# Number of points modulo q is a stable birational invariant

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’).

References.

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.

# The Larsen–Lunts theorem

The Larsen–Lunts theorem is one of the most beautiful theorems I know. But first, let me recall some definitions.

Definition. The Grothendieck ring of varieties over a field is the free abelian generated by (formal) symbols for a variety over (which I will take to mean a geometrically reduced, separated scheme of finite type over ), subject to the relations

whenever is a closed immersion and . It becomes a ring by setting (exercise: show that this is well-defined). The class is called the Lefschetz motif.

Remark. Recall that a rational map is a morphism defined on some dense open . Varieties with rational morphisms form a category, and is called a birational map if it is an isomorphism in this category. We say that and are birational if there exists a birational map . If and are integral, this is equivalent to the equality of function fields .

We say that and are stably birational if is birational to for some . This is equivalent to the existence of an isomorphism

There are examples of stably birational varieties that are not birational.

Definition. Write for the set of stable birational classes of smooth proper varieties over . To avoid confusion, I shall denote the class of by . This set becomes a commutative monoid by setting (again: show that this is well-defined).

Theorem. (Larsen–Lunts) Let . There exists a unique ring homomorphism

such that for any smooth proper , the image of is . Moreover, the kernel of is the ideal generated by .

Proof (sketch). The map is constructed by induction on the dimension. For smooth proper , it is clear what should be (namely ). If is smooth, we can find a smooth compactification (using resolution of singularities). Then we set , where the right-hand side is defined by the induction hypothesis.

To check that it is independent of the compactification chosen, we need a strong form of weak factorisation: any two compactifications differ by a series of blow-ups and blow-downs along smooth centres disjoint from . Now if is the blow-up along a smooth centre with exceptional divisor , then is a -bundle over for some ; thus and are stably birational. Now well-definedness of the map on lower-dimensional varieties proves independence on the smooth compactification.

Finally if is singular, we simply set . After some further checks (like additivity and multiplicativity), this finishes the construction of .

Now clearly , since , and . Conversely, let . We can write any as

for certain smooth proper (we again use resolution here). Since is the free algebra on , we conclude that and after renumbering. Thus it suffices to consider the case for and smooth proper and stably birational (to each other). We may replace by since their difference is , which is in the kernel. Thus, we may assume and are birational.

Now by weak factorisation, we reduce to the case of a blow-up in a smooth centre . Let be the exceptional divisor, which is a -bundle over . Thus and differ by a multiple of , since .

Remark. The hard part of the theorem is the definition of the map. In order to define for not necessarily smooth and proper, we need to assume resolution of singularities (for this, a very mild version of resolution suffices). To check that it is independent of choices, we need the weak factorisation theorem (which in turn uses a very strong version of resolution of singularities). The computation of the kernel again uses resolution of singularities and weak factorisation.

This is why we restrict ourselves to . I suspect that it is also fine for arbitrary algebraically closed fields of characteristic .

Corollary. Let and be smooth proper. Then and are stably birational if and only if .

Proof. Since is the free algebra on , we have if and only if and are stably birational. The result is now immediate from the theorem.

Remark. If we knew weak factorisation (without knowing resolution), then one implication would follow immediately: if and are stably birational, then for some . Clearly is divisible by , so we may assume . Now by weak factorisation, a birational map factors as a chain of blow-ups and blow-downs along smooth centres, so we reduce to that case. But if has exceptional divisor , then is a -bundle over for some , hence is divisible by .

However, for the other implication there is no direct proof even if we knew weak factorisation.

In my next post, I will address a statement in positive characteristic (where neither resolution of singularities nor weak factorisation are currently known) that is related to the corollary (but much weaker).