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July 28, 2010

Zeta Functions: Dedekind Versus Hasse-Weil

Posted by John Baez

I’m a bit confused. It serves me right: I’ve been trying to learn too much math from Wikipedia articles.

I think I understand the Dedekind zeta function of a number field. You can write it as a product over prime ideals in the ring of algebraic integers for that field… and you can group that product to make it into a product over ordinary primes. This is called the ‘Euler factorization’ of the zeta function.

I also have some rough understanding of the Hasse–Weil zeta function of an algebraic variety defined over an algebraic number field. It too has an Euler factorizatoin. For each ordinary prime we get a factor called a local zeta function, and for the primes that are unramified there’s an easy formula for this local zeta function. But the Wikipedia article mumbles rather portentuously when it comes to the ramified primes, emitting smoke and lightning but not (as far as I can tell) a precise formula…

Namely, it says:

A more refined definition became possible with the development of étale cohomology; this neatly explains what to do about the missing, ‘bad reduction’ factors. According to general principles visible in ramification theory, ‘bad’ primes carry good information (theory of the ‘conductor’). This manifests itself in the étale theory in the Ogg–Néron–Shafarevich criterion for good reduction; namely that there is good reduction, in a definite sense, at all primes pp for which the Galois representation ρ\rho on the étale cohomology groups of VV is ‘unramified’. For those, the definition of local zeta function can be recovered in terms of the characteristic polynomial of

ρ(Frob(p)), \rho(\operatorname{Frob}(p)) \, ,

Frob(p)\operatorname{Frob}(p) being a Frobenius element for pp. What happens at the ramified pp is that ρ\rho is non-trivial on the inertia group I(p)I(p) for pp. At those primes the definition must be ‘corrected’, taking the largest quotient of the representation ρ\rho on which the inertia group acts by the trivial representation. With this refinement, the definition of the zeta function Z(s)Z(s) can be upgraded successfully from ‘almost all’ pp to all pp participating in the Euler product.

What I’d like are two things:

  • A precise ‘formula’, in as lowbrow terms as possible, for the local zeta function at a ramified prime, and
  • a little story about why we don’t see such elaborate hand-wringing about ramified primes in the case of the Dedekind zeta function… which should be a special case, no?

Also: what’s a good place to learn about the Hasse–Weil zeta function, including the nuances that arise at ramified primes?

I guess I’d prefer a reference that doesn’t say “étale cohomology” every other sentence: I like sheaves and sites and all that jazz, but right now I’m sort of looking to compute the Hasse-Weil zeta functions for some incredibly simple examples, as simply as possible. But I want to get the ramified primes right.

Posted at July 28, 2010 7:01 AM UTC

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Re: Zeta Functions: Dedekind Versus Hasse-Weil

I think the standard definition of the Euler factor at pp is that you take the characteristic polynomial of Frobenius at pp 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 pp (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 KK is a finite Galois extension of Q\mathbf{Q}, then the \ell-adic cohomology of Spec(K)\mathrm{Spec}(K) is trivial in positive degrees and is just the regular representation of Gal(K/Q)\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 KK is not Galois over Q\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 Q\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 XX over Z\mathbf{Z}, rather than varieties over Q\mathbf{Q}. Then the good definition of the zeta function would just be the Euler product of the local zeta functions of the mod pp reductions X pX_p (aka ‘fibers over pp’) of XX. 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 XX over Z\mathbf{Z} means, because all you’re using is the fibers X pX_p. (But if you pick a random family of varieties X pX_p, each defined over Z/pZ\mathbf{Z}/p\mathbf{Z}, then it will almost never come from a variety over Z\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 pX_p look like they’re all mod pp reductions of a single variety, you’ll probably be OK.)

Posted by: James on July 28, 2010 11:15 AM | Permalink | Reply to this

Re: Zeta Functions: Dedekind Versus Hasse-Weil

Thanks, James! That helps quite a bit. But still…

I think the standard definition of the Euler factor at pp is that you take the characteristic polynomial of Frobenius at p on the inertia-invariant subspace of the \ell-adic cohomology of the variety.

You take it… and then… that polynomial is the Euler factor? Or maybe its inverse?

The Wikipedia article was way too vague for me: it just said “the definition of local zeta function can be recovered in terms of the characteristic polynomial”.

Another thing: I begged respondents to avoid saying ‘étale cohomology’. You did this, but you said ‘\ell-adic cohomology’ instead. Are you hinting that this sort of thing is unavoidable at the current state of understanding? I.e., maybe the only way people know to define the local zeta function at a ramified prime is to look at the characteristic polynomial of the Frobenius acting on the inertia-invariant subgroup of some sort of cohomology group… and, umm, do something with that?

