The n-Category Café

I think the standard definition of the Euler factor at $p$ is that you take the characteristic polynomial of Frobenius at $p$ on the inertia-invariant subspace of the $\ell$-adic cohomology of the variety. Here’s a bit of explanation. The $\ell$-adic cohomology naturally comes with a linear action of the absolute Galois group, and the inertia subgroup means the inertia subgroup for any prime over $p$ (it doesn’t matter which), so when you take invariants, you get a well-defined Frobenius endomorphism. Also by ‘characteristic polynomial’, I mean in the graded sense, so you actually get a ratio of two polynomials.

It’s then an exercise to show that the Hasse-Weil zeta function agrees with the Dedekind zeta function. The key thing to know is that if $K$ is a finite Galois extension of $\mathbf{Q}$, then the $\ell$-adic cohomology of $\mathrm{Spec}(K)$ is trivial in positive degrees and is just the regular representation of $\mathrm{Gal}(K/\mathbf{Q})$ in degree zero. Then you need a bit of basic algebraic number theory and representation theory of finite groups. (NB I didn’t check it when $K$ is not Galois over $\mathbf{Q}$. It’s probably true and the proof is probably essentially the same.)

There are actually some problems with the Hasse-Weil zeta function at bad primes. For instance the zeta function of a product of varieties (over $\mathbf{Q}$) is not determined by the zeta functions of their factors. For disjoint unions, it’s OK. Probably, the right way to solve this is by working with varieties $X$ over $\mathbf{Z}$, rather than varieties over $\mathbf{Q}$. Then the good definition of the zeta function would just be the Euler product of the local zeta functions of the mod $p$ reductions $X_p$ (aka ‘fibers over $p$’) of $X$. For a bit more on this, see this.

With this definition, you can even forget about etale cohomology. In fact, you don’t even really have to know what a variety $X$ over $\mathbf{Z}$ means, because all you’re using is the fibers $X_p$. (But if you pick a random family of varieties $X_p$, each defined over $\mathbf{Z}/p\mathbf{Z}$, then it will almost never come from a variety over $\mathbf{Z}$. And in that case, the zeta function you get won’t have any good properties, because the fibers might not have anything to do with each other. But as long the varieties $X_p$ look like they’re all mod $p$ reductions of a single variety, you’ll probably be OK.)

It might be worth rewriting the Dedekind picture in the more general Hasse-Weil language. I realize this is the opposite of what you asked for, but I don’t know how to do what you asked for. There’s going to be a trade off between efficiency and being categorically correct. I’m going to try to stay on the efficient side.

Let $K$ be a finite extension of $\mathbb{Q}$, and let $X = \mathrm{Spec} \mathcal{O}_K$. Let $G$ be the Galois group of “the” Galois closure of $K$, and $H$ the stabilizer of $K$.

So the $\overline{\mathbb{Q}}$ points of $X$ form a finite set, of cardinality $\deg K$. The group $G$ acts on $X(\overline{\mathbb{Q}})$, and the stabilizer of any point is conjugate to $H$. For any prime $p$, $X(\overline{\mathbb{F}_p})$ is also a set of size $\deg K$. It comes with a natural action of the Frobenius $\mathrm{Frob}_p$.

Noncanonically, here is how this data fits together. For an unramified prime $p$, there is a conjugacy class $\Phi_p$ in $G$, and a bijection between $X(\overline{\mathbb{Q}})$ and $X(\overline{\mathbb{F}_p})$, so that the action of any element in the conjugacy class $\Phi_p$ goes to $\mathrm{Frob}_p$.

For a ramified prime, there are two subgroups $I_p$ and $D_p$ of $G$, so that $I_p$ is normal in $D_p$ and $D_p/I_p$ is cyclic with a canonical generator $\Phi_p$. More precisely, all of this data exists up to common conjugacy in $G$. The points $X(\overline{\mathbb{F}_p})$ are in bijection with $I_p \backslash X(\overline{\mathbb{Q}})$, with the $\mathrm{Frob}_p$ action gong to the $\Phi_p$ action.

I realize that I have been very sloppy about canonicalness here. But let me try to push ahead. So far, I have discussed how to talk about $X(\overline{\mathbb{Q}})$ and $X(\overline{\mathbb{F}_p})$ in terms of groups acting on finite sets. Now, I want to turn this into statements about $\zeta$ functions and representation theory.

Let $V_p$ and $V_{\mathbb{Q}}$ be short for $H^0(X \times \overline{\mathbb{Q}})$ and $H^0(X \times \overline{\mathbb{F}_p})$. These are vector spaces of dimension $\deg K$, with actions of $G$ and $\mathrm{Frob}_p$ respectively. Again noncanonically, $V_{\mathbb{Q}}$ is the permutation representation of $G$ on $G/H$. For an unramified prime $p$, $V_{p}$ is that representation restricted to the cyclic subgroup $\langle \Phi_p \rangle$.

In terms of $\zeta$ functions, the Euler factor at $p$ is $\det ( 1- \Phi_p p^{-s})$.

