I have been discussing the Weil conjectures. In this post, I want to discuss one of the main difficulties in their proof. You should be able to follow this even if you have not been reading the earlier posts.
The main claim of the Weil conjectures can be summarized as follows:
Theorem Let be a smooth projective variety of dimension over . Then there are complex numbers , for and , and obeying , such that
The analogous claim in characteristic zero, which we proved last time, is
Theorem Let be a smooth projective variety over , with the hyperplane class. Let be an endomorphism of such that . Then there are complex numbers as above, such that
The way we proved this was to essentially establish:
Theorem Let , and be as above. Then there is a positive definite Hermitian inner product on such that is a unitary operator.
The required theorem then follows from the Lefschetz trace formula, and the fact that the eigenvalues of a unitary matrix have norm .
If we wanted to prove the characteristic theorem in the same way, we might hope to follow the same process: define , place a Hermitian structure on it, and prove a Lefschetz trace formula.
There are cohomology theories with Lefschetz trace formulas. But they do not have coefficients in . Their coefficient ring is (etale cohomology) or the ring of Witt vectors (various -adic theories).
In particular, while we can talk about the eigenvalues of , there will be no natural way to embed these eigenvalues in . There is an unnatural way: If we could prove that the characteristic polynomial of has coefficients in , then we could consider the roots of the same polynomial in . It would be enough to prove that these complex roots have norm . But it is unclear how to show that the polynomial has coefficients in , let alone how to bound the roots. In particular, it is unclear what could be the analogue of a Hermitian structure for a vector space.
In future posts, we will tackle these problems. For now, below the fold, I reproduce an argument of Serre showing that there is no way to define a reasonable cohomology theory for finite characteristic varieties, with coefficients in . (And therefore, not in any subring of either.)
An example of Serre
Let , for a prime which is . Let denote one of the square roots of in . Let be the elliptic curve over (including the point at infinity.)
Let be the automorphism of , and let be . Then , and . Let be the non-commutative ring . So , should such a thing exist, should be an -module, and a dimensional -vector space.
But is isomorphic to the quaternions. (Take and .) And any module for the quaternions has dimension divisible by , a contradiction.
The intriguing thing is that there is dimensional complex representation of . The action of and are given by
Every element of has integral trace and determinant in this representation. For any , we have
and the eigenvalues of have norm . It is just as if had a representation in which rescales a positive definite Hermitian form!
Those of you who like tensor categories may like to start thinking about how to axiomatize this example; it will come back when we get to motives.