Dynamical Days

Opening remarks by the host of Dynamical Days, Dr. Carlo Pagano.

Propagation of Zariski Dense Orbits in Arithmetic Dynamics.

Let X/K be a smooth projective variety defined over a number field, and let f : X --> X be a morphism defined over K. We consider statements of the following sort: ”If there is one point P in X(K) whose forward f-orbit is Zariski dense, then after taking a finite extension of K, there are many such points.” For example, a weak interpretation of ”many” is that X(K) is not the union of finitely many (grand) f-orbits, while a strong interpretation of ”many” is that every set of representatives for the Zariski dense grand f-orbits is itself Zariski dense. We prove various such statements for various classes of varieties and maps, including projective spaces, abelian varieties, and surfaces. (Joint work with Hector Pasten)

Local heights and the arithmetic of small points.

Local height functions play a key role in adelic approaches to problems concerning points of small canonical height. In the setting of abelian varieties, these were first introduced by Néron; later work generalized these to polarized dynamical systems and beyond. This talk will discuss ways in which local heights have been used successfully in studying small points, as well as some questions and limitations surrounding these functions. As time allows, I will discuss potential avenues of future exploration.

Arboreal Galois groups with colliding critical points.

Abstract: Let f ∈ K(z) be a rational function of degree d ≥ 2 defined over a field K (usually Q), and let x0 ∈ K. The backward orbit of x0, which is the union of the iterated preimages f−n(x0), has the natural structure of a d-ary rooted tree. Thus, the Galois groups of the fields generated by roots of the equations fn(z) = x0 are known as arboreal Galois groups. In 2013, Pink observed that when d = 2 and the two critical points c1, c2 of f collide, meaning that fm(c1) = fm(c2) for some m ≥ 1, then the arboreal Galois groups are strictly smaller than the full automorphism group of the tree. We study these arboreal Galois groups when K is a number field and f is either a quadratic rational function (as in Pink’s setting over function fields) or a cubic polynomial with colliding critical points. We describe the maximum possible Galois groups in these cases, and we present sufficient conditions for these maximum groups to be attained.

Root Extraction and Roots of Unity in Iterated Extensions.

I’ll present some results on a class of rational functions whose associated iterated extensions contain all m-power roots of unity for some fixed m > 1. This leads to applications on the depth of ramification in these iterated extensions and to a case of the Andrews-Petsche conjecture on abelian arboreal Galois representations. I’ll also discuss some generalizations to maps in higher dimension.

Abelian arboreal representations.

In 2020 Andrews–Petsche proposed a classification of all the abelian dynamical Galois groups coming from polynomials over number fields. More precisely, they conjectured that the only examples should be the ones coming from Chebischev and power polynomials.

I will present joint and ongoing work with Andrea Ferraguti where we focus on
the case of unicritical polynomials. In this realm, we reduced the conjecture to the
postcritically finite case and proved it in the following cases:
• When the critical orbit is periodic, over any number field.
• When the number field has degree at most 2 over Q.
• When the number field is totally real.
• When the degree of the polynomial is sufficiently large as a function of the degree of the number field.
• For all but finitely many basepoints, as long as the polynomial lies outside of a finite exceptional set Ud depending only on the degree. The methods are a combination of group theory, height estimates, rational/integral point on explicit curves and local class field theory.

On the morning of June 6th, you are invited to join us for coffee and an open problems session in room LB 921.04. During this gathering, we will serve coffee and facilitate interactions between attendees and speakers, allowing for the discussion of intriguing mathematical problems in number theory.

After this session, the conference will resume in room MB 3.270.

Dynamical irreducibility of polynomials modulo primes

In this talk we look at polynomials having the property that all compositional iterates are irreducible, which we call dynamical irreducible. After surveying some previous results (mostly over finite fields), we will concentrate on the question of the dynamical irreducibility of integer polynomials being preserved in reduction modulo primes.

More precisely, for a class of integer polynomials f, which in particular includes all quadratic polynomials, and also trinomials of some special form, we show that, under some natural conditions, the set of primes p such that f is dynamical irreducible modulo p is of relative density zero. The proof of this result relies on a combination of analytic (the square sieve) and diophantine (finiteness of solutions to certain hyperelliptic equations) tools, which we will briefly describe.

Infinite orbits in certain elliptic surfaces, Ax Schanuel, and ramification in the Legendre family.

Motivated by work of Cantat-Dujardin, we study orbits by translations in K3 surfaces with two elliptic fibrations. We prove in particular that all orbits are infinite away from a proper Zariski-closed subset.

Among the tools, beyond the Pila-Wilkie counting, are a new version of Ax-Schanuel by Bakker-Tsimerman and a(n old) theorem of Shioda on ramification of sections of the Legendre family. This is joint work with Corvaja and Tsimerman.

Bad reduction of PCF maps.

Abelian varieties admitting complex multiplication have everywhere potentially good reduction. The analogy between CM and PCF is already slightly undermined by the ease with which one writes down PCF polynomials with a place of incorrigibly bad reduction, but still a question arises to which we seem to have no clear answer: how bad can the reduction of a PCF map be? We revisit this question, with an emphasis on the significant differences between polynomials and other rational functions.

