 Welcome to the last lecture. It is a kind of a course where you always feel that one can do a little more or not a little much more, but anyway one has to put a stop at some point. So, here is what I am going to do today. I think I said this last time. I will very quickly describe the bridge Sminert and Bayer conjecture and then finally, wind up by showing you how elliptic curves over complex numbers are equivalent to torus. So, let us start with that recalling the our discussion earlier that over elliptic curves the zeta function that we define the and there is a analog corresponding power series. So, that power series being modeller is essentially equivalent to the zeta function having a functional equation and that functional equation was symmetric along the line. So, the corresponding version of Riemann hypothesis for this would be that all the zeros of this zeta function lie on this line real z equals 1 and as I already said very little is known about this fact. In fact, something very simple as something that you might expect that something much simpler is also completely unknown which is. So, we just ask the following question how does zeta 1 behave is it 0 is it non-zero and if it is 0 what is the order of that 0. So, answer and that conjecture which I referred to actually refers to this behavior. So, what is the following it actually describes the behavior of zeta 1 and what is known is this is the fact that if you look at this group of points on the elliptic curve group of rational points on this elliptic curve and this group has a fairly simple structure and the structure is like this that this group of points is isomorphic to the following structure. There is a set of points which is called the torsion point which are points of finite order. So, which means that if you add the point to itself a finite number of times you get the 0 which is the infinity. Then there is this image of integers z and z to the r means the disjoint sum of r copies of z. So, this group essentially is a disjoint sum of r plus 1 subgroups first one of them is a torsion subgroup which is group of all points of finite order and then there are r copies of z and this number r is not fixed it can be 0 it can be 1, but it is finite. So, this is known about the group of rational points on the elliptic curve. So, now we come back to this conjecture Now, if you recall the what is the zeta function of the corresponding curve defined this is equal to summation n greater than equal to 1 a n divide by n to the z where a n in a sense measures the number of n is prime then a n is number of points on this curve modulo p and for other non composite numbers there is some way that you can define. So, a this numbers a sub n are in a loose sense counting the points on this curve over fields of finite characteristic or rings of finite characteristic. Now, let us come to this conjecture. So, this conjecture says not 1, but if you look at the zeta z as a power series around z equals 1 then how the behavior of this is the following says that the power series of zeta equals 1 goes like z minus 1 to the r times a constant and this constant also the conjecture describes very precisely. I am only going to describe two numbers in this constant overall constant that occur one is simple that is the number of the size of this torsion subgroup. So, that is the denominator in the numerator there is this number. So, absolute value of omega 2 and this number will this value will come to in the moments time when we talk about elliptic curve over complex number then this number will be defined and then there are there is another some other numbers here which relate to certain other quantities associated with this elliptic curve plus the higher power. So, forgetting the constants the main point here is that zeta 1 has a 0 of order r where r is precisely the number of copies of it in the group of rational points which of course means that if r is 0 then zeta 1 is non 0. So, 0 if and only if r is greater than 0 and the order is exactly. Now, this is conjecture again is a fairly recent origin it was done in 70s 1970s not very old conjecture very interesting thing is that the point this conjecture at the time this conjecture was defined nothing was known about elliptic curve in this for example, the fact that the zeta function that is why the functional equation is itself of recent vintage 1990s at that point not even clear whether the functional equation here and there was no reason to believe really why the why it should have a 0 and the order should be precisely r. So, what these guys did was they did extensive computer experiments calculating for specific curves the 0's and the order of 0's and then trying to look at the corresponding rational group and order of the copy number of copies and that is how they using some more intuition plus this they formulated it remains completely open and in addition to Riemann hypothesis this is also identified as one of the 7 millennium problem. So, that is all I will say about this conjecture now let us come to the other topic elliptic curves were complex. Now, one thing that should become immediately clear is that the geometric picture we have in mind for elliptic curves that there is this two independent curves one is a close curve and other is infinite curve that no longer holds over complex because that reason we there is two independent curves is precisely that there are some imaginary solutions in the in between region. So, real self that would not exist, but the moment you go over complex with the all the solutions do exist. In fact, for every value of y there will be precisely three values of x. So, that geometric picture will go away and interestingly and one may wonder what is the geometric picture that comes out of once you look at over complex number then interesting part is that is actually a torus, but one has to do a little bit of work in making this correspondence. So, exactly what I am going correspondence would be is the following. So, let us first look at the torus. The torus is a sort of a circular thing which is actually