Arithmetic Geometry - A Conference in Honor of Hélène ESNAULT on the Occasion of Her 70th Birthday

22 avr. 2024, 09:00 → 26 avr. 2024, 16:00 Europe/Paris

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

Arithmetic Geometry
April 22-26, 2024
at IHES - Centre de conférence Marilyn et James Simons

On the occasion of Hélène Esnault's 70th birthday, Marco D'Addezio, IRMA Strasbourg, Kay Rülling, Univ. Wuppertal, and Tanya Srivastava, IIT Gandhinagar, organize a conference in her honor from April 22 to 26, 2024.

This conference centers around the mathematical contributions and interests of Hélène Esnault. It aims at bringing together mathematicians with diverse backgrounds, providing a platform to exchange their ideas and foster new collaborations. 

Registration deadline: January 31, 2024

Invited Speakers:

• Tomoyuki Abe, IPMU - University of Tokyo  
• Yves André, IMJ-PRG  
• Emelie Arvidsson, University of Utah
• Bhargav Bhatt, IAS - Princeton University & University of Michigan  
• Ana Caraiani, Imperial College London  
• Dustin Clausen, IHES 
• Johan De Jong, Columbia University  
• Michael Groechenig, University of Toronto  
• Lars Hesselholt, Nagoya University & University of Copenhagen 
• Katharina Hübner, Goethe-Universität Frankfurt  
• Moritz Kerz, Universität Regensburg      
• Marc Levine, Universität Duisburg-Essen  
• Daniel Litt, University of Toronto   
• Alexander Petrov, Harvard University  
• Claude Sabbah, École polytechnique  
• Peter Scholze, MPIM - University of Bonn   
• Atsushi Shiho, University of Tokyo  
• Carlos Simpson, Université Nice-Sophia Antipolis
• Vasudevan Srinivas, SUNY, Buffalo
• Jakob Stix, Goethe-Universität Frankfurt  

Scientific Committee: Marco D'Addezio (IRMA Strasbourg), Marcin Lara (Goethe Universität-Frankfurt), Simon Pépin Lehalleur (Radboud Universiteit Nijmegen), Kay Rülling (Universität Wuppertal), Annette Werner (Goethe Universität-Frankfurt), and Lei Zhang (Sun Ya-Tsen Univ. Zhuhai)
 

Hélène Esnault is a mathematician specializing in algebraic and arithmetic geometry. She obtained her PhD in 1976 at the University of Paris VII under the direction of Lê Dũng Tráng. She then completed her habilitation at the University of Bonn in 1985. Afterwards, she was a Heisenberg scholar at the MPI in Bonn and maître de conférence in Paris VII. In 1990, she became a full professor at the Universität Duisburg-Essen. In 2012, she moved to Berlin as the first Einstein Professor at the Freie Universität Berlin, where she became emerita in 2019. She continued her mathematical work and accepted a visiting professor position in 2019 at IAS Princeton. From there, she moved back to Europe in 2020. In the fall 2022, she held the Eilenberg Chair at Columbia University and is currently a part-time professor at Copenhagen University and a Faculty Associate at Harvard University. 

As a mathematician, she published more than 135 research articles with 45 coauthors covering a wide range of topics. In the following, only a small extract of her influential oeuvre is mentioned. In the 1980’s, she found together with Eckart Viehweg a new method to prove vanishing theorems in algebraic geometry. At the end of the 1990s, she gave with Vasudevan Srinivas and Viehweg the first general construction of an Albanese variety for singular projective varieties over an algebraically closed field. At the beginning of the 2000s, she proved that a Fano variety over a finite field has a rational point, answering positively a conjecture of Lang-Manin. She furthermore showed that a smooth projective variety over a local field with a regular model has a rational point in the special fiber if the étale cohomology of the generic fiber has coniveau 1. With Spencer Bloch and Pierre Berthelot, she proved that Serre’s Witt vector cohomology of a singular proper variety in positive characteristic is the slope <1 part of rigid cohomology, generalizing results of Bloch and Illusie in the smooth case. A spectacular result is her proof with Vikram Mehta in 2010 of a conjecture by Gieseker, which says, there are no non-constant stratified bundles on a geometrically simply connected smooth projective variety over a perfect field of positive characteristic. She proved with Bloch and Moritz Kerz a p-adic infinitesimal version of the Fontaine-Mazur conjecture. A result that recently has drawn much attention is her joint work with Michael Groechenig on the integrality of certain rigid local systems, which was conjectured by Carlos Simpson.

