 Today's lecture is part of a series of lectures on elliptic functions, and it will be about theta functions. This particular theta function is called sigma. I have no idea why it is called sigma rather than theta. I mean, I sort of imagined someone got really confused about the Greek alphabet at some point. But the motivation for it as follows. First of all, let's recall the functions we had in previous lectures. We had the Weierstrich's p function, which has two poles, or rather a pole of order two, and two zeros in a fundamental domain. And it's periodic, so p of z plus lambda equals p of z. Here lambda is in our lattice lambda, which you recall is a subset of the complex numbers. Then in another lecture, we talked about the Jacobi elliptic functions, which were s, n, c, n, and d, n. And in a fundamental domain, these had one pole and one zero, except they're not quite periodic. They satisfy f of z plus lambda equals plus or minus f of z. So they're sort of periodic up to this extra fudge factor here. And this is the sort of price we have to pay for having just one pole and one zero instead of two. We lose periodicity. So it would be really nice if we could have just one zero and no poles in a fundamental domain, because that would make it much easier to construct functions in terms of their zeros. And this function does exist, and is the sigma function that we're going to talk about today. And it's going to have some sort of function equation that looks like this, where this is some sort of new fudge factor, and the problem is to figure out what this is. First of all, it can't just be a constant depending on lambda, because we saw earlier that if f of z plus lambda is equal to c of lambda f of z, this implies the number of poles is equal to the number of zeros. So the price we pay for having just one zero and no pole is that f is going to be even less periodic than it was before. So how do we find a function with one zero in a fundamental domain and no poles? Well, first of all, we have to decide where to put the zeros. Well, it's pretty obvious we put the zeros on the lattice L. Then in particular, we should have a zero at L. And we've got a function p of z, just poles of order two on L. So the idea is we want to start with p and go to our sigma function, and we want to convert poles of order two to zeros of order one. So how do we convert a pole of order two into a zero of order one? So here we've got a pole of order two, and we want to convert it into a zero of order one. And there's an obvious way to do this. We can multiply this by z cubed. Well, this is no good. This only works for the zero at z equals zero. I mean, we'll work for that, but for all the other zeros and all the other lattice points, multiplying by z cubed will just turn them into a slightly more complicated pole. Well, there's another way to convert z to the minus two to z to the minus one. As follows, we recall that the integral of z to the n d z becomes z to the n plus one over n plus one. So that's great. We can raise the exponent by one by integrating. So you think, well, we just have to raise the exponent by three, so we just integrate three times. So let's try and do this. So we go z to the minus two, and then we integrate z to the minus two, and we get minus z to the minus one. That's good. And then we integrate z to the minus one. And we get, well, it says we should have something time z to the zero, but we don't because this is only for n not equal to minus one. And if n equals minus one, we sort of get minus log of z. And if we try integrating this, we get some sort of a mess and we don't get z to the one. So integrating three times doesn't work, but that doesn't really matter because if you've got log of z, you know how to convert that into z. All we have to do is to apply x of minus, well, we have to get rid of that minus sign. We do x of minus something, then we get z. So we can convert a pole of order two into a zero of order one. We just have to integrate twice and then exponentiate. And we have to be a little bit careful about this minus sign here. So this gives us our definition of sigma z. It's going to be roughly x of minus the integral of the integral of p of z. I'm being a little bit vague about constants of integration and where we're integrating from and so on, but it doesn't really matter too much. And now we have to think about periodicity of this. And the trouble is, you remember, when we integrate something that's periodic, we don't necessarily get something that's periodic. So I've got to draw my p's there. p of z plus lambda is equal to p of z. Now if we integrate the integral of p of z plus lambda, it's going to be the integral of p of z plus some constant. And if we integrate twice, this will be double integral of p of z. I'm missing out the dz's, which is very sloppy, but whatever. And then this will be some constant times z plus some constant, where this constant might depend on lambda. And then if we take the exponential of this, this is equal to the exponential of that plus, sorry, times x of something linear in z, where this depends on lambda. So this gives us the form of the periodicity of the sigma function. Well, I haven't said what these constants are, but you can sort of sit down and work them out and you'll find that sigma of z plus omega one is minus x of two eta one times z plus omega one times sigma z, where two eta one is sort of what you get by