 This talk will be about the Bay conjectures, about the zeta function of a curve over a finite field or more generally a variety over a finite field. These functions are closely analogous to the Riemann zeta function. So I'll start by recalling some properties of the Riemann zeta function. So the Riemann zeta function is defined to be zeta of s, which is sum over n greater than or equal to one of one over n to the s, which is one over one to the s plus one over two to the s and so on. And the first basic property it has is it has an Euler product. So this is equal product over all primes p of one over one minus p to minus s. And the equivalence of these is easy to prove because you just expand out this product as, well, one over one minus two to the minus s will be one over one plus one over two to the s plus one over two to the two s and so on. And then for p equals three you get a similar factor and then another one for five and so on. And if you multiply all these out, you see that this is just a sum of all integers, positive integers of one over n to the s because every positive integer can be written as a product of prime powers in a unique way. So you select one term from this and one term from this and one term from this and that's corresponds to the prime power decomposition. So this identity is equivalent to unique factorization. In the fundamental theorem of arithmetic. The next property of the zeta function is somewhat deeper. So it has the following properties. First of all, it has an analytic continuation to all complex numbers. It's holomorphic except for a pole at s equals one of order one and residue one. Secondly, as Riemann and possibly Euler discovered, it also has a functional equation and the easiest way to write the functional equation is to have a slightly modified zeta function. So we write zeta star of s is pi to the minus s over two times gamma s over two times zeta of s. And this modified zeta function satisfies a very easy functional equation, even like that. And zeta star has poles at s equals zero and one and otherwise is holomorphic. The third property is the famous Riemann hypothesis. I guess I shouldn't really call this a property because no one's managed to prove it yet, which says that all zeros of zeta star have real part of s equal to a half. So the original function zeta of s also has zeros at s equals minus two, minus four. And so these counts out with poles of gamma of s over two. So zeta star, all the zeros are known to have real part between zero and one and the Riemann hypothesis says they all have real part exactly a half. So that's the most notorious open problem in mathematics. Well, the Riemann zeta function has a generalization to the zeta function of a number field k. So k is going to be a number field. For example, it might be the field of quotients of say the Gaussian integer z of i. And the Riemann zeta function of a number field is defined in a very similar way to the zeta function of the integers. It's sum over all divisors or positive divisors is a one over norm of D to the s where D is a divisor of k or rather strictly speaking should be a divisor of the ring of integers of k, but I won't worry too much about that. And the divisor is a linear combination sum over Ni Pi, this is sum over B where the Ni are integers almost all zero and Pi are prime ideals of the ring of integers are of the number field k. And the divisor has a sort of unique factorization into linear combination of prime divisors and just as for the zeta function of the integers, this implies we get an oil of product, which is a product over all primes of the number field of one over one minus norm of P to minus s. So this is over prime ideals of the number field rather than primes because the number field in general doesn't have unique factorization but its divisors do have unique factorization. I should tell you what the norm of an ideal is. So the norm of a prime is just the number of elements of R modulo, the prime ideal and the norm of a divisor is defined multiplicatively. So this is norm of D1 times norm of D2. In fact, for number fields, the divisors happen to correspond to the non-zero primes, sorry, so the non-zero ideals of the number fields. And so you can think of this as being a sum over all non-zero ideals, if you like. And then the norm of any ideal is again, just given by the number of elements of R modulo the ideal. So that describes the zeta function of a number field and it has very similar properties to the Riemann zeta function. For instance, it has a functional equation relating zeta of s to zeta of one minus s. And this is a very interesting functional equation that you can spend an entire lecture course talking about but I'm going to, at least for today, I won't be saying much more about it. So now I want to discuss art in, found a generalization of the zeta function of a number field. So he defined the zeta function of a curve over finite field. And this curve will usually be implicitly be non-singular. So the motivation for this is that there's a very close analogy between curves over finite fields and algebraic number fields, which was noted in the 19th century. So Artin decided to try and find out what the analog of the zeta function of a number field was. So all you do is you sort of take a ring of integer of a number field and if you have any property of the ring of integers of a number field, you try and find the corresponding property of the coordinate ring of the curve. And conversely, if you've got some property of curves, you can try and