Ronald van Luijk

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.

All are modulo 6 congruent to the same surface, the same one as the one I started with in the first series of examples. See that page for a little more explanation of what is happening here. Let's say that one is given by F=0. Below are 35 examples, one for each of the 35 quartic monomials in four variables. Four each monomial M, I compared the expected contribution from the real place to the Peyre constant for the two surfaces given by F+6M and F-6M. The one with the highest contribution is given here. All surfaces here have geometric Picard number 1.

The surface given by F=0 contains infinitely many rational points on the curves given by x=0 and w=0. If M is divisible by x or w then the surface given by F+-6M=0 will contain infinitely many points on the corresponding hyperplane intersection as well, so we leave out all those points. This could explain (if there is an explanation needed at all, actually) why we see relatively few points of very low height, i.e., the graph often starts off with smaller slope than it reaches eventually. Indeed, points with very small height are more likely to lie on the curves that we left out.

In fact, even when the monomial M is not divisible by x, the hyperplane section x=0 on the surface given by F+-6*M = 0 may still have smaller genus. It turns out that the genus of this hyperplane section is zero when M is divisible by x or not by w^2, in which case we leave out the points on it, and it has genus at least 2 otherwise, in which case we do not leave them out.

Similarly for the hyperplane section given by w=0. It has genus 0 when M is divisible by w. It has genus 1 for M = +-X^3Y, +-X^3Z, -Y^4, +-XY^3, Y^4, +-Y^3Z, +-YZ^3, +-Z^4, XZ^3, -xz^3, and +-x^4 and the corresponding elliptic curves have rank 1 in the first 5 cases, and rank 2 in the next 10, rank 3 for m = -xz^3, and in the last 2 cases I actually can not find any points, so perhaps it is not even an elliptic curve... We leave out the points in this hyperplane section unless we explicitly mention otherwise. For the remaining 6 monomials (or twelve, +-m) the section has genus 0 in which case we have left out all points on it.

In almost all cases the naive constant that is a product of local densities gives the slope of the line in the graph. In some case there was clearly a correction needed, which was always less than 1, namely either equal to 2/3 or 3/4. In a few cases neither of these correction factors seemed obvious, but also the factor 1 was not obviously givving the right thing. Im these cases the correction was very close to 1 and two lines are given, one with correction factor 1, and one with a correction factor close to 1. This phenomenon could be explained by a combination of the error term (still) being larger than we often see and the fact that the product of local densities coming from finite places is hard to approximate, so this can be off in some cases. We computed the product over all primes up to 1000. However, one explanation could also be the one in the next paragraph.

We already noted that in the beginning we get fewer points than expected. At first I thought this could be explained by the fact that we are leaving out some hyperplane sections and that affects the number of points of lower height more that the number of those of larger height. However, of course it could also be that there is some error term that does not go to zero. In some graphs it certainly looks like the error terms gets very small, but we are still looking only at a relatively small number of points. The growth could for instance look something like

N(B) ~ a*log B + O(sqrt{log B}) Certainly functions like c*x + a*sqrt{x+b} + d have graphs that look a lot like the graphs we see here! Actually, I do not mean to be as bold as to guess the size of the error term. I am merely trying to say that the fact that I hope that asymptotically the function N(B) is linear in log B in these cases, that does not mean that the graph should have a line as an asymptote, even though some graphs make it tempting to believe that there might be.

Also the oscillating behavior may lead us to believe in some cases that the slope of the graph of the number of points does not diverge to our constant. However, there are also enough graphs where this seems to happen at lower height and then it straightens out again, so the fact that the graph temporarily wonders away from the line is not at all a reason to start disbelieving the conjecture-to-be.

The section given by x=6y has genus 0. Graphs with and without points on that section.

In this picture and the next two we see an oscillating behaviour that steadies out (perhaps only temporarily?). If we only had the information of the number of points up to height somewhat less than we do now, it might have looked like the growth was getting steeper than we think now. This also means that in graphs to come, the oscillating behavior might give us the wrong impression. What's more important to notice is how close the growth resembles not just a line, but the rate implied by the naive constant that is the product of local densities in all cases on this page.

Am I just dreaming, or do I see the effects of two independent periodic oscillations, one about four times the size of the other, that sort of cancel each other out at the end?

Hmm, does the number of points seem to grow faster than linear here? Could there be a hidden curve of genus 1? Or is this just something very similar to the case above?

Should I ask the same question here?

A perfect example of why the assymptotics could be something like 7.03*(log B) - a*sqrt(log B) + o(sqrt(log B)), as the slope gets closer and closer to 7.03. I am actually not trying to say anything about the size of the error term, just that the graph may not have a line as an asymptote (as mentioned above).

The curve given by w=0 has genus 1 and contains 24 points of height at most 100000. Here are both the graph with and without those points.

See comment at first picture with two lines of relatively close slope.

The section w=0 has genus 1 and contains 21 points of height at most 100000. Graphs both with and without them.

The section w=0 has genus 1 and contains 11 points of height at most 100000. Graphs both with and without them.

The curve given by w=0 has genus 1 and rank 2. Both graphs with and without these points are given.

The curve given by x=6z has genus 0. Both graphs with and without the points on that curve.

The curve given by w=0 has no nonsingular points to we didn't leave any out (i.e., left in the singular point that isn't on x=0).

Sorry for the three lines... The intersection with w=0 has genus 1 and rank 2, with 6 points of height at most 100000. Both graphs with and without are given. The graph with the points on that curve matches up with the line with slope 2/3*5.08 very nicely. However, I'd rather see the graph without those points line up. A correction factor other than the 2/3 for the graph with the points that would make this graph without the points also match up nicely would be 5/9, but since 5/9 has height so much higher than all other correction factors I have had to use on this page, I also included a line with slope 1/2, which doesn't seem to match up equally nicely, but that could easily be because we don't really have that many points.