Local-global principles: arithmetic statistics and obstructions

Methods for solving polynomial equations in integers and rationals have been sought and studied for more than 4000 years. Sometimes it is easy to see that a polynomial equation admits no 'global' (meaning integral or rational) solution. For example, if the equation has no solution in the real numbers, then it clearly has no integer solution. The reason that looking for real solutions is easier is because the real numbers have a desirable property called completeness, which relates to the fact that the real numbers form a continuum with no gaps. It is the discrete nature of the integers which makes them difficult to deal with. By viewing the integers within other exotic number systems (called the p-adic numbers) that also enjoy the property of completeness, we can avail ourselves of other ways to rule out existence of integer solutions to polynomial equations. If the equation has no p-adic solution then it has no integer solution. But what if the equation has solutions in the field of real numbers and in all the p-adic fields? Does this mean it has a rational solution? If we have a family of equations where the answer to this question is yes, then we say the Hasse principle holds for that family. For example, the Hasse principle holds for quadratic forms. This means that determining whether a quadratic form has an integer solution is easy. However, there are equations of degree 3 and higher for which the Hasse principle fails. This leads to some natural questions, such as: How often does the Hasse principle fail? Why does it fail?

This research addresses both of these questions for certain families of equations. To answer the first question, we will fix a family of equations which can be enumerated in a meaningful way. We will then determine whether the Hasse principle can fail for any equation in the family. For those equations where failures can occur, we will calculate an algebraic object which measures the severity of the failure and determines the precise local conditions which are responsible for the failure. The most difficult step will be to calculate what proportion of the equations in the family give failures. This will tell us whether failure is, as we hope, a rare occurrence in the family. If failures are rare, then a randomly chosen equation in the family will satisfy the Hasse principle and determining whether it has a global solution is equivalent to checking whether it has real and p-adic solutions. The latter calculation can be performed in finite time, whereas no general algorithm exists for determining whether a polynomial equation has an integer solution.

The second question concerns obstructions to local-global principles such as the Hasse principle. The most important known obstruction is the Brauer-Manin obstruction. There are several challenges to be overcome in order to understand the consequences of the Brauer-Manin obstruction for a family of varieties. One must calculate the Brauer group, which is the algebraic object quantifying the obstruction. Then one must calculate the obstruction given by each element of the Brauer group. Finally, one must determine whether the Brauer-Manin obstruction suffices to explain all failures of local-global principles in the family. The second part of this project will push the boundaries of our current understanding of each step in this process.

This proposal describes a programme of research in number theory. Number theory is one of the most high-impact areas of pure mathematics in terms of its applications to cryptography and security. In particular, it can claim credit for the development of elliptic curve cryptography, which is a public key encryption system that has been widely used for over a decade by big players such as the USA National Security Agency and Microsoft. More recently, in search of higher levels of security, cryptographers have been developing public key cryptosystems based on a family of surfaces called Kummer surfaces. A substantial part of the research in this proposal aims for a better understanding of the arithmetic of these surfaces, which may have an impact on Kummer cryptography in the future. Moreover, it may be that another aspect of the research programme ends up having a real-world impact. The point is that it is hard to predict the impact of research in pure mathematics: 19th century researchers studying elliptic curves would certainly be amazed at how much we rely on them today for internet transactions and national security.

By wide dissemination of the results through publication in international journals, open access repositories, and talks for broad audiences as well as specialists, I will do my best to maximise the chances of finding pathways to applications for the research described in this proposal. I have links to the cryptography community through my work with Kristin Lauter (Principal Researcher and Research Manager for the Cryptography group at Microsoft Research) and others, and I will continue to collaborate with people outside my research area. I will maintain pages on arXiv, Research Gate and Google Scholar, as well as my personal website, to increase the visibility of my work.

Through teaching, mentoring and outreach, this research will have a positive impact on several generations of younger mathematicians. For example, I will lead a research project on a related topic at Women in Numbers Europe 3. The project will introduce young women to this field of research and enable them to extend their networks of collaborators and mentors through working together on the project with women from a variety of countries and career stages. I will lead master classes as part of the Reading Scholars programme, which is a programme for Year 12 students that gives priority to those students who have no parental experience of Higher Education and aims to introduce them to university study. I will also speak in the Reading Undergraduate Research Seminar which is an extracurricular seminar that aims to introduce undergraduates to more advanced topics and current research. Many of the students attending the seminar go on to apply for postgraduate study.

Inspiring and encouraging more students from diverse backgrounds to study mathematics will not only be of benefit to them as individuals but also to the UK economy and society. Employers in sectors such as including pharmaceuticals, national security, engineering and the low carbon economy will all benefit significantly from an increase in the number of mathematically skilled graduates. In turn, society will benefit from the innovations these industries bring in terms of medical treatments, information security, infrastructure and energy. Increased productivity and the accompanying tax revenues will enable investment in infrastructure and services for the benefit of UK society as a whole.