integrating via Stryce's p function over a period. So you see this is linear in z, as it ought to be with some funny constants. So what this says is that sigma isn't quite periodic, but if you move it by a period, it's still the function of the same shape. It's just sort of got being rescaled by this factor here. So you can think of sigma as looking like a sort of two-dimensional ray of waves, and the waves are all the same shape, but they sort of get bigger and bigger as you go away from the origin or maybe smaller and smaller, depending which direction you go in. And we know that sigma has zeros on L and nowhere else, and you can easily check that sigma is odd. So sigma of minus z is minus sigma of z. Well, that gives us a zero at a point a, but now you notice that sigma of z minus a has zeros at L plus a. So we can put zeros anywhere we want and similarly one over sigma of z minus a has poles at L plus a. So this makes it really easy to construct functions with zeros and poles wherever we want. All we do is we take sigma of z minus a1, sigma of z minus a2, up to sigma of z minus am, and divide it by sigma of z minus b1, all the way up to sigma of z minus bm. And this obviously has zeros at L plus a1, L plus a2 and so on, and poles at L plus b1 and so on. And we can ask, when is it periodic? Well, if we change z to z plus omega1, then this function gets multiplied by all the fungifactors of the sigmas. So it gets multiplied by the product of minus x of 2 eta1 times z minus a1 plus omega1, divided by the product of the same thing for the b's, minus x of 2 eta1 times z minus b1 plus omega1. And we want this to be one, and it's periodic if it satisfies the following two conditions. First of all, the number of zeros must be equal to the number of poles. So this causes the factors involving z to cancel out. And secondly, we want the sum of the zeros should be equal to the sum of the poles. So whenever a function satisfies these two conditions, we can find a periodic function with those zeros and poles. In fact, we can do it slightly more generally, or we need the sum of the ai minus sum of the bi is in L, because if this is true, we can add some element of L to one of the ai's or the bi's in order to obtain this condition. If the a's and the bi's just satisfy this, then this function isn't quite periodic. You have to sort of change the ai's or the bi's by something in L. Well, now I want to discuss this fudge factor, as we get a little bit more. So suppose more generally, we want to have functions with f of z plus lambda is c lambda z times f of z or lambda in L. So when this is one, we get the vi stress function. If it's plus or minus one, we get Jacobi functions. And if it's sort of x to something linear in z, we get the sigma function. And we can ask what other possibilities are there? Well, we want f of z plus lambda plus mu to be equal to f of z plus lambda plus mu. And this multiplies it by c of lambda of z times, sorry, c lambda of z times c mu z plus lambda. It's very easy to get these the wrong way around when there's about a one and two chance I have. So this is not quite the same as c lambda z times c of lambda plus mu of z. And this multiplies it by c lambda of z times, the same as c lambda z times c mu of c, which you might guess we've got this extra sort of twiddly factor in here. So if it satisfied this condition, then c would just be a sort of homomorphism from something to another else to another. This condition is called the one co-cycle condition. And there are things called two co-cycles and three co-cycles and so on, but we won't worry about those this lecture. So generally, if we write down a one co-cycle which is a collection of c lambda satisfying this condition, the collection of things c lambda is called a line bundle. Actually, if you look at the official definition of a line bundle, it's something a lot more complicated than this, but that's because people want to define line bundles over lots of complicated spaces. If you're just working on the complex numbers modulo and lattice L, then you can get away with this very simple condition for a line bundle. It's just a collection of functions depending on lambda satisfying this one co-cycle condition. You can easily reduce this to the case of c omega one of z and c omega two of z because the co-cycle condition implies that everything else can be written in terms of this. However, there are some non-trivial relations between these. For example, we want c of omega one plus omega two of z should be the same as c of omega two plus omega one of z. For example, if we put c of omega one z equals e to the az plus b and c omega two of z equals e to the cz plus d, then you see you can't choose a, b, c and d to be any complex numbers because this condition here implies you get this condition that e to the c omega one is equal to e to the a omega two if I've done the arithmetic correctly. So you can write down one co-cycle by writing down these two functions of z, but there are some compatibility conditions they have to satisfy. And if you've got two line bundles, you can do various operations from them. For instance, if you've got one line bundle, so we're looking at functions of fz plus lambda equals c lambda z times f of z and suppose you've got another line bundle. So let's call this one d. So we want d lambdas tell us how g transforms. Now we can ask how does f times g transform? Well, this is equal to c