# The Weil conjectures : Curves

Our goal for today is to prove the following theorem:

Theorem 1: Let $X$ be a projective algebraic curve of genus $g$ and $F$ an endomorphism of degree $q$. Let $H^*$ be a reasonable cohomology theory. Then the action of $F^*$ on $X$ has eigenvalues which are algebraic integers, with norm $q^{1/2}$.

For those who know the term, “reasonable cohomology theory” means “Weil cohomology theory”.

The consequence of this theorem, which does not mention cohomology, is

Theorem 2: Let $X$ be a projective algebraic curve of genus $g$ and $F$ an endomorphism of degree $q$. Then there are algebraic integers $\alpha_1$, $\alpha_2$, …, $\alpha_{2g}$ with norm $q^{1/2}$ such that

$\displaystyle{ \# \mathrm{Fix}(F^k) = q^k - \sum \alpha_i^k +1 }$.

In a previous post, we established this in characteristic zero, by putting a positive definite hermitian structure on $H^1(X, \mathbb{C})$ such that $q^{-1/2} F^*$ became unitary. But, as I discussed last time, we can’t define $H^1(X, \mathbb{C})$ when $X$ has characteristic $p$. Instead, $H^1(X)$ will be defined over some other field of characteristic zero, like $\mathbb{Q}_{\ell}$. We will therefore need to know that the eigenvalues of $F$ are algebraic integers before we can even make sense of the statement that they have norm $q^{1/2}$.

It is possible to take the proof I present here and strip it down to its bare essentials, to give a proof of Theorem 2 which doesn’t even mention cohomology. See Hartshorne Exercise V.1.10. I am going to do the opposite; I will go slowly and focus on what each step is proving about $H^1(X)$. The essential argument here is Weil’s, although I have modernized the presentation.

# Characteristic zero analogues of the Weil conjectures: higher dimension

In our previous post, we proved

Theorem Let $X$ be a smooth projective curve over $\mathbb{C}$ and $F$ an endomorphism of degree $q > 0$. The eigenvalues of $F$ on $H^1(X)$ have norm $q^{1/2}$.

Today, we would like to generalize this to varieties of higher dimension. The obvious guess is

Nontheorem Let $X$ be a smooth projective variety, over $\mathbb{C}$, of dimension $d$. Let $F$ be an endomorphism of $X$ of degree $q^d$. The eigenvalues of $F$ on $H^r(X)$ have norm $q^{r/2}$.

This is not a theorem! I believe it is Serre who first figured out how to fix and prove this result. That is the topic of today’s post.

# Characteristic zero analogues of the Weil conjectures: Curves

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 $p$. 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 $X$ be a smooth projective curve over $\mathbb{C}$ and $F$ an algebraic endomorphism of degree $q > 0$. The eigenvalues of $F$ on $H^1(X)$ obey $|\alpha|=q^{1/2}$.

# Christol’s theorem and the Cartier operator

Let’s suppose that we want to compute $2^n \mod p$, and we have already been given $n$ written out in base $p$ as $n = \sum n_i p^i$. Here $p$ is a small prime and we want to do this conversation repeatedly for many $n$‘s.

Remember that $2^p \equiv 2 \mod p$ and thus $2^n \equiv 2^{n_0+n_1 p + n_2 p^2 + \cdots + n_{\ell} p^{\ell}} \equiv 2^{n_0} 2^{n_1} \cdots 2^{n_{\ell}} \mod p$. So, start with $1$, multiply it by $2^{n_0}$, then by $2^{n_1}$, then by $2^{n_2}$ and so forth. When you get to the end, read off $2^n$.

We can precompute the effect on $\mathbb{F}_p$ of multiplying by $2^k$, for $0 \leq k \leq p-1$. Then we can compute $2^n$ just by scanning across the base $p$ representation of $n$ and applying these precomputed maps to the finite set $\mathbb{F}_p$.

The precise way to say that this is a simple, one-pass, process is that it is a computation which can be done by a finite-state automaton. Here is the definition: let $I$, $S$ and $O$ be finite sets (input, states and output), and $s \in S$ (the start). For each $i \in I$, let $A(i)$ be a map $S \to S$. We also have a map $r:S \to O$ (readout). Given a string $(i_0, i_1, \ldots, i_{\ell})$ in $I^{\ell}$, we compute $r( A(i_{\ell}) \circ \cdots A(i_1) \circ A(i_0) (s))$. So our input is a string of characters from $I$, and our output is in $O$.