I was hoping for more of a ‘counting points’ style definition of the local zeta function, maybe with some cleverly ‘corrected’ way to count points. Can the stuff about characteristic polynomial of the Frobenius be translated into a language that sounds more like that?

Posted by: John Baez on July 29, 2010 12:59 PM | Permalink | Reply to this

Re: Zeta Functions: Dedekind Versus Hasse-Weil

I guess I read the question, but then didn’t answer it till a few hours later, by which time I had forgotten the instructions. If Matt’s more detailed explanation below doesn’t do the trick, then I can try again. (By the way, the reference I use for the definition of the Hasse-Weil zeta function is Serre’s paper cited in the Wikipedia article.)

Posted by: James on July 29, 2010 11:58 PM | Permalink | Reply to this

Re: Zeta Functions: Dedekind Versus Hasse-Weil

As an exercise, you can try writing the Hasse-Weil zeta function of projective nn-space over Q\mathbf{Q} in terms of the Riemann zeta function. Or if you prefer point counting to etale cohomology, you can apply the variety-over-Z\mathbf{Z} version of the HW zeta function (explained in more detail by Matt below) to projective nn-space over Z\mathbf{Z}, which for the present purposes you can just think of as the union of the projective nn-spaces over Z/pZ\mathbf{Z}/p\mathbf{Z} for all pp. You’ll get the same answer either way because projective nn-space over Z\mathbf{Z} is a sufficiently nice integral model of projective nn-space over Q\mathbf{Q}. (If you had done something silly and let the fiber over 1313 be a degenerate projective space, then your Euler factor at 1313 would be different in the two cases.)

Posted by: James on July 30, 2010 12:11 AM | Permalink | Reply to this

Re: Zeta Functions: Dedekind Versus Hasse-Weil

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 KK be a finite extension of \mathbb{Q}, and let X=Spec𝒪 KX = \mathrm{Spec} \mathcal{O}_K. Let GG be the Galois group of “the” Galois closure of KK, and HH the stabilizer of KK.

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

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

For a ramified prime, there are two subgroups I pI_p and D pD_p of GG, so that I pI_p is normal in D pD_p and D p/I pD_p/I_p is cyclic with a canonical generator Φ p\Phi_p. More precisely, all of this data exists up to common conjugacy in GG. The points X(𝔽 p¯)X(\overline{\mathbb{F}_p}) are in bijection with I p\X(¯)I_p \backslash X(\overline{\mathbb{Q}}), with the Frob p\mathrm{Frob}_p action gong to the Φ p\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(¯)X(\overline{\mathbb{Q}}) and X(𝔽 p¯)X(\overline{\mathbb{F}_p}) in terms of groups acting on finite sets. Now, I want to turn this into statements about ζ\zeta functions and representation theory.

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

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

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

And, the key point, the Euler factor is still given by det(1Φ p| V pp s)\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.

Posted by: David Speyer on July 29, 2010 3:19 PM | Permalink | Reply to this

Re: Zeta Functions: Dedekind Versus Hasse-Weil

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

To count points on XX, you have to choose a model, 𝒳\mathcal{X}, for XX over \mathbb{Z}. And this choice can change the answer! For example, suppose that XX is an elliptic curve with split multipliciative reduction at pp. If you choose a model where the fiber over pp is a nodal cubic, then there are pp points (11 node and p1p-1 smooth points); if you choose a model where the fiber is a Neron kk-gon then there are kpkp 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 KK, rather than the whole of 𝒪 K\mathcal{O}_K. (For example, working with [12]\mathbb{Z}[\sqrt{12}] inside [3]\mathbb{Q}[\sqrt{3}], rather than the correct [3]\mathbb{Z}[\sqrt{3}]. This would throw off my point counts at 22). But, when the dimension of XX is 22 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?

Posted by: David Speyer on July 29, 2010 3:51 PM | Permalink | Reply to this

Re: Zeta Functions: Dedekind Versus Hasse-Weil

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 Q\mathbf Q, but instead think of it as a notion for varieties over Z\mathbf Z, then the local factors are given by point counting: namely you form the zeta-function of the fibre X pX_p for each prime pp — which is a point-counting thing , then replace the variable TT in the zeta-function by p sp^{-s} (if you use the form of the zeta-function for varieties in char. pp which has a variable TT; if you use the definition which already has p sp^{-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 Q\mathbf Q (or more generally an algebraic number field, but let me stick to the case of Q\mathbf Q), and wants to define its Hasse–Weil zeta-function. Ideally, one would want to define some sort of “minimal model” of XX over Z\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 Z\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 XX 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 Q \mathbf Q_{\ell}-vector space with an action of the Galois group of Q¯\overline{\mathbf Q} over Q\mathbf Q. Its dimension is just the same as the usual (singular, say) cohomology of XX over the complex numbers with Q\mathbf Q (or R\mathbf R, or C\mathbf C) coefficients.