Hélène Esnault has mentored about 25 PhD students and even more  postdocs. In recognition of her seminal contributions, she was awarded the Doisteau-Blutet prize of the Academy of Sciences in Paris in 2001, jointly with Eckart Viehweg  the Leibniz Prize in 2003 and received the Cantor medal of the German Mathematical Society in 2019. She received multiple honorary doctorates and was a member of various significant committees, including the Fields Medal Committee of the ICM 2018, the Structure Committee ICM 2022, the Shaw Prize Committee 2021-2020 and 2021-2025, and the Infosys Prize Committee 2023. Additionally, she has served on the editorial boards of several prestigious journals, including the Duke Mathematical Journal (since 1995), Mathematische Annalen (1998–2010), Mathematical Research Letters (since 2007), Algebra and Number Theory (since 2007 as a founding editor), Memoirs of the European Mathematical Society (since 2023), and Acta Mathematica (since 2023).

 

The conference receives partial support from the GRK 2240: Algebro-geometric Methods in Algebra, Arithmetic and Topology

lundi 22 avril

09:30 Registration & Welcome coffee

1 Finite Type Properties of (Tame) Fundamental Groups

We are interested in finite generation or finite presentation of fundamental groups as topological profinite groups. Our knowledge of group theoretic properties of étale fundamental groups relies traditionally on Riemann's existence theorem (in char 0) and Grothendieck's specialization map (for the transition to char $p$). But not all varieties lift to characteristic 0. Building on recent results by Esnault, Shusterman and Srinivas for smooth projective varieties in char $p$, we are going to explain in the talk how to generalize finite presentation to arbitrary proper varieties (joint work with Lara and Srinivas). Furthermore, we introduce an adic tameness condition and discuss finite generation/presentation of tame fundamental groups for rigid analytic spaces. The second part is joint work with Achinger, Lara and Hübner.

Orateur: Prof. Jakob Stix (Goethe-Universität Frankfurt)

Vidéo

11:30 Coffee break / Discussion

2 Pure Local Systems Over Local Fields

In joint work with Hélène we study certain pure $l$-adic local systems on varieties over $p$-adic local fields which are analogs of variations of pure Hodge structures. In the talk I will explain an approach via tilting to the most basic open problems in this setting: analogs of limiting mixed Hodge structures and purity of cohomology for a curve.

Orateur: Prof. Moritz Kerz (Universität Regensburg)

Vidéo

13:00 Lunch break

3 Various Remarks on the Donagi-Pantev Program for Construction of Hecke Eigensheaves

Donagi and Pantev set out a program for the construction of the parabolic logarithmic Higgs sheaves associated to Hecke eigensheaves in the geometric Langlands correspondence. One of the main features is that their spectral varieties over $\mathrm{Bun}_G$ are Hitchin fibers viewed birationally as subvarieties of $T^*(\mathrm{Bun}_G)$. We'll discuss various aspects of this construction: cases where it is known, the difficulties that can
arise, and relationships with the geometry of the Hitchin moduli space.

Orateur: Prof. Carlos Simpson (Université Nice-Sophia Antipolis)

Vidéo

16:00 Coffee break / Discussion

4 Different Notions of Tameness Revisited

For an étale morphism $f:Y \to X$ of schemes over a base $S$ there are different approaches to define what it means that $f$ is tame. Behind all of them lies the intuition that the induced morphism of compactifications $\bar{f} : \bar{Y} \to \bar{X}$ is tamely ramified along the boundary $\bar{Y} \setminus Y$ (in an appropriate sense). Many of the tameness definitions work with valuations without relying on the choice of a compactification. Kerz and Schmidt compare these different notions of tameness in their article “On different notions of tameness” mainly working with compactifications. The disadvantage of this approach is that they need to assume resolution of singularities in order to obtain nice compactifications. In my talk I want to present work in progress with Michael Temkin that approaches the problem purely valuation theoretic by using nonachimedean geometry. As a consequence we can drop the assumption on resolution of singularities. The heart of the project lies in a careful study of the geometry of adic curves over an arbitrary affinoid field (of higher rank) and of the wild locus of an étale morphism of such curves.

