 So, let us talk about elliptic curves over q. Actually, yesterday we did part of it. So, E y square equals q plus h rational numbers. So, we say prime p is good if p does not divide and we define the zeta function for the elliptic curve in the rational field as the product over p p good of simply 1 minus a p p to the power minus z plus p to power 1 minus 2. That times something else which is a product over bad prime. Just for the sake of completeness let me give you what that is the product over bad prime of 1 minus a p p to the power minus z. Very simple, but a p is not well defined for bad prime and actually it is not defined in the same way for bad primes. So, this a p is actually without going into too much details it is either minus 1 or 0 or plus 1 depending on what is the type of badness. So, the type of badness could be there is a repeated root which is 2 roots repeat all 3 roots repeat right. So, depending on what kind of badness is minus 1 0. So, this is the zeta function over for the elliptic curve over rational and like I said last time this is closely connected with the with the form of last theorem amongst other things of course. So, what is the connection? So, let us start with that connection it is depends on the type of badness. So, when that of course, I just have the enumerated 2 types 2 roots repeat and all 3 root repeat, but within that there are classification. So, form of last theorem is I guess all of you know what this is there is x to the n plus y to the n plus z to the n for n greater than equal to 2. So, how does the proof go for this? So, I will just provide a very brief proof sketch assume a to the n plus b to the n plus c to the n for number integers a b and c whose g c d is 1. So, they do not divide otherwise you can clear out denominator and simplify this and n greater than equal to 6 for n less than 6 we already know the solution. So, y n equals 3 4 and 5 one can prove using very simple method that there are no solutions. So, these are proved by Fermat himself I think Fermat proved it for n equals 3 proved it for n equals 3 and 4 5 are like generalization. So, that I think now consider this elliptic curve which is a very simple and it is defined using this solution one of this one solution to the form of this equation a b and c we just take n b and use make a to the n and b to the n or a to the n and minus b to the n b the 2 roots of the right hand now for this curve elliptic it is a discriminant discriminant is same as the 4 a q plus 27 b square and if you remember that that is non zero if and only if it has no repeated roots. So, an alternative definition of discriminant in terms of roots is. So, that is discriminant delta f it is product of root differences if any 2 roots repeat then product is 0 and discriminant is 0 and it is actually easy to show that for an elliptic curve the discriminant the definition I gave is equal to product of different roots. So, what is the product of different roots for this curve well one is 0 one root is 0 the other root is a to the n the third root is minus b to the n. So, the difference between a to these 2 roots is a to the n first 2 roots 0 minus a to the n 0 minus b to the n that is b to the n and a to the n minus or the third one is will be minus b to the n minus a to the n minus a to the n minus b to the n. So, this is equal to a to the n b to the n a to the n plus b to the n and because a b c is the solution of that form of the equation. So, a to the n plus b to the n equals c to the n. So, I can write it as a b c to the a b c is an integer. So, this is showing that the discriminant of this particular elliptic curve is n th power t f. So, what does it mean well it means many things, but one of the things that it means is the following theorem that if delta f this is a more general theorem which I am specializing for the case of this elliptic curve. If delta f is l th power integer then it has a point that is this theorem is conditional. Let us keep it as it is since it is a n th power of an integer. So, it has a point of order n. Point of order n is there is a group associated with the elliptic curve which is a group of rational point this is a rational elliptic curve and point of order l simply means a point such that if you add the point to itself n time you get 0 or infinity that identity and no smaller number of addition will give you infinity. So, that is precisely point of order. So, the power of discriminant in relates to the order of a particular point. And then there is a theorem another theorem that if the f is modular. So, if the curve is modular which I will define soon the curve is modular then it does not have a point of order greater than equal to 6. So, if we can prove that f is modular whatever that means then we are done because then we say that it cannot have a point of order greater than equal to 6. So, therefore, either all points have order less than equal to 5 or have infinite order and keep on adding the points will never get infinity. So, it has a point of cannot have a point of order greater than equal to 6 then the discriminant of f cannot be 6th or higher power of an integer by this theorem which means in turn that a to the n plus b to the n equals c to the n there is no such solution a, b, c cannot be 6th and we have proved that. So, that is the connection. So, this whole part of these things were known already, but this was in this connection was put together by Gerrard Frey late 80s I think he proved this. Now, I am hiding away certain details actually strictly speaking the statements I am making are not true, but approximately true. There are some small, small twists then one has to give even the discriminant definition changes by there you have to divide it by certain power of 2, but we will not get into that because I just mix the whole thing without adding anything to our understanding. So, the challenge at this point was can we prove f to be modular. So, what is what does it mean for a curve to be modular? So, let us go back to elliptic curves and let us go back to those zeta function. So, if you recall zeta e z over of q if you remember last time we wrote it as also as a n by n to the right with certain multiplicative properties of a n this came out of the product form of it which you expanded to get this form. So, this is one series we associate with an elliptic curve and like I said may be a couple of lectures ago there is another series we can associate with elliptic curve which is a natural power series. So, let us define that to be f e z q instead of rising z to the end I am going to write slightly differently this is the Fourier series form of the power series. We can derive it naturally from power series by whatever the variable in power series replace that variable by e to the 2 pi i x and then we have this Fourier series form. So, these two you can see them in this coefficients are really common to these two series and therefore, there is some relationship clearly between these. Now, we so far looked at the zeta function. So, let us focus on this alternative form the Fourier series form of this can you identify some interesting properties there is one very simple, but nice properties and the name Fourier series itself gives a hint to the periodicity this function f is periodic what is the period one is any period of this function certainly divides one because f of z plus 1 what is that n greater than equal to 1 a n e to the 2 pi i n z plus 1 which is n greater than equal to 1 a n e to z then e to the since n is integer e to the pi i n 1. So, this is a periodic function and that is a natural Fourier series. So, that is one very obvious property of this function there is another very interesting property which is see this function clearly is the way it is defined. Now, let us look at what should I say where is this function defined this function may or may not for various points on the complex plane this function may or may not be defined. Suppose, z is on the upper half of complex plane which means the real the imaginary part is positive. So, let z be alpha plus i beta and beta being greater than then what is f z a n e to the 2 pi i n alpha plus i beta. So, if you look at the absolute value of z f z this is less than equal to sum over n absolute value of a n actually a n is an integer. So, it does not matter it is a positive integer times well this goes away and this is again always positive this is less than equal to summation n greater than equal to 1 what is the upper bound on a n like order n. The a p is p plus 1 plus minus 2 square root p and the multiplicative property show that a p q is a p times a q. So, this is essentially order p q and the a p square although I am not given the definition one can show is order p square. So, this would be order n a n would be order n divided by e to the 2 pi beta this clearly converges because the denominator grows far more rapidly. The same argument can be used to show that when you are in the lower half of complex plane then this diverges because then this would be positive and that will really shoot up no matter what the coefficient do to you and on the real line may or may not converge depending on how these coefficients are set up. So, that is the sort of the structure of the space where f is defined essentially upper half of complex plane. Now, comes an interesting transformation on the upper half of complex plane which is called the Mobius transformation. So, here plus is upper half of complex tau z goes to A B C D determinant of A B C D. So, this is a well these matrices 2 by 2 matrices whose determinant is one they form a group under multiplication and this is a very well known group called the symmetric linear group. So, essentially this SL 2 is operating on z. So, tau z is simply tau is simply SL 2 operating on the complex number in this form. The interesting thing is that this maps upper half of complex plane to upper half of complex plane tau z slightly simple a z plus b times c bar