A Conference in Arithmetic Algebraic Geometry in Memory of Jan Nekovář

9 oct. 2023, 09:00 → 13 oct. 2023, 16:00 Europe/Paris

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

Arithmetic Algebraic Geometry

Jan Nekovář was a mathematician who worked and has made many significant and seminal contributions to algebraic geometry. He left us prematurely in November 2022. Here is a tribute note (in French) by Pierre Colmez.

In his memory, Anna Cadoret, IMJ-PRG, Wiesława Nizioł, IMJ-PRG, and Sarah Zerbes, ETH Zürich, organize a conference in arithmetic algebraic geometry at IHES from October 9th to 13th, 2023.

This is an international conference in arithmetic geometry, aimed at covering some of the most recent advances in the field with a focus on the areas Jan Nekovář was interested in.

 

Invited Speakers:

• Tomoyuki Abe, IPMU - University of Tokyo  
• Spencer Bloch, University of Chicago  
• Ana Caraiani, Imperial College London - University of Bonn
• Henri  Darmon, McGill University  
• Vesselin Dimitrov, Institute for Advanced Studies  
• Matthew  Emerton, University of Chicago  
• Veronika  Ertl, Universitat Regensburg  
• Hélène Esnault, Frei Universität Berlin  
• Olivier Fouquet, Université de Franche-Comté 
• Alexander Goncharov, Yale University  
• Giada Grossi, CNRS, Université Sorbonne Paris Nord      
• Jie Lin, Universität Duisburg-Essen  
• David  Loeffler, University of Warwick   
• Akhil Mathew, University of Chicago   
• James Newton, University of Oxford  
• Alexander Petrov, Harvard University  
• Alena Pirutka, New York University  
• Tony Scholl, University of Cambridge   
• Yunqing Tang, University of California - Berkeley  
• Jacob Tsimerman, University of Toronto  

 

 

IHES thanks all the donors of the 2021 Friends of IHES Gala for making this conference possible.

lundi 9 octobre

09:30 Welcome coffee & Registration

1 Crystallinity Properties and Rigid Flat Connections Revisited

We generalise a theorem on the existence of Frobenius isocrystal and Fontaine-Laffaille module structures on rigid flat connections to the non-proper setting. The proof is based on a new strategy of a point-set topological flavour, which allows us to produce a purely $p$-adic statement and thereby to avoid complex-analytic methods.

Joint with Michael Groechenig.

Orateur: Prof. Hélène Esnault (Frei Universität Berlin)

Vidéo

11:00 Coffee break

2 Iwasawa Theory for Asai Representations and the Adjoint of a Modular Form

I will report on an ongoing project with Giada Grossi and Sarah Zerbes in which we construct Euler systems for the Galois representations appearing in the cohomology of Hilbert modular surfaces, and relate these to special values of Asai L-functions. I will also explain an application of these results to the symmetric square of a modular form.

Orateur: Prof. David Loeffler (University of Warwick)

Vidéo

12:15 Lunch break

3 The Arithmetic of Power Series and Applications to Irrationality

In this talk, we will discuss various irrationality and linear independence problems including the irrationality of 2-adic zeta value at 5. The proofs use an arithmetic holonomicity theorem, the special case of which was used in the proof of the unbounded denominators conjecture; arithmetic holonomicity theorems have also been studied in the recent work of Bost and Charles. This is a joint work in progress with Frank Calegari and Vesselin Dimitrov.

Orateur: Prof. Yunqing Tang (University of California - Berkeley)

15:00 Coffee break

4 BSD and Estimates for Class Groups of Number Fields

(Joint with A.Shankar)

We consider question of bounding (Fixed) torsion in class groups of number fields: Given two integers $m,n$, what is the smallest constant $e_{m,n}$ such that $\#Cl(K)[m]$ is bounded by $|Disc(K)|^{e_{m,n}+o(1)}$ for all number fields $K$ of degree $m$? Conjecturally $e_{m,n}=0$. Unconditionally, the bound of 1/2 follows directly from the Brauer-Siegel Theorem and this is the best we can do in most cases, though there are improvements in existing cases. We discuss a method to bound these quantities by re-interpreting the torsion in the class group as a finite Selmer group, and embedding it into an appropriate motive. We prove that if one assumes GRH + Refined BSD, one can prove the bound $e_{5,n} = 1/4$ independently of what $n$ is. We also show that this can be made unconditional in the function field case, where even though BSD is open, in a sense, one still has the refined BSD formula.
The method requires us to find appropriate motives with suitably trivial modular Galois representations, and one of our hopes in giving this talk is to encourage the audience to aid us in finding such motives!

