The study of Diophantine equations was initiated by the Greeks and has led to the development of arithmetic geometry during the past centuries. In modern language, one is interested in rational points on projective varieties. The main questions are: Given a projective variety, under which conditions does it contain rational points / infinitely many rational points, what can be said about the structure of the set of rational points, how are they distributed, when does the Hasse principle hold, when does weak approximation hold?

Our project deals with the quantitative behaviour of rational points, i.e. we are interested in the number of rational points of bounded height. This question is basically solved for curves. The next natural step is to investigate this problem for surfaces. Most relevant in this regard are Del-Pezzo surfaces because, roughly speaking, they contain many rational points. There is an important conjecture relating the quantitative behaviour of rational points with the geometry of the surface due to Manin. Currently, the resolution of Manin's conjecture is a dynamic field of research. We are mainly interested in the following instances of Del Pezzo surfaces: 1) degree 1 of singularity type E8, 2) degree 2 of singularity E7, 3) degree 3 of singularity type D5. Manin's conjecture has been resolved for the two last-mentioned Del-Pezzo surfaces by Baier and Browning and by Browning and Derenthal, respectively. However, it seems feasible to prove stronger versions with a power saving in the error term which allows to continue the associated height zeta function meromorphically to the left of the line Re s = 1. The study of the first-mentioned Del-Pezzo surface of degree 1 seems hard, but a resolution of Manin's conjecture for this case is not completely out of scope and would present a breakthrough. At least, a proof of a lower bound with the correct order of magnitude seems feasible in this case. 

Although emphasis is put on the above-mentioned concrete problems, the project is open to a much wider range of problems regarding the quantitative behaviour of rational points on varieties. The idea is to use tools from analytic number theory, such as exponential sums, to attack problems of this type. Therefore, the applicant should have some knowledge of number theory. Some background in arithmetic geometry would be welcome though not absolutely necessary.

(i) T. D. Browning, Quantitative Arithmetic of Projective Varieties, Progress in Mathematics 277, Birkhauser 2009.
(ii) M. Hindry, J. H. Silverman, Diophantine Geometry: An Introduction, Graduate Texts in Mathematics, Springer 2000.
(iii) M. Huxley, Area, Lattice Points, and Exponential Sums, London Mathematical Society Monographs, Oxford Science Publications 1996.
(iv) S.W. Graham, G. Kolesnik, Van der Corput's Method of Exponential Sums, Lecture Note Series 126, London Mathematical Society 1991.