Arithmetic and Diophantine Geometry, via Ergodic Theory and o-minimality

8 sept. 2025, 03:15 → 12 sept. 2025, 16:30 Europe/Paris

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

Arithmetic and Diophantine Geometry, via Ergodic Theory and o-minimality    
A Conference in Honor of Emmanuel Ullmo's 60th Birthday    
September 8-12 2025 at IHES - Marilyn and James Simons Conference Center    
How to get to IHES

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This conference aims to honor the spectacular contributions of Emmanuel Ullmo to arithmetic geometry and centers around his mathematical contributions and interests. It will cover and highlight recent stage of development on topics such as Diophantine geometry, ergodic theory, Hodge theory, and arithmetic dynamics.

 Registration deadline: August 31, 2025

Speakers:

• Fabrizio Andreatta (Univ. degli Studi di Milano)
• Uri Bader (Weizmann Inst. & Univ. of Maryland)
• Benjamin Bakker (Univ. of Illinois, Chicago)
• Gregorio Baldi (CNRS, IMJ-PRG)
• Yves Benoist (CNRS, Univ. Paris-Saclay)
• Anna Cadoret (IMJ-PRG, Sorbonne Univ.)
• François Charles (ENS Paris)
• Laura DeMarco (Harvard Univ.)
• David Fisher (Rice Univ.)
• Javier Fresán (IMJ-PRG, Sorbonne Univ.)
• Philipp Habegger (Univ. of Basel)
• Philippe Michel (EPFL)
• Hee Oh (Yale Univ.)
• Naomi Sweeting (Princeton Univ.)
• Yunqing Tang (Caltech & UC Berkeley)
• David Urbanik (IHES)
• Xinyi Yuan (BICMR, Peking Univ.)
• Mingjia Zhang (IAS & Princeton Univ.)
• Shouwu Zhang (Princeton Univ.)
• Wei Zhang (MIT)
  
  
  

 

Organizing committee:     
Ahmed Abbes (CNRS, IHES), Jennifer Balakrishnan (Boston Univ.), Ziyang Gao (UCLA), Marc Hindry (Univ. Paris-Cité), Fanny Kassel (CNRS, IHES), Bruno Klingler (Univ. Humboldt zu Berlin), Yuri Tschinkel (NYU and Simons Foundation)

 

FINAL_POSTER_ULLMO_60.pdf

lun. 8 septembre

09:15 → 12:15 Morning Chair: Hélène Esnault

09:15 Registration and welcome coffee

09:55 Opening address

10:00 The Arithmetic of Power Series and Applications to Irrationality

We will discuss a new approach to prove irrationality of certain periods, including the value at 2 of the Dirichlet L-function associated to the primitive quadratic character with conductor -3. Our method uses rational approximations from the literature and we develop a new framework to make use of these approximations. The key ingredient is an arithmetic holonomy theorem built upon earlier work by André, Bost, Charles (and others) on arithmetic algebraization theorems via Arakelov theory. This is joint work with Frank Calegari and Vesselin Dimitrov.

Orateur: Yunqing Tang (Caltech & UC Berkeley)

Vidéo

11:00 Coffee break

11:15 Integral Points and Affineness in Arakelov Geometry

I will describe results old and new regarding finiteness and infiniteness of integral points on arithmetic schemes, subject to analytic conditions. I will explain how the language of A-schemes and the use of theta-invariants, as developed in joint work with Bost, provides a geometric framework for this kind of questions, and how it relates to notions of affineness in Arakelov geometry.

