Selmer groups, arithmetic statistics, and parity conjectures.

This project, based in number theory but spanning algebraic geometry, topology, and probability theory, is about improving our understanding of two major conjectures: the Cohen--Lenstra heuristics, and the Birch and Swinnerton-Dyer conjecture, the latter one of the `millennium prize problems'.

The Cohen--Lenstra heuristics concern class groups, mysterious objects studied already by Gauss over 200 years ago, which measure the failure of certain generalised integers to admit a unique factorisation into primes. Whilst notoriously hard to understand in specific examples, in the 1980s Cohen and Lenstra proposed a radical alternative approach to studying them, predicting that their behaviour in families could be modeled by a random process. Subsequently, this principle has proven effective for understanding many related objects, leading to the field of `arithmetic statistics' in which Bhargava was awarded the Fields medal in 2014. This project aims to study in this way certain other groups ubiquitous in number theory: K-groups of rings of integers. These are natural `higher analogues' of class groups and, like class groups, play a central role in a remarkable link between arithmetic and analysis; their order appears in special value formulae for certain complex analytic functions called Dedekind zeta functions. Whilst arguably more mysterious than class groups (determining completely the K-groups of the integers would solve the famous Kummer--Vandiver conjecture for example) there is evidence that they too can be modeled by random processes. I aim to initiate a systematic study of these objects from a statistical point of view, extending the Cohen--Lenstra heuristics to K-groups of rings of integers of imaginary quadratic fields, and leverage new breakthroughs to prove a big piece of this, improving significantly our understanding of these important objects.

The remarkable link between analysis and arithmetic alluded to above again manifests itself in the second of the conjectures central to this project, the Birch and Swinnerton-Dyer conjecture. This concerns the arithmetic of abelian varieties, certain geometric objects whose points naturally form a group. This structure distinguishes them amongst other geometric objects and has placed them at the forefront of research in modern number theory, algebraic geometry, and cryptography. For example, both Faltings's resolution of the Mordell conjecture and Wiles's proof of Fermat's last theorem made crucial use of abelian varieties, despite the problems not initially appearing to involve them.

Attached to an abelian variety are two fundamental objects of a very different nature. One, the rank, is a measure of how many rational points the abelian variety has and is arithmetic in nature. The other object is the L-function, and is complex analytic. The Birch and Swinnerton-Dyer conjecture predicts a striking relationship between the two: the order of vanishing of the L-function at its critical point should equal the rank. This conjecture was made in the 1960s and has been a focal point for research ever since. One remarkable consequence is the parity conjecture: a certain easily computable analytic quantity, the root number, should determine whether the rank is odd or even. This alone is often sufficient to predict when the equations defining an abelian variety have infinitely many solutions, and has ramifications for many ancient problems. Indeed, a proof of the validity of this criterion would settle many important cases of the Congruent Number Problem, dating back to at least the 17th century. The second major aim of this proposal is to draw on new techniques introduced in my recent work to establish a variant, the 2-parity conjecture, for large classes of abelian varieties of arbitrary dimension, in doing so providing evidence that the Birch and Swinnerton-Dyer conjecture extends in the expected way to this setting, where almost nothing is known.