Quantitative arithmetic geometry

A Diophantine equation is a polynomial equation where one is interested in finding solutions in the whole numbers. Mathematicians have been fascinated by such equations since antiquity. Often simple to state problems require very difficult tools to solve (one of the most famous examples of this being Andrew Wiles's celebrated proof of Fermat's last theorem). Moreover such equations, whilst originally viewed as nothing but a curiosity, have found important applications in modern times to information security and cryptography.

Given a Diophantine equation, a fundamental problem is to determine whether a solution actually exists. This problem in itself is very difficult. Things get even more interesting if one has a *family* of Diophantine equations (given by varying the coefficients of the equations, say). In this case one would like to understand the distribution of equations in the family with a solution. This is a very popular modern topic, with Manjul Bhargava being awarded the Fields Medal in 2014 for his work on such problems (this is a kind of mathematician's version of the Nobel prize).

The project concerns problems of this type. Here there is a conjecture due to Jean-Pierre Serre, a famous French mathematician, on the distribution of Diophantine equations in certain families with a solution (namely plane conics). We will answer some cases of Serre's problem, as well as extending Serre's original framework to more general problems.

A famous theorem of Erdos and Kac also states that a "random" integer n has approximately log log n prime factors (in a precise probabilistic sense). We will obtain analogues of this probabilistic result in the setting of families of Diophantine equations, where we ask for the number of primes p for which a given equation is not soluble modulo p.

This research proposal is concerned with theoretical aspects of pure mathematics. Academics all over the globe working in number theory and neighbouring areas will have new tools at their disposal for tacking problems, coming from probability, harmonic analysis and geometry. They will also have new research problems to work on, guided by the problems addressed in the project, and will help guarantee that the UK retains its international reputation for high quality research.

In the long term, the general public has the potential to benefit from my research. In modern times number theory has played a key role in cryptography and information security, which have had huge impacts on society and continue to do so. New ideas in number theory often contribute to such applications in unexpected and unpredictable ways. My track record as a former Heilbronn fellow makes me well placed to exploit any such applications which may arise.

Some of the results will also be of interest to a much wider audience. For example some of the objectives concern explicit Diophantine equations. As is often the case with such problems, the objectives and results will be easy to state, but will require deep techniques to prove. These objectives should be of interest to anyone with even a passing interest in Diophantine equations. I plan to get the general public more involved in mathematics by being available to give public lectures about my work, for example at the big national science festivals "The Big Bang" and "Pint of Science", as well as on the popular YouTube channel "Numberphile".