# Algebraic, Complex and Arithmetic Dynamics

Date & Time

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

Organized by Laura DeMarco (Northwestern University) and Mattias Jonsson (University of Michigan), the goal of this symposium is to formulate a vision for future developments in complex, algebraic, and arithmetic dynamics. One concrete objective is to produce high quality expository material presenting the methods, recent results, and goals of these active research areas. Another is to identify key problems and directions of research connected with the study of height functions and complexity growth in algebraic dynamical systems.

Discussion topics include:

• Degree growth of rational maps.
• The growth of heights along orbits.
• Height functions on parameter spaces.

• Agenda & Slides

#### SUNDAY

 8:30 - 9:30 PM Dinner at La Salle

#### MONDAY

 7:30 - 9:45 AM Breakfast at La Salle 10:00 - 11:00 AM Charles Favre | Heights of Polynomial Dynamical Pairs 11:00-11:30 AM Break 11:30 - 12:30 PM Joseph Silverman | Dynamical Degrees and Arithmetic Degrees: History, Conjectures and Future DirectionsView Slides 12:30 - 1:30 PM Lunch at La Salle 1:30 - 4:30 PM Discussion & Recreation* 4:30- 5:00 PM Tea 5:00 - 6:00 PM Holly Krieger | The Arakelov-Zhang Pairing and Dynamical Heights 6:00 - 6:15 PM Break 6:15 - 7:15 PM Yohsuke Matsuzawa | Some Topics from Kawaguchi-Silverman Conjecture 8:00 - 9:30 PM Dinner: Kaminstüberl Fondue Restaurant

#### TUESDAY

 7:30 - 9:45 AM Breakfast at La Salle 10:00 - 11:00 AM Curtis McMullen | Billiards and the Arithmetic of Non-Arithmetic GroupsView Slides 11:00-11:30 AM Break 11:30 - 12:30 PM Simion Filip | Dynamics on K3s: Berkovich and Tropical Versions 12:30 - 1:30 PM Lunch at La Salle 1:30 - 4:30 PM Discussion & Recreation* 4:30- 5:00 PM Tea 5:00 - 6:00 PM Junyi Xie | The Betti Foliation, the Canonical Height and the Geometric Bogomolov Conjecture 6:00 - 6:15 PM Break 6:15 - 7:15 PM Problem Session 8:00 - 9:30 PM Dinner

#### WEDNESDAY

 7:30 - 9:45 AM Breakfast at La Salle 10:00 - 11:00 AM Shou-Wu Zhang | Admissible Pairing of Algebraic Cycles 11:00 - 11:30 AM Break 11:30 - 12:30 PM Jason Bell | Some Dynamical Problems Motivated by Questions in Noncommutative Algebra 12:30 - 1:30 PM Lunch at La Salle 1:30 - 4:30 PM Discussion & Recreation* 4:30- 5:00 PM Tea 5:00 - 6:00 PM John Lesieutre | Numerical Dimension Revisited 6:00 - 6:15 PM Break 6:15 - 7:15 PM Nicole Looper | Uniform Boundedness, Equidistribution and the Arithmetic of Dynamically Small PointsView Slides 7:30 - 8:15 PM Dinner 8:30 - 9:30 PM Concert: Messiaen: Quatuor pour la fin du temps

#### THURSDAY

 7:30 - 9:30 AM Breakfast 9:45 - 2:00 PM Guided Hike to Partnach Gorge 2:00 - 3:00 PM Lunch at Ganesha Restaurant in the Retreat 3:00 - 4:30 PM Recreation & Discussion 4:30 - 5:00 PM Tea 5:00 - 6:00 PM Ad Hoc Talk 6:00 - 6:15 PM Break 6:15 - 7:15 PM Reference Collection 8:00 - 9:00 PM Dinner

#### FRIDAY

 7:30 - 9:45 AM Breakfast at La Salle 10:00 - 11:00 AM De-Qi Zhang | Equivariant Minimal Model Program with a View Towards Algebraic and Arithmetic DynamicsView Slides 11:00-11:30 AM Break 11:30 - 12:30 PM Keiji Oguiso | Inertia Groups, Decomposition Groups and Smooth Projective Varieties with Nonfinite Generated Automorphism Groups 12:30 - 1:30 PM Lunch at La Salle 1:30 - 4:30 PM Discussion & Recreation* 4:30- 5:00 PM Tea 5:00 - 6:00 PM Problem Session 6:00 - 6:15 PM Break 6:15 - 7:15 PM Matt Baker | On Some Unlikely Coincidences 8:00 - 9:30 PM Dinner: Ganesha Restaurant in the Retreat

#### LOCATIONS

 SESSIONS Pavilion located at the Schloss Elmau Retreat MEALS La Salle unless otherwise noted TEA & DISCUSSION Pavilion located at the Schloss Elmau Retreat EXCURSION Meet in Schloss Elmau Lobby SATURDAY DEPARTURE Meet in Schloss Elmau Lobby

*Participants may explore the hotel property and its surrounding areas as well as engage in informal discussion with other participants.