find the corresponding property of number fields. So we define the zeta function of a curve C to be the sum over all divisors of C of one over norm of D to the S, just as we did for algebraic number fields. And a divisor as before is just a linear combination of prime divisors. And the prime divisor you have to be a little bit careful about, it's not quite a point on a curve. So prime divisor is a point on the curve over the algebraic closure of the field up to Galois conjugacy, roughly speaking. Here we're going to take K to be a finite field. So this will be the algebraic closure of the finite field. And again, just as before, divisors can be uniquely written as a sum of prime divisors. So this becomes an Euler product where this is a product over all prime divisors of the curve, which are not quite points on the curve. And we get this Euler product here. And just as before, the norm of P is the number of elements in the coordinate ring of an affine curve modulo, the prime ideal. Actually you have to modify this a little bit because we're also going to look at projective curves in a moment. So let's work out the simplest non-trivial case just to see what's going on. So here we're going to take C to be the affine line over K, and let's suppose this is a finite field of order Q, which is going to be some sort of prime power. And now prime divisors of this are easy to work out because the coordinate ring is just K of X. And now prime divisors correspond exactly to irreducible polynomials with leading coefficient one. So it's going to be something like X to D plus C D minus one X to D minus one up to C zero. So this is going to be some polynomial F of X and prime divisors correspond to irreducible polynomials and arbitrary divisors correspond to arbitrary polynomials with leading coefficient one, arbitrary non-zero polynomials. So this is in K of X over our finite field K. And we can work out the norm of this divisor. So the norm will just be the number of elements of K of X modulo this, which is obviously just Q to the power of D, where D is the degree. And now our zeta function will be sum over D greater than or equal to zero. So D greater than or equal to zero just means if D is sum of Ni Pi then means Ni of greater than or equal to zero. So the sum over one over norm of D minus S is now easy to work out because we just sum over all degrees greater than or equal to one. And for a divisor of degree D we get Q D minus S because that's the norm and there's a minus S. However, there are lots of devices of degree D. In fact, we can see the number of devices of degree D is just Q to the D. So here we have the number of divisors of degree D and this bit here is the norm of a divisor of degree D. And now this is really easy to work out. It's just a geometric series. So this is one over one minus Q to the one minus S. So that's the zeta function of an affine line. Now, instead of looking at the affine line we can also look at the projective line. So let's start with the affine line A1 and change it to the projective line which is equal to A1 union a point at infinity. And the point of infinity is just another divisor of degree one. So you have to find a more subtle definition of a divisor you can't just say it's a consequence of an irreducible polynomial but we won't worry about that. So let's work out the zeta function of the projective line. Well, this is going to be a product over all divisors of one over one minus norm of the prime divisor minus S and this will be, this is easy to work out because it's just the zeta function of the affine line of S. And this accounts for all prime divisors other than the one at infinity but we should also multiply it by one plus one over Q to the S plus one over Q to the two S and so on. So this term here is what you get from the prime divisor at infinity. And this is just one over one minus Q to the one minus S times one over one minus Q to the minus S. That's another geometric series. And now let's compare this with the Riemann's zeta function. So the Riemann's zeta function is just zeta of S and we have to multiply this by a fudge factor of infinity in order to get a nice functional equation. So we should multiply this by pi to the minus S over two times gamma S over two. So you see here, this thing here corresponds to line P one, whereas this bit here just corresponds to the affine line A one and this bit here corresponds to Z or rather to the spectrum of Z. I guess it would be slightly better to say. And this whole thing here corresponds to, well, nobody really knows what it corresponds to some sort of completion of spectrum of Z. So just as we go from the affine line to the projective line by adding a point, we should go from the spectrum of Z to something or other by adding something or other and nobody has really figured out what this something or other is. We can sort of quite often guess what it looks like. For instance, we know what the Euler factor corresponding to it looks like because this is what gives us a nice functional equation. And somehow this lack of knowledge of how to complete spectrum of Z. There's one school of thought that says this is the reason why we have so much trouble proving the Riemann hypothesis. If anyone could find a really good understanding of how you compactify Z, whatever that means we could prove the Riemann hypothesis and everything else. So anyway, that's a bit of a problem. So anyway, now you know, if you add in this extra fudge factor to the Riemann zeta function we get a nice functional equation. And the same thing