This is the homepage for the workshop on Galois representations, Integral points, and Unlikely intersections held in Mainz from April 9th until April 11th. The speakers are
- Laura Capuano (Oxford),
- Philipp Habegger (Basel),
- Aaron Levin (Michigan), and
- Daniel Litt (IAS, University of Georgia).
- Kang Zuo (Mainz)
All talks will take place in the Hilbertraum. (Fifth floor of the mathematics building.)
The tentative schedule is as follows. Abstract and titles are below.
Tuesday April 9th
Wednesday April 10th
Thursday April 11th
Laura Capuano. Unlikely intersections in families of abelian varieties
Abstract. The Zilber-Pink conjectures on unlikely intersections predict the behaviour of subvarieties of families of (semi)abelian varieties and more generally of Shimura varieties when intersected with "special" subvarieties of the ambient space. These conjectures generalise many well-known conjectures/results such as Manin-Mumford, Mordell-Lang and André-Oort. In this course, I will present recent results in this framework obtained in a series of joint papers with F. Barroero for curves in families of abelian varieties defined over a number field (the case of families of abelian varieties of relative dimension 2 is nothing but the "relative version" of Manin-Mumford and has been previously proved in a series of papers by Masser and Zannier).
The proof of these results follows the now well-estabilished Pila-Zannier strategy, first introduced by the two authors in 2008 to give an alternative proof of Raynaud's Theorem (formerly the Manin-Mumford conjecture for abelian varieties), which combines theorems about counting rational points of bounded height in o-minimal structures (Pila-Wilkie and subsequent generalizations) with other diophantine ingredients. I will also present an application of these unlikely intersections results to the study of the solvability of a function field variant of the classical Pell equation.
Philipp Habegger. Unlikely Intersections, height bounds, and applications to rational points on curves
Abstract: This course contains an introduction to the Conjecture on Unlikely Intersections, sometimes also called the Zilber-Pink Conjecture. This conjecture generalizes many classical results in diophantine geometry such as Faltings's Theorem (the Mordell Conjecture), Raynaud's Theorem (the Manin-Mumford Conjecture), and the André-Oort Conjecture, solved now in many cases after work of Daw, Klingler, Pila, Orr, Tsimerman, Ullmo, Yafaev, and others. Important tools in this area is the point counting strategy going back to an idea of Zannier and tools from the theory of heights.
I will present joint work with Ziyang Gao comparing the Néron-Tate and Weil heights in a one parameter family of abelian varieties. As an application we give new estimates for the number of rational points on a one-parameter family of curves of genus at least 2. The latter is more recent joint work with Vesselin Dimitrov and Ziyang Gao.
Aaron Levin. Integral points on algebraic varieties
Abstract: We will give an overview of the subject of integral points on algebraic varieties, starting from introductory examples and definitions, and ending with topics drawn from current research in the area. Topics will include points on curves, effective methods, higher-dimensional results, points of bounded degree, and related tools (primarily from Diophantine approximation).
Flying to Mainz
The nearest airport is Frankfurt Airport. It's about 20 minutes from the central station of Mainz by train.
Getting to the math department
There are three possibilities.
- You can walk from the central station. The walk is about 35 minutes.
- From the central station (Mainz Hauptbahnhof) take a bus or tram to Friedrich von Pfeifferweg. From there, the mathematics building (Staudingerweg 9) is about 7 minutes walking.
- From the central station (Mainz Hauptbahnhof) take a bus or tram to Universität. From there, the mathematics building (Staudingerweg 9) is about 20 minutes walking.
Organizer: Ariyan Javanpeykar