• 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

For a smooth and projective variety X over a global field of dimension n with an adelic polarization, Zhang proposes canonical local and global height pairings for two cycles Y , Z of pure codimension p, q satisfying p + q = n + 1. Zhang will discuss some applications to algebraic dynamical systems and Shimura varieties.

• Participants
 Matthew Baker Georgia Institute of Technology Jason Bell University of Waterloo Nguyen-Bac Dang Stony Brook University Laura DeMarco Northwestern University Jeff Diller University of Notre Dame Charles Favre École Polytechnique Simion Filip Institute for Advanced Study and Clay Mathematics Institute Patrick Ingram York University Mattias Jonsson University of Michigan Shu Kawaguchi Doshisha University Holly Krieger University of Cambridge John Lesieutre Pennsylvania State University Nicole Looper University of Cambridge Yohsuke Matsuzawa University of Tokyo Myrto Mavraki Northwestern University Curtis McMullen Harvard University Keiji Oguiso University of Tokyo Matthew Satriano University of Waterloo Joseph Silverman Brown University Tom Tucker University of Rochester Junyi Xie Université de Rennes Shou Wu Zhang Princeton University De-Qi Zhang National University of Singapore
• AV

Participants will have access to a projector and screen for computer-based talks and blackboards for those who prefer to give board-based talks.

High-speed Internet access is available.

• Wednesday Excursion

On Wednesday symposium activities are shortened for a guided hiking tour from Schloss-Elmau through the Partnach Gorge. The hike difficulty is moderate and will take approximately 3 hours.

If you will participate in the hike, hiking boots or shoes appropriate for traversing rocky/wet terrain, warm clothing (e.g. a sweater), sun protection (e.g. light cap) and waterproofs (e.g. raincoat or umbrella) are strongly recommended.

A small satchel containing a light snack and water is provided, but feel free to bring your own backpack in which to carry a camera and other items that may be useful to you.

• Travel

### Air

The foundation will coordinate and purchase air and train travel to the symposium. Travel specifications can be provided by clicking the registration link above.

Participants arriving via plane will arrive via Munich International Airport (MUC) Airport.

### Train

The closet train station to Schloss Elmau is in Klais. Those traveling by train should book their tickets directly and request reimbursement at the conclusion of the meeting. Should you require booking assistance, please contact Meghan Fazzi.

All participants are required to arrive on Sunday prior to the meeting’s start and attend the entire week. No accommodations will be made for partial participation.

### Passports

Passports are required.

### Local Ground Transportation

#### Arrival at Munich International Airport

The foundation will arrange for your transfer from MUC airport to Schloss Elmau via a private car. After claiming your luggage at baggage claim you will proceed through customs and to the exit where a uniformed driver holding a Simons Symposia sign will greet you.

Travel time from Munich to Schloss Elmau is 90 minutes.

#### Arrival at Klais Train Station

The foundation will arrange for your transfer from Klais train station to Schloss Elmau via a hotel shuttle.

Travel time from Klais to Elmau is approximately 10 minutes.

#### Departure from Schloss Elmau

The day prior to your departure you will receive a departure letter advising on your departure time to the airport or train station via private car. Please meet your driver in the hotel lobby ten minutes prior to your noted departure time. If for any reason your flight departure time changes, please visit the front desk to update hotel staff.

• Hotel

#### Schloss Elmau Luxury Spa & Cultural Hideaway

In Elmau 2, 82493 Krün
Germany
Tel: +49 (0) 8823 18-0
www.schloss-elmau.de

Check-in time is at 4:30 PM
Check-out time is 11:30 AM

The meeting concludes on Friday evening. Participants have the option to depart on Saturday following the conclusion of the meeting or depart on Sunday at no additional cost. Please note your desired departure date when registering.

#### Dress Code

Business casual clothing should be worn during the symposium.

The weather can change very quickly; please bring cold-weather clothing appropriate for spring in the mountains.

• Meals