two cycles in it one is a big cycle and one is a small. Now, as a geometric figure is simple enough and we can view it as follows if you make it take the complex plane and identify some parallelogram in the one end of which lying at you that is also not important it is a parallelogram. Now, what you do is you cut out this parallelogram from the plane fold this. So, think of it as a paper then fold this this side folding on to this side and joining there. So, it makes a cylinder and then you take the cylinder and fold it on the along that x along the x axis and join this two cycles together that becomes a torus. So, the effect here on this piece here is as if we are identified this line with this line by folding and this line that is a geometric that is one way of forming a torus. It is a nice way for our purpose because it gives an understanding of how do we start from a complex plane. We already know some ways of twisting and turning complex plane in this various Riemann surfaces. So, this actually is another Riemann surface another way of forming a Riemann surface from a complex plane. Now, what I promised you was that I will elliptic curves I will show the elliptic curves over complex numbers to be equivalent to torus. In order to do that what I will need to do is to give homomorphism actually an isomorphism from the points on a torus to the points on a elliptic curve. And this isomorphism should preserve the group operation what is the group operation as a good point we know the group operation on elliptic curve what is the group operation on torus. Well, the group operation on torus says again can be easily defined if you use this picture. So, the points on torus will be the points inside this though one thing that we have to be careful about is when we look at this piece we should the points in the torus are counted by taking these two edges in and other two edges out. So, given any two points on this torus or on this piece what is the natural addition operation that we can define well these are two complex numbers that add the two complex numbers that gives you to another complex numbers and very natural addition. But, then it also needs to be inside the same piece, but this addition of these two may fall out of this. So, say addition of these two is somewhere here then wrap it back. So, just think of this take the another copy of the same piece stick it there and wherever that this location is just pull it back here. So, in sense you are going double modular modular in this direction as well as modular in this direction. So, whichever way you come back to this. So, that is a natural way of defining addition of this and that is it. So, that is this it is easier to see that the points on this piece or strip not strip this here. These are closed under or form a group under such an addition operation with the identity being the 0 that is why I said we will start with one end we will keep and 0. And we will show that the mapping there is a isomorphism from the torus to the elliptic curve with preserve this group. Exactly. So, that is the perfect observation here. So, essentially when you look at this structure can think of it as an lattice on the complex plane. So, if you look at consider this let us say this as omega 1 and this as omega 2. Then you can define a lattice to be m omega 1 plus n omega 2 m n in z. So, set of all points that come out of such linear combinations from the lattice. So, every point at the corners of such pieces will be in a lattice and then the torus can just be viewed as complex numbers co-scented with this lattice. So, L is a group it is easy to see L is a group it is a commutative group. So, it can be just co-scented set of group of complex numbers and the radiation with this lattice. That group that you get that is a group operation defined on this. Now, I want to give a mapping from this structure which is a torus to an elliptic curve. Now, in mapping there I will go will not only be an isomorphism of group structure, but it will also be a meromorphic function. So, it is all possible nice properties you can imagine and in order to define that what I will do is I will go back to this structure here and define a function on the over the entire complex plane. I mean one thing that becomes very obvious is if this function which is from complex plane to complex number complex number it has to make sense over this function from C slash L. Then it has to be periodic in that is if whatever value it takes at this point it must take the same value at all point corresponding points and all the pieces here. So, this will have there is a two dimensional grid here each one has a corresponding point to this here and at each one of those points function value will have to be the same. So, this function is obviously a periodic function. In fact, it is more than periodic because it is a periodic is just typically along one. So, that is f of z is f of equal to f of k plus z then it is periodic with period k, but this function actually has two periods. So, if this is omega 1 omega 2 then we must have f of z equals f of z plus omega 1 and f of z plus omega 2 for all such functions are called w period. So, that is the minimum we will need, but as I said we will do more we will actually make this f meromorphic function. But the moment you put the condition of meromorphism on the such a function which is w periodic a number of interesting things happen. So, let me quickly summarize that. So, there is an associated lattice L with this w periodic function which is essentially all integer linear combination of the two periods and f is a fundamental parallel pipe at associated lattice which is essentially that defining structure which one point at 0, one edge at 0 that is important. So, this theorem says that if you sum over all points z in f and look at the residues of each one of this point, residues of f each one of this point and sum is 0. Recall what is the residue of f at a point which it is the if you look at the power series expansion or Lorentz series expansion of f at that point there is a coefficient of power