspects of the $p$-adic Locally Analytic Langlands Correspondence IV

The goal of these lectures is to discuss some aspects of the geometrization of the $p$-adic Langlands program (more precisely, the part of it which deals with locally analytic representations on topological $\mathbb{Q}_p$-vector spaces). The lectures by Hellmann (based on joint work with Hernandez and Schraen and with Heuer) will focus on the « Galois » side, introducing moduli stacks of (variants of) $p$-adic Galois representations and on the study of their geometry. The lectures of Le Bras (based on joint work with Anschütz, Rodriguez Camargo, Scholze) will focus on the « automorphic » side, explaining how to realize locally analytic representations as sheaves on a variant of Fargues-Scholze’s stack of $G$-bundles on the Fargue-Fontaine curve.

Orateur: Arthur-César Le Bras (Université de Strasbourg)

Vidéo

12:15 Buffet-lunch

17 Uniqueness and Functoriality of Igusa Stacks

I will introduce a perspective on Igusa stacks that can be interpreted as providing a uniformization of the $p$-adic Shimura variety. Using deformation theory and $p$-adic Hodge theory, I will discuss how this uniformization can be pinned down uniquely. As a consequence, we can deduce that Igusa stacks are canonical objects that are unique.

Orateur: Dongryul Kim (Stanford University)

Vidéo

15:00 Coffee break

18 $p$-adic Motives and Special Values of Zeta Functions

In 1966, Tate proposed the Artin–Tate conjectures, which describe the special values of zeta functions associated to surfaces over finite fields. Building on this, and assuming the Tate conjecture, Milne and Ramachandran formulated and proved analogous conjectures for smooth
proper schemes over finite fields. However, the formulation of these conjectures already relied on other unproven conjectures.
In this talk, I will present an unconditional formulation and proof of these conjectures. The approach relies on the theory of $F$-Gauges, a notion introduced by Fontaine–Jannsen and further developed by Bhatt–Lurie and Drinfeld, which has been proposed as a candidate for a theory of $p$-adic motives. A central role is also played by the notion of stable Bockstein characteristics, which will be introduced in the talk.

Orateur: Shubhodip Mondal (Purdue University)

Vidéo