For each prime pp, and each degree ii of cohomology, we take the invariants under I pI_p (the inertia group at pp); this will be the whole thing if pp 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 pFrob_p. We take the char. poly. of Frob p 1Frob_p^{-1} (the exponent 1-1 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)P_{i,p}(T). (Remember that ii is the degree of cohomology.)

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

(There are more conjectures that need to hold to make sure this makes sense, since a priori the coefficients of the P i,pP_{i,p} are \ell-adic numbers, but we are plugging in the complex number p sp^{-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 sp^{-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 ζ\zeta-function corresponds to the case XX 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 Q\mathbf Q by a number field KK, there are two ways to proceed. (1) One can just work over KK, and replace the rational primes pp considered above by primes qq of KK. (2) One can apply restriction of scalars to think of XX over KK as a variety XX' over Q\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 XX'.

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

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

Now XX' won’t have good reduction everywhere; it has bad reduction at primes pp that are ramified (concretely, because at these primes, not matter what α\alpha you choose, some roots of the corresponding ff will come together mod pp). 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 pp, the Euler factor of the zeta-function of KK has smaller degree (i.e. degree less than [K:Q][K:\mathbf Q]) than at the unramified primes.

Posted by: Matthew Emerton on July 29, 2010 4:37 PM | Permalink | Reply to this

Re: Zeta Functions: Dedekind Versus Hasse-Weil

Matthew wrote:

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 \mathbb{Q}, but instead think of it as a notion for varieties over \mathbb{Z}, then the local factors are given by point counting.

Holy moly! That point hadn’t penetrated my thick skull! You just hit the nail on the head. Since I’m going at this from a peculiarly autodidactic approach, it had never fully sunk in that people seeking zeta functions start with varieties defined over \mathbb{Q} and then need to contend with the fact that these have different ‘models over \mathbb{Z}’!

Wow. That seems insanely risky, like lowering your airplane to the ground and then checking to see if there’s a runway there. Exciting, and impressive if you succeed, to be sure. But you see, personally, I’ve been thinking about zeta functions for schemes defined over \mathbb{Z}, or in fact even simpler entities, namely functors

CommRingSet CommRing \to Set

(What do people usually call these? ‘Functors of points’? That sounds curiously awkward as a name for an entity on its own.)

So thanks for straightening me out. Your explanation is beautifully pellucid.

And more thanks, too, to James and David! It’ll take me a while to fully absorb everything you guys are saying, because I’m just a novice at this business. You see, I’m writing a paper with James Dolan on zeta functions, and I want to make contact with the usual stuff, but we’re starting from a pretty unorthodox approach. However, thanks to your help, stuff is starting to make more sense.

One more question, for now: do people have a standard name for the simple sort of ‘counting points’ zeta function in the case of varieties over \mathbb{Z}? Is it called a ‘Hasse–Weil zeta function’ or something else?

Posted by: John Baez on July 30, 2010 7:11 AM | Permalink | Reply to this

Re: Zeta Functions: Dedekind Versus Hasse-Weil

Dear John,

It’s hard to match Matthew’s nice summary, but you might enjoy at least the beginning of this article I wrote for a summer school in 2006.

Posted by: Minhyong Kim on July 30, 2010 6:10 PM | Permalink | Reply to this

Re: Zeta Functions: Dedekind Versus Hasse-Weil

Dear John,

The setting of varieties over Z\mathbf Z (or, in more official language, finite type schemes over Z\mathbf Z) was first considered by people like Serre and Tate in the 60s (I think this is the right decade; of course they were building on earlier work and ideas of Hasse and Weil, among others), perhaps a little before the fancier definitions via etale cohomology were completely worked out (these emerged in the mid-to-late 60s; but note that the flow of ideas in this period is particularly hard to follow, since there were many very clever people (Deligne, Dwork, Grothendieck, Serre, Tate, and then Langlands, also Shimura, as well as Weil, and others too) thinking about these very deep ideas at roughly the same time, from closely related but sometimes subtly different points of view.)

In any event, the reason for bringing up Serre and Tate is that there is a nice article of Serre, called “Zeta and LL-functions”, from 1965 (in volume II of his collected works), where he discusses precisely the functions that in this thread we have been calling “Hasse-Weil zeta functions for varieties over Z\mathbf Z”. He simply calls them “zeta-functions”, so perhaps you could just use this terminology, and refer to his paper to legitimize your choice of terminology. (This paper is also a very pleasant read; the same is true of the later, 1969/70 paper referenced in wikipedia, but that one is more technical.)

Posted by: Matthew Emerton on August 3, 2010 4:02 AM | Permalink | Reply to this

Re: Zeta Functions: Dedekind Versus Hasse-Weil

I just looked over Serre’s 1965 article again, and on p. 85 he says that the concept that we have been calling “Hasse–Weil zeta-functions for varieties over Z” was in fact the original definition of Hasse and Weil. So it seems perfectly fine to call them Hasse–Weil zeta-functions as well.

