Talks

Andrei Yafaev: Overview of the Pila-Zannier approach to the Andre-Oort conjecture

We will give an overview of the approach to the André-Oort conjecture due to Pila and Zannier. We will emphasize a crucial ingredient — the problem of bounding below the degrees of special points on Shimura varieties, which is the main object of this workshop.

Gal Binyamini: Point counting: Pila-Wilkie and beyond

In the first talk I’ll give an overview of the area of point-counting, from the Pila-Wilkie theorem to some conjectures and results about sharper polylogarithmic counting theorems. In particular I will formulate the counting result for leafs of foliations over number fields (used in our joint work with Schmidt and Yafaev to deduce Galois lower bounds for special points). I will also explain how this applies in the context of Shimura varieties.

In the second talk I’ll explain the main ideas from differential algebra and complex geometry that go into the proof of the counting result for foliations. At the end I will also discuss some results and conjectures for p-adic and positive characteristic spaces, and how they relate to the proofs in the classical situation.

Harry Schmidt: Counting rational points and lower bounds for Galois orbits

In joint work with Binyamini and Yafaev we prove Galois lower bounds for special points depending on their Weil height. Before our work such a bound was known by using the isogeny estimates of Masser and Wüstholz which are restricted to abelian varieties. In our proof we need to count algebraic points on a sufficiently large subset of the graph of a covering map of a Shimura variety. I will explain how the polynomial dependence on the degree of the algebraic points enters the counting argument and the connection to the poly-log dependence on the height in the bounds. 

Teruhisa Koshikawa: Heights of pure motives

I will talk about heights of pure motives generalizing the Faltings height of an abelian variety, with some focus on the work by Kazuya Kato and myself several years ago. While our work is not used in the proof of the André-Oort conjecture, I hope to indicate a picture behind it.

Jonathan Pila: Overview of the Abelian case

This talk will give an overview the proof of the André-Oort conjecture in the abelian case, highlighting the role of the Faltings height of special points and the Colmez conjecture. We then discuss the role of height bounds more generally in the Zilber-Pink conjecture.

Jacob Tsimerman: Heights and Tori

We introduce an intrinsic height for automorphic vector bundles on 0-dimensional Shimura varieties, study their functorial properties, and explain how the height bounds in the Abelian case imply a whole host of more general height bounds. We then assume a good height theory for automorphic vector bundles on arbitrary Shimura varieties, and explain how to finish the proof.

Ananth Shankar: Construction of Heights using p-adic Hodge theory

We explain how to use Liu-Zhu’s (and Diao-Lan-Liu-Zhu’s) p-adic Riemann-Hilbert correspondence to metrize automorphic line bundles on Shimura varieties. We will also relate this to a metric that arises from crystalline local systems (we have crystillinity due to work of Esnault-Groechenig).

Hélène Esnault & Michael Groechenig: Strongly rigid local systems and crystalline representations

The first half of this talk is devoted to introducing various notions of rigidity for local systems on complex projective varieties as well as Simpson’s motivicity conjecture. In the second half we will explain how this picture relates to a Fontaine-Mazur type conjecture and we will establish a Frobenius property for certain rigid local systems.