In Shioda cycles, II and Shioda cycles, I we discussed how almost any arithmetic family of surfaces produces an infinite family of instances of the Tate conjecture for divisors of surfaces over finite fields. In this post we’ll see how to produce explicit equations for surfaces where the Tate conjecture is open.

Namely, a while back I wrote a computer program that computes the matrix of geometric frobenius on H^2_{cris, prim}(X) for a quasi-smooth hypersurface in a weighted projective space over a prime field using an algorithm due to Kiran Kedlaya. It is quite usable, except that only I can parse its output since I didn’t bother to write documentation. One of the things I like about it is that it works for any quasi-smooth hypersurface in any weighted projective 3-space (of course in most cases the run time is too large). For example, look at the degree 92 hypersurface

X : x^11y + x^5z^2w + x^2yzw^2 + xy^3zw + xz^5 + y^5z + w^4 = 0

in the weighted projective space P(7, 15, 17, 23) over F_2. The characteristic polynomial of Frobenius on the primitive middle cohomology is

x^9 + 2x^8 + x^7 – 2x^6 – 4x^5 + 8x^4 + 16x^3 – 32x^2 – 256x – 512

which has x = 2 as a root. Hence the Picard group of this surface should have rank 2 according to the Tate conjecture.

One of the things I tried was to write a bunch of scripts running through all possible quasi-smooth surfaces for a given collection of combinatorial data. Moreover, I wrote a program that looked for (very) low degree curves lying on the surfaces; this often finds a cycle if the Tate conjecture predicts one (presumable because I only looked at cases of low degree and with few monomials). But the example above is a case where my (silly) cycle finder didn’t find one. Can you find one?

Another explicit example is the hypersurface of degree 91 in P(7, 11, 16, 25) over F_2 defined by

x^13 + x^6y^3z + x^4y^2zw + x^3y^2z^3 + x^2yzw^2 + y^6w + yz^5 + zw^3 = 0

which has characteristic polynomial of frobenius on the primitive part of H^2 given by

x^12 + 2x^10 – 4x^9 – 14x^8 + 12x^7 – 64x^6 + 48x^5 – 224x^4 – 256x^3 + 512x^2 + 4096

and 2 is a root of this polynomial with multiplicity 2. Hence the Picard group should have rank 3. My cycle finder program suggests looking at the curve defined by

x^5y^2 + x^2yz^2 + z^2w = 0 and x^3z^3 + x^2y^5 + y^4w = 0

but even if this works (i.e., is independent of the hyperplace class) we still have to find yet another cycle in order to finish the proof of the Tate conjecture for this surface.

In other words, with current technology, there is no effort involved in making explicit examples where we know the Tate conjecture predicts something nontrivial. Even if we assume the Tate conjecture, we don’t know how to get our hands on these cycles. When you compute the matrix of Frobenius on the crystalline cohomology (as in Kiran’s algorithm) you are actually performing some polynomial operations such as raising to the pth power, taking derivatives, and (occasionally) dividing by p. In some sense these computations are “proving” cycles should exist. This motivates the idea, explained by previous two posts, that similar computations could provide hints as to _where_ to find the cycles too.