We started inviting speakers for this conference about one year ago, with the clear goal of having an on-site conference, but without making a definitive decision, at that time, whether we would also have talks given remotely. In the summer of 2022, several of the invited speakers approached us and asked whether online talks would be possible, for various reasons; in fact, for some of them traveling is still de facto impossible because of Covid restrictions.

It was a tough decision, with good arguments either way, but in the end we decided to follow a strict policy of having only on-site talks.

Regrettably, therefore, the following people whom we had invited and who had accepted to speak at the conference, could in the end not give a talk at Darmstadt: Bhargav Bhatt, Hélène Esnault, Moritz Kerz, Ruochuan Liu, Yifeng Liu.

## Schedule

Monday, Oct. 3 Tuesday, Oct. 4 Wednesday, Oct. 5 Thursday, Oct. 6 Friday, Oct. 7
8:30
Registration
9:00
Registration
9:00
Pauli

9:00
Pilloni
10:00
Rapoport
9:30
Fintzen
10:30
Tamme
9:30
Gee
10:30
Hellmann
11:30
Anschütz
11:00
Ziegler
11:45
Heuer
11:00
Pappas
11:45
Nicaise
14:30
Registration
14:30
Le Bras

Excursion

15:15
Caraiani
16:00
Fargues

Excursion
15:00
Temkin

16:45
Zhu
17:15
Scholze

Excursion
16:30
Habegger

18:30
Get together

19:30
Conference dinner

All talks will be 60 minutes long.

## Speakers

Johannes Anschütz
Ana Caraiani
Laurent Fargues
Jessica Fintzen
Toby Gee
Philipp Habegger
Eugen Hellmann
Ben Heuer
Moritz Kerz
Arthur-César Le Bras
Johannes Nicaise
George Pappas
Sabrina Pauli
Vincent Pilloni
Michael Rapoport
Peter Scholze
Georg Tamme
Michael Temkin
Xinwen Zhu
Paul Ziegler

## Abstracts

#### Johannes Anschütz: $\mathbb C_p$-semilinear Galois representations of $p$-adic fields via Cartier-Witt stacks

This talk will report on joint work in progress with Arthur-César Le Bras and Ben Heuer. The aim of this project is a possible description of the category of $\mathbb{C}_p$-semilinear Galois representations of a p-adic field $K$ via vector bundles on Cartier-Witt stacks of rings of integers of finite extensions of $K$.

#### Ana Caraiani: On the modularity of elliptic curves over imaginary quadratic fields

I will survey how to prove the modularity of elliptic curves defined over the rational numbers, as pioneered by Wiles and Taylor-Wiles and completed by Breuil, Conrad, Diamond and Taylor. I will also mention the case of elliptic curves defined over real quadratic fields, more recently completed by Freitas, Le Hung and Siksek. I will then explain why the case of imaginary quadratic fields is qualitatively different from the previous ones. Finally, I will discuss joint work in progress with James Newton, where we prove a local-global compatibility result in the crystalline case for Galois representations attached to torsion classes occurring in the cohomology of locally symmetric spaces. This has an application to the modularity of elliptic curves over imaginary quadratic fields, which also builds on recent work of Allen, Khare and Thorne.

#### Laurent Fargues: Extension du domaine de la lutte

I will explain how to define an extended Kottwitz set that contains the usual one and whose basic elements allows us to reach all inner forms of a given reductive $p$-adic group, even if its center is not connected. This relies on the work of Kaletha. In particular I give a geometric interpretation of its gerb in terms of the curve. One can then define an extended stack of $G$-bundles that contains the one we defined and studied with Scholze as a closed / open substack and whose points are the extended Kottwitz set. At the end this allows us to formulate a geometrization conjecture for any reductive group even if it's not an extended pure inner form of its quasisplit inner form.

#### Jessica Fintzen: Representations of $p$-adic groups

A fundamental problem on the representation theory side of the Langlands program is the construction of all (irreducible, smooth, complex or mod-$\ell$) representations of $p$-adic groups. I will provide an overview of our understanding of the representations of $p$-adic groups, with an emphasis on recent progress, including some mod-$\ell$ results and joint work with Kaletha and Spice that introduces a twist to the story.

#### Toby Gee: An introduction to the categorical $p$-adic Langlands program

I will attempt to explain what the categorical $p$-adic Langlands program is, and how it connects to more classical questions about congruences between modular forms.

#### Philipp Habegger: Torsion Points on Families of Abelian Varieties

A subvariety of an abelian variety does not contain a Zariski dense subset of points of finite order, except for a special class of subvarieties. This is the Manin-Mumford Conjecture and was proved by Raynaud in the 1980s. Over 10 years ago, Masser and Zannier obtained the first results for curves in the relative setting of a family of abelian varieties. I will speak on work in progress together with Ziyang Gao in higher dimension. I will describe how the Pila-Zannier counting strategy works, the relevance of definability results of Peterzil-Starchenko in o-minimal geometry, and height bounds.

