Schedule for: 20w5206 - Algebraic Dynamics and its Connections to Difference and Differential Equations (Online)

A brief welcome and introduction by BIRS Staff

Adamczewski and Bell proved in 2017 a 30-year old conjecture of Loxton and van der Poorten, asserting that a Laurent power series, which simultaneously satisfies a p-Mahler equation and a q-Mahler equation for multiplicatively independent integers p and q, is a rational function. Similar looking theorems have been proved by Bezivin-Boutabaa and Ramis for pairs of difference, or difference-differential equations. Recently, Schafke and Singer gave a unified treatment of all these theorems. In this talk we shall discuss a similar theorem for (p,q)-difference equations over fields of elliptic functions. Despite having the same flavor, there are substantial differences, having to do with issues of periodicity, and with the existence of non-trivial (p,q)-invariant vector bundles on the elliptic curve.

Please turn on your cameras for the "group photo" -- a screenshot in Zoom's Gallery view.

We prove the following theorem. Let f be a dominant endomorphism of a projective surface over an algebraically closed field of characteristic 0. If there is no nonconstant invariant rational function under f, then there exists a closed point whose orbit under f is Zariski dense. This result gives us a positive answer to the Zariski dense orbit conjecture for endomorphisms of projective surfaces. We define a new canonical topology on varieties over an algebraically closed field which has finite transcendence degree over Q. We call it the adelic topology. This topology is stronger than the Zariski topology and an irreducible variety is still irreducible in this topology. Using the adelic topology, we propose an adelic version of the Zariski dense orbit conjecture, which is stronger than the original one and quantifies how many such orbits there are. We also prove this adelic version for endomorphisms of projective surfaces, for endomorphisms of abelian varieties, and split polynomial maps. This yields new proofs of the original conjecture in the latter two cases.

By a Schwarzian differential equation, we mean an equation of the form $S_{\frac{d}{dt}}(y) +(y')^2 R(y) =0,$ where $S_{\frac{d}{dt}}(y)$ denotes the Schwarzian derivative and $R$ is a rational function with complex coefficients. The equation naturally appears in the study of automorphic functions (such as the modular $j$-function): if $j_{\Gamma}$ is the uniformizing function of a genus zero Fuchsian group of the first kind, then $j_{\Gamma}$ is a solution of some Schwarzian equation. In this talk, we discuss recent work towards the proof of a conjecture/claim of P. Painlev\’e (1895) about the irreducibility of the Schwarzian equations. We also explain how, using the model theory of differentially closed fields, this work on irreducibility can be used to tackle questions related to the study of algebraic relations between the solutions of a Schwarzian equation. This includes, for example, obtaining the Ax-Lindemann-Weierstrass Theorem with derivatives for all Fuchsian automorphic functions.

I will discuss some unlikely-intersection problems for curves in elliptic surfaces, defined over a number field. We introduce a new tool: an equidistribution theorem for heights associated to R-divisors, extending known results for Q-divisors of Chambert-Loir, Thuillier, and Yuan (2008). As applications, we obtain a new proof of a result of Barroero and Capuano (2016) and prove a case of a conjecture of Zhang (1998). This is joint work with Myrto Mavraki.

The Dynamical Mordell-Lang Conjecture predicts the structure of the intersection between a subvariety $V$ of a variety $X$ defined over a field $K$ of characteristic $0$ with the orbit of a point in $X(K)$ under an endomorphism $\Phi$ of $X$. The Zariski dense conjecture provides a dichotomy for any rational self-map $\Phi$ of a variety $X$ defined over an algebraically closed field $K$ of characteristic $0$: either there exists a point in $X(K)$ with a well-defined Zariski dense orbit, or $\Phi$ leaves invariant some non-constant rational function $f$. For each one of these two conjectures we formulate an analogue in characteristic $p$; in both cases, the presence of the Frobenius endomorphism in the case $X$ is isotrivial creates significant complications which we will explain in the case of algebraic tori.

For a fixed group G, we study the model theory of actions of G by field automorphisms. The main question here is to characterize the class of groups G for which the theory of such actions has a model companion (a first-order theory of "large" actions). In my talk, I will discuss several classes of groups G in this context. The case of finite groups is joint work with Daniel Hoffmann ("Existentially closed fields with finite group actions", Journal of Mathematical Logic, (1) 18 (2018), 1850003). The case of finitely generated virtually free groups is joint work with Özlem Beyarslan ("Model theory of fields with virtually free group actions", Proc. London Math. Soc., (2) 118 (2019), 221-256). The case of commutative torsion groups is joint work with Özlem Beyarslan ("Model theory of Galois actions of torsion Abelian groups", arXiv:2003.02329).

I will present in detail a new twist in the subject of arithmetic algebraization theorems. It comes out of a joint work in progress with Frank Calegari and Yunqing Tang on irrational periods, and bears also a relation to a variation by Zudilin around the classical Polya-Bertrandias determinantal criterion for the rationality of a formal function on the projective line. Time permitting, I will sketch an application to an irrationality proof of the 2-adic avatar of $\zeta(5)$.

This work is a collaboration with B. Adamczewski (ICJ, France), T. Dreyfus (IRMA, France) and M. Wibmer (Graz University of Technology, Austria). In this talk, we will consider pairs of automorphisms $(\phi,\sigma)$ acting on fields of Laurent or Puiseux series: pairs of shift operators, of $q$-difference operators and of Mahler operators. Assuming that the operators $\phi$ and $\sigma$ are "independent", we show that their solutions are also "independent" in the sense that a solution $f$ to a linear $\phi$-equation and a solution $g$ to a linear $\sigma$-equation are algebraically independent over the field of rational functions unless one of them is a rational function. As a consequence, we settle a conjecture about Mahler functions put forward by Loxton and van der Poorten in 1987. We also give an application to the algebraic independence of $q$-hypergeometric functions. Our approach provides a general strategy to study this kind of questions and is based on a suitable Galois theory: the $\sigma$-Galois theory of linear $\phi$-equations developed by Ovchinnikov and Wibmer.

Consider a complex projective surface $X$, with a non-abelian free group $G$ acting faithfully and regularly on $X$. It may happen that $G$ has infinitely many periodic orbits: this is the case when $X$ is an abelian surface and all torsion points are $G$-periodic. In this talk, I will describe recent results obtained with Romain Dujardin aiming at a complete classification of all such examples. The main players will be canonical heights, arithmetic equidistribution, and rigidity results in ergodic theory.