# Algebraic, Complex and Arithmetic Dynamics (2019)

Date & Time

Organizers:
Laura DeMarco, Northwestern University
Mattias Jonsson, University of Michigan

In this symposium series, we aim to formulate a vision for future developments in complex, algebraic and arithmetic dynamics. The theme of this first symposium was Heights and Complexity in Algebraic Dynamics. We discussed arithmetic and algebraic features of dynamics on algebraic varieties. Important questions relate to the values and computations of dynamical degrees and heights; intersection (and unlikely intersection) problems in families of maps; and uniform bounds on dynamically  distinguished subvarieties, independent of the map. The specific goals of the week were to (1) pose questions, (2) determine where we are today with the already-existing big questions in the field, and (3) present the guiding principles and new methods that underlie our approaches to these problems. There were 12 research talks, one overview talk, two problem sessions and a reference collection session.

• Report

#### Research talks

Jason Bell gave a talk with the title “Some dynamical problems motivated by questions in noncommutative algebra,” illustrating that problems in algebraic dynamics sometimes arise from unexpected areas of mathematics. Results about abelian varieties have given rise to many interesting conjectures in arithmetic and algebraic dynamics. Charles Favre explained joint work with Thomas Gauthier on a version of the dynamical André-Oort conjecture, as formulated by Baker and DeMarco. The title was “Heights of polynomial dynamical pairs.” K3 surfaces play an important role in both geometry and dynamics. In his talk “Dynamics on K3s: Berkovich and tropical versions,” Simion Filip explained a non-Archimedean versions of results on the complex dynamics on K3 surfaces. Holly Krieger explained joint work with DeMarco and Ye, in which they use the Arakelov-Zhang height pairing and quantitative equidistribution results to give uniform bounds on the number of common preperiodic points for certain families of rational maps. The title was “The Arakelov-Zhang pairing and dynamical heights.” In his talk “Numerical dimension revisited,” John Lesieutre explained how dynamics can be used to construct divisors with weak positivity properties that in fact give counterexamples to conjectures (and theorems) in birational geometry. Nicole Looper gave a talk, “Uniform boundedness, equidistribution and the arithmetic of dynamically small points,” explaining how celebrated conjectures in Diophantine geometry implies conjectures in arithmetic dynamics, such as the uniform boundedness conjecture by Morton and Silverman for unicritical polynomials. The Kawaguchi-Silverman conjecture predicts a relation between the growth of degrees and heights under iteration. In his talk “Some topics from Kawaguchi-Silverman conjecture,” Yohsuke Matsuzawa described recent progress. Curt McMullen gave a talk with the title “Billiards and the arithmetic of nonarithmetic groups,” in which he presented work on the Hecke groups G_n, with particular emphasis on the (nonarithmetic) group G_5, the matrices this contains, and its connection to billiards.

Keiji Oguiso explained work analyzing whether or not the automorphism group of a smooth projective variety is finitely generated. The title was “Inertia groups, decomposition groups and smooth projective varieties with nonfinite generated automorphism groups.” In his talk, “The Betti foliation, the canonical height and the geometric Bogomolov conjecture,” Junyi Xie explained recent joint work with Cantat, Gao and Habegger, in which they prove the geometric Bogomolov conjecture over (possibly higher-dimensional) function fields of characteristic zero. The equivariant Minimal Model Program aims to classify (weakly) polarized endomorphisms of projective varieties. De-Qi Zhang gave a talk titled “Equivariant Minimal Model Program, with a view toward algebraic and arithmetic dynamics” in which he discussed recent progress, together with some applications. The height pairing on curves introduced by Shouwu Zhang has played a crucial role in arithmetic dynamics and other fields. In his talk, “Admissible pairing of algebraic cycles,” Zhang proposed a higher-dimensional generalization, together with dynamical applications.

#### Other talks and activities

Joe Silverman gave an overview of what is known about the growth of heights and degrees under iteration of a rational self-map of a normal projective variety defined over a number field. This talk served as natural starting point for the talks of Matsuzawa and D.Q. Zhang.

We organized two problem sessions. A wide range of open problems and possible new directions were proposed at these sessions. We will type up notes and post them to the organizers’ webpage.

In order to take stock of the current state of knowledge and to help researchers in the future, we organized a session where the participants collected references, detailing progress on a number of important problems and conjectures, such as dynamical degree computations, the dynamical Manin-Mumford problem or the dynamical Mordell-Lang problem. We assigned a group of participants to each problem and asked them to write down a list of key references with a very brief description of the content for each one.

Finally, the format of the symposium gave ample opportunities for informal discussions, explanations of technical points as well as philosophical ideas. These kinds of interactions often form the embryo of future research, and we are confident that many new results and collaboration will result from this first meeting in the symposium series.

