Rational Points on Varieties

I will be working in the general area of arithmetic geometry and number theory, specifically looking at Brauer-Manin obstructions to certain Diophantine equations. A Diophantine equation is a polynomial equation in some number of variables and generally a number theorist is interested in seeking solutions in the integers or the rationals. Explicitly computing these points can be difficult, instead we attempt to answer if there exists any such points and if so how many. We will do this by looking at the geometric properties of these Diophantine equations and seek if any obstructions exist to there being a integer or rational solution. Moreover the aim this project is to look at certain classes of Diophantine equations (such as log K3 surfaces).

Diophantine equations have the property of if they have a rational solution the one can look "locally" i.e. check if it has p-adic solutions for some prime p. This works perfectly well for conics, where the Hasse-Minkowski principle states that having p-adic solutions for all primes is both a necessary and sufficient condition. However the Hasse-Minkowski theorem does not hold in general for example if we look at homogenous spaces that occur through performing decent on elliptic curves, these present diophantine equations which have p-adic solutions for all primes but the set of rational solution is empty. To study such failures we will use the Brauer-Manin obstruction.

One can also predict the asymptotic behaviour of a rational points on diophantine equations, for example the Manin conjecture predicts the precise asymptotic behaviour of the number of rational points of bounded height. We will aim to make progress in certain cases of this conjecture. Our research is not bounded by working over the field of rationals; we will extend this by studying solutions over number fields.

Working within number theory and arithmetic geometry may seem inapplicable to the untrained eye however this has direct applications to the field of cryptography. Many of the tools developed within number theory are applied in cryptography such as elliptic curve cryptography, RSA and many more. This research furthers results within number theory which can provide tools for future research for example in post quantum cryptography. Further to this using topology and algebraic geometry in the area of data science has been an increasingly intense area of research, by using various techniques in algebraic geometry we lay foundations for applications of similar techniques used by us. Furthermore we are contributing to the wider framework of pure mathematics research, this is essential for the progression of number theory and to allow others to work we have developed.

My funding body (EPSRC) has aims of further expanding research areas within their wide portfolio. My research would contribute to the continued work of EPSRC and will hopefully supply them with high quality research.