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

This is the one of the two surfaces that I searched for points on in my thesis. Every surface in this series (i.e, on this page) is modulo 6 equivalent to this surface. This implies that these surfaces have geometric Picard number 1. For each of these surfaces I computed a naive constant that could be close to the asymptotic slope of the graph, which represents the number of points on the surface versus the logarithm of the height. This naive constant is the product of local densities as in Emanuel Peyre's constant for del Pezzo surfaces. For this surface the contribution for the real place is approximately 10.05. There are no bad primes below 1000, and the product of the contributions of all the primes below 1000 is approximately 0.78. The product is around 7.8. In the graph we also see a line whose slope is 3/4*7.8. Its equation is given below the graph, just as the equation for the surface. This rational number of low height, 3/4, is the reason why I call the constant "naive". In general there may also be contributions from the Brauer-Manin obstruction to weak approximation, or other reasons, such as constant coming from the configuration of the effective or other cones, though in the case of geometric Picard number 1, that should not be much, if anything. This is why the slope in the equation is broken down as a product of a rational number and the computed naive constant. The same will hold for all surfaces below. Sometimes there will be more than one line. If only one of the equations for these lines is given, then the other one is just y=cx, where c is the naive constant. The surfaces presented here were obtained by randomly adding 6*m, -6*m, or 0*m to the equation of this first surface for each quartic monomial m besides w^4. I left out w^4 to ensure that there was at least one point on the surface, namely [0:0:0:1]. The growth of the number of points on this surface is so close to linear that I don't think any comment is necessary, besides the fact that all points on the curve given by w=0 are taken out as there are infinitely many. Unless mentioned otherwise specifically, in the below cases we did not leave out any points.

The results below are not as striking as the first graph. This is mainly because not as many points are found. This is mostly due to the fact I chose the distribution of -6,0,6 for the coefficients of the monomials to be (2/5, 1/5, 2/5, I believe). This gives many nonzero coefficients, all bigger than the coefficients in the original equation, resulting in a lower contribution from the real place. Of course this is not at all a waste of effort, as if we want to see that the growth of the number of rational points depends on this constant, we better examine examples among which this constant varies, including ones where this constant is small. These examples include some of those. In some other series, however, I will restrict to families with higher real contribution, or only pick out examples with a relatively high real contribution.

By the way, all rational numbers on this page that the naive constants appear to be off by are among 1 (with multiplicity 15), and 1/2, 2/3, 3/4, 4/3, 3/2, 7/4, 2 (all with multiplicity at most 4), as tallied among all the drawn lines (sometimes two per graph). I am not sure how to interpret the fact that the rational number that the naive constant seems to be off by can be bigger than 1 in this case of geometric picard number 1. Perhaps this means in those cases we see many more rational points for low height than we would expect from the correct asymptotics, making the growth seem faster than it really is. Indeed, from these graphs with few points one can not deduce much. However, there are a lot of graphs, and in each the growth we see is very close (up to a factor of at most 2) with the expectation coming from the naive constant.

Indeed if we look at graphs with more points, such as in the second series, then none of the low-height fractions are bigger than 1, and they are in fact either 2/3 or 3/4 (at least in that series).