Orateur: Prof. Katharina Hübner (Goethe-Universität Frankfurt)

Vidéo

18:00 Concert Yves André

mardi 23 avril

09:00 Welcome coffee

5 The Bloch-Esnault-Kerz Fiber Square

A theorem of Bloch-Esnault-Kerz published in 2014 states that the formal part of the Fontaine-Messing $p$-adic variational Hodge conjecture holds for schemes smooth and proper over an unramified local number ring. The theorem states that a class in the rational $p$-adic Grothendieck group of the special fiber admits a lifting to the rational $p$-adic continuous Grothendieck group of the formal completion along the special fiber if and only if the image of its crystalline Chern class under the de Rham-crystalline comparison isomorphism lies in the appropriate part of the Hodge filtration. In a paper also published in 2014, Beilinson generalized the equivalence of the relative rational $p$-adic $K$-theory and cyclic homology, implicit in the Bloch-Esnault-Kerz paper. As much else, this work, was greatly clarified by the Bhatt-Morrow-Scholze unification of $p$-adic Hodge theory and topological cyclic. Indeed, Antieau-Mathew-Morrow-Nikolaus showed that Beilinson's equivalence is given by the map of horizontal fibers in a square in which the map of vertical fibers is an equivalence by the Nikolaus-Scholze Tate-Orbit-Lemma. In this talk, I will recall how said cartesian square appears from the Nikolaus-Scholze Frobenius of $\mathbb{Z}$ and explain a proposal by Clausen for how it may lead to a definition of the Hodge-Tate period map that does not require any calculational input.

Orateur: Prof. Lars Hesselholt (Nagoya University & University of Copenhagen)

Vidéo

10:30 Coffee break / Discussion

6 On Secondary Invariants and Arithmetic Rigidity

A complex local system on a space $S$ gives rise to "secondary" Chern classes in $H^{2p-1}(S; \mathbb{C}/\mathbb{Z}(p))$, refining the usual "primary" Chern classes in $H^{2p}(S;\mathbb{Z}(p))$. In fact, Esnault in a survey article describes four methods of defining such classes, of which 3 are proved to be equivalent by means of her "modified splitting principle". I will explain how to show that the remaining 1 out of 4 definitions, that of Cheeger-Simons, agrees with the others. Then, changing gears, I will describe some arithmetic analogs of the phenomenon of rigidity of secondary Chern classes. This has bearing on another question from Esnault's article, and leads us to some motivic speculations.

Orateur: Prof. Dustin Clausen (IHES)

Vidéo

7 Finiteness Questions for Étale Coverings with Bounded Wild Ramification at the Boundary

We will consider étale coverings $ f : Y {\rightarrow} X$ of varieties over an algebraically closed field in characteristic $p$ > 0 (with some further restrictions on the boundary ramification, in the non-proper case). This talk will give an overview of some work done with Hélène Esnault and others over the last few years, as well as open problems, related to this theme.

Orateur: Prof. Vasudevan Srinivas (SUNY, Buffalo)

Vidéo

13:15 Lunch break

8 Donaldson-Thomas Invariants: Classical, Motivic, Quadratic and Real

Let $X$ be a smooth projective 3-fold over the complex numbers. Following work of Thomas, Behrend-Fantechi, and others, one has a virtual fundamental class in the Chow group of 0-cycles on the Hilbert scheme of dimension 0, length $n$ subschemes of $X$, the degree of which is the $n$th Donaldson-Thomas invariant of $X$. Now take $X$ over an arbitrary field $k$. We have developed a construction of virtual fundamental classes with values in an arbitrary motivic cohomology theory. An example of such, a "quadratic" analog of the Chow groups, is the cohomology of the sheaf of Witt rings, which leads to a refinement of the classical DT-invariants to quadratic DT-invariants with values in the Witt ring of quadratic forms over $k$. We will discuss some developments and conjectures for these refined DT invariants, including some computations of the signature of these invariants due to Anneloes Viergever.

Orateur: Prof. Marc Levine (Universität Duisburg-Essen)

Vidéo

16:00 Coffee break / Discussion

9 Characteristic Cycle and Pushforward

Characteristic cycle for $l$-adic sheaf was introduced by T. Saito after the existence of singular support by Beilinson. This measures the ramification of the sheaf, and can be viewed as a vast generalization of Swan conductor. Various compatibility with cohomological operation had been verified by Saito and Beilinson, but the compatibility of pushforward along proper morphism has been left open. In this talk, I wish to discuss this compatibility.

