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 and the second series of examples. See those pages for more explanation of what is happening here.
On the examples for the second series we had lots of points, but the number of points of low height appeared to be growing a lot slower than expected. Here we check if that was because we often threw out the hyperplane sections x=0 and w=0, as they often had genus 0 or 1. In this series we chose examples where that doesn't happen. As said before, all surfaces are congruent modulo 6 to the same surface, let's say that one is given by F=0. For all triples m1, m2, m3 of monomials of degree 4 and triples a1,a2,a3 in {-1,0,1}, I computed the genus of the hyperplane sections given by w=0 and x=0 of the surface given by F = a1*m1 + a2*m2 + a3*m3. For all cases for which both genera are at least 2, I computed the contribution of the real place to the naive constant that is the product of local densities. Below are all 14 examples for which that contribution exceeded 6.
In all cases on this page I used the product of local densities for the slope of the line in the graph directly, without any nontrivial factor. Only in one case did I leave out the points on one particular curve.
Not all graphs are equally convincing, sometimes possibly because of oscillatory effects, but all cases are certainly consistent with linear growth (picard number 1) with some oscillating term in the error.