happens here. We get a functional equation for the zeta function of the affine line provided we add in the factored infinity. So we recall zeta of P1 of S is one over one minus Q to minus S times one over one minus Q to one minus S. And this satisfies the functional equation, zeta of one minus S is equal to zeta of S, except it's not quite, we get an elementary fudge factor. So, you know, this is just a very elementary function which doesn't really matter very much. And we also notice that this zeta function of the curve P1 is a rational function of the variable T which is Q to the S. In particular, it's periodic in S because it's invariant if you change S to S plus two pi i over log of Q. So we can also ask what are its poles? Well, it says poles at S equals zero one just as the Riemann zeta function does. Well, except that's not quite true because it also has extra poles because it's periodic. So we should really add two pi i n over log of Q and these are its poles. And it has zeros at nowhere. So it does sort of satisfy an analog of the Riemann hypothesis in that all its zeros are real part of half, but that's a completely uninteresting statement because it doesn't actually have any zeros. So Riemann hypothesis is kind of vacuously true. So that's what the affine line does. What about other curves? Well, this gets a lot more interesting. So it turns out that the zeta function of a curve C is very closely related to the number of points of C over the finite field of order Q to the M. So this is an extension of degree M of our original finite field, which was just F of Q. And the number of points of C over the finite field of order Q to the M is usually noted by N sub M. And you can relate these to divisors over C because a prime divisor of C of degree D doesn't actually necessarily give you a point of C unless D is equal to one. What it does is it gives you D points defined over F of Q to the M whenever D divides M. So if you know all the prime divisors and their degrees, you can use this to work out how many points the finite field has, sorry, how many points this curve has defined over any finite field. And you can relate this to the zeta function as follows. So we know that the logarithm of zeta of S, that's the zeta of the curve, is equal to sum over all primes. Now we take the logarithm of the factor for the, of the Euler factor. And if we take the logarithm of the Euler factor, we get this. So this is just the Euler product written in a sort of additive way. So this is really a sum over all powers of prime divisors in some sense. And we can rewrite this as sum over all P and M of Q to the minus degree of P M S over M by, because we know what the norm is. And if we rearrange this a little bit, we can write this as sum over all M of N M times Q to the minus M S over M where we rearrange this using this fact that N M is equal to a certain sum over, all divisors whose degree divides M. So here is, this is sometimes used as an alternative way of defining the zeta function of a curve. We define it to the exponential of this series here. And if you see it defined like this, it's a little bit mysterious because it's unclear why you put this M in here and why you put a logarithm in there. And the reason why you put an M in there and the logarithm in there is that this makes this very closely analogous to the usual Riemann zeta function. So they proved various properties of this function, well, Vay and Schmidt and Hasson and so on. So it's sometimes convenient to write this in terms of Z of T. So we sometimes write it in terms of a variable T where T is equal to P minus S. And then it has the following properties. First of all, Z of T is rational in T. So this is a sort of analogous to saying the usual Riemann zeta function can be analytically continued. This can not only be analytically continued, but you even get a rational function. In fact, it has the form, the following form P of T over one minus T, one minus Q of T, where this is some polynomial. So we work this out for the projective line. I should say I'm taking C to be non-singular and projective. If it has singularities or if it's not projected, the zeta function is a little bit messier to write down. And furthermore, P of T is a polynomial of degree two G, where G is equal to the genus of the curve C. Now the Riemann zeta function has a functional equation. So of course, this will have a functional equation and its functional equation says that zeta C of one minus S is equal to Q to the one minus G times one minus two S of zeta of S. So here I'm writing the functional equation in terms of zeta rather than Z, because this makes it look much more like the functional equation of the Riemann zeta function, except that we've got this, an extra minor fudge factor appearing here. And the third and deepest property is the Riemann hypothesis, which for curves was proved by André Baye. It's one of the results he's most famous for. And this says that the polynomial P of T, suppose we factorize it into a product of its two G roots. So well, I guess omega i is actually the inverse of the roots. Then it says that omega i has absolute value equal to Q to minus a half. So this is the analog of the Riemann hypothesis. And the reason it's called an analog of the Riemann hypothesis is that we recall that T is equal to Q to the S, sorry, that should be a Q, that should be a Q to the power of S. And this condition here just says the real part of S is equal to a half, which is the exact analog of the usual Riemann hypothesis. So how do you prove these? Well, I