Orateur: Prof. Tomoyuki Abe (IPMU - University of Tokyo)

Vidéo

mercredi 24 avril

09:00 Welcome coffee

10 Vanishing Theorems for the Irregular Hodge Filtration

I will give an overview of recent advances concerning the irregular Hodge filtration (introduced by Deligne 40 years ago) and I will focus on Kodaira vanishing theorems similar to those of Saito for mixed Hodge modules.

Orateur: Prof. Claude Sabbah (École polytechnique)

Vidéo

10:30 Coffee break / Discussion

11 The Non-Abelian $p$-Curvature Conjecture

The classical Grothendieck-Katz $p$-curvature conjecture gives an arithmetic criterion for the solutions to an algebraic linear ODE to be algebraic functions. We formulate a version of the $p$-curvature conjecture for certain non-linear ODEs arising from algebraic geometry (for example, the Painlevé VI equation or the Schlesinger system), which implies the classical conjecture, and prove it for "Picard-Fuchs initial conditions." The proof is inspired in part by Katz's resolution of the classical p-curvature conjecture for Picard-Fuchs equations, and in part by Esnault-Groechenig's recent resolution of the classical conjecture for rigid $\mathbb{Z}$-local systems. This is joint work with Josh Lam.

Orateur: Prof. Daniel Litt (University of Toronto)

Vidéo

12 Motives and (Super-)Representation Theory: Principles and Case Studies

I shall outline how existence and shape of motives can sometimes be (not only predicted but) established using abstract motivic Galois theory, bypassing concrete constructions of algebraic cycles.

Orateur: Prof. Yves André (IMJ-PRG)

Vidéo

13:15 Lunch break

15:00 Free Afternoon

jeudi 25 avril

09:00 Welcome coffee

13 Characteristic Classes of Étale Local Systems

Given an étale $\mathbb{Z}_p$-local system of rank $n$ on an algebraic variety $X$, continuous cohomology classes of the group $\rm{GL}_n(\mathbb{Z}_p)$ give rise to classes in (absolute) étale cohomology of the variety with coefficients in $\mathbb{Z}_p$. These characteristic classes can be thought of as $p$-adic analogs of Chern-Simons characteristic classes of vector bundles with a flat connection. For a smooth projective variety over complex numbers, Reznikov proved that the usual Chern-Simons classes in degrees $>1$ of all $\mathbb{C}$-local systems are torsion. It turns out that characteristic classes of étale $\mathbb{Z}_p$-local systems on algebraic varieties over non-closed fields are often non-zero even rationally. In particular, if $X$ is a smooth variety over a $p$-adic field, and the local system is de Rham, then its characteristic classes are related to Chern classes of the graded quotients of the Hodge filtration on the associated vector bundle with connection. This relation can be established through considering an analog of Chern classes for vector bundles on the pro-étale site of $X$. This is a joint work with Lue Pan.

Orateur: Prof. Alexander Petrov (Harvard University)

Vidéo

10:30 Coffee break / Discussion

14 Towards an Eichler-Shimura Decomposition for Ordinary $p$-adic Siegel Modular Forms

There are two different ways to construct families of ordinary $p$-adic Siegel modular forms. One is by $p$-adically interpolating classes in Betti cohomology, first introduced by Hida and then given a more representation-theoretic interpretation by Emerton. The other is by $p$-adically interpolating classes in coherent cohomology, once again pioneered by Hida and generalised in recent years by Boxer and Pilloni. I will explain these two constructions and then discuss joint work with James Newton and Juan Esteban Rodríguez Camargo, very much in progress, that aims to compare them.

Orateur: Prof. Ana Caraiani (Imperial College London)

15 The Frobenius Action on the De Rham Moduli Space

In a suitable mixed characteristic setting, the moduli stack of flat vector connections can be endowed with a Frobenius-pullback operation. This talk is devoted to the properties of this map, which yields amongst other things a new construction of the $F$-isocrystal structure for rigid flat connections. This is joint work with Hélène Esnault.

