 that this talk is part of a series on the vague conjectures. So in previous talks, I've discussed the vague conjectures for curves and sketched how you can prove them using the Riemann-Roch theorem. And in the previous talk, I discussed vase calculation of the zeta function of a firma hypersurface, which was how he came up with the vague conjectures. In this talk, I'm going to say a little bit about how they're proved in higher dimensions. So we're going to start with the left shet's fixed point formula, which at first sight seems to have nothing to do with the vague conjectures, but in a moment we'll see how it applies. Now in algebraic topology, this says the number of fixed points of a map phi from a manifold to itself is equal to a sum, alternating sum over the cohomology groups of a trace of phi on the ith cohomology group. So it's a very nice formula from algebraic topology. I should notice that M should be compact, and the fixed points of phi should be isolated, and they should all have multiplicity equal to one because there's a way of defining the multiplicity of a fixed point. So anyway, now let's write NM is the number of fixed points of phi to the M, the nth power of phi. So we're just going to iterate phi. We can think of this as a sort of dynamical system when we're counting the number of fixed points of the dynamical system. And this is just equal to a sum over i and j of minus one to the i times alpha ij to the M, where the alpha ij are the eigenvalues of phi on the ith cohomology group of M. And that sort of just follows immediately from this because a trace is a sum over eigenvalues. And now we can write down the sum, sum over M of NM, t to the M over M, which should look very familiar to anyone who's looking at the previous lectures because this is more or less the formula for the zeta function of a curve, the opto exponentials. And this is just sum over i and j and M of minus one to the i times alpha ij to the M, t to the M over M. Let's just follow this immediately from this formula. And this is equal to sum over i and j of minus one to the i times log of one minus alpha ij times t. Maybe there should be F minus there. So this means that if we put z of t to be the exponential of this, so it's x with sum of NM, t to the M over M, this is just equal to the product of one minus alpha ij t to the minus one to the i or maybe minus minus one to the i. So this is equal to p one of t times p three of t and so on divided by p naught of t, p two of t and so on, where p i of t is almost the characteristic polynomial of phi on the ith cohomology group. Okay, it's not quite the characteristic polynomial because you have to change t to one over t and multiply by power of t or something, but who cares? And now you notice by the way that this is exactly the formula for the zeta function of an algebraic variety, except that NM is the number of points defined over a certain finite field. By the way, we can also notice that we have something called Poincare duality, which relates p i of t to p dimension of M minus i of t because the ith cohomology group of M is sort of dual to the dimension M minus ith cohomology group of M. And from this, you can get a functional equation for z of t, which I wrote out in a previous lecture, so I'm not going to bother writing it out again. So more precisely to make this formula, the zeta function of a finite field, we take phi to be the Frobenius. So you recall the Frobenius automorphism just takes a point with coordinates x1 up to xk to x1 to the q up to xk to the q, where we're working over a finite field with q elements. And this just follows because f to the k consists of exactly the points in the algebraic closure with x equals x to the q. So the number of fixed points, and M of phi to the M is equal to the number of points defined over the finite field of order q to the M. So if we take phi to be the Frobenius, then Lefchitz formula gives exactly the zeta function of the variety. And furthermore, this is going to give us rationality because this is obviously rational. It's just a product of characteristic polynomials. And moreover, we're going to get the functional equation for the zeta function coming from Poincare duality. So in order to do this, there's just one little problem we need to solve, which is problem. How do we define the cohomology of v with coefficients in z for v over a finite field? So we can do it for a variety of the complex numbers, say, just by using ordinary singular cohomology. But does this work over a finite field? We'll know. It turns out, well, we can define the cohomology groups with integer coefficients of the variety. We can give this, say, we can give v, say, the Zariski topology. If we do that, the groups H, I and V with coefficients in z are really badly behaved. In fact, they're usually zero. And they're quite useless for proving anything whatsoever about v. So the problem is how do we define the cohomology groups of v with integer coefficients? Well, it turns out we can't for a reason I would explain in a moment. But we can get pretty close. So growth and Dick sort of analysed this problem and came up with this amazing