• Agenda & Slides/Notes

#### MONDAY

 10:00 - 11:00 AM Charles Favre | Heights of Polynomial Dynamical Pairs View Notes (PDF) 11:30 - 12:30 PM Joseph Silverman | Dynamical Degrees and Arithmetic Degrees: History, Conjectures and Future DirectionsView Slides (PDF) 5:00 - 6:00 PM Holly Krieger | The Arakelov-Zhang Pairing and Dynamical Heights View Notes (PDF) 6:15 - 7:15 PM Yohsuke Matsuzawa | Some Topics from Kawaguchi-Silverman Conjecture View Notes (PDF)

#### TUESDAY

 10:00 - 11:00 AM Curtis McMullen | Billiards and the Arithmetic of Non-Arithmetic GroupsView Slides (PDF) 11:30 - 12:30 PM Simion Filip | Dynamics on K3s: Berkovich and Tropical Versions View Notes (PDF) 5:00 - 6:00 PM Junyi Xie | The Betti Foliation, the Canonical Height and the Geometric Bogomolov Conjecture View Notes (PDF) 6:15 - 7:15 PM Problem Session

#### WEDNESDAY

 10:00 - 11:00 AM Shou-Wu Zhang | Admissible Pairing of Algebraic Cycles View Notes (PDF) 11:30 - 12:30 PM Jason Bell | Some Dynamical Problems Motivated by Questions in Noncommutative Algebra View Notes (PDF) 5:00 - 6:00 PM John Lesieutre | Numerical Dimension Revisited View Notes (PDF) 6:15 - 7:15 PM Nicole Looper | Uniform Boundedness, Equidistribution and the Arithmetic of Dynamically Small PointsView Slides (PDF) 8:30 - 9:30 PM Concert: Messiaen: Quatuor pour la fin du temps

#### THURSDAY

 9:45 - 2:00 PM Guided Hike to Partnach Gorge 5:00 - 6:00 PM Ad Hoc Talk 6:15 - 7:15 PM Reference Collection

#### FRIDAY

 10:00 - 11:00 AM De-Qi Zhang | Equivariant Minimal Model Program with a View Towards Algebraic and Arithmetic DynamicsView Slides (PDF) 11:30 - 12:30 PM Keiji Oguiso | Inertia Groups, Decomposition Groups and Smooth Projective Varieties with Nonfinite Generated Automorphism Groups View Notes (PDF) 5:00 - 6:00 PM Problem Session 6:15 - 7:15 PM Matt Baker | On Some Unlikely Coincidences
• Abstracts

Jason Bell
University of Waterloo

Some Dynamical Problems Motivated by Questions in Noncommutative Algebra

Bell will give an overview of some of the dynamical questions that arise when studying the representation theory of algebras in noncommutative projective geometry, highlighting some of the results already obtained in this direction.

Charles Favre
École Polytechnique

Heights of Polynomial Dynamical Pairs

Favre will discuss some of our progress with Thomas Gauthier in the problem of unlikely intersection in polynomial dynamics (over a one-dimensional base defined over a number field). Our results lead to further insights into the dynamical Andre-Oort conjecture.

Simon Filip
Institute for Advanced Study & Clay Mathematics Institute

Dynamics on K3s: Berkovich and Tropical Versions

Filip will start by recalling the basic facts about algebraic automorphisms of K3 surfaces. He will then explain how to extend some of the results to the non-archimedean setting and discuss the resulting dynamical systems. The interesting part of the dynamics has an explicit, elementary description in terms of tropical geometry. Filip will end with some questions and discuss possible applications.

Holly Krieger
University of Cambridge

The Arakelov-Zhang Pairing and Dynamical Heights

For any two rational maps of the Riemann sphere with algebraic coefficients, the Arakelov-Zhang pairing of their canonical heights provides an arithmetic measure of the dynamical distance between the two maps. Krieger will discuss how this pairing can be used, together with quantitative equidistribution, to provide bounds on points of small height for both maps, as done in recent joint work with DeMarco and Ye. Krieger will highlight some of the many open questions about the behavior of this pairing in moduli.

John Lesieutre
Pennsylvania State University

Numerical Dimension Revisited

The Iitaka dimension of a line bundle $$D$$ on a projective variety $$X$$ is the dimension of the image of the rational map given by $$|mD|$$ for large and divisible $$m$$. The Iitaka dimension is not a numerical invariant of $$D$$, and there are several approaches to constructing a “numerical dimension,” which should be an analogous invariant depending only on the numerical class of $$D$$. Lesieutre will discuss some divisors of dynamical origin whose behavior with respect to these invariants is pathological and which provide counterexamples to some conjectures from birational geometry. The examples hinge on a sort of dynamical positivity property, which also arises in arithmetic contexts. Lesieutre will then pose some related problems about degree growth on varieties with large groups of pseudoautomorphisms.

