Computations on the Manin Conjectures for K3 surfaces, by Ronald van Luijk

All graphs below represent the number of points on some smooth quartic in P^3 of bounded height B, versus the logarithm of the B. They are supposed to support a hopefully-one-day-to-be conjecture.

I started with in the first series and the second series and the third series of examples. See those pages for more explanation of what is happening here.

For this series I followed a suggestion of Noam Elkies to find more surfaces with small coefficients and Picard number 1 in a very cheap way. The way I obtained a K3 surface with Picard number 1 in the first place was by constructing a surface that was right modulo both 2 and 3. By applying an automorphism to the surfaces in the reduction before taking a lift using the Chinese remainder theorem, we obtain many more surfaces with Picard number 1 as well. The nice thing about these, as opposed to just picking different lifts of the very same surface modulo 6, is that the surfaces are all given by equations with small coefficients. This makes it more likely that the contribution to the real part of the coefficient in the conjectured asyptotics is not very small.

For "fair" statistics I applied only an automorphism to the surface modulo 3. I applied all 3^6 upper triangular matrices with F_3-entries and ones on the diagonal. Of these 3^6, there were 87 where the real contribution exceeded 7. For each of these I computed the genus of the hyperplane sections of the surface given by w=0 and z=0. For each I also did the same for all surfaces obtained by adding +-6M for some monomial. There were just over 500 where the genus of both intersections was at least 2. Among these there were 35 for which the real contribution exceeded 7. Those 35 are given here.

In a few cases the points on the curve given by x+2z=0 were left out. This is indicated underneath the graph. The cases where I use a nontrivial correction factor or where I added some other comment come first.