 This lecture is part of a series of lectures on elliptic functions and in today's lecture I'm going to be talking about the functions Studied by Jacobi which are usually denoted by SN, CN and DN And these functions are kind of a bit out of fashion. I mean books on elliptic functions in the 19th century were all about Jacobi's functions, whereas They seem to be almost unused in pure math these days. I mean Lang wrote a quite large book on elliptic functions Last century and doesn't even mention the Jacobi functions So I'm not going to go into too much detail about Jacobi functions Instead I'm going to use some mainly as an excuse to discuss line bundles over an elliptic curve And so I'll just quickly recall Basics of elliptic functions. We have a lattice L In the complex numbers and our elliptic function is periodic in L So f of z plus lambda is equal to f of z whenever lambda is in L And as we saw in previous lectures f must have some poles And the sum of the residues of the poles of f That's in a fundamental domain of course must be equal to zero As you can see you remember we just integrate f The z round the boundary of a fundamental domain and this is zero because it cancels by periodicity So if we want to find the simplest sort of functions We might try having one pole But one pole doesn't work because if we had just one pole of order one Then the residue would have to be zero so it wouldn't exist So there are two possibilities. We can either one pole of order two And residue zero or we can have two poles of order one where one has residue minus the other And roughly speaking this is the approach taken by Weierstrice Which we discussed in the previous lecture and this is the approach taken by Jacoby Which we're going to discuss today And Weierstrice's approach is easier for the following reason Suppose we've got one pole in a fundamental domain where do we put it? Well it's pretty obvious where to put it The pole is going to be at the point z equals zero Or more generally for z in any lattice and any other choice to just be stupid But if we've got two poles where are we going to put them? Well first of all there's a constraint because some of the poles If there are poles at z1 and z2 then z1 plus z2 Must be in the lattice as you remember from lecture two So inside a fundamental domain Here we've got a fundamental domain We've got to choose two different points Whose sum is in the lattice and if you think about it there's no really good chronicle way of doing this z1 or z2 can't be zero because then the other would have to be zero without a double pole at zero And similarly you see that the other obvious choice would be these half lattice points And these don't work either because again z1 would have to be equal to z2 So we've got to somehow choose two points That sum up to zero inside this fundamental domain As you can see there are lots of ways of doing this And that doesn't seem to be any way that's picked out for being rather special And this is one of the main reasons why Jacobi's approach to elliptic functions is a bit of a mess compared to Weierstrich's approach Because you've kind of got to break the symmetry of your problem by choosing a pole somewhere What Jacobi did in fact was he chose two points of order four like that But there are lots of other ways of doing that I mean you could choose these two points here or you could choose these two points here and so on So you've got to break symmetry And this breaking of symmetry means that everything gets more complicated Because you've got to write out everything several times instead of just once because you've lost the symmetry Well there's actually a way of dealing with the Jacobi functions that doesn't avoid breaking the symmetry Or at least not quite so much So there's a solution We can have just one pole of order one in the fundamental domain Well I just said you can't have just one pole of order one if you've got a periodic function So you have to modify the periodicity slightly So the function is going to be not quite periodic And for an example of this let's look at the function sin of z So sin of z plus 2 pi is equal to sin of z So it's got a period 2 pi But instead we could say sin of z plus pi is equal to minus sin of z So what we have is we have here a fudge factor And the function has period pi Except it's not quite a period it's periodic up to this fudge factor And we can do the same or doubly periodic functions So what we want is f of z plus lambda is going to be fz times c lambda Where c lambda is going to be some element of complex numbers And now we want f of z plus lambda plus mu should be equal to f of z of mu plus lambda Obviously which gives you c lambda times c mu is equal to I guess this should also be f of lambda z plus lambda plus mu So this gives us c lambda c mu equals c mu c lambda equals c lambda plus mu So c lambda is actually homomorphism from the lattice L To the non-zero complex numbers And we're going to look at the case when c lambda is equal to plus or minus 1 So this is the simplest case and obviously c is determined by its values on omega 