We can say that our example above shows that $(n_{\ell}, \ldots, n_1, n_0) \mapsto 2^{\sum n_i p^i} \mod p$ is computable by a finite-state automaton. (In our example, the sets $I$, $S$ and $O$ all have cardinality $p$, but I do not want to identify them.)

This is a special case of an amazing result of Christol et al: Let $f_n$ be a sequence of elements of $\mathbb{F}_p$. Then $(n_{\ell}, \ldots, n_1, n_0) \mapsto f_{\sum n_i p^i}$ can be computed by a finite-state automaton if and only if the generating function $\sum f_n x^n$ is algebraic over $\mathbb{F}_p(x)$!

We have just explained the case $\sum f_n x^n = 1/(1-2x)$. The reader might enjoy working out the cases $1/(1-x-x^2)$ (the Fibonacci numbers) and $(1 - \sqrt{1-4x})/2x$ (the Catalans).

In this post, I will use Christol’s theorem as an excuse to promote the Cartier operator, an amazing tool for working with differential forms in characteristic $p$.

# Grothendieck’s letter

Recently on meta.mathoverflow.net, Harry Gindi pointed out that Laszlo’s webpage for an edition of SGA 4 now contains the message

Alexandre Grothendieck a malheureusement souhaité que cessent les travaux de réédition de SGA. Les pages qui étaient consacrées sont donc closes.

It has since come to light that this request came in the form of a letter, which has been circulating in the French mathematical community for the last month. I include here a link to a typed version of that letter, vouching for neither its authenticity or accuracy, along with my pathetic attempt at translating it, for those few whose French is even worse than mine. Feel free to suggest better translations (I’ll incorporate them here).

Declaration of intent of non-publication

I do not intend to publish or republish any work or text of which I am the author, in any form whatsoever, printed or electronic, whether in full or in excerpts, texts of personal nature, of scientific character, or otherwise, or letters addressed to anybody, and any translation of texts of which I am the author. Any edition or dissemination of such texts which have been made in the past without my consent, or which will be made in the future and as long as I live, is against my will expressly specified here and is unlawful in my eyes. As I learn of these, I will ask the person responsible for such pirated editions, or of any other publication containing without permission texts from my hand (beyond possible citations of a few lines each), to remove from commerce these books; and librarians holding such books to remove these books from those libraries.

If my intentions, clearly expressed here, should go unheeded, then the shame of it falls on those responsible for the illegal editions, and those responsible for the libraries concerned (as soon as they have been informed of my intention).

Written at my home, January 3, 2010,

Alexandre Grothendieck.

UPDATE: I have replaced the letter with a corrected version sent to me by Prof. Illusie, and made some changes to the translation, suggested by several people.
UPDATE: You can now see the original handwritten letter..

# When fine just ain’t enough

If you use sheaves to study differential geometry, one of the basic lemmas you’ll want is the following: Let $X$ be a smooth manifold and let $\mathcal{E}$ be a sheaf of modules over $C^{\infty}(X)$. (For example, $\mathcal{E}$ might be the sheaf of sections of a vector bundle.) Then all higher sheaf cohomology of $\mathcal{E}$ vanishes.

The proof of this theorem is basically homological algebra plus the existence of partitions of unity. This gives rise to a slogan “when you have partitions of unity, sheaf cohomology vanishes.” One way to make this definition precise is through the technology of fine sheaves.

As Wikipedia says today, “[f]ine sheaves are usually only used over paracompact Hausdorff spaces”. That means they are not used when working with the Zariski topology on schemes, for example. When I started digging into this, I realized there were good reasons: The technology of fine sheaves (and the closely related technology of soft sheaves) does not include the scheme theory cases which we would want it to.

However, there are theorems of the form “when you have partitions of unity, sheaf cohomology vanishes” on schemes and on complex manifolds. I put up a question at MathOverflow asking whether there were better formulations that included these examples, but I probably didn’t formulate it well. I think spelling out all my issues would be too discursive for MathOverflow, so I’m bringing it over here.

# Chromatic Stable Homotopy Theory and the AHSS

Now that we’ve all gotten over the excitement surrounding the new iPad, I wanted to talk about something else which I actually find very exciting (unlike the iPad). This semester Jacob Lurie is giving a course on Chromatic Homotopy Theory. This is a beautiful picture which relates algebraic topology and algebraic geometry. Hopefully with Jacob at the helm we’ll also see the derived/higher categorical perspective creeping in. This seems like a great opportunity the learn this material “in my heart”, as my old undergraduate advisor used to say.

