I am interested in the idea of a cohomological interpretation of the Riemann hypothesis (suggested by Deninger/Connes).

I am a beginner in étale cohomology, and I would like to ask the following

Question. Why does étale cohomology not offer a cohomological interpretation of the local zeta function of the scheme $\operatorname{Spec} \mathbb{Z}$ in terms of the (étale?) cohomology of the associated topological space?

I would appreciate any answer as well as a reference. I understand this question may be a bit annoying since the obstructions should supposedly be immediate for if otherwise this would probably be well known. Nonetheless, I am curious and do not know who to ask.

  • 6
    $\begingroup$ Coming from someone who knows very little about these things, I believe the main obstacle is absence of any (obvious) notion of Frobenius. $\endgroup$
    – Wojowu
    Mar 22 at 13:36
  • 11
    $\begingroup$ Etale cohomology works only with $\ell$-adic coefficients, while zeta functions take complex values. Over finite fields, you can cheat and choose an isomorphism $\overline{\mathbb Q}_\ell\cong \mathbb C$, but this could only work as zeta functions are in fact algebraic functions (of $p^{-s}$). $\endgroup$ Mar 22 at 14:54
  • $\begingroup$ What do you mean by "associated topological space"? I'd guess that space is the same for all curves over countable algebraically closed fields so it probably doesn't know all that much. $\endgroup$
    – Faris
    Mar 26 at 11:06

One can't give a complete answer to this question without first understanding how etale cohomology does give a cohomological interpretation of the zeta function in the function field case.

Let $X$ be a smooth projective curve over a finite field $\mathbb F_q$. Then

$$ \zeta_{X}(s) = \frac{ \det (1 - \operatorname{Frob}_q q^{-s}, H^1(X_{\overline{\mathbb F_q}}, \mathbb Q_\ell))}{\det (1 - \operatorname{Frob}_q q^{-s}, H^0(X_{\overline{\mathbb F_q}}, \mathbb Q_\ell)) \cdot \det (1 - \operatorname{Frob}_q q^{-s}, H^2(X_{\overline{\mathbb F_q}}, \mathbb Q_\ell))}$$

It is crucial that, to get this formula, one passes to the curve $X_{\overline{\mathbb F_q}}$ obtained by base-changing to the algebraic closure. The reason this is important to note is that there is no analogue of "the algebraic closure of the base finite field" after replacing $X$ with $\operatorname{Spec} \mathbb Z$, at least not in the usual world of schemes.

An optimistic guess is that it suffices to take the etale cohomology of $\operatorname{Spec} \mathbb Z$. If that were true, then it would presumably likewise suffice in the function field world to take the etale cohomology of $X$, without base-changing to the algebraic closure. What happens when we do this?

We can see this using the long exact sequence

$$ \to H^i(X, \mathbb Z_\ell) \to H^i (X_{\overline{\mathbb F_q}}, \mathbb Z_\ell) \to H^i (X_{\overline{\mathbb F_q}}, \mathbb Z_\ell) \to$$

where the third arrow denotes the map $\operatorname{Frob}_q-1$. Examining this exact sequence and the known description of the action of Frobenius on $H^0$ and $H^2$, we see that

$$H^0 (X, \mathbb Z_\ell) = \mathbb Z_\ell$$ $$ H^1 (X, \mathbb Z_\ell)= \mathbb Z_\ell$$ $$H^2(X, \mathbb Z_\ell) = H^1(X_{\overline{\mathbb F_q}}, \mathbb Z_\ell)/ (1 - \operatorname{Frob}_q $$ $$H^3(X, \mathbb Z_\ell) = \mathbb Z_\ell/ (q-1)$$

The only interesting cohomology group here is $H_2$, which is a finite abelian $\ell$-group. Its order is the maximum power of $\ell$ dividing the determinant of $\operatorname{Frob}_q-1$ acting on $H^1$. Equivalently, this is the maximum power of $\ell$ dividing the residue of the zeta function of $X$ at $s=1$.

Combining this information for all primes $\ell$, we have a cohomological interpretation of the residue of the zeta function at $s=1$ (at least up to a power of $p$).

Similarly, over rings of integesr of number fields, not necessarily $\mathbb Z$, we can obtain a cohomological intepretation of the residue of the zeta function. This cohomology group turns out to be dual to the $\ell$-part of the class group, so this is just the Dirichlet class number formula.

There is one last thing you can do here. The field of functions on $X_{\overline{\mathbb F_q}}$ is a Galois extension of the field of functions on $X$, with Galois group $\prod_p \mathbb Z_p$. We can just pick an extension of $\mathbb Q$ with Galois group $\prod_p \mathbb Z_p$, or even just $\mathbb Z_p$, pretend that this is $X_{\overline{\mathbb F_q}}$, and take etale cohomology of its ring of integers.

This, modulo technical details, is the approach of Iwasawa theory, which gives a cohomological interpretation of $p$-adic $L$-functions. Some $p$-adic $L$-functions are related to special values of the Riemann zeta function at negative integers, so there is, in a sense, a cohomological interpretation of these specific values.

You will note some synchrony here with Peter Scholze's point in the comments that etale cohomology is only suitable for constructing $\ell$-adic $L$-functions, and is only able to construct complex-analytic ones in the function field setting by a trick using the fact that the $L$-functions are polynomials.


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