Orateur: Prof. Jacob Tsimerman (University of Toronto)

Vidéo

mardi 10 octobre

09:30 Welcome coffee

5 On Geometric Construction of Artin Representation

A be a regular local ring whose residue field is of characteristic $p$. Let $G$ be a finite group which acts on $A$. Under certain condition, Serre defined an integer-valued function on $G$, and conjectured that the function is the character of a $Q_l$-rational representation of $G$. We discuss this conjecture in the equal characteristic situation.

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

Vidéo

11:00 Coffee break

6 Plectic Cohomology

I will discuss joint work with Jan Nekovár on the plectic cohomology of mixed Shimura varieties and stacks.

Orateur: Prof. Tony Scholl (University of Cambridge)

Vidéo

12:15 Lunch break

7 Uniform Bounds for Chow Groups

In this talk, we will discuss uniform bounds for the torsion subgroup in the Chow group of cycles of codimension two for families of varieties over number fields. This is joint work with F. Charles.

Orateur: Prof. Alena Pirutka (New York University)

Vidéo

15:00 Coffee break

8 The Galois Group of the Category of Mixed Hodge-Tate Structures

This talk is based on the joint work with Guangyu Zhu.

The category of rational mixed Hodge-Tate structures is canonically equivalent to the category of finite-dimensional graded comodules over a graded commutative Hopf algebra H over Q. The latter is the algebra of functions on the Galois group of the category.
Since the category has homological dimension 1, the Hopf algebra H is isomorphic to the commutative graded Hopf algebra given by the tensor algebra of the direct sum of over n>0 of C/Q(n), placed in the degree n, with the shuffle product.
However this isomorphism is not natural, e.g. does not work in families. We give a natural explicit construction of the Hopf algebra H.
Generalizing this, we define a Hopf dg-algebra, describing a dg-model of the derived category of variations of Hodge-Tate structures on a complex manifold X. Its cobar complex is a dg-model for the rational Deligne cohomology of X.
Here is an application. Periods of weight n variations of mixed Hodge-Tate structures are multivalued functions, e.g. the weight n polylogarithms. We define refined periods. They are single-valued, and take values in the tensor product of the multiplicative group of complex numbers and n-1 copies of the abelian group of complex numbers.
We also consider a p-adic variant of the construction which starts from Fontaine's crystalline / semi-stable period rings and produces graded / dg Hopf algebras, related to the p-adic Hodge theory.

Orateur: Prof. Alexander Goncharov (Yale University & IHES)

Vidéo

16:15 Coffee break

9 Kummer Extensions, with Applications to Generalized Cross-Ratios, Functions on Hilbert Schemes, and the Gross-Zagier Conjecture

Kummer extensions are extensions of Hodge structure of the form 0 -> Z(1) -> K -> Z(0) -> 0. The group of such extensions is isomorphic to C^x. I will show how to construct such extensions in two situations, one geometric and the other arithmetic. The arithmetic work is joint with Jeanine Van Order.
The geometric construction begins with a smooth projective variety P over the complex numbers. We are given algebraic cycles A and B on P. We assume A and B are homologous to 0 (Betti cohomology) and have disjoint supports. We assume further dim B=codim_P(A)-1 and H^{2codim A-1}(P)=(0). (Ex. P=P^1, A=a-a', B=b-b' disjoint 0-cycles). With these assumptions, the assumed cohomological vanishing means that the height biextension associated to the height <A,B> degenerates, yielding a Kummer extension. The extension class in C^x can be thought of as a generalized cross-ratio of A and B. In particular, the construction yields functions on the appropriate Hilbert schemes.
The arithmetic construction grows from work of A. Mellit who proved some special cases of Gross-Zagier in his thesis in 2008. One starts with a smooth, projective variety X of dimension n. One is given a motivic cohomology class in CH^p(X,1) = H^{2p-1}_M(X,Z(p)) and an algebraic cycle class in H^{2n-2p+2}(X,Z(n-p+1)). The (higher) Abel Jacobi class associated to the cycle in CH^p(X,1) corresponds to a Hodge extension 0 -> H^{2p-2}(X,Z(p)) -> V -> Z(0) -> 0. Then multiplication by the cycle class pushes out the extension to yield a Kummer extension 0 -> Z(1) -> K -> Z(0) -> 0.

Orateur: Prof. Spencer Bloch (University of Chicago)

Vidéo

18:00 Cocktail

mercredi 11 octobre

09:00 Welcome coffee

10 Eigenvarieties and Completed Cohomology

In this talk I will review some of the connections between eigenvarieties —- that is, p-adic families of finite slope automorphic eigenforms —- and ideas in locally analytic p-adic cohomology representation theory.