Orateur: François Charles (ENS Paris)

Vidéo

12:15 → 14:00 Lunch

14:00 → 16:25 Afternoon Chair: Ricardo Menares

14:00 Infinitely Many Non-Hypergeometric Local Systems

The Bombieri-Dwork conjecture predicts that an irreducible differential operator with a G-function solution comes from geometry, that is, encodes how periods vary in a pencil of algebraic varieties. This conjecture is completely open for operators of order at least 2. At the beginning of the 90s, Dwork proposed a strategy to establish the conjecture for G-operators of order 2, which would consist in proving that they are all pullbacks by a correspondence of some Gauss's hypergeometric differential operator. Sporadic counterexamples to this expectation were found by Kraamer (1996) and Bouw-Möller (2010). I will present a joint work with Josh Lam and Yichen Qin where we prove that most G-operators of order 2 coming from geometry are not pullbacks of hypergeometric differential operators. A key ingredient to construct infinitely many counterexamples will be a theme dear to Emmanuel: the André-Pink-Zannier conjecture for Shimura varieties, in the cases recently established by Richard and Yafaev.

Orateur: Javier Fresán (IMJ-PRG, Sorbonne Univ.)

Vidéo

15:00 Break

15:15 Intersection Cohomology of Shimura Varieties

The intersection cohomology of Shimura varieties is related to automorphic representations through the comparison with L^2-cohomology and the work of Borel-Casselman. I will explain joint work in progress with Ana Caraiani and Linus Hamann, where we study the intersection cohomology of Shimura varieties through Igusa stacks.

Orateur: Mingjia Zhang (IAS & Princeton Univ.)

mar. 9 septembre

09:30 → 12:15 Morning Chair: Laurent Clozel

09:30 Welcome Coffee

10:00 The Dynamical Schinzel-Zassenhaus Conjecture and the Transfinite Diameter of Trees

In 2019, Dimitrov proved the Schinzel-Zassenhaus Conjecture. Harry Schmidt and I extended his general strategy to cover initial first dynamical variants of this conjecture. One common tool in both results is Dubinin's Theorem on the transfinite diameter of a hedgehog, which is a star-shaped tree in the plane.

In this talk, I will report on joint work in progress with Schmidt. We find new upper bounds for the transfinite diameter of finite topological trees. These trees are constructed using the Hubbard tree of a postcritically finite polynomial and reflect its dynamical properties. As a consequence, we can prove lower bounds for the Call-Silverman (or canonical) height for a class of postcritically finite polynomials.

Orateur: Philipp Habegger (Univ. of Basel)

Vidéo

11:00 Break

11:15 Elliptic Surfaces, Equidistribution, and Bifurcations

In joint work with Mavraki a few years ago, we studied the arithmetic intersection numbers of sections of elliptic surfaces, defined over number fields. One consequence was a Bogomolov-type extension (and new proof) of a theorem of Barroero and Capuano, addressing a case of the Zilber Pink conjectures. I will describe the underlying geometric features of this problem and formulate related open questions about families of maps (dynamical systems) on $P^1$.

Orateur: Laura DeMarco (Harvard Univ.)

Vidéo

12:15 → 14:00 Lunch

14:00 → 17:45 Afternoon Chair: Pascal Autissier

14:00 Baily–Borel Compactifications of Period Images and the b-Semiampleness Conjecture

We address two questions related to the semiampleness of line bundles arising from Hodge theory. First, we prove there is a functorial compactification of the image of a period map of a polarizable integral pure variation of Hodge structures for which a natural line bundle extends amply. This generalizes the Baily--Borel compactification of a Shimura variety, and for instance produces Baily--Borel type compactifications of moduli spaces of Calabi--Yau varieties. We prove more generally that the Hodge bundle of a Calabi--Yau variation of Hodge structures is semiample subject to some extra conditions, and as our second result deduce the b-semiampleness conjecture of Prokhorov--Shokurov. The semiampleness results crucially use o-minimal GAGA, and the deduction of the b-semiampleness conjecture uses work of Ambro and results of Koll\'ar on minimal lc centers to verify the extra conditions geometrically. This is joint work with S. Filipazzi, M. Mauri, and J. Tsimerman.

Orateur: Benjamin Bakker (Univ. of Illinois, Chicago)

Vidéo

15:00 Break

15:15 Applications of Unlikely Intersections to Integral Geometry

We explain how a theory of infinitesimal period maps can be used to "transfer" Hodge-theoretic unlikely intersection results from characteristic zero to positive characteristic for sufficiently large primes p. We survey several results of this type and give an idea of the methods used to prove them. Partially joint work with Ziquan Yang.

