Asymptotic properties of global fields

Our talk deals with the behaviour of arithmetic data (class number, number of places of given degree) in families of global fields. During the first part of our talk, we will motivate this study with questions related to sphere packings and coding theory. After that, we will recall the classical Brauer–Siegel theorem that precisely describes the behaviour of the product of the class number by the regulator in families, and then give some generalizations and applications. For this purpose, we will introduce Tsfasman–Vladuts invariants of infinite global fields.
In the second part of our talk, we will first explain Schmidt"s $K(pi,1)$ property and make use of it in order to construct nice families of global fields. Then, if time permits, we will give another related context where this property plays a major role. Finally, we will raise some open questions and explain why they are interesting and difficult.
Доклад будет прочитан на английском языке.