Orateur: Prof. Matthew Emerton (University of Chicago)

Vidéo

10:30 Coffee break

11 Sato-Tate for Bianchi Modular Forms

I'll discuss potential automorphy results which imply the Ramanujan and Sato-Tate conjectures for regular algebraic cuspidal automorphic representations of GL(2) over an imaginary quadratic field. This generalizes results in the "10 author paper" which restricted to cohomological weight 0, and is joint work with George Boxer, Frank Calegari, Toby Gee and Jack Thorne. New ingredients include an application of my work with Ana Caraiani on local-global compatibility, pieces of the cohomology of Dwork hypersurfaces previously investigated by Lie Qian, and a generic reducedness result for mod p fibres of local deformation rings.

Orateur: Prof. James Newton (University of Oxford)

Vidéo

11:45 Coffee break

12 Kolyvagin’s Conjecture and Iwasawa Theory

Let E be a rational elliptic curve and p be an odd prime of good ordinary reduction for E. In 1991 Kolyvagin conjectured that the system of cohomology classes derived from Heegner points on the p-adic Tate module of E over an imaginary quadratic field K is non-trivial. I will talk about joint work with A. Burungale, F. Castella, and C.Skinner, where we prove Kolyvagin's conjecture in the cases where an anticyclotomic Iwasawa Main Conjecture for E/K is known. Moreover, our methods also yield proof of a refinement of Kolyvagin's conjecture expressing the divisibility index of the Heegner point Kolyvagin system in terms of the Tamagawa numbers of E. One of the proof’s key ingredients, which I will focus on during the talk, is a refinement of the Kolyvagin system argument for (anti-cyclotomic) twists of E studied by Jan Nekovář.

Orateur: Prof. Giada Grossi (CNRS, University Sorbonne Paris Nord)

Vidéo

13:15 Buffet lunch

jeudi 12 octobre

09:30 Welcome coffee

13 Shimurian Generalizations of Truncated Barsotti-Tate Groups (after V. Drinfeld)

In a recent article, Vladimir Drinfeld proposed (using prismatic F-gauges) new "Shimurian" analogs of the stack of n-truncated Barsotti-Tate groups. I will give an overview of Drinfeld’s work and the relation to the prismatic Dieudonné theory of Anschütz and Le Bras, and explain some work in progress on proving algebraicity of these stacks.

Orateur: Prof. Akhil Mathew (University of Chicago)

Vidéo

11:00 Coffee break

14 Conjectures on L-functions for Varieties Over Function Fields and Their Relations

(Joint work with T. Keller, Groningen, and Y. Qin, Berkeley)

We consider versions for smooth varieties $X$ over finitely generated fields $K$ in positive characteristic p of several conjectures that can be traced back to Tate, and study their interdependence. In particular, let $A/K$ be an abelian variety. Assuming resolutions of singularities in positive characteristic, I will explain how to relate the BSD-rank conjecture for $A$ to the finiteness of the $p$-primary part of the Tate-Shafarevich group of $A$ using rigid cohomology. Furthermore, I will discuss what is needed for a generalisation.

Orateur: Prof. Veronika Ertl (Universitat Regensburg)

Vidéo

12:15 Lunch break

15 On de Rham Cohomology in Characteristic p

I will discuss two topics related to de Rham cohomology of algebraic varieties in characteristic p: (1) how the stacky approach to p-adic cohomology theories developed by Drinfeld and Bhatt-Lurie (or the approach of Ogus-Vologodsky via the sheaf of differential operators) can be thought of as equipping the de Rham complex with additional structures not explicitly visible otherwise, which have consequences such as degeneration of the Hodge-to-de Rham spectral sequence for F-split and quasi-F-split smooth varieties; (2) how the discrepancy between the Steenrod operations on de Rham and Hodge cohomology leads to examples of varieties over $F_p$ that lift to $Z_p$ but have a non-degenerate (logarithmic) Hodge-to-de Rham spectral sequence.

Orateur: Prof. Alexander Petrov (Harvard University)

Vidéo

15:00 Coffee break

16 On the Cohomology of Shimura Varieties with Torsion Coefficients

I will survey results concerning the cohomology of Shimura varieties with torsion coefficients from the past few years. I will discuss the geometry of the Hodge-Tate period morphism, including a recent generalization of Igusa varieties to Igusa stacks due to Mingjia Zhang. Then I will contrast the original approach of computing cohomology with torsion coefficients due to myself and Peter Scholze, which relies on the trace formula, with more recent approaches due to Teruhisa Koshikawa, Linus Hamann, and Si Ying Lee, who rely on deep local results. Finally, I will explain how, by combining the two approaches, one can obtain a new instance of local-global compatibility.