Orateur: David Urbanik (IHES)

Vidéo

16:15 Break

16:45 A Tribute to Emmanuel's Contributions to the Zilber–Pink Conjecture

We will survey several of Emmanuel’s influential results over the past two decades concerning the André–Oort conjecture and the broader Zilber–Pink conjecture. Beginning with the framework of Shimura varieties, we will gradually transition toward more general settings involving arbitrary variations of Hodge structures. This shift marks a movement from the arithmetic study of CM points to a more geometric perspective. We will highlight some of the consequences of this viewpoint, including results obtained through various collaborations with Emmanuel in the past five years.

Orateur: Gregorio Baldi (IHES)

Vidéo

18:00 → 19:00 Cocktail party

mer. 10 septembre

09:00 → 15:00 Morning Chair: Martin Orr

09:00 Welcome Coffee

09:30 Determinant Values on Irrational Lattices

We study the value-distribution problem of det on an irrational lattice $L < M_n(\mathbb R)$: how are the values of det on L distributed on $\mathbb R$? In a recent joint work in progress with Wooyeon Kim, we obtain quantitative results toward this question; for n=2, this amounts to a quantitative version of the Oppenheim conjecture for quadratic forms of signature (2,2), as studied by Eskin-Margulis-Mozes (2005).

Orateur: Hee Oh (Yale Univ.)

Vidéo

10:30 Break

10:50 Higher Property T, Banach Representations and Applications

Gromov conjectured that the $L^p$-cohomology of simple groups vanishes below the rank.
Farb conjectured a fixed point property for actions of lattices in such groups on CAT(0) cell complexes of dimension lower than the rank.
Both conjectures follow from a new cohomological vanishing result, which could be seen as a Banach version of higher property T.
In my talk I will survey the subject and explain the new contribution.
Based on a joint work with Saar Bader, Shaked Bader and Roman Sauer.

Orateur: Uri Bader (Weizmann Inst. & Univ. of Maryland)

Vidéo

11:50 Break

12:00 Convolution and Square on Abelian Groups

We will construct functions on finite abelian groups whose convolution square is proportional to their square. For that, we will interpret the abelian group as a subgroup of an abelian variety with complex multiplication, and use the modularity properties of their theta functions.

Orateur: Yves Benoist (CNRS, Univ. Paris-Saclay)

Vidéo

13:00 → 14:00 Lunch

jeu. 11 septembre

09:30 → 12:15 Morning Chair: Umberto Zannier

09:30 Welcome Coffee

10:00 A Quantitative Version of the Uniform Mordell Conjecture

The celebrated Mordell conjecture proved by Faltings asserts that the number of rational points on a curve of genus greater than one over a number field is finite. A deep uniform upper bound on the number of rational points follows from Vojta's inequality and the recent works of Dimitrov-Gao-Habegger and Kuhne. In this talk, I will introduce an explicit version of this uniform bound. This is a joint work with Jiawei Yu and Shengxuan Zhou.

Orateur: Xinyi Yuan (BICMR, Peking Univ.)

Vidéo

11:00 Break

11:15 Heights of Ceresa and Gross—Schoen Cycles

In this talk, I will survey recent work on the heights of Gross—Schoen and Ceresa cycles, as well as their relation to the triple product L-series.

Orateur: Shouwu Zhang (Princeton Univ.)