When $p$ is ramified, the vector space $V_p$ is smaller than $\deg K$ dimensional. Specifically, it can be viewed as the subspace of $V$ on which $I_p$ acts trivially. (Or the quotient? With all the noncanonicalness around, I don’t see a principled way to decide.) Since $I_p$ is normal in $D_p$, it is well defined to talk about the action of $\Phi_p$ on $V_p$. We’ll write $\Phi_p|_{V_p}$ to emphasize that this action only makes sense on a subspace of $V$.

And, the key point, the Euler factor is still given by $\det ( 1- \Phi_p|_{V_p} p^{-s})$. As I understand it, the situation for higher varieties is that the Euler factor for ramified primes HAS to be defined in this representation theoretic way, because people don’t know a point counting way to think about it.

Some speculative thoughts: Let $X$ be a smooth variety over $\mathbb{Q}$. The Hasse-Weil $\zeta$ function only depends on $X$, and in particular on the Galois action on $H^*(X \times \overline{\mathbb{Q}})$.

To count points on $X$, you have to choose a model, $\mathcal{X}$, for $X$ over $\mathbb{Z}$. And this choice can change the answer! For example, suppose that $X$ is an elliptic curve with split multipliciative reduction at $p$. If you choose a model where the fiber over $p$ is a nodal cubic, then there are $p$ points ($1$ node and $p-1$ smooth points); if you choose a model where the fiber is a Neron $k$-gon then there are $kp$ points.

With elliptic curves, there is a canonical “best” model; the Neron model. Even when dealing with number fields, I could obstinately insist on working with an order in $K$, rather than the whole of $\mathcal{O}_K$. (For example, working with $\mathbb{Z}[\sqrt{12}]$ inside $\mathbb{Q}[\sqrt{3}]$, rather than the correct $\mathbb{Z}[\sqrt{3}]$. This would throw off my point counts at $2$). But, when the dimension of $X$ is $2$ or higher, there is no best choice of model.

So, any point counting interpretation of the Hasse-Weil zeta function would have to be something which didn’t see the choice of model. Such objects come up in the theory of motivic integration. Maybe some of the experts know whether there is a way to see the Hasse-Weil zeta function in this context?

Dear John,

What James is alluding to at the end of his post is that if you don’t try to define the Hasse–Weil zeta-function for varieties over $\mathbf Q$, but instead think of it as a notion for varieties over $\mathbf Z$, then the local factors are given by point counting: namely you form the zeta-function of the fibre $X_p$ for each prime $p$ — which is a point-counting thing , then replace the variable $T$ in the zeta-function by $p^{-s}$ (if you use the form of the zeta-function for varieties in char. $p$ which has a variable $T$; if you use the definition which already has $p^{-s}$ in it, then leave it alone!), then multiply all of these together.

This is not the conventional way that things are done, though. Typically, one has a variety over $\mathbf Q$ (or more generally an algebraic number field, but let me stick to the case of $\mathbf Q$), and wants to define its Hasse–Weil zeta-function. Ideally, one would want to define some sort of “minimal model” of $X$ over $\mathbf Z$, and then apply the formalism of the preceding paragraph. But the theory of bad reduction of varieties over number fields is not well-understood (what one would like would be “semi-stable reduction”-type theorems, which are closely related to the problem of resolution of singularities. In positive char., one doesn’t yet have a general analogue of Hironaka’s theorem on resolution of singularities, and the situation in mixed characteristic (i.e. for varieties over $\mathbf Z$) is just as bad, if not worse).

So, rather than using concrete geometry at bad primes, and then point counting, one uses the crutch of etale cohomology. (This is a frequently used crutch — often there is concrete geometry that one would like, but which is unknown, but the cohomological avatar of the geometry is somewhat better understood; think of the Hodge conjecture, say. As always seems to be the case, though, a price is paid for taking this short-cut: you’ll see below that I have to invoke various conjectures for everything to make sense in the end.)

The recipe is as follows: one takes the $\ell$-adic cohomology of $X$ for some prime $\ell$ (conjecturally, it shouldn’t matter which one, although I don’t think this is known in general — another failure of our concrete geometric knowledge, and another sign that we are using a crutch!). This gives a finite-dimensional $\mathbf Q_{\ell}$-vector space with an action of the Galois group of $\overline{\mathbf Q}$ over $\mathbf Q$. Its dimension is just the same as the usual (singular, say) cohomology of $X$ over the complex numbers with $\mathbf Q$ (or $\mathbf R$, or $\mathbf C$) coefficients.

For each prime $p$, and each degree $i$ of cohomology, we take the invariants under $I_p$ (the inertia group at $p$); this will be the whole thing if $p$ is a prime of good reduction, but can be (and often is) smaller otherwise. On this space we have an action of the Frobenius $Frob_p$. We take the char. poly. of $Frob_p^{-1}$ (the exponent $-1$ appears for technical reasons, to make everything work out well when you compare with the point-counting picture at good reduction primes), call this $P_{i,p}(T)$. (Remember that $i$ is the degree of cohomology.)