Orateur: Prof. Ana Caraiani (Imperial College London)

Vidéo

16:15 Coffee break

17 Hidden Symmetries of the Work of Jan Nekovář

I will give an outline of the research work of Jan Nekovář, emphasizing the conceptual principles that Jan pursued throughout his career and which unify his contributions, from the constructions of Euler systems to parity results for L-functions of automorphic representations of $GL2$ to the study of Galois action on quaternionic Shimura varieties.

Orateur: Prof. Olivier Fouquet (Université de Franche-Comté)

Vidéo

vendredi 13 octobre

09:00 Coffee break

18 Period Relations for Arithmetic Automorphic Periods on Unitary Groups

Given an automorphic representation of a unitary group, one can define an arithmetic automorphic period as the Petersson inner product of a deRham rational form. Here the deRham rational structure comes from the cohomology of Shimura varieties. When the form is holomorphic, the period can be related to special values of L-functions and is better understood. In this talk, we formulate a conjecture on relations among general arithmetic periods of representations in the same L-packet and explain a conditional proof.

Orateur: Prof. Jie Lin (Universität Duisburg_Essen)

Vidéo

10:30 Coffee break

19 Square roots from $GSp_4$ or $GL(2) x GL(2)$

"If the formal square root of an abelian surface over Q looks exceedingly like an elliptic curve, it has to be an elliptic curve."
We discuss what such a proposition might mean, and prove the most straightforward version where the precise condition is simply that the L-function of the abelian surface possesses an entire holomorphic square root. The approach follows the Diophantine principle that algebraic numbers or zeros of L-functions repel each other, and is in some sense similar in spirit to the Gelfond– Linnik–Baker solution of the class number one problem.
We discuss furthermore this latter connection: the problems that it raises under a hypothetical presence of Siegel zeros, and a proven analog over finite fields. The basic remark that underlies and motivates these researches is the well-known principle (which is a consequence of the Deuring–Heilbronn phenomenon, to be taken with suitable automorphic forms f and g): an exceptional character $\chi$ would cause the formal
$\sqrt{L(s,f)L(s,f\bigotimes_{}\chi)}L(s,g)L(s,g\bigotimes\chi)$ to have a holomorphic branch on an abnormally big part of the complex plane, all the while enjoying a Dirichlet series formal expansion with almost-integer coefficients. This leads to the kind of situation oftentimes amenable to arithmetic algebraization methods. The most basic (qualitative) form of our main tool is what we are calling the "integral converse theorem for $GL(2)$," and it is a refinement of the unbounded denominators conjecture on noncongruence modular forms.
This talk will be partly based on a joint work with Frank Calegari and Yunqing Tang.

Orateur: Prof. Vesselin Dimitrov (Institute for Advanced Study)

Vidéo

11:45 Coffee break

20 Mock Plectic Points (joint work with Henri Darmon)

The Ihara group acts by Möbius transformations on the Poincaré upper half plane and on Drinfeld's p-adic counterpart and acts discretely on the product. The resulting quotient can be envisaged as a “mock Hilbert modular surface”, following a suggestive terminology of Barry Mazur. This leads to a fruitful dictionary in which Oda’s conjecture on periods of Hilbert modular surfaces corresponds to the exceptional zero conjecture of Mazur-Tate Teitelbaum proved by Greenberg-Stevens, and where complex ATR points, conjecturally defined over class fields of somewhat complicated reflex fields, correspond to Stark-Heegner points over ring class fields of real quadratic fields.

According to a striking prediction of Nekovar and Scholl, the CM points on genuine Hilbert modular surfaces should give rise to “plectic Heegner points” that encode non-trivial regulators attached, notably, to elliptic curves of rank two over real quadratic fields. I will describe the analogy between Hilbert modular surfaces and their mock counterparts, with the aim of transposing the plectic philosophy to the mock Hilbert setting, where the analogous plectic invariants are expected to lie in the alternating square of the Mordell–Weil group of certain elliptic curves of rank two over Q. The consistency of this prediction with an “anti-cyclotomic exceptional zero conjecture” that Massimo Bertolini and I formulated in the previous century may provide some oblique evidence for the Nekovar-Scholl philosophy.

Orateur: Prof. Michele Fornea (CRM Barcelona)

Vidéo

13:15 Buffet lunch