This post is about a (very weak) shadow in characteristic of the Larsen–Lunts theorem. See my previous post for the statement and sketch of the proof of Larsen–Lunts.
Remark. In characteristic , we do not even know the weakest form of resolution of singularities (e.g. find a smooth proper model for any function field). Thus, we certainly do not know the Larsen–Lunts theorem. However, we can still try to prove corollaries (and if they fail, we know that resolution must fail).
Today, I want to talk about the following statement:
Theorem. (Ekedahl) Let . Let and be smooth proper varieties, and assume and are stably birational. Then .
Remark. This would follow immediately from Larsen–Lunts if we knew a sufficiently strong form of resolution of singularities. Indeed, the map
given by counting -points modulo factors through since . Hence, by Larsen–Lunts, it factors through .
It turns out that the theorem is true without assuming resolution of singularities, and the proof is due to Ekedahl (although in his paper he never explicitly states it in this form). The reader should definitely check out Ekedahl’s article (see references below), because his proof is more beautiful than the one I present here, and actually proves a bit more.
We will need one fairly deep theorem:
Theorem. Let be a variety of dimension over . Let be an eigenvalue of Frobenius on . Then and are both algebraic integers.
The first part (integrality of ) is fairly well-known. For the second part (integrality of ), see SGA 7, Exp. XXI, Corollary 5.5.3(iii).
The statement that appears in Ekedahl’s article is the following:
Theorem. (Ekedahl’s version) Let . Let and be smooth connected varieties (not necessarily proper!), and assume and are birational. If is an eigenvalue of Frobenius on which is not an eigenvalue on , then is divisible by .
This statement should be taken to include multiplicities; e.g. a double eigenvalue for which is a simple eigenvalue for is also divisible by . By symmetry, we also get the opposite statement (with and swapped). Thus, the eigenvalues (with multiplicities) that are not divisible by are the same for and .
Proof. We immediately reduce to the case where is an open immersion, with complement . We have a long exact sequence for étale cohomology with compact support:
If is an eigenvalue on some , then is an algebraic integer (see above). Hence, for any valuation on with , we have . We conclude that the eigenvalues for which some valuation is on and agree. Hence, by Poincaré duality, the eigenvalues of and for which some valuation is agree. These are exactly the ones that are not divisible by .
The theorem I stated above immediately follows from this one:
Proof. Since , we may replace by . Thus, we can assume and are birational; both of dimension .
By the Weil conjectures, we know that
where the inner sum runs over all eigenvalues of Frobenius. If we reduce mod , then we only need to consider eigenvalues that are not divisible by . By Ekedahl’s version of the theorem, the set (with multiplicities) of such are the same for and .
Historical remark. Although the theorem above was essentially proven in 1983 (but not explicitly stated), a separate proof for threefolds appeared in a paper by Gilles Lachaud and Marc Perret in 2000. It uses Abhyankar’s results on resolution of singularities, and is much closer to the proof of Larsen–Lunts than Ekedahl’s proof was. In 2002, Bruno Kahn provided a different proof for the general case using some (fairly advanced?) motive machinery (‘almost without cohomology’).
Torsten Ekedahl, Sur le groupe fondamental d’une variété unirationelle. Comptes rendus de l’académie des sciences de Paris, Serie I: mathématiques, 297(12), p. 627-629 (1983).
Bruno Kahn, Number of points of function fields over finite fields. arXiv:math/0210202
Gilles Lachaud and Marc Perret, Un invariant birationnel des variétés de dimension 3 sur un corps fini. Journal of Algebraic Geometry 9 (2000), p. 451-458.