And with most of our principal bloggers distracted by MathOverflow, it also seems like a good time to experiment with new media. So here’s the plan so far:

• During lectures I’m going to be live-TeXing notes, which I’ll flush out and post to my website. (Jacob’s also posting his own notes!)
• In addition, I’ll try to post blog articles (like this one) about related topics or topics I find interesting/confusing.
• There might be a little MO action thrown in for fun.
• The offshoot is that today I want to talk a little about chromatic homotopy and about the Atiyah-Hirzebruch Spectral Sequence.

# Residues and Integrals

This post is about a computation every algebraic geometry student should do, but that none of my courses covered. Let $X$ be a smooth, projective curve over $k$. Then $H^1(X, \Omega)$ is a one dimensional $k$-vector space. If you’ve read Hartshorne III.7 carefully, you’ll remember that there is a canonical isomorphism: $Tr: H^1(X,\Omega) \to k$. Explicitly, let $p$ and $q$ be two points of $X$; consider the open cover $(X \setminus \{ p\}) \cup (X \setminus \{ q\})$ of $X$ and let $\omega$ be a holomorphic $1$-form on $X \setminus \{ p, q\}$. Let $c$ be the cocycle $X \setminus \{ p,q \} \mapsto \omega$. Then $Tr$ sends the cocycle $c$ to the residue, at $p$, of $\omega$. (A good question which I might ask on a qual one day: Why is it OK that this is asymmetric in $p$ and $q$?)

On the other hand, suppose that $k = \mathbb{C}$. Then $H^1(X, \Omega)$ is isomorphic to $H^{1,1}(X)$. An element of $H^{1,1}(X)$ is a $\overline{\partial}$-closed $(1,1)$ form, modulo $\overline{\partial}$-exact $(1,1)$-forms. But, because $X$ only has two real dimensions, this simplifies: Every $2$ form is a $(1,1)$-form and every $2$-form is closed because there are no $3$-forms. So an element of $H^{1,1}(X)$ is a $2$-form modulo $\overline{\partial}$-exact $2$-forms. It turns out that this is the same as a $2$-form modulo $d$-exact two forms. In other words, $H^{1,1}(X)$ is the same as the deRham cohomology group $H^2(X, \mathbb{C})$ that we learn about in differential geometry. And we know a canonical map $H^2(X, \mathbb{C}) \to \mathbb{C}$: Take the integral!

The point of this post is to compute the relation between $Tr$ and $\int$. I invite you to try it yourself, then meet me on the other side to see if we got the same answer.

UPDATE: I claimed earlier that this was easy to show $H^{1,1} \cong H^2$ for curves. As Akhil Mathew points out, it seems to only be easy to show that there is a well defined surjection $H^{1,1} \to H^2$. Since I only need the map to exist, I’ll leave it at that for now.

# A (partial) explanation of the fundamental lemma and Ngo’s proof

I would like to take Ben up on his challenge (especially since he seems to have solved the problem that I’ve been working on for the past four years) and try to explain something about the Fundamental Lemma and Ngo’s proof.  In doing so, I am aided by a two expository talks I’ve been to on the subject — by Laumon last year and by Arthur this week.

Before I begin, I should say that I am not an expert in this subject, so please don’t take what I write here too seriously and feel free to correct me in the comments.  Fortunately for me, even though the Fundamental Lemma is a statement about p-adic harmonic analysis, its proof involves objects that are much more familiar to me (and to Ben).  As we shall see, it involves understanding the summands occurring in a particular application of the decomposition theorem in perverse sheaves and then applying trace of Frobenius (stay tuned until the end for that!).

First of all I should begin with the notion of “endoscopy”.  Let $G, G'$ be two reductive groups and let $\hat{G}, \hat{G}'$ be there Langlands duals.  Then $G'$ is called an endoscopic group for $G$ if $\hat{G}'$ is the fixed point subgroup of an automorphism of $\hat{G}$.  A good example of this is to take $G = GL_{2n}$, $G' = SO_{2n+1}$.  At first glance these groups having nothing to do with each other, but you can see they are endoscopic since their dual groups are $GL_{2n}$ and $Sp_{2n}$ and we have $Sp_{2n} \hookrightarrow GL_{2n}$.

As part of a more general conjecture called Langlands functoriality, we would like to relate the automorphic representations of $G$ to the automorphic representations of all possible endoscopic groups $G'$.  Ngo’s proof of the Fundamental Lemma completes the proof of this relationship.

# The Fundamental Lemma

So, Noah came up to New England a few days ago, and at some point over dinner, the topic of Fields Medal candidates came up. Neither of us had any good ideas (sorry, anonymous grad student) but I mentioned that I had heard Bao Châu Ngô’s name quite a bit. The conversation then went roughly like this: Continue reading