Orateur: Prof. Michael Gröchenig (University of Toronto)

13:15 Lunch break

16 Analytic Prismatization

Prismatic cohomology is a unifying $p$-adic cohomology of $p$-adic formal schemes. Motivated by questions on locally analytic representations of $p$-adic groups and the $p$-adic Simpson correspondence, an extension of prismatic cohomology to rigid-analytic spaces (over $\mathbb{Q}_p$ or over $\mathbb{F}_p((t))$ has been sought. We will explain what form this should take, and our progress on realizing this picture. This includes a degeneration from the analytic Hodge-Tate stack underlying the $p$-adic Simpson correspondence to a similar (analytic) stack related to the Ogus-Vologodsky correspondence in characteristic $p$. This is joint work in progress with Johannes Anschütz, Arthur-César le Bras and Juan Esteban Rodriguez Camargo.

Orateur: Prof. Peter Scholze (MPIM - University of Bonn)

Vidéo

16:00 Coffee break

17 Local Systems and Higgs Bundles in $p$-adic Geometry

The classical Corlette--Simpson (CS) correspondence relates local systems on complex varieties to Higgs bundles; it is highly transcendental in nature. Its characteristic $p$ counterpart surprisingly turns out to be purely algebraic: Bezrukavnikov identified de Rham local systems on a smooth variety $X$ over $\mathbb{F}_p$ with Higgs bundles twisted by a natural $\mathbb{G}_m$-gerbe on the cotangent bundle $T^*X$. By trivializing the gerbe over suitable loci in $T^*X$ using additional choices, Ogus--Vologodsky then recovered an honest CS correspondence (i.e., with untwisted Higgs bundles). In this talk, I'll explain that this story has an exact analog for a smooth rigid space $X$ over a perfectoid $p$-adic field: (generalized) local systems identify with Higgs bundles twisted by a natural $\mathbb{G}_m$-gerbe on $T^*X$, and honest CS correspondes (as studied by many authors in the last 2 decades) can be recovered by trivializing the gerbe over suitable loci in $T^*X$.

This is joint work in progress with Mingjia Zhang, and is inspired by recent work of Heuer.

Orateur: Prof. Bhargav Bhatt (IAS - Princeton University & University of Michigan)

Vidéo

vendredi 26 avril

09:00 Welcome coffee

18 Vanishing Theorems in Positive Characteristic

Starting from the seminal book of Hélène Esnault and Eckart Viehweg on vanishing theorems my talk will be centered around vanishing theorems in positive characteristics. The Kodaira and Kawamata—Viehweg vanishing theorems are incredibly useful in Complex geometry but fail in general to be true over fields of positive characteristics. It was long expected that this failure would be pathological and that these theorems still would be true for some important classes of varieties, such as log Fano varieties. It turns out that starting from dimension two there are log Fano varieties which contradict Kodaira vanishing. However, the known constructions have the dimension of the Fano variety increasing with the characteristic of the base field. One could therefore ask if in any given dimension log Fano's satisfy this vanishing theorem in large enough characteristic depending on the dimension? In this direction, joint with Fabio Bernasconi and Justin Lacini we proved that the Kawamata—Viehweg vanishing theorem holds on log del Pezzo surfaces over a perfect field of characteristic $p$>5.

Orateur: Prof. Emelie Arvidsson (University of Utah)

Vidéo

10:30 Coffee break / Discussion

19 Weight Filtration on Log Crystalline Site

Let $p$ be a prime. For a family of simple normal crossing log varieties on which $p$ is nilpotent, we construct a filtered complex on certain log crystalline site which gives rise to the weight filtered $p$-adic Steenbrink complex defined by Mokrane and Nakkajima when we project it to the Zariski site.

Orateur: Prof. Atsushi Shiho (University of Tokyo)

Vidéo

20 Integrality of the Betti Moduli Space

This is a report on joint work with Hélène Esnault. Let $X$ be a smooth projective variety over the complex numbers $\mathbb{C}$. Let $M$ be the moduli space of irreducible representations of the topological fundamental group of $X$ of a fixed rank $r$. Then $M$ is a finite type scheme over the spectrum of the integers $\mathbb{Z}$. We may ask whether $M$ is pure over $\mathbb{Z}$ in the sense of Raynaud-Gruson, for example we can ask if the irreducible components of $M$ which dominate ${\rm Spec}(\mathbb{Z})$ actually surject onto ${\rm Spec}(\mathbb{Z})$. We will explain what this means, present a weak answer to this question, apply this to exclude some abstract groups as the fundamental groups of smooth projective varieties over $\mathbb{C}$, and we discuss what other phenomena can be studied using the method of proof.

Orateur: Prof. Johan de Jong (Columbia University)

Vidéo

13:15 Lunch break