New frontiers of the Manin conjecture

Mathematics is undeniably the universal language of science and nature, whose processes are often governed by equations. The study of rational or integral solutions to Diophantine equations is a subject that is both ancient and difficult, having commanded our attention since the time of the ancient Greeks nearly 2000 years ago. It has had profound interactions with a host of subject areas, ranging from algebraic geometry to complex analysis via mathematical logic and everything in between. At its core this project uses analytic and numerical methods to probe a central conjecture about the quantitative distribution of rational solutions to Diophantine equations. After linear and quadratic equations, for which we have a reasonably complete picture, one is naturally led to examine the arithmetic of equations involving polynomials of degree 3 and 4. In dimension 2 the Manin conjecture is central in Diophantine geometry; it emerged in the 1990s and purports to describe precisely the distribution of rational points on the surface described by the polynomial equations. The first phase of the project will produce substantial numerical data for such families of cubic equations, in order to test whether there is any structure in the oscillatory error terms apparent in the associated counting functions. The second phase of the project will investigate the extent to which the philosophy of the Manin conjecture can be extended to higher degree (K3) surfaces, where there is very little evidence - both numerical and theoretical. Finally, the project will prove new cases of the Manin conjecture for some special varieties which are expected to contradict the original Manin conjecture, due to the presence of thin sets of rational points.

The study of rational or integral solutions to Diophantine equations goes back to the ancient Greeks, nearly 2000 years ago. It has had profound interactions with a host of subject areas, ranging from algebraic geometry to complex analysis via mathematical logic and everything in between. This project used analytic and numerical methods to probe a central conjecture about the quantitative distribution of rational solutions to Diophantine equations, and to formulate a new conjecture about the distribution of integer solutions to cubic polynomials in three variables. Beginning with the former, the Manin conjecture is a central part of Diophantine geometry. It emerged in the 1990s and purports to describe precisely the distribution of rational points on algebraic varieties that are described by polynomial equations with integer coefficients. One of the main achievements of the project was a full resolution of the Manin conjecture for a class of three dimensional Fano varieties on which there exist surprising "thin sets" of rational points that need to be handled with special care. This project also studied the behaviour of the Manin conjecture on average, resulting a proof of the conjecture (with very strong error terms), for 100% of homogeneous polynomials of degree d in n variables, provided that n>d. The Manin conjecture completely breaks down when looking at integer points, rather than rational points. Inspired by recent work of Booker and Sutherland about which integers can be represented as the sum of three cubes of integers, a major goal of the project was to develop a new conjecture about the density of integer solutions on affine cubic surfaces. This combined sophisticated tools from harmonic analysis, together with extensive numerics to shed light on this difficult topic, and resonates with recent a lot of recent activity around the arithmetic of "log K3" surfaces.