In the last few days I tried (unsuccessfully) to find some “new” supersingular surfaces by computation. Please read the previous post to see why one might want to find these surfaces. Anyway, one of the things that I have to show for this are some distributions of Newton Polygons (NPs) in the data. Here is an example:

The sequence of numbers at the top mean the following: We are looking at computations of NPs on H^2_{prim} of randomly chosen quasi-smooth surfaces over F_11 defined by an equation in weighted projective space of the form

W^2 = F(X, Y, Z)

where X, Y, Z have weights 13, 15, 19, the polynomial F is homogeneous of degree 184, and W has degree 184/2 = 92. Summing up the integers in the first column we see that we did a run of 3999 experiments and we got NP counts as shown.

The table suggests that the primitive Hodge numbers of these surfaces are h^{0, 2} = 2, h^{1, 1} = 3, and h^{2, 0} = 2 as is indeed the case. All possible NPs occur in the table, except for 4/3,4/3,4/3,1,2/3,2/3,2/3. The table suggests that the NP 2,2,1,1,1,0,0 happens generically and 2,3/2,3/2,1,1/2,1/2,0 happens in codimension 1 because 11 * 302 is almost 3616. Next, we expect the NPs 2,1,1,1,1,1,0 and 5/3,5/3,5/3,1,1/3,1/3,1/3 to happen in codimension 2. In fact, the whole table is strangely consistent with the known theory of NP jumps, except that 1,1,1,1,1,1,1 occurs too often.

Why is this strange? Well, because the equations cutting out the NP strata typically have high degree (polynomial in p) and hence we cannot expect *any* good behaviour of point counts over F_p (only when we work over F_q for a high power of p can we expect such a thing). The same happens for other experiments (see below). For smaller primes the behaviour is less regular; I think this happens because of the limited sample space.

Please let me know if you have any kind of guess as to why this should be!

PS: How did I compute these tables? To get the Frobenius polynomials I used a program I wrote a long time ago. More precisely, to replicate the results above you have checkout the double branch which computes Frobenius matrices of double covers of weighted projective planes. In each case I ran this program on random inputs repeatedly. You can find the outputs produced in this github repository.