The $\zeta$-function is then defined to be $$\prod_{p,i}P_{i,p}(p^{-s})^{(-1)^i}.$$ It might help to break the product up into two products: $$\prod_{p} \prod_i P_i(p^{-s})^{(-1)^i}.$$ The inner product is finite(it just runs over $i = 0$ to $2 dim(X)$), and its gives the Euler factor for each prime $p$. The outer product is now over all primes, and is the Euler product for the $\zeta$-function of $X$.

(There are more conjectures that need to hold to make sure this makes sense, since a priori the coefficients of the $P_{i,p}$ are $\ell$-adic numbers, but we are plugging in the complex number $p^{-s}$! What happens is that at the good reduction primes, by using the Lefschetz fixed point formula, one finds that the Euler factors at good reduction primes are just given by point counting, and in particular have integral coefficients rather then just $\ell$-adic ones, so it is okay to plug in $p^{-s}$. This is conjectured to be true at bad primes too, but is not know in general, I think. It is related to the independence-of-$\ell$ conjecture I mentioned above.)

Finally, the Dedekind $\zeta$-function corresponds to the case $X$ is a point, which has good reduction everywhere (!), which is why one doesn’t need to tell this story.

One last (even more) technical remark, related to the preceding paragraph: if one replaces $\mathbf Q$ by a number field $K$, there are two ways to proceed. (1) One can just work over $K$, and replace the rational primes $p$ considered above by primes $q$ of $K$. (2) One can apply restriction of scalars to think of $X$ over $K$ as a variety $X'$ over $\mathbf Q$. (Restriction of scalars, which sometimes goes under the even lengthier name of “Weil restriction of scalars” is just algebraic-geometry speak for the process via which you can think of a complex manifold as also being a real manifold of twice the dimension.) One can then apply the above story to $X'$.

It turns out that processes (1) and (2) give the same answer. When you were collecting your terms in the Dedekind $\zeta$-function of $K$ for all the $q$ of $K$ dividing a given rational $p$, you were implicitly moving from point-of-view (1) to point-of-view (2).

Now if we take a point over $K$, and take point-of-view (2), we get a variety over $\mathbf Q$ which is a finite union of points ($[K:\mathbf Q]$ many), which are however not defined over $\mathbf Q$. Concretely, if $K = \mathbf Q(\alpha)$ and $f(X)$ is the minimal polynomial of $\alpha$ over $\mathbf Q$, then $X'$ is the variety $f(X) = 0$.

Now $X'$ won’t have good reduction everywhere; it has bad reduction at primes $p$ that are ramified (concretely, because at these primes, not matter what $\alpha$ you choose, some roots of the corresponding $f$ will come together mod $p$). So the above story does apply! (But of course, in the zero-dimensional case one doesn’t need to use all the cohomological machinery.)

How do you see this concretely? Well, at the ramified $p$, the Euler factor of the zeta-function of $K$ has smaller degree (i.e. degree less than $[K:\mathbf Q]$) than at the unramified primes.

After posting my preceding comment, I opened Minhyong’s article, to discover that it exactly recalls the basic definitions from Serre’s 1965 paper, and then goes on to discuss much more (including the issue of varieties over $\mathbf Q$ vs. schemes over $\mathbf Z$). I highly recommend it!

Dear John,

It occurred to me that I should have said something about *why* people work with varieties over $\mathbf Q$ rather than $\mathbf Z$, and why they attempt to define the Hasse–Weil zeta function in this context.

The point is that there is a good reason to expect that the variety over $\mathbf Q$ should canonically determine *all* its Euler factors, including those at the bad primes, without having an explicit model over $\mathbf Z$ in hand, as I will explain in a moment. (Of course, at primes of good reduction, these Euler factors will be given by point counting, but even at bad reduction primes there should be something absolutely canonical, which is what the etale cohomology definition hopes to capture.)

The good reason is as follows: one expects, when you have these canonical Euler factors, that the Hasse–Weil zeta function should have a beautiful, Riemann zeta functionesque functional equation under $s \mapsto d+1 - s$ (where $d$ is the dimension of the variety over $\mathbf Q$; or if you prefer, you can think of $d+1$ as being the absolute dimension of the variety — i.e. we include one more dimension because Spec $\mathbf Z$ has dimension one).

Now if you have a zeta function with such a functional equation (just involving Gamma factors, powers of $\pi$, and such) and you modify the Euler factor at a finite number of primes, you are going to screw things up; your modified zeta function won’t have quite such a nice functional equation. So, given that we *do know* what to do at all the good reduction primes (which is to say, at all but finitely many primes), the requirement of having a good functional equation (assuming it can be satisfied — and the conjecture that it can be satisfied in general is one of the biggest open conjectures in number theory) pins down the Euler factors at *every* prime.

What’s more, for various (well-motivated) reasons, we expect that the etale cohomology definition does give these correct Euler factors.

Paulo Almeida wrote a very nice intro on Connes’ ideas conc. number theory and RH.

Although I’m far from being a specialist of number theory, I would like to know whether the Hasse-Weil zeta function of a rationnal elliptic curve is always a primitive function of the Selberg class (possibly modulo some good renormalization) or not.
Thank you in advance.