idea of etal cohomology. So this is an analogue of singular cohomology for varieties over any field. If you're working with a variety of the complex numbers, then etal cohomology is pretty close to being singular cohomology. So let's just summarise what the problem is that the problem is that aren't enough open sets in the Zariski topology. So suppose we've got a complex manifold. It has a complex topology. And we can define singular cohomology. And this gives cohomology groups of our manifold with constant coefficients. For instance, it might be integer coefficients. And we can also define sheaf cohomology, which occurs in chapter three of Hart-Schrung that we haven't quite covered yet, but never mind. And this gives good cohomology groups of V with coefficients in a quasi-coherent sheaf. Now, if we've got a variety over some field K and we use the Zariski topology, then this gives terrible groups for HIV with constant or Z coefficients. Rather surprisingly, it gives good sheaf cohomology groups. So HIV with coefficients in a quasi-coherent sheaf still works fine, even if you just use the Zariski topology. This was an extraordinary discovery by Seher. It must have been a bit of a surprise to everybody at the time because the Zariski topology seemed to be too weak to do anything like sheaf cohomology, but it turns out that it's got enough. So the problem that the growth in Dick Ruh is the problem why this isn't working is very simple. There are too few open sets in the Zariski topology. So the problem is how do we add in enough extra open sets? So there are two key ideas in this. So the first idea is first of all, we should add new open sets to the Zariski topology. And growth in Dick's really amazing idea is that open sets do not need to be subsets of V. So classically we think of an open set as being a subset of a space V, so V might be a variety. And what growth in Dick did was he replaced this as a map from some set U to V where this is not necessarily injective. And the question is what properties should this map have? Well, in the etal topology, this should be something called an etal map. What does etal mean? Well, it's a sort of analog of being a local homeomorphism. So for the complex topology etal means more or less local homeomorphism. And the inclusion of an open set into V is certainly etal because that's obviously local homeomorphism. But you can get lots of other examples. So the simplest non-trivial example of a local homeomorphism is just the map from the non-zero complex numbers to the non-zero complex numbers which take Z to Z squared. And we can of course do this for any field where you can take the map Z to Z squared and for a KF field not a characteristic two, this will in fact be an etal map. To do that, you have to give the definite precise definition of an etal map which I'm not going to do today. So, well, that seems to be a bit of a problem if open sets are not subsets of your space V but this doesn't really matter because the open sets, let's call these funny open sets, the etal open sets where we have maps from U, I to V form a category because if we've got an open set U, I mapping to V and an open set U, J mapping to V, we can define a morphism to be a map from U, I to U, J making this commute. So we can recall a pre-sheaf is a functor, a contravariant functor from open sets to sets. Well, since these funny, more funny, these funny more generalized sorts of open sets still form a category, we can still define a pre-sheaf on these open sets. Now, you can also define sheaves and to do this and to work at all their properties takes rather long time. So what I'm going to do is I'm going to omit about 500 pages of explanation and just say this ends up with a reasonable cohomology theory over any field. So this takes an awful lot of work to set up, but somehow the one key idea you need is to take open sets that aren't subsets. By the way, you might ask, why are we taking etal maps? Well, in the early days when these sorts of problems were being investigated, people actually investigated quite a lot of other different sorts of maps. For instance, you can look at flat maps and I think there are more than a dozen different variations of these. And etal maps turn out to behave better than most of the others, although some of the others are also useful. I mean, using flat maps or smooth maps sometimes works quite nicely. Anyway, using etal maps gives the closest correspondence with the classical singular cohomology theory. So we can define these things called etal cohomology groups of a variety V with coefficients in Z. Unfortunately, these are still bad. So after doing this 500 pages of work, we seem to have discovered it's all a complete waste because these groups are still no use. And there's actually no good way to define cohomology groups with integer coefficients for a variety over a finite field for a reason pointed out by Seher as follows. Suppose you take V to be a super singular elliptic curve. These are explained in chapter four of Hart-Shorn's book. Then the cohomology V with coefficients in Z should be Z squared if it behaves nicely because that's what it is