6
$\begingroup$

It was already known to Weil that a sufficiently reasonable cohomology theory for algebraic varieties over $\mathbb{F}_p$ would allow for a possible solution to the Weil conjectures.

It was also understood that such a cohomology theory could not take values in vector spaces over either the rational numbers $\mathbb{Q}$ or the $p$-adic numbers $\mathbb{Q}_p$. The classical argument that I know is that the cohomology of a supersingular elliptic curve should admit an action by a quaternion algebra structure of the endomorphism group of the elliptic curve, and representation theory rules out such an action. (Side question: Due to whom is this reasoning, and was it known to Weil's generation?)

This reasoning does not, however, rule out the possibility of cohomology with coefficients in $\mathbb{Q}_\ell$ with $\ell \neq p$, and indeed $\ell$-adic cohomology would end up being found.

Question. Were there any other historical reasons to believe that looking at cohomology valued in $\mathbb{Q}_\ell$-vector spaces would lead to something fruitful?

N.B. This question was posted on MathSE, but I got advised to post it here instead.

$\endgroup$
4
  • 6
    $\begingroup$ The argument with supersingular elliptic curves and quaternion algebras is due to Serre, I believe. $\endgroup$ Commented Jan 30, 2020 at 14:03
  • 1
    $\begingroup$ (See 1.7, p. 315 in Grothendieck's "Crystals and the de Rham cohomology of schemes", "As Serre has pointed out...") $\endgroup$ Commented Jan 30, 2020 at 14:31
  • 7
    $\begingroup$ It is not true that etalé cohomology with coefficients in $\mathbb{Q}_\ell$ is a "good" cohomology. $\ell$-adic cohomology is something more involved that just "cohomology with coefficients": its is the projective limit of some with finite coefficients (and tensoring with $\mathbb{Q}$). $\endgroup$
    – Xarles
    Commented Jan 30, 2020 at 14:53
  • 1
    $\begingroup$ @Xarles That is of course what I meant. It takes values in $\mathbb{Q}_\ell$-modules, anyway. I'll edit the question accordingly. $\endgroup$ Commented Jan 30, 2020 at 15:55

2 Answers 2

6
$\begingroup$

One historical reason for considering $\ell$-adic cohomology, not completely disconnected from the example you introduce, is that for a curve over a field, we get a natural Galois representation by taking the $\ell$-adic Tate module of the Jacobian (i.e., the projective limit of $\ell$-power torsion). Furthermore, if such a curve is defined over a subfield of the complex numbers, then the rank of the Tate module is equal to the rank of the classical degree 1 cohomology of the complexified curve. We now know that the Tate module is naturally dual to étale cohomology in degree 1.

One might then reasonably hope for a similar relationship between the higher degree classical cohomology of higher dimensional varieties and certain $\ell$-adic Galois representations naturally attached to the variety.

$\endgroup$
3
$\begingroup$

I think an important motivation for the $\ell$-adic theory comes from the Riemann existence theorem/the Grauert–Remmert theorem. This says that a finite (topological) covering space $Y \to X$ of a normal complex variety can again be equipped with an algebraic structure, which is more or less what you need to prove to obtain $$\pi_1^{\operatorname{\acute et}}(X) = \widehat{\pi_1^{\operatorname{top}}(X(\mathbf C))}.$$ So finite covering spaces can be detected using the étale topology (except it's not really a topology, but that's not stopping Grothendieck!).

In particular, you expect to get a good theory of étale cohomology with finite coefficients. But to run the arguments that Weil was dreaming of, you need characteristic $0$ coefficients, so what do you do? Just take a limit!

I think that's really the explanation of why the adic formalism enters the picture. But it doesn't quite explain why it's different at the prime $p$. There are a few ways to look at this:

  • Serre's argument shows that a $\mathbf Q_p$-valued Weil cohomology theory cannot exist (over any ground field $k$ containing $\mathbf F_{p^2}$; so in particular for $k = \bar{\mathbf F}_p$).
  • The basic results on $\ell$-adic étale cohomology take the Galois cohomology of function fields of curves over algebraically closed fields as a starting point, and these behave differently at the prime $p$.
  • Already for elliptic curves, the $p$-adic Tate module behaves a little different from the $\ell$-adic one.

In the end it doesn't really matter, because all they needed was one Weil cohomology theory.

$\endgroup$
9
  • $\begingroup$ Serre only shows that there is no $\mathbf{Q}$-linear Weil cohomology, at least if we restrict to algebraic varieties over a field which contains $\mathbf{F}_{p^2}$. If we work over $\mathbf{F}_p$, crystalline cohomology actually is a $\mathbf{Q}_p$-linear Weil cohomology, and I am not aware of an argument which rules out the existence of $\mathbf{Q}$-linear one. Your assertion that "it does not matter" can also be discussed: independence of $\ell$ problems have a long history which seem to point in a rather different direction. $\endgroup$ Commented Jul 3, 2020 at 12:17
  • 1
    $\begingroup$ It might also be worth mentionning that the standard conjectures together with tannakian arguments imply the existence of a $\bar{\mathbf{Q}}$-linear Weil cohomology. $\endgroup$ Commented Jul 3, 2020 at 12:20
  • $\begingroup$ @Denis-CharlesCisinski thanks for your comments. Serre's argument depends on representations of the quaternion algebra over $\mathbf Q$ that splits at $p$ and $\infty$, so it equally rules out $\mathbf Q_p$- and $\mathbf R$-valued Weil cohomology theories (as soon as $k \supseteq \mathbf F_{p^2}$). $\endgroup$ Commented Jul 3, 2020 at 21:41
  • $\begingroup$ Also, the original question is about "historically, why $\mathbf Q_\ell$", and my point is that if you're trying to prove the Weil conjectures you only need one cohomology theory. (Of course I am aware of the interesting "independence of $\ell$" questions ― I even have a paper on it!) $\endgroup$ Commented Jul 3, 2020 at 21:43
  • $\begingroup$ Crystalline/rigid cohomology for $k$-varieties take values in $W(k)$. So $\mathbf{Q}_p$ is a possible field of coefficient for $k=\mathbf{F}_p$. $\endgroup$ Commented Jul 4, 2020 at 7:38

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .