Objectives

Arithmetic statistics concerns the distribution of arithmetic objects. This perspective of studying objects in a family, rather than individually, has helped to improve our understanding of many fundamental objects in number theory, algebra, and geometry. In this programme grant, we will study the following problems:

• counting rational points of bounded height on varieties and stacks, for example cubic surfaces, and in particular Manin's conjecture and its generalisations;

• counting number fields of bounded discriminant, or other orderings, and in particular Malle's conjecture and variants;

• the distribution of class groups of global fields and the Cohen–Lenstra heuristics;

• the distribution of ranks and of Selmer groups of abelian varieties, and particularly variants of Goldfeld's conjecture and the Poonen–Rains heuristic;

• the distribution of Galois module structures in families, such as unit groups of number fields or Mordell–Weil groups of abelian varieties.

We will explore the numerous, often subtle connections between these problems, and will study connections to themes such as the Brauer–Manin obstruction and local–global principles more generally. One overarching theme of the programme is the geometry and arithmetic of stacks and connections to the above problems.

The programme also has the following themes.

Computation: Many of the great leaps in number theory have been made through computation. This includes numerous advances in Malle's conjecture and the Cohen–Lenstra heuristics. Our programme will have several computational components, particularly regarding the generation of new data which will be included with our project partner the LMFDB.

Cryptography: Numerous objects in arithmetic statistics are of fundamental importance to information security, for example class groups and elliptic curves. Understanding the statistics of such objects is fundamental for both testing the security of such schemes, but also for developing new ones. Our programme will develop cryptographic applications using our new framework in arithmetic statistics in collaboration with our project partners, the CANARI team in Bordeaux and the COSIC team in Leuven.