Unlikely Intersections in arithmetic and dynamical systems

``A few years ago, Gaël Rémond formulated a conjecture on lower bounds for the height on tori and abelian varieties, which generalizes and unifies Lehmer's conjecture and its relative variants, with the goal of studying its applications to the Zilber–Pink conjecture. A very specific case of this conjecture allows one to locate the points of small height of an algebraic group with coordinates in an extension of a number field obtained by adjoining the saturated closure of a finitely generated subgroup.

Recently, Pottmeyer established a necessary group-theoretical condition for the conjecture to hold and proved it in the case of tori. I will present joint work with G. A. Dill, in which we extend this result by showing that the condition also holds for split semi-abelian varieties."

Abstract: ``While the Zilber-Pink conjecture remains open, there have been interesting recent developments. Work of Barroero-Dill for Shimura varieties with simple associated adjoint group, and Klingler and Tayou for more general settings, prove the conjecture when the varieties are geometrically generic. In this talk I will present joint work with V.Aslanyan and G. Fowler where we prove a somewhat similar (but different) result for the case of powers of the modular curve. Our methods differ from those of Barroero-Dill and Klingler-Tayou, instead building upon the joint work of Pila and Scanlon, which relies on the model theory of differential fields, as well as using arithmetic input in the form of Modular Mordell-Lang."

Abstract: ``A classical result of Mann implies that, for a given n, there are, excluding trivialities, only finitely many n-tuples of linearly dependent roots of unity. A modular analogue of this result, where roots of unity are replaced by CM-points on the modular curve Y(1), was obtained by Pila as a consequence of his proof of the André-Oort conjecture for Y(1)^n. Unlike Mann's result, Pila's result is ineffective. In this talk, I will present an effective version of Pila's result in the special case that the CM-points correspond to elliptic curves with endomorphism orders which are maximal."

Hang Fu (Basel):

Title: Dynamics of quadratic polynomials and rational points on a curve of genus 4

Abstract: ``Let f_t(z)=z^2+t. For any rational number z, let S_z be the set of rational numbers t such that z is preperiodic for f_t. In this talk, we will discuss a uniform result regarding the sizes of S_z over rational numbers z. In order to do it, we will also discuss how to find the set of rational points on a specific curve of genus 4. This is a joint work with Michael Stoll."

Title: Effective height bounds for very unlikely intersections in abelian varieties

Abstract: ``André and Bombieri's Hasse principle for relations between values of G-functions can be used to obtain effective height bounds for certain very unlikely intersections in an abelian variety A (i.e. intersections between a curve C and algebraic subgroups of A of large codimension). This in turn implies effective height bounds for rational points in some curves. In this talk, I will discuss explicit calculations of these bounds."

This talk is about an application of combinatorial algebraic geometry to complex/arithmetic dynamics. The n-th Gleason polynomial G_n is a polynomial in one variable with Z-coefficients, whose roots correspond to degree-2 polynomials with an n-periodic ramification point. Per_n is an affine algebraic curve, defined over Q, parametrizing degree-2 rational maps with an n-periodic ramification point. Two long-standing open questions in complex dynamics are:

(1) Is G_n is irreducible over Q?

(2) Is Per_n connected?

We show that if G_n is irreducible over Q, then Per_n is irreducible over C, and is therefore connected. In order to do this, we find a Q-rational smooth point on a projective completion of Per_n — this Q-rational smooth point represents a special degeneration of degree-2 self-maps.