1 and omega 2 So there are four possibilities We could have c omega 1 and c omega 2 could be plus 1, plus 1, plus 1, minus 1, minus 1, plus 1, minus 1, minus 1 Now this case is where the functions are periodic and we've already covered that But you see that there are now three more cases And these are going to correspond to the three Jacobi functions And now you notice if a function is not quite periodic in this way Then it still has periods but the periods are going to be slightly bigger than omega 2 So here the periods are specified by omega 1 and omega 2 in this case But here you see omega 2 changes the function by minus 1 So if you have 2 omega 2 then the function is going to be periodic And similarly this is periodic with respect to 2 omega 1 and omega 2 And this is periodic with respect to 2 omega 1 and omega 1 plus omega 2 So here these are the three index 2 sub lattices of L And one of the annoying things about Jacobi functions is the lattice they're periodic under keeps changing I mean so SN is periodic under some lattice and CN is periodic under some different lattice And this is very annoying But if you don't think of them as being periodic functions But just functions that are periodic up to sine then the lattice stays the same each time So this makes it a little bit easier to think about them You can ask why is C lambda equal to plus or minus 1 And the answer is there's no good reason We can take C lambda is an nth root of unity for some n And then we get n squared possibilities for the different sorts of functions And this means that instead of getting you know when n was 2 we got 2 squared possibilities One of which is periodic so all together we would get n squared minus 1 elliptic functions So Jacobi was doing the case n equals 2 of this and and you can do other other values of n But it just gets a bit messy I mean you would have you know if n is 3 you would get 8 elliptic functions instead of just 3 Actually when I say there's no good reason that's not quite true There is actually a reason why n equals 2 is rather better than the other cases that n equals 2 corresponds to something called the symmetric line bundles The line bundles that are isomorphic to their dual and that does make things a little bit easier Well let's take a quick look at some properties of these not quite periodic functions So we're going to have f of z plus omega i is equal to ci times f of z So you recall for periodic functions we have the number of poles is equal to the number of zeros And this still holds for this case here and you can prove it in much the same way you remember We can count the number of poles minus the number of zeros by doing this funny integral here as in complex analysis And we take the integral around the boundary of fundamental domain like that The bits on the top and the bottom cancel out and the bits on the left and the right cancel out so that's zero A slightly more interesting case was for periodic functions we had the sum of the zeros minus the sum of the poles And for periodic functions you remember this was in the lattice L And the way we got it was by looking at the integral 1 over 2 pi i times the integral of z f prime of z over fc dz Well the problem is if the function is not quite periodic then the left and the right intervals no longer quite cancel out So what we get is something like omega 2 times the integral from 0 to omega 1 of f prime of z over fc dz Minus omega 1 times some integral and this is the logarithm of f of z and it might change by log of c1 And if c1 is an nth root of unity this might be some integer of the form m over n times 2 pi i So here we're taking c1 to the n equals 1 We're taking n equals 2 in our case And if we work this out we find that this is actually not in L So this is not in zero it's some element of c modulo L It's not necessarily the zero element It's in 1 over n L modulo L if c i to the n is equal to 1 for some integer n which is the most useful case So instead of the sum of the zeros minus the sum of the poles being zero modulo L It's some fixed non-zero element of this lattice depending on this quasi periodic case So let's look at the case of the Jacobi functions In this case c i is equal to plus or minus 1 So the sum of the poles minus the sum of the zeros is equal is something in half L And you see that there are four possibilities if here's a fundamental domain And there are four elements of c modulo L such that two of that is in L which are like this And one of these doesn't count that's just the periodic case where you can't have a function with one pole So we can have functions and so these are almost periodic functions with a pole at one of these three points So these four points So the sum of the pole minus the sum of the zeros is at one of these four points So we've got a way can we get a function with just one pole of order one And the answer is yes we can and where are you going to put this pole Well it's obvious where you're going to put this pole if you've got a function with a pole you put it at the origin You wish in fact Jacobi decided not to push it at the