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.
I started with examples with Picard number 1 in the first series and the second series and the third series and the fourth series of examples. See those pages for more explanation of what is happening here.
For this series I took 20 examples of picard number 2. They all originate from the same example. I take F1:=-2*x^3-2*x*y^2-x*y*w-2*x*z*w-x*w^2+y^3-y^2*w+2*z^3+w^3, F2:=x*y+x*z+y*z, and F3:=x*y+x*z-2*y*z. I went rhrough all combinations f = w*(F1+5M1)-(F2+5*M2)*(F3+5*M3) with M_i appropriate monomials, as well as f = w*(F1+5M1+5*M2)-(F2)*(F3) with M1 =/= M2 and selected all where the contribution from the real place exceeded 9. Then given the fact that many of these were of the first form with M2=0 and M3=xw, I also went through those of the form w*(F1+5M1+5*M2)-(F2)*(F3+5*x*w) exceeding 9. There are 61 total. I computed the points of height at most 100,000 for 20 of these, shown below.
The reduction at 5 of all of these surfaces is always the same and has Picard number 2, so all these surfaces have Picard number 2. The Neron-Severi lattice is spanned by the hyperplane class H and and a conic, with intersection numbers H.H=4, H.C = 2, and C.C=-2, and discriminant -12.
The parabolas all of the form y = 1/2*c*x^2+d*x+e, where c is the product of local contributions, e.g. (1-1/p)^2 * #X(F_p) /p^2 for nonarchimedean primes of good reduction. Although the surfaces are chosen to have a high contribution to c from the real place, the contribution at 2 and 3 is often quite a bit smaller than 1. The constants d and e are picked just by eyeballing to fit the data. Not much time or anything fancy was used there.
