FOUVRY73

This course is an introduction to some topics in arithmetic statistics, centred around class groups of quadratic fields. We will start by recalling some basic background on class groups, and then move on to some classical results, including Davenport--Heilbronn. We will also discuss some other cases where aspects of the distribution of class groups have been studied, introducing some of the standard tools in the area as they arise.

Given an algebraic variety \(V\) defined over a number field \(k\), a natural question is to study its set of \(k\)-rational points \(V(k)\).
(Q1) Is it empty or not? If it is empty, what are the obstructions to the existence of a rational point?
(Q2) Is it finite or infinite? In the latter case, can we say something more quantitative?
(Q3) The problem of deciding whether or not a given variety has a rational point is difficult. Can we say something about this problem on average in families?
We will explain, during this mini-course, how analytic number theory can help us answer some of these questions. In particular, we will discuss a conjecture of Manin, which predicts the asymptotic behaviour of the number of rational points of height at most \(B\) on smooth Fano varieties as \(B\) goes to infinity. We will also discuss a conjecture of Loughran and Smeets regarding the number of varieties in families that have a rational points or a point everywhere locally. Finally, based on recent works, we will explain how to tackle such conjectures in two instances, highlighting two different approaches : the first based on Birch's circle method and the second one based on a descent method combined with the geometry of numbers.

Random multiplicative functions provide a model for certain functions of number theoretic interest, like Dirichlet characters. They can also be important tools for proving results about those functions, as well as an interesting probabilistic object in their own right.
In these lectures, I will try to introduce and motivate random multiplicative functions; explain some ways of thinking about them, which have proved to be useful in various recent works; and present details of some proofs. At the end I hope to also have time to explain how one can (sometimes) transfer results from random multiplicative functions to deterministic ones.

The course will survey the theory of trace functions, from their origin in the study of exponential sums over finite fields to current developments. We will focus on providing intuition and useful statements for applications (in particular convenient "black box" versions of Deligne's Riemann Hypothesis over finite fields), and highlight many applications of the theory to analytic number theory.