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

The point of these pages is to put together the results of a series of computations I have done on the growth of the number of points on K3 surfaces. In the first series, the second series, and the third series, all surfaces are given as quartic surfaces in P^3 given by an equation of the form f_0 = 6h, where f_0 is some fixed polynomial and the reduction modulo 2 and 3 garantees that the geometric Picard number is indeed equal to 1. The points were found with an algorithm based on a p-adic version by Michael Stoll of an algorithm of Noam Elkies. The examples in the fourth series are obtained in almost the same way. All these have Picard number one. Here is the first series with Picard number 2.

My plan is to do a lot more examples, in order to state a well-supported conjecture that on some (Zariski open?) subset of any K3 surface (perhaps excluding scary cases such as those containing infinitely many elliptic curves or infinite automorphism groups?) the number of rational points of height at most B grows like c*(log B)^b, where c is a constant and b is the rank of the Picard group.

Disclaimer: The comments written here and on the linked pages are written quickly and not very carefully yet, but anybody who is interested in this topic probably does not need much more background information.

Each picture you will see contains one or more graphs of points corresponding to the number of points on different open subsets of the K3 surface given by the equation below the graph. The picture also contains one or more lines of which the equations are given below the graph. These equations are of the form r*c*x+d, where c is the product of local densities, r is some rational number, hopefully of low height, and d is some constant. If an equation is missing, then it is just c*x.

In the first series of examples I took h to be the sum of quite a few random monomials of degree 4. In the second series I just took h to be one monomial. The latter case yields much higher contributions from the real place to a potential candidate for the constant c (or part of it), as in Peyre's constant for del Pezzo surface (a product of local densities). And indeed, in the second series we see a lot more points. Unfortunately, in the last case I very often had to leave out the coordinate planes x=0 and w=0 as f_0 was constructed for those planes to have genus 0 on the surface given by f_0=0. It is possible that this is why the graphs grow much slower at low height, as points of low height are more likely to lie on those hyperplane sections. For this reason the next series I do of surfaces of this form, I will only pick surface where the hyperplane sections given by x=0 and w=0 are smooth (and thus have genus 3).

Types of examples to do in the future:

• Surfaces of the form f_0=6h with h containing few monomials (high real density) and smooth plane sections x=0 and w=0.
• Surfaces in P^3 with different reductions modulo 2 and 3.
• Surfaces in P^3 with geometric Picard number 2 and 3 and the whole Picard group defined over Q.
• Surfaces in P^3 with higher geometric Picard number but low Picard number over Q.
• Intersections of 3 quadrics in P^5 (need to rewrite my algorithm for searching for points).
• Double covers of P^2 ramified over sextics (need to rewrite my algorithm for searching for points).
• Quartic surfaces in P^3 of the form f(x,y) = g(z,w). An algorithm by Dan Bernstein works really well for this kind of surfaces. The geometric Picard number is at least 10, which I why I have first focused on other surfaces, as a higher geometric Picard number might influence the constant c in ways that are not so easy to get our hands on.