origin for complicated historical reasons He actually put the pole here at omega 2 over 2 and this is another way in which Jacobi's elliptic functions have kind of got messed up Jacobi put the pole in the wrong place So the pole of s n c n d n is at omega 2 over 2 and not 0 for some weird reason So what's going on here is the poles and the zeros look like this So here's a fundamental domain Omega 1, omega 2, 0 and the functions all have a pole this point here So that's s n c n d n all have a pole here And then the sum of the poles minus the sum of the zeros must be in a half else So there are three possibilities for 0 and s n is equal to 0 at this point here And c n is equal to 0 at this point here and d n is equal to 0 at this point here By the way I should say the periods are usually called 2k and 2k prime Instead of omega 1 and omega 2 when people do Jacobi functions As I mentioned earlier one of the annoying things is when there are several completely different systems on notation When people do Jacobi functions they use a different period lattice from people who do viastrice functions Which is a utter pain when you're trying to compare the two sorts of functions Well so we want to find functions with a pole of order 1 here and zeros of order 1 at one of these three points So let's find existence and relation to the viastrice function So at the moment we haven't yet actually shown that the Jacobi functions exist Well existence is quite easy if I take the viastrice function of z and subtract rho of omega 2 over 2 This has a zero of order 2 at omega 2 over 2 and a pole of order 2 at zero And it's got no other zeros and poles So because all its zeros and poles have even order we can take its square root and get a single valued function So this is now single valued I mean usually when you take the square root of something you get a horrible mess whenever the function vanishes because you get a branch point But if all zeros of order 2 then taking a square root just gives you a zero of order 1 And it has a pole of order 1 at zero and it's periodic up to sine So if we define this function here then we see that f of z plus omega 2 is equal to minus f of z And f of z plus omega 1 is equal to f of z So we pick up a minus sign because we're sort of taking a square root which puts in minus ones at various points This is almost but not quite one of the Jacobi functions You see the Jacobi function has a s n has a zero here and this one has a pole at the origin So in fact this function is sort of linear in 1 over s n And some constant plus a constant times s n I'm not going to put the constants in because I can never remember them And the other problem is people use different conventions and different notations So whichever constants I put in someone's going to tell me I'm wrong So that constructs the function 1 over s n So you can obviously just take an inverse and get the function s n and you can get c n and d n by similar formulas So that shows the existence of the Jacobi functions with the properties I've stated And then you remember that all elliptic functions for L are given by either by a stress p function or its derivative Let's take functions with the only poles at zero And all elliptic functions are given by quotients of these and there's a similar theme for Jacobi functions If you take all elliptic functions for 2L they're given by poles only at zero They're given by c of n s, c s and d s And I'd better explain this notation so n s is equal to 1 over s n c s is equal to c n over d n where these are Jacobi's functions and d s is equal to d d n over s n So these are the functions these are three functions with poles at The origin and there's also a relation between these functions here you remember there's this differential equation Satisfied by Vistris's function which is minus 4 p plus g So p cubed plus g2 p plus g3 Similarly, there are some relations between these three functions In fact, if you look at n s squared, c s squared and d s squared These all have a double pole at zero and no other poles so they're all linearly dependent on each other So we should really quotient this out by various linear combinations of these These are usually written in terms of the Jacobi functions s n c n and d n as follows so s n squared plus c n squared equals 1 And k squared s n squared plus d n squared is also equal to 1 By the way, you notice this looks rather like the formula for sine squared plus cosine squared equals 1 In fact, sine and cosine are kind of degenerate cases of s n and c n if you allow one of the periods w2 to tend to infinity This seems to be one of the reasons why Jacoby put the pole in such a funny place He was trying to make his functions look like sine and cosine And sine and cosine definitely don't have poles at the origin So if you want your functions look like sine and cosine, you'd better put the poles somewhere else We can also use this to give several embeddings of elliptic curves For instance, Weistreist's function we saw maps the curve c over l to a cubic in the projective plane Similarly, by mapping a point z to n s z c s c and d s c, we map the lattice c over l to a degree 4 curve in e cubed And this again shows that Weistreist is a little bit easier than Jacoby because a degree 3 curve in a 2 dimensional plane is a little bit easier to handle than a degree 4 curve in space I'll just finish with some historical remarks about why Jacoby chose such funny conventions So we can say why did Jacoby put the poles at omega 2 over 2? Well, as I said, he had three functions s, n, c, n and d, n And these are actually short for sine of the amplitude, cosine of the amplitude and derivative of the amplitude Where the amplitude was a certain function that Jacoby thought was really fundamental Well, you can see what the amplitude is because d n is its derivative So the amplitude of z is equal to the integral from something to z of d n of z d z The problem is this function is rather a mess, you see it's not quite periodic So amplitude of z plus 2 omega 1 is equal to the amplitude of z plus some mysterious constant of integration And similarly the amplitude of z plus omega 1 plus omega 2 is equal to the amplitude of z plus another constant So the first place it's not periodic, it's only periodic, adding a constant The other problem is it has lots of logarithmic singularities Because d n is poles of order 1 and when you integrate those you get logarithms So the amplitude is a multi-valued function of logarithmic singularities and branch points everywhere It's really quite a mess, I've got a picture of it here So here's a sort of picture of the amplitude value of the amplitude function These sort of black patches here are where the function has a branch point and becomes multi-valued if you go around these points here And as you can see the function really looks like a bit of a mess So my feeling is that the amplitude function is probably best forgotten about it It's mainly of interest for historical reasons that it's how Jacobi actually originally came across elliptic functions You can define it explicitly not using d n, in fact this is the way Jacobi originally did it You write z is equal to the integral from 0 to phi of d theta over square root of 1 minus k squared sine squared theta And then phi is equal to the amplitude of z So the amplitude was really a bit of a mess First of all you've got this rather weird looking integral And secondly it's not given by the value of this interval, it's given by the inverse of this interval The amplitude is what you have to integrate up to in order to get the number z So you're dealing with this sort of weird inverse of an integral I think this kind of illustrates the well-known saying that pioneering work is really clumsy The first person to do something quite often gets things in a rather clumsy way Because they're doing something the first time and don't know what should be going on And then later on other people come along and clean up the area So Weistreist came along and gave a much cleaner version of elliptic functions But we should still give the credit to the person who came first and did things in this rather clumsy way So Jacobi's elliptic functions like the Weistreist functions have absolutely masses of identities I'll just show you very quickly show you a few of them Again wish even Watson is absolutely pages and pages of these You see there are all these identities there are so many identities that they write s1 instead of sn of z And it just goes on for pages and pages I don't see any point in going through any of these identities You get things like addition formulas which are rather similar to the addition formula for Weistreist functions and so on But all of these identities or most of them can be proved in a sort of mechanical way just by checking where the poles of both sides of the identity are Because if two elliptic functions have the same poles they must be the same up to a constant Jacobi didn't seem to have this technique I mean one of Jacobi's problems is that he was developing elliptic functions before complex analysis had really been developed So he simply didn't have things like Leuville's theorem And instead of giving these very simple proofs of identities just by checking the poles He gave rather complicated proofs by sort of explicitly calculating both sides and showing they were the same So to summarize the difference between Weistreist and Jacobi is Weistreist was looking at sections of an order one line bundle By sections I mean Meromorphic sections and an order one line bundle is just a fancy way of saying a complex valued function Whereas Jacobi the Jacobi functions are sections of the three order two line bundles This is all over the elliptic curve C modulo L So Weistreist and Jacobi are the cases n equals one and n equals two of looking at line bundles of order n As I mentioned earlier you don't have to stop at n equals two if you really want to you can do sections of the nine or eight order three line bundles And while away many happy hours writing down masses of identities between them but I really don't see the point Okay next lecture will probably be about theta functions of lattices