#### Eugen Hellmann: On the K-theory of the stack of L-parameters for $GL_n$

The categorical version of the local Langlands conjecture of Fargues-Scholze asserts, in the case of $GL_n$, that there is an equivalence between a certain category of $\ell$-adic sheaves on the stack of rank $n$ vector bundles on the Fargues-Fontaine curve and a certain category of coherent sheaves on the stack of $n$-dimensional Weil-Deligne representations. We will give some evidence for this conjecture by showing that there is an isomorphism between the K-groups (in particular: on Grothendieck groups) of these categories. In particular we will (rather explicitly) compute the K-theory of the stack of $n$-dimensional Weil-Deligne representations.

#### Moritz Kerz: The Picard-Lefschetz formula for normal crossings

In the study of semi-stable degeneration of Lefschetz pencils one is led to a generalization of the classical Picard-Lefschetz formula for certain perverse sheaves on normal crossing spaces. In the talk I will recall the formalism of nearby cycle and vanishing cycle functors and I will explain how Hodge theory allows one to obtain the normal crossing Picard-Lefschetz formula. Joint work with A. Beilinson and H. Esnault.

#### Arthur-César Le Bras: A Fourier transform for Banach-Colmez spaces

I will explain how to define an $\ell$-adic Fourier transform for Banach-Colmez spaces and will discuss some examples of it. Joint work with J. Anschütz.

#### Johannes Nicaise: Irrationality of complete intersections

This talk is based on joint work with John Christian Ottem and builds upon earlier work with Evgeny Shinder. I will explain a degeneration technique that allows to prove irrationality of complete intersections by induction on dimension and/or degree. It is based on a version of the nearby cycles functor for birational types of algebraic varieties. The talk will focus on some concrete applications, including complete intersections of quadrics.

#### George Pappas: $p$-adic shtukas and integral models of Shimura varieties

I will discuss the problem of constructing integral models of Shimura varieties that are suitably “canonical” and describe related results. These are obtained using Scholze’s theory of p-adic shtukas and a connection between (global) Shimura varieties and corresponding local Shimura varieties. This is joint work with M. Rapoport.

#### Sabrina Pauli: Arithmetic enrichments in enumerative geometry

Results from motivic homotopy theory allow to answer questions in enumerative geometry over an arbitrary base field. In the talk I will explain how this works and give some examples.

#### Vincent Pilloni: Higher Hida theory

Modular forms are degree zero cohomology of certain invertible sheaves on modular curves. One is often led to consider also higher cohomology of automorphic vector bundles on Shimura varieties. We try to define and understand the integral coherent cohomology for Siegel modular varieties (and the Hecke action). If we restrict to the ordinary part, this is possible. One key ingredient is the cohomology with partial support of the ordinary Igusa variety. This is joint work with G. Boxer.

#### Michael Rapoport: $p$-adic uniformization of Shimura curves

Cherednik discovered that a Shimura curve attached to a quaternion algebra over a totally real field $F$ admits $p$-adic uniformization by the Drinfeld $p$-adic upper half plane if it is ramified at some $p$-adic place. This observation is now almost 50 years old and there is finally the hope for a published proof. One of the main difficulties is that outside the case $F=\mathbb Q$ the Shimura curve does not represent a moduli problem of abelian varieties. This difficulty disappears when the multiplicative group of the quaternion algebra is replaced by the group of unitary similitudes of a binary hermitian space, but then other interesting problems arise. I will discuss work of Boutot-Zink, of Kudla-Rapoport-Zink, and of Scholze-Weinstein.

#### Georg Tamme: $K$-theory and regularity

Algebraic $K$-theory of rings is not $\mathbb{A}^1$-invariant in general. However, it is so on regular rings. Vorst conjectured a certain converse to this statement. In characteristic 0, his conjecture was settled by Cortiñas-Haesemeyer-Weibel. In the talk I will explain how significant advances in algebraic K-theory and trace methods that have been achieved in recent years allow some progress towards this conjecture in positive characteristic. Based on joint work with Moritz Kerz and Florian Strunk.

#### Michael Temkin: Functorial resolution of singularities: comparison of the methods

Until very recently we knew different descriptions of essentially the same method to canonically resolve singularities (in characteristic zero), but in the past few years a few new methods were found. Perhaps, this is already a good time to compare the methods, point out principles which are common to all of them, and pin down what was the main ingredient which changes from one method to another and essentially determines the method. In my talk I will suggest such a systematization and will give a panoramic view of the known desingularization algorithms.

#### Xinwen Zhu: Isogenies of mod $p$ CM abelian varieties via main theorem of complex multiplication

The main theorem of complex multiplication describes how automorphisms of $\mathbb C$ act on CM abelian varieties and their torsion points. I will explain an extension of this theory to describe the action of $p$-quasi-isogenies between mod $p$ reductions of CM abelian varieties on prime-to-$p$ torsion points. Joint work with Liang Xiao.

#### Paul Ziegler: A Tannakian formalism for Bruhat-Tits buildings

By Goldman-Iwahori, the Bruhat-Tits building of the general linear group $\operatorname{GL}_n$ over a local field k can be described as the set of non-archimedean norms on the vector space $k^n$. I will explain how via a Tannakian formalism this can be generalized to a concrete description of the Bruhat-Tits building of an arbitrary reductive group. This also gives a description of the functor of points of Bruhat-Tits group schemes.