This post won’t make sense if you haven’t read Shioda cycles, I.

Let X be a hypersurface of even degree d in **P**^3_{F_p} such that the determinant of the geometric frobenius acting on H^2 has a positive sign. Assuming the Tate conjecture (which we will do throughout this post), we can find our Shioda cycle by listing all the low degree curves in **P**^3_{F_p} and for each of them checking whether the curve lies on X and if so whether it gives a Shioda cycle. Now although this is a common recipe for finding a Shioda cycle if it should exist, it isn’t the kind of pattern I am looking for. (Moreover, you’d be hard pressed to argue that this recipe is uniform over all primes p because after all the lists will change with p.)

Now, I have a suggestion for a recipe that could work (which only means I can’t prove it doesn’t work). I am not saying or conjecturing that it does work, although I do have some very special cases where I can show that it works (basically families of surfaces related to families of abelian surfaces). A while ago I wrote a preprint about this (you can find it on my web page), but I think I can explain it here in a few sentences.

Namely, suppose that F = F(X_0, …, X_3) ∈ **Z**[X_0, X_1, X_2, X_3; A_I] is the universal polynomial of degree d, i.e., the coefficients A_I of F are variables where I = (i_0, i_1, i_2, i_3) with i_0 + i_1 + i_2 + i_3 = d. For every collection of values a = (a_I), a_I ∈ F_p we obtain a hypersurface X(a) in **P**^3_{F_p} by setting A_I equal to a_I in F. Now, suppose that we have a polynomial

G(X_0, …, X_3, Y_0, …, Y_3) ∈

Z[X_i; Y_j; A_I]

For each a = (a_I) as above we can consider the intersection of X(a) with

G(X_0, …, X_3, X_0^p, …, X_3^p)|_{A_I = a_I} = 0

i.e., we replace Y_j by X_j^p and A_I by a_I. Let’s call this intersection Z(a). Then my suggestion is to look for a Shioda cycle among the *irreducible components* of Z(a). In other words, given the even integer d, is there a polynomial G as above, such that, if X(a) is a surface which should have a Shioda cycle, then one of the irreducible components of Z_a is a Shioda cycle?

Actually in my write-up I (a) only require this to work in most of the cases where we expect a Shioda cycle, and (b) allow G also to depend on more variables which get replaced by X_j^{p^n}.

You might think it would be more natural to consider a system of polynomials such as G and ask them, after being mangled as above, to actually cut out a Shioda cycle. It seemed to me at the time of writing the preprint that this might be too strong a requirement, but I actually do not know how to disprove even this statement.

There are variant constructions we could use, e.g., we could allow variables Z_{ij} that get replaced by

(1/p)[(X_i + X_j)^p – X_i^p – X_j^p] mod p

if you know what I mean. The _meta_ question I have is whether anything like this can be true? Can you think of a (heuristic) argument showing this cannot work?

For example, if you could show that the (minimal) degrees of Shioda cycles tends to infinity rapidly with p then we would get a contradiction. However one can prove, assuming the Tate conjecture is true, an upper bound of the degree of Shioda cycles occuring in the family (unfortunately I don’t remember the shape of the formula I got when I worked it out) which shows this kind of argument won’t contradict my suggestion.

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