A heuristic for a Manin-type conjecture for K3 surfaces

In the late 1980's Manin conjectured the asymptotic growth of the number of rational points up to bounded height in a suitable open subvariety of any Fano variety. Since his original conjecture, much work has been done in (dis)proving several cases and many modifications and refinements have been proposed. My upcoming PhD thesis partly concerns the 'next case' in dimension 2: that of K3 surfaces. Using the circle method, I was able to find heuristics for diagonal quartic surfaces that agree with numerical data produced by my supervisor Ronald van Luijk. In my talk I will explain the method, sketch the proof, and discuss its limitations, along with explaining some problems that need to be overcome before a reasonably strong conjecture can be stated.