1/4 Arithmetic Statistics | Melanie Matchett Wood, Harvard University

This course will be an introduction to arithmetic statistics, and start by explaining the motivating questions of the field. Given a finite group G, how many number fields with Galois group G are there with discriminant bounded by X, asymptotically in X? As we vary a number field in a family, what distribution on class groups do we obtain? What is the distribution of local properties of the number field, such as whether a rational prime is split in that number field? As we vary an elliptic curve in a family, what is the distribution of Selmer groups and of ranks? These questions all lead to further generalizations as well. We will discuss the known results, and give an introduction to some of the methods that have been successful so far, including geometry of numbers and analytic methods. We will explain the important conjectures, and some open questions where it is not even known what to conjecture. We will introduce the random linear algebraic models that help support and inform many of the conjectures. We will also highlight places where conjectures have been informed or supported by computation and places where new computations would have the potential to advance the field.