Shioda cycles, I

Let X be a smooth projective geometrically irreducible surface over F_p where p is a prime. (For example a smooth hypersurface in P^3_{F_p}.) The second betti number of X is the dimension over Q_l of the \’etale cohomology group H^2 = H^2(\bar{X}, Q_l) where l is a prime different from p and where \bar{X} is the base change of X to the algebraic closure of F_p. Since X is projective the first chern class of an ample divisor is a nonzero element of H^2 which is an eigenvector for the action of the geometric frobenius of X with eigenvalue p. Assume that the second betti number of X is even. (For example a smooth hypersurface in P^3_{F_p} of even degree.) Since the geometric frobenius of X acts by multiplication with p^2 on H^4 and since the nondegenerate pairing H^2 x H^2 —> H^4 is compatible with the action of frobenius we see that the eigenvalues of geometric frobenius on H^2 which do not equal to +p or -p have to pair up: if λ occurs so does p^2/λ (with the same multiplicity). Since the second betti number is even, we conclude that, besides the eigenvalue p we found above, there is at least one more eigenvalue p or -p. Moreover, which of the two cases occurs depends on the sign of the determinant of the geometric frobenius acting on H^2.

In the situation above the Tate conjecture predicts the rank of the Picard group of X is equal to the multiplicity of p as a generalized eigenvalue of the geometric frobenius acting on H^2. (Of course this is just a very special case of the Tate conjecture.) Thus in the situation above there is a “50% chance” that the Picard group is strictly bigger than Z. In fact, in a paper of Nick Katz and myself we proved a precise version of this statement in some cases. Perhaps the most straightforward of these is the case of hypersurfaces:

Fix d even, d ≥ 4. In this case the percentage of smooth hypersurfaces X of degree 4 in P^3 over F_p where the multiplicity of p of an eigenvalue of geometric frobenius on H^2(X) is 2 tends to 50% as p tends to infinity.

One concludes that for p large enough at least 49% of the smooth hypersurfaces X have Picard number 2 (according to the Tate conjecture). Similarly, in at least 49% of the cases the Picard number is (proveably) 1. Finally, there is a remaining 2% of cases where the Picard number could be larger than 2.

Let’s say a Shioda cycle is an effective divisor on one of our X’s above which is independent of the hyperplane class in Pic and whose existence is predicted via the Tate conjecture by the parity considerations above. Now, if you assume the Tate conjecture, then there exist a lot of Shioda cycles (they exist on roughly 50% of all hypersurfaces over F_p of even degree). What I want to know is this:

Is there a pattern in these cycles?

More precisely, as we vary X in the family, could it be that there is a common recipe for the construction of a Shioda cycle for most of the 50% of the X’s where we expect one? I’ll say a bit more about this in a future post.

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