Nicole Looper
University of Cambridge

Uniform Boundedness, Equidistribution and the Arithmetic of Dynamically Small Points

In this talk, Looper will discuss a uniform boundedness theorem for unicritical, along with the relevant tools from, Diophantine geometry. Looper will also discuss connections to other results concerning points of small canonical height relative to polynomials.

Yohsuke Matsuzawa
University of Tokyo

Some Topics from Kawaguchi-Silverman Conjecture

Matsuzawa will talk about some topics from Kawaguchi-Silverman conjecture, which asserts that arithmetic degrees of Zariski dense orbits under self-rational maps are equal to the dynamical degree of the map. The situation is completely different whether the self-rational map is a self-morphism of a projective variety or not. Tools from birational geometry are very helpful for self-morphisms, but it seems these are not sufficient to understand the arithmetic of self-rational maps. We have to know the behavior of the height function associated with the indeterminacy locus of the self-map. Matsuzawa will discuss recent progress on the conjecture for self-morphisms of projective varieties, and also mention algebraically stable self-rational maps.

Curtis McMullen
Harvard University

Billiards and the Arithmetic of Non-Arithmetic Groups

The classical Hecke groups $$G_n$$ in $$SL_2(R)$$, also known as the (2,n,infinity) triangle groups, are non-arithmetic for most n: no simply criterion is known for describing the matrices they contain. McMullen will discuss new insights into these groups arising from their connection with billiards in polygons and totally geodesic curves in moduli space.

Keiji Oguiso
University of Tokyo

Inertia Groups, Decomposition Groups and Smooth Projective Varieties with Nonfinite Generated Automorphism Groups

The so-called ‘Coble problem’ concerning complexities of the decomposition group and inertia group of the special smooth rational curve on a classic complex Coble surface is a long-standing problem which is still open. In this talk, Oguiso will attempt to negatively answer another long-standing problem — the finite generation problem of the automorphism group of a smooth projective variety of any dimension $$\ge 2$$ over an algebraically closed field, under the assumption that the base field is not an algebraic closure of a finite field or not of characteristic $$2$$. In our construction, the decomposition group of a smooth rational curve of some special K3 surface, which is closely related to classical Coble surfaces, and the arithmetic of the base field play essential and delicate roles, as Oguiso will explain. It also turns out that the inertia group in our construction has rich complex dynamics. This talk is partly based on Oguiso’s joint work with Coung-Tien Dinh and is also much inspired by the works of Lesieutre and Dolgachev.

Joseph Silverman
Brown University

Dynamical Degrees and Arithmetic Degrees: History, Conjectures and Future Directions

Silverman will give an overview of various quantities that are used to measure the complexity of an algebraic dynamical system $$f: X\to X$$, including the dynamical degree $$\delta(f)$$, which gives a coarse measure of the geometric complexity of the iterates of $$f$$, the arithmetic degree $$\alpha(f,P)$$, which gives a coarse measure of the arithmetic complexity of the orbit of $$P\in{X}(\overline{\mathbb{Q}})$$, and various versions of the canonical height $$\hat{h}_f(P)$$ that provide more refined measures of arithmetic complexity. Emphasis will be placed on open problems and directions for exploration.

Junyi Xie
Université de Rennes

The Betti Foliation, the Canonical Height and the Geometric Bogomolov Conjecture

With Cantat, Habegger and Gao, Xie will prove the geometric Bogomolov conjecture over a function field of characteristic zero. This generalizes recent work of Habegger and Gao, who proved the geometric Bogomolov conjecture over a function field of a curve of characteristic zero.

De-Qi Zhang
National University of Singapore

Equivariant Minimal Model Program with a View Toward Algebraic and Arithmetic Dynamics

Zhang will elaborate the notion of ‘int-amplified’ endomorphism f of a normal projective variety X, a property weaker than ‘polarized’ yet preserved by products. Zhang will show that the existence of such a single f guarantees that every Minimal Model Program (MMP) is equivariant w.r.t. a finite-index submonoid of the whole monoid SEnd(X) of all surjective endomorphisms of X. Applications of the equivariant MMP are discussed: Kawaguchi-Silverman conjecture on the equivalence of arithmetic and dynamic degrees, and characterization of a subvariety with Zariski dense periodic points. Some parts are based on joint work with Cascini and Meng.

Shou-Wu Zhang
Princeton University