lambda of z times d lambda of z times fg of z. So this is now a new one co-cycle called the tensor product of line bundles. So line bundle is just a collection of fudge factors telling you how something fails to be periodic and a tensor product of line bundles just says that if you've got a function for one line bundle and a function for the other it tells you the fudge factor that you need for the product of these functions. And of course you can take inverses by taking the inverse of c lambda. There's another thing we can do. We can take f of z times h of z where this is any non-vanishing function. And then we can ask how does this transform? Well, this is equal to c lambda of z times hz plus lambda divided by h of z times f of z h of z. And this is a one co-cycle called... Sorry, that should be f of z plus lambda and h of z plus lambda is equal to that times f of z h of z. And this thing here is called a one co-boundary. And this thing here is a new co-cycle. However, you see it's not... I mean, it's giving you a new line bundle but it's not really that different from the old line bundle because you can get from sections of one to sections of the other just by multiplying by h of z. So we say this is equivalent to the line bundle of c lambda. And classifying all line bundles is sort of a hopeless task. I mean, you more or less have to classify all complex functions. However, we're really only interested in classifying all line bundles up to equivalence. So we want problem, classify line bundles up to this sort of equivalence. And this can be done in two steps. I'm not going to go into details. So the first step is to show we can take c lambda equals x of something linear in z. So we can reduce any line... I mean, you could have a line bundle where the c lambdas was something really, really complicated, some large polynomial in z or some even worse function, but they're all equivalent to something where this is just linear in z. And then you can just classify these by doing a little bit of elementary algebra. And I'll just tell you what the answer you get is that the line bundles form a group and it's just isomorph to c modulo l. So the complex numbers modular lattice times the integers. So these are the so-called degree zero cos cycles, optical equivalence. And this integer here is called the degree of the cos cycle. So this entire group here is sometimes called the Picard group or sometimes the Picard scheme. And rather confusingly, this bit here is just the c over l is sometimes called the Picard variety. And if you want to know what this one is called, it's sometimes called the Neuron-Severi group, but we don't need to worry about that. So what is the meaning of c over l and z? So suppose we're given a line bundle. That means we've got numbers c lambda. And what we do then is take a meromorphic section. So we want f of z plus lambda equals c lambda of z times f of z. Then we look at the poles and the zeros. So the number of zeros minus the number of poles is equal to the degree, which is an element of this z here. And the sum of the zeros minus the sum of the poles is going to be a well-defined element of c over lambda, sorry, c over l, which gives us this bit here. So if we've got a line bundle, we can tell what its image is in this group taking a meromorphic section and looking at the zeros and the poles, and that gives us these two invariants. And as I said, any line bundle is going to be equivalent to one of these by bump by this process. So let's just see a few examples. This means that we can arrange all elliptic functions and all theta functions and everything else in a sort of neat array just by keeping track of what element of this group c modulo l times z therein. So let's give a few examples. So suppose I take the element 0 in c over l and 0 in z. Well, this just gives us our usual periodic functions. And typical examples would be the vice-stress p function and its derivative. Or we could take an element in half l. You remember, there are four elements of order 2 in c modulo l. We could take 0 here. And if we do this, we get the Jacobi functions s, n, c, n, d, n. If we take 0 here and degree 1, that means we want some of the poles minus the sum of the zeros to be 0. And there should be one more 0 than pole. And we had an example of this, which is just the sigma function that we discussed earlier this lecture. Or we could take an order 2 element of c over l and 1 here. And these we get the Jacobi theta functions, which I haven't discussed yet. But I will probably be discussing them in a lecture coming up fairly soon. Well, suppose you want an arbitrary element a in here and 1 in here. Well, then we could just take sigma of z minus a. Suppose you want an arbitrary element a here and degree 0. Well, we can do that just by taking sigma of z minus a and dividing out by sigma of z. Well, suppose you want something 0 in c over l and degree minus 3. Well, here we can just take sigma of z to the minus 3, which just has a pole of order 3 and 0 and so on. So the fact that every, if you want to summarize the entire theory of elliptic functions and theta functions in the shortest possible space, all you do is you say that the Picard group is c over l times z. This describes, essentially describes all the possible sorts of elliptic functions and theta functions and everything else you want. Okay, as I said, next lecture will probably be on these Jacobi theta functions.