plan to prove some of them in a few later lectures. So this property here follows from the Riemann rock theorem. This property here follows from the Riemann rock theorem. And this property here follows from the Riemann rock theorem. Actually, it wasn't realized that the third property follows from the Riemann rock theorem for quite a long time. The original proofs of this by Andre Ve used some rather difficult higher dimensional algebraic geometry, but step out of and Bombieri a few decades ago, discovered rather than everyone's surprise that if you're really clever, you can actually use the Riemann rock theorem to deduce even the Riemann hypothesis. So the Riemann hypothesis says something about the number of points over a finite field. So if we write log of Z of T, we can write this as we expand out the oil of product. It's one plus T squared over two plus T cubed over three coming from the factor one over one minus T plus QT plus Q squared T squared over two and so on coming from one over one minus QT. And then we get plus omega one T and so on coming from one of the factors of P of T and we get omega two T and so on. And since this is equal to N one T plus N two T squared over two and so on, we see that the number of points of C over F Q to the M, which is NM, is satisfies NM minus one minus Q to the M is less than or equal to two G times Q to the M over two. So this is because this is the absolute value of omega I and this term here is the number of terms omega I and this one here is just coming from this coefficient here and the Q to the M comes from this coefficient here and the various omega I's come from here and here and here and so on. So the number of points is one plus Q to the M plus various powers of the omega I. So the powers of the omega I are bounded by this term. So the Riemann hypothesis says that we gives us a very tight error bound for the number of points of a curve over a finite field. Conversely, if you can prove this error bound for the number of points, that's equivalent to the absolute value of the omega I being given in Bayes theorem. Now I'll just mention the Vey conjectures in higher dimensions. So for curves, they're not the Vey conjectures, they're the Vey theorem. What Vey did was he calculated the zeta function for Fermat hypersurfaces. These are hypersurfaces that form X1 to the N1 plus X2 to the N2 and so on plus XK to the NK equals zero. Well, actually he didn't calculate the Riemann zeta function because that hadn't been defined at the time. I mean, he used his calculation of the number of points to define the zeta function. So he defined it to be log by log of zeta of t equals sum of NM t to the M over M, where this is now the number of points of the curve, sorry, of the, of the, of variety over a finite field of order q to the M just as for curves. And then he've conjectured based on his calculations for Fermat hypersurfaces that first of all, zeta of t is rational in t. So this was proved by Dwork in about 1960. Next he conjectured a functional equation which says the number of points of the curve should be equal to plus or minus q to the N chi over two t to the chi z of t where N is the dimension, chi is the Euler characteristic, whatever that is. One of the problems is to define the Euler characteristic of a, of a variety. And thirdly, he conjectured the Riemann hypothesis of the Euler characteristic of the Euler characteristic of the Euler characteristic of the Riemann hypothesis which says that zeta of t should not only be rational but it should have the following form. So it should be a product of certain polynomials divided by certain other polynomials. And two of these factors are easy to figure out. This one here is one minus t. This one here is one minus q to the N t. And his key conjecture was that if you write p i of t as a product of one minus omega to the ij t then omega ij always has absolute value q to the i over two. So this is the Riemann hypothesis or the Vey conjecture for a variety over finite field. So this part of it was proved by Grotendick using his theory of L-series and Eladic cohomology. And this, the deepest part of it was proved by Deline in the early 1970s. So I'd finished just by drawing a picture of what the zeros of this function look like in terms of s. So this is the Riemann hypothesis in terms of the variable t. In terms of s, what it says is the poles and zeros of zeta v of s look like the following. So what we do is we draw the points 0, 1, 2, 3. Let's say the dimension is 3. Then the poles and zeros look like the following. So first of all, here are the poles. There are poles there and there. And there may also be some poles at various other points. But the poles always are going to have integer coefficients. So these are the poles. And it also has some zeros. And the zeros all have integer part plus a half. So it might have zeros there and there and so on. And now you notice there's a functional equation which sort of says this is symmetric. So the functional equation kind of switches things like this. So this pink arrow stands for the functional equation, which means if you've got a zero here, you've also got a zero here. And therefore you've got a zero there because the function is real. So the Riemann hypothesis that all the zeros have real part a half or 3 over 2 or 5 over 2. And all the poles have real part 0, 1, or 2, or 3. OK, so that's enough for the introduction. And next lecture, I will probably say a bit about how you use the Riemann rock theorem to prove some of the easier parts of these conjectures.