Posted by: Matthew Emerton on August 3, 2010 6:56 AM | Permalink | Reply to this

Re: Zeta Functions: Dedekind Versus Hasse-Weil

Dear Minhyong and Matthew -

Thanks again, both of you!

In any event, the reason for bringing up Serre and Tate is that there is a nice article of Serre, called “Zeta and L-functions”, from 1965 (in volume II of his collected works), where he discusses precisely the functions that in this thread we have been calling “Hasse-Weil zeta functions for varieties over Z”. He simply calls them “zeta-functions”, so perhaps you could just use this terminology, and refer to his paper to legitimize your choice of terminology.

Great — that’s exactly what I need now: legitimacy.

I see the NUS library lacks Serre’s collected works, but has the original book containing this paper:

  • Jean-Pierre Serre, Zeta and LL functions, Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), Harper and Row, 1965, pp. 82–92.

In fact it has two copies! So, I’ll go over there and get one now.

I just looked over Serre’s 1965 article again, and on p. 85 he says that the concept that we have been calling “Hasse–Weil zeta-functions for varieties over Z” was in fact the original definition of Hasse and Weil. So it seems perfectly fine to call them Hasse–Weil zeta-functions as well.

Interesting — thanks yet again.

Minhyong’s paper is interesting too, though I’m focused on the most elementary part…

Posted by: John Baez on August 3, 2010 7:12 AM | Permalink | Reply to this

Re: Zeta Functions: Dedekind Versus Hasse-Weil

Dear John,

You’re very welcome. If memory serves, the original book that Serre’s article appeared in is a (physically, as well as mathematically) very nice little book; I hope you enjoy it.

Best wishes,

Matthew

Posted by: Matthew Emerton on August 3, 2010 7:51 AM | Permalink | Reply to this

Re: Zeta Functions: Dedekind Versus Hasse-Weil

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 Q\mathbf Q vs. schemes over Z\mathbf Z). I highly recommend it!

Posted by: Matthew Emerton on August 3, 2010 4:14 AM | Permalink | Reply to this

Re: Zeta Functions: Dedekind Versus Hasse-Weil

Dear John,

It occurred to me that I should have said something about *why* people work with varieties over Q\mathbf Q rather than Z\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 Q\mathbf Q should canonically determine *all* its Euler factors, including those at the bad primes, without having an explicit model over Z\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 sd+1ss \mapsto d+1 - s (where dd is the dimension of the variety over Q\mathbf Q; or if you prefer, you can think of d+1d+1 as being the absolute dimension of the variety — i.e. we include one more dimension because Spec Z\mathbf Z has dimension one).

Now if you have a zeta function with such a functional equation (just involving Gamma factors, powers of π\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.

Posted by: Matthew Emerton on August 14, 2010 7:56 AM | Permalink | Reply to this

Re: Zeta Functions: Dedekind Versus Hasse-Weil

Matthew wrote:

It occurred to me that I should have said something about *why* people work with varieties over Q\mathbf{Q} rather than Z\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 Q\mathbf{Q} should canonically determine *all* its Euler factors…

Wow, that’s cool. Thanks for pointing that out: some things that seem obvious to the experts would never occur to me, because they’re superficially ‘too good to be true’.

By the way, we’re continuing this zeta function conversation in a somewhat different vein over in the comments to week300, and I raised a question you might know how to answer.

Once, when we were trying to drum up a list of 21st-century ‘neo-Hilbert problems’, Noam Elkies suggested including the Riemann Hypothesis, and wrote:

If I remember right, Riemann’s original Hypothesis was on Hilbert’s list; but we have many more zeta and L-functions now, and are still arguably no closer to a proof even for Classic RH.

N.B. I do not include the Langlands program, Beilinson conjectures etc. on this short list because they are subsumed by the most general arithmetic Riemann hypotheses; but even a proof of the Riemann and Artin conjectures for zeta and L-functions of number fields would be a significant legacy of 21st-century math.

I replied saying that I hadn’t realized the Langlands program had been “subsumed”. He said it had… but I guess I still don’t understand what he meant. What are the “most general arithmetic Riemann hypotheses” of which he speaks?

Posted by: John Baez on August 16, 2010 11:34 AM | Permalink | Reply to this

Re: Zeta Functions: Dedekind Versus Hasse-Weil

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

Posted by: Thomas on October 8, 2010 1:46 PM | Permalink | Reply to this

Re: Zeta Functions: Dedekind Versus Hasse-Weil

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.

Posted by: Sylvain JULIEN on October 31, 2010 11:06 PM | Permalink | Reply to this

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