Non-Archimedean Motives I

• Timo Richarz (TU Darmstadt)

Room: Centre de conférences Marilyn et James Simons 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.

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

• Eugen Hellmann (Universität Münster)

Room: Centre de conférences Marilyn et James Simons 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.

Igusa Stacks I

• Mingjia Zhang (IAS, Princeton)

Room: Centre de conférences Marilyn et James Simons 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.

Non-Archimedean Motives II

• Alberto Vezzani (Università degli Studi di Milano)

Room: Centre de conférences Marilyn et James Simons 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.

Higher Pushforwards in Rigid Cohomology via Motives

• Veronika Ertl (Université de Caen)

Room: Centre de conférences Marilyn et James Simons 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.)

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

• Arthur-César Le Bras (Université de Strasbourg)

Room: Centre de conférences Marilyn et James Simons 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.

Igusa Stacks II

• Ana Caraiani (Imperial College London)

Room: Centre de conférences Marilyn et James Simons 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.

Igusa Stacks III

• Mingjia Zhang (IAS, Princeton)

Room: Centre de conférences Marilyn et James Simons 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.

Divisors on Fargues-Fontaine Curves

• Johannes Anschütz (Université Paris-Saclay)

Room: Centre de conférences Marilyn et James Simons 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 $Q_p$.

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

• Gabriel Dospinescu (CNRS, Université de Clermont-Auvergne)

Room: Centre de conférences Marilyn et James Simons 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.

Non-Archimedean Motives III

• Timo Richarz (TU Darmstadt)

Room: Centre de conférences Marilyn et James Simons 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.

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

• Eugen Hellmann (Universität Münster)

Room: Centre de conférences Marilyn et James Simons 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.

Igusa Stacks IV

• Ana Caraiani (Imperial College London)

Room: Centre de conférences Marilyn et James Simons 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.

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

• Kalyani Kansal (Imperial College London)

Room: Centre de conférences Marilyn et James Simons 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.

Non-Archimedean Motives IV

• Alberto Vezzani (Università degli Studi di Milano)

Room: Centre de conférences Marilyn et James Simons 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.

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

• Arthur-César Le Bras (Université de Strasbourg)

Room: Centre de conférences Marilyn et James Simons 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.

Uniqueness and Functoriality of Igusa Stacks

• Dongryul Kim (Stanford University)

Room: Centre de conférences Marilyn et James Simons 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.

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

• Shubhodip Mondal (Purdue University)

Room: Centre de conférences Marilyn et James Simons 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.