over the complex numbers. And we want these cohomology groups to behave like singular cohomology. On the other hand, if V is a super singular elliptic curve over a finite field, the endomorphism group of V is a quaternion algebra. And this does not act in any interesting way on Z squared, whereas the endomorphism group of V should act on its cohomology groups. So there's simply no good way of defining cohomology groups for varieties over a finite field with coefficients in the integers. So what can we do about this? Well, we could try taking coefficients in a finite field say Z over PZ where P is the characteristic. These are still bad. Or in desperation, we could try taking coefficients with coefficients in over modulo L where L is a prime not equal to P. I don't know why the letter L is chosen for this. It's just sort of traditional. And these turn out to be good. They sort of behave like the cohomology groups over complex numbers with coefficients taking mod L. So you can ask, why exactly does the theory work modulo of prime not equal to P but not for the integers or mod P? Well, the key point is that you can look at the following exact sequence. Here, this is the Lh roots of one. And this is some field. I guess that should be one not zero. And this just maps X to X to the L. And the point is that this gives you essentially a nice map. This turns out to be an exact sequence of sheaves that you can start applying sheaf cohomology to. You notice if you take L equal to P, this is not a very nice map because we're then looking at the map taking X to X to the P for a field of characteristic P, which is notorious for being inseparable and so on. So we need to take L not equal to P in order for this to be nice. For instance, if L was equal to P, this would be giving you the P roots of unity mod P, which is a bit of a funny group, like it only has one element. And then the Lh roots of unity are more or less the same as the integers modulo L, at least if you're working over an algebraically closed field. However, there's still a problem. We want groups over a field of characteristic equal to zero. And this doesn't have characteristic zero, it is characteristic L. So what can we do about that? Well, what we can do is the following. What we do is we notice that we can also take the cohomology group of V with coefficients in some, any power of L. So these are still well-behaved groups. And now we can define the L-added cohomology groups to be the inverse limit over N of the etal cohomology groups of V with coefficients in Z modulo L to the NZ. And these are modules over the L-adic integers. And finally, we can define the L-adic cohomology groups of V with coefficients in QP to be QP tensor over ZP with HL of I with V with ZP. And these are modules over the P-adic integers and give you nice cohomology groups over a field of characteristic zero. There's one awful warning. There's a sort of standard blunder everybody makes in this theory, so I'll just tell you. This inverse limit over N of HI etal with V with coefficients in Z over L to the NZ is not equal to HI of etal of V with coefficients in the inverse limit of Z over L to the NZ. These groups here are kind of badly behaved. And the notation is really confusing because this group here sort of looks as if it means this group here because ZL is the inverse limit of this, but it's not equal to that. It's actually equal to this group. So this is very confusing and misleading notation. This doesn't mean the cohomology of V with coefficients in the chief CL. It means the inverse limit of the cohomology of V with coefficients in Z modulo L to the NZ. Anyway, these groups here finally give you well-behaved cohomology groups over a field of characteristic zero. So Grothendick was able to use this to construct the zeta function for varieties over finite fields, and he was able to prove two of the V conjectures, the rationality and the function equation of the zeta function. In fact, the rationality had really been proved a couple of years earlier by Bourke who really annoyed everybody by finding a proof that made no use of any sort of V cohomology. So I've just finished by mentioning a few books you can find out more about etal cohomology. The very first collection of notes on etal cohomology is probably these notes by Michael Artin on Grothendick topologies, and that's still worth reading. In fact, these notes are actually easier than most of the later ones because they were written when people were still struggling about how to construct etal cohomology theories. You can actually see some of the different ideas that people were trying. The classical introduction to etal cohomology by people who didn't want to read several thousand pages of SGA is these notes on cohomology etal by Deline and several others, and the first 60 or 70 pages are Deline's description of how to construct etal cohomology theory. Finally, if you want to know how these are used to prove the vague conjectures, there are several books on them, in particular this book by Freitag and Kiel defines etal cohomology and actually shows you how to prove the Riemann hypothesis using etal cohomology.