The geometry of preperiodic points in families.

Recently, Gao and Habegger established the ‘Relative Manin-Mumford Conjecture’ for families of abelian varieties. This result came shortly after works of Dimitrov–Gao–Habegger and Kuhne proving (in particular) that the ‘Manin–Mumford Conjecture’ holds uniformly. In the dynamical setting, in works with DeMarco and with Schmidt, we established some instances of uniformity in Zhang’s dynamical Manin- Mumford conjecture. In a related context, Gauthier–Taflin–Vigny proved that postcritically finite endomorphisms of P^k of degree d ≥ 2 are sparse when k ≥ 2, in contrast with the situation when k = 1. More recently, Ji–Xie proved the ‘dynamical Andr´e–Oort conjecture’ of Baker-DeMarco in the case of curves.

In this talk we will discuss a conjecture that generalizes all the aforementioned results (and conjectures). Inspired by the Relative Manin-Mumford conjecture, its aim is to characterize the subvarieties of S × P^N containing a Zariski-dense set of preperiodic points for a family of endomorphisms of P^N, parametrized by a quasi-projective complex variety S. The characterization is given in terms of the non-vanishing of a power of the invariant Green current associated to the family of maps. We will give examples where the equivalence is known to hold.

A dynamical Schanuel theorem

Let f : Pm → Pm be a morphism of degree d ≥ 2 defined over a number field K of degree n. Let h denote the standard height on Pm(K), and let hf denote the canonical height attached to f. Given a real number x, we wish to determine (1) the number of points P in f(Pm(K)) such that h(f(P)) < x, and (2) the number of points P in Pm(K) such that hf (P) < x. It turns out that the answer to both questions depends on the quantity N(x) := {P ∈ Pm(K) : h(f(P)) < x}. In this talk, we will illustrate the connection between (1), (2), and N, and then sketch a proof that N(x) is asymptotic to a constant cK(f) times en(m+1)x/d, giving a decomposition of cK(f) as a product of local factors over all places of bad reduction of f.

Invariant rational functions and Cartesian products of algebraic dynamical systems.

Let X be a complex algebraic variety equipped with a dominant rational map φ : X → X. We introduce a quantity called the stable algebraic dimension, which is the maximum of the number of new algebraically independent invariant rational functions on (X×Y, φ×ψ), as we range over (Y, ψ). We show that this birational invariant is bounded above by the maximum dimX′ where (X, φ) → (X′, φ′) is a dominant rational equivariant map and φ′ is part of an algebraic group action on X′ and that if (X, φ) has no nonconstant invariant rational functions then these two quantities agree. As a consequence, it is deduced that if some cartesian power of (X, φ) admits a nonconstant invariant rational function, then the second cartesian power already witnesses this. This is joint with Rahim Moosa and Matt Satriano.

Abelian dynamical Galois groups over the rationals.

Andrews and Petsche conjectured in 2020 that the dynamical Galois group of a polynomial over a number field cannot be abelian, unless the polynomial is conjugate to a power map or (plus or minus) a Chebishev polynomial. In this talk, we will explain how it is possible to use ideas from unlikely intersection theory to prove that the conjecture holds true when the base field is Q. Moreover, we will explain how to use heights to prove that if the dynamical Galois group of any rational function over a number field is abelian, then the rational function needs to be post-critically finite. This is joint work with A. Ostafe and U. Zannier.

Landen dynamics and special values of G-functions

We introduce a method for proving the irrationality of certain special values of G-functions based on properties of the Landen iterative layering of the modular tessellation map. This includes the two-variables G-function f(x, y) = log(1−x)log(1−y) at the special arguments of the form x = 1/n and y = 1/m, where m and n are integers whose ratio is within a millionth of 1. The proofs are based on the method of ”rational” arithmetic holonomy bounds and go towards characterizing the G-series with {0, 1,∞} singularities and denominator type [1, . . . , n]^2. We also formulate an inquiry towards a hypothetical ”integral” version of the holonomy bounds, which would cover various further linear independence proofs on special values of certain G-functions at arguments of the form x = 1/n, under an analogy with the scheme of Andre’s reproof of the Siegel-Shidlovsky theorem. This is based on a joint new work with Frank Calegari and Yunqing Tang, whose main application is a proof of the Q-linear independence of 1, ζ(2), and the Dirichlet L-function special value L(2, χ−3).

Bounded geometry for PCF points in Md.

The geometry of the postcritically finite (or PCF) locus in moduli spaces Md (of maps on P^1 with degree d) is constrained by their number-theoretic properties. In joint work with Myrto Mavraki and Hexi Ye, we provide uniform bounds on configurations of PCF points in families of subvarieties in Md. As an example consequence, we deduce that the irreducible components of the Pern(0) curves in M2 (which are conjectured by Milnor to be irreducible) have degrees growing to infinity with n.

Closing remarks by the host of Dynamical Days, Dr. Carlo Pagano.