Vidéo

12:15 → 14:00 Lunch

14:00 → 16:15 Afternoon Chair: Jonathan Pila

14:00 Proportionality and the Arithmetic Volumes of Shimura Varieties and the Moduli of Shtukas

The volume of a locally symmetric space is essentially a product of special values of zeta functions. More generally, Hirzebruch's proportionality theorem (extended by Mumford) tells us how to integrate any Chern class polynomial on a locally symmetric space. We prove an analog and extension of these results to function fields, where locally symmetric spaces will be replaced by the moduli space of Drinfeld Shtukas with multiple legs, and special values of zeta functions will be replaced by a linear combination of their derivatives of various order. Over number fields, we formulate a conjecture and prove it in a couple of new cases, relating the arithmetic volume, defined by the Arakelov degree of certain Hermitian metrized line bundle, to the special values of derivatives of suitable Artin L-functions.
This is joint work with Tony Feng and Zhiwei Yun

Orateur: Wei Zhang (MIT)

Vidéo

15:00 Break

15:15 On the Bloch–Kato Conjecture for Some Four-Dimensional Symplectic Galois Representations

The Bloch–Kato Conjecture predicts a relation between Selmer ranks and orders of vanishing of L-functions for Galois representations arising from etale cohomology of algebraic varieties. In this talk, I’ll describe results towards this conjecture in ranks 0 and 1 for the self-dual Galois representations that come from Siegel modular forms on GSp(4) with parallel weight (3, 3); these contribute to cohomology of classical Siegel threefolds. The key step in the proof is a construction of auxiliary ramified Galois cohomology classes, which then give bounds on Selmer groups. The ramified classes come from level-raising congruences and the geometry of special cycles on Shimura varieties.

Orateur: Naomi Sweeting (Princeton Univ.)

Vidéo

ven. 12 septembre

09:00 → 13:00 Morning Chair: Chris Daw

09:00 Welcome coffee

09:30 On Two mod p Period Maps: Ekedahl–Oort and Fine Deligne–Lusztig Stratifications

Consider a Shimura variety of Hodge type admitting a smooth integral model S at an odd prime $p > 3$. Consider its perfectoid cover $S(p^\infty)$ and the Hodge-Tate period map introduced by A. Caraiani and P. Scholze. We compare the pull-back to $S(p^\infty)$ of the Ekedahl-Oort stratification on the mod p special fiber of S and the pull back to $S(p^\infty)$ of the fine Deligne-Lusztig stratification on the mod p special fiber of the flag variety which is the target of the Hodge-Tate period map. If time allowa, an application to the non-emptiness of Ekedhal-Oort strata is provided.

Orateur: Fabrizio Andreatta (Univ. degli Studi di Milano)

Vidéo

10:30 Break

10:50 Tate Locus - Conjectures and Results

Let k be a field and X a geometrically connected variety over k. The Tate or degeneracy locus of a l-adic local system on X is the etale counterpart of the Hodge locus of a VHS. While in the last decade tremendous progresses have been made in understanding the latter thanks to, in particular, techniques from o-minimality, much less is known about the former. I will review the main conjectures (and mention briefly some applications) about this locus when k is a number field, and explain what we can currently prove. This should include joint works with Jakob Stix and Akio Tamagawa.

Orateur: Anna Cadoret (IMJ-PRG, Sorbonne Univ.)

Vidéo

11:50 Break

12:00 A Split Version of the Mixing Conjecture and Applications

The mixing conjecture was proposed by Venkatesh and myself 20 years ago and postulates that pair of CM-points multiplicatively connected equidistribute on products of locally homogeneous space associated to forms of $PGL_2$.
It was established by Khayutin, for sequences of fundamental discriminant splitting two given primes and under the assumption that there is no Landau-siegel zero,using measure classification results of Einsieldler-Lindenstrauss. There are currently promising efforts by Blomer, Brumley and Radziwill to remove the splitting conditions and weaken the Landau-Siegel zero assumption using purely methods from analytic number theory.

The split version (for the split quadratic algebra) concerns the distribution of multiplicatively pairs of Hecke points of large modulus.
It has now been established unconditionally by Blomer and myself in the prime modulus case and by Assing in general.

In this lecture, I will discuss the proof of (of the split version of) the mixing conjecture as well as some recent application of the methods involved to the generation of Hecke fields by algebraic L-values (joint work with Blomer, Burungale and Min).

Orateur: Philippe Michel (École Polytechnique Fédérale de Lausanne (EPFL))

Vidéo

13:00 → 14:00 Lunch