of 1 over z coefficient of 1 over z sum of all the residues is 0. Second the same sum over also points in the n f the order of f at z the sum of that is what is the order of f at z well if f as a 0 at z then the order is the smallest power of smallest non-zero power of the power series at z and if f as a pole at z then it is the largest negative power. So, if sum over the orders is also 0 then order is defined to be 0 residue and order both are defined to be 0 if there is residue is given now obviously 0 but order is also defined. Third another very interesting property that f is a surjective map not only surjective map it has the interesting property that it given any value on the complex plane f takes a value z w for exactly l values of z counting multiplicities. This is another way of saying that f z minus w has a 0 of order l. So, for it is not necessary that there are l distinct values of z taking value 0 here because for one particular value of z the order of 0 may be higher. So, if you just add up the order of all the 0's for all z of f z minus w that is exactly where l is precisely the sum of the order of f at all the poles inside f. So, look at the order of all the poles of f inside f sum them up they are all negative of course and then take minus of that that is and the proof is fairly simple this just uses the standard Cauchy's theorem. So, if you look at this what is the sum equal to by Cauchy's theorem this is simply integral over delta f boundary of f and the boundary of f is defined to be 0 to omega 1 plus omega 1 to omega 1 plus omega 2 plus omega 1 plus omega 2 to omega 2 and this is equal to let us go to the next page. Now, integral from omega 1 to omega 1 plus omega 2 omega 1 plus omega 2 of f w d w you can do a change of variable and transform it from a integral from 0 to omega 2 of f w plus omega 1. Now, since omega 1 is a period of f it is same as f w d. So, it is I can write this as 0 to omega 2 and exactly the same fashion I can eliminate omega 2 from here. So, it goes omega 1 to 0 which is of course, there is a small point here which I am skipping over that I am not am I assuming implicitly here that on the boundary of f there is no pole of boundary of capital f there is no pole of function f then this argument goes through there is a pole then you cannot really run through the integral line integral to that. So, what one needs to do is as you come to a pole you just make a tiny circle around it. Now, by periodicity if there is a pole on one edge there will be a corresponding pole on the other parallel edge there. So, whatever circle you do you same circle you do that. So, again they will cancel out each other and then you will get this here. Similarly, if you look at the second one some of this again this is straight forward because this again by this is by Cauchy's integral formula and that this is equal to we derived this the residues of f prime over f is precisely the orders of the vanishing either 0 or pole with positive negative sign. So, that is this now f prime over f is also a W periodic function with the same period why if f is period W periodic that is it there is a period. If you look at f prime it will be again the same W periodic function because it is the values remain the same. So, as you move across the one piece to the next piece you will again get the same derivatives all over the. So, f prime is also W periodic with the same periods. So, f prime over f is also W periodic with the same periods and therefore, this integral we just showed this is 0 for a for any W periodic function integral was 0. So, this would be 0 and third one was to is to show that given. So, you just consider f z minus W this function W is fixed. So, this function is also W periodic with the same periods since it is W periodic with the same periods if you look at the fundamental parallelepiped which again the same fundamental parallelepiped will hold will be exist for this also. All you are doing is you are shifting every value at every point in for f by this amount W. So, it remains W periodic with the same fundamental parallelepiped. If you focus inside the fundamental parallelepiped the sum of this is going to be let us just use this the second one. Some of the orders of this function inside fundamental parallelepiped is 0. Now, there are two possibilities. Possibility one is that it has no poles if this function has no poles then because it is W periodic function f has no poles anywhere. In fact, f is bounded. Now, if f is bounded I should have gone back and modify this and f is not constant because if f is constant now obviously, it is W periodic and obviously, this part 3 is false because it takes only one value everywhere. I should also say z in f. So, that is how many values how many z inside f are there for which you attain the value W because again if it is since it is periodic then there are infinity many values on which it take particular value that is not your interest in. So, consider f z minus W then this is W periodic is the same period. So, summation sum of orders is this is 0. Now, if f has no poles then it is constant and if f does have a pole then let us say we are already saying that L is the sum of all the orders of poles. So, sum of the orders of pole is L then by 2 sum of the orders of 0 must also be L. In fact, sum of the orders of poles is minus L sum of orders of 0 must be L which is another way of saying that the number of 0's of this function inside the fundamental pyrolyphiped are L counting the last one. So, now L is the sum of the orders of all the poles or minus L is the sum of the orders minus L is the sum of the orders of all the poles minus L because orders of poles are negative. If f has no poles then inside the fundamental pyrolyphiped the value of f is bounded because it is a compact set and if its value is bounded by eroticity it remains bounded everywhere. Now, f a function which is whose value and f is of course f is meromorphic. So, this is an again a whole theorem we proved that a meromorphic function whose value is bounded over the entire complex then is actually constant. So, now coming back to this to say the sum of the orders of f is 0 the sum of the orders of f at all the points inside the fundamental pyrolyphiped is 0. Now, if you just separate these the points poles and 0's. So, poles add up to minus L. So, 0's will also the orders 0's will also add up to L. So, what is the meaning of this that there are certainly 0's of f in the fundamental pyrolyphiped further their orders add up to L. So, applying this to that to this function it means that there are points z inside the fundamental pyrolyphiped on which this function is 0 in the f z minus w. So, it is a w periodic function with f capital f sorry. It is the same problem. So, the same problem because this result that 1 and 2 hold for any w periodic function which is meromorphic and as a fundamental pyrolyphiped f. So, this f z minus w also falls in the same category. So, the point the theorem 2 a part of the part 2 of the theorem holds for this function as well fine. Now, poles of this function are same as poles of f. So, it borrow this poles from f, but 0's of course, it does not borrow from f 0's are other locations. But, what is guaranteed is that sum of the orders of 0's is exactly. So, that is a kind of function we need if we have to define a map from a torus to complex plane. So, what kind of maps can exist which satisfy this? Certainly fairly non-trivial in order to satisfy all these properties. So, there are these very nice wire stresses this probably no anybody an expert in Greek alphabet does not matter. What? I do not know. Yeah, that is right, but it is w l does not matter let us call it whatever let us call it rho as long as we are clear what it means. So, let l be a lattice then define rho z to be. So, this is 1 over z square plus sum over all lattice points except 0, because in 0 this will get messed up 1 by z minus omega square minus 1 over 1. So, here is a nice theorem about this says that first ensures the convergence of f is well defined at every non lattice point of course, there is a pole second is that rho z is equal to rho minus z and rho z plus omega equal to rho z. So, it is and this is for all w omega in it. So, this is w periodic for the lattice over the lattice and third is this is even more interesting that any w periodic function over l as this form it is a rational function which involves rho and rho prime. So, it is a rational function in rho and rho prime over complex. Any rational function over rho and rho prime is obviously w periodic and this part says that that is all there is nothing more. And of course, rho has a pole of order 2 that is also fairly obvious just look at this definition. Any point in the lattice there is a pole of order 2 to do a little bit of work to ensure that the rest of the sum in this series converges, but that follows essentially from part 1. The proof of all of this is fairly straight forward not part 3 which I am not going to prove. In part 1 I am not going to prove where I will give a very brief sketch. So, if you look at to start by considering any point z that absolute value of z is less than absolute value of omega for all lattice points except 0. Then for that a particular z if you consider this expression I can rewrite this as and this I am going to now use the fact that absolute value of z by omega is less than 1 by using this property. And therefore, I can expand it as a power series and the power series is of this square 1 minus x whole square that if you remember the power series of this is equal to. So, you get this and just swap over the sum z 1 by z square. So, just rearranging sum essentially you get this power series well it is a Lorentz series with the coefficient here which is in sum over all lattice points except 0 of 1 over omega to the n plus 2. This actually one can show converges and it converges to the value which is called the Eisenstein series. So, this is the n plus 2 th Eisenstein series defined for this sum is an Eisenstein. And simple fact about this is that let me just rewrite first few terms of this 1 by z square plus if you will consider n equals 1 then this is g 3 if you consider g 3 the sum is 0 that is g 3 is 0 why g 3 is 0. Because if omega is in the lattice then minus omega is also in the lattice and. So, if you have an odd power of omega sitting here then they will just cancel each other out that the g odd is 0. So, this actually would be the smallest value this will survive is 3 g 4 z square plus next 1 by z to the 4 right 5 g 6 z to the 4. So, that is the power series expansion of rho around z equals 0. So, what was because 0 is in the lattice so on linear combinations of 0. So, if omega 1 is in the lattice omega 2 is in the lattice then all linear combination integer linear combinations. So, basically that shows that it converges converges essentially comes out of this fact that you have just do a bit of manipulation you get convergence, but I am I will skip over that I am mostly interested in this and now I am ready to give this map which I promise the chromataurus to an elliptical. So, this is rho z expanded in a Lorentz series around z equals 0 where it has a double pole. Now, consider the derivative of this rho prime of z if you just look at what is the rho prime of z going to be minus 2 over z cube plus 6 g plus g 6 z cube plus now let us do a bit more work what is rho cube of z and will only focus on those powers of z which are negative or 0 the rest we will just ignore. What are the non what are the negative less than equal to 0 powers of rho cube that exists cube of this square of this times this that is also negative square of this times this is 0 no cube it cube the whole anything else. So, that is it there is no there are only there are only 3 terms cube of this square of this times this and square of this times this any other term will be higher power. So, what are those terms so you get 1 over z to the 6 that is a cube plus square of this times this there is a 3 choose 2 times this there are 3 times this times this is 9 by z square plus square of this times this which is 15 g 6 plus the rest and square rho prime.