The quest to prove the Weil conjectures drove algebraic geometry throughout the middle of the twentieth century. It was understood very early that a proof should involve creating a theory of cohomology for varieties in characteristic . This theory, known as étale cohomology, was developed by Grothendieck and his collaborators. Near the end, there was a period where étale cohomology was established but the hardest of the conjectures, the Riemann hypothesis, was not proved. Several mathematicians* proposed a path which would require proving results that were new even in the complex setting; results now known as the standard conjectures. That route was not taken; instead, Deligne found a different proof with its own insights and innovations.
This is the first of a series of posts sketching how the route through the standard conjectures would have gone. There is of course nothing original here; the usual sources are Kleiman, Grothendieck and Serre. I will try to write in a very classical way; I won’t even leave characteristic zero for the first two posts. And there will be tensor categories before the end!
In today’s post, we will prove the following theorem.
Theorem 1 Let be a smooth projective curve over and an algebraic endomorphism of degree . The eigenvalues of on obey .
Counting fixed points
By the Lefschetz trace formula, Theorem 1 has the following consequence:
Theorem 2 There are complex numbers , …, with norm such that
When we get around to proving a characteristic version of this theorem, we will be able to apply this with a curve over , and the Frobenius endomorphism. Then the fixed points of are the valued points of . So Theorem 2 will then be the Riemman hypothesis for curves.
One caveat: We need to count fixed points with multiplicity. For example, the automorphism of has a doubly fixed point at infinity. In the case of the Frobenius, all fixed points are multiplicity one.
I should acknowledge at the outset that there are easier ways to prove Theorem 1 in the complex case. If the genus of is or higher, then Riemann-Hurwitz shows that must be an automorphism, which must have finite order by a result of Hurwitz. So the result simply says that a matrix of finite order has eigenvalues with norm . Genus is easy, and we will see soon how to handle the case of genus . But I will present a harder proof, which will be closer to the route which works in any characteristic.
Let’s begin with the case where has genus . Theorem is already nontrivial; the homotopy classes of endomorphisms of are described by integer matrices. It is not obvious that the topology can’t be realized by a complex map.
Not obvious, but not that hard. is isomorphic to for some lattice . Any automorphism of lifts to multiplication by on for some . The lattice is naturally identified with . The action of on is multiplication by , with eigenvalues and . We have , so .
Now, let be a complex curve of genus . We first try to replicate the above argument; although we will have to modify it in the end.
The cohomology breaks up as a direct sum , where are the cohomology classes which can be represented (in deRham cohomology) by closed forms. There are no coefficients on the right hand side because only makes sense with complex coefficients. Composing
we get a rank lattice in a -dimensional complex vector space, and acts on everything.
Unfortunately, this isn’t enough. We get that is an element of which preserves a lattice, but that doesn’t restrict its eigenvalues very strongly. There exist examples** of curves for which there are symmetries of the pair which do not obey Theorem 1.
Here is the solution. We define a Hermitian inner product on by
This is easily shown to be positive definite. We note that
where the middle equality is because acts by on .
So is a unitary operator on ! The eigenvalues of a unitary operator have norm , and this completes the proof.
* These ideas appear in print in the papers of Grothendieck, Kleiman and Serre which I link above. But Grothendieck and Kleiman both imply that what they are recording was common knowledge already.
** Let be the genus two curve . This curve has an automorphic of order ; multiplying by a primitive fifth root of one. The action of on has eigenvalues , , and , where is a primitive fifth root of . Then preserves the splitting and the lattice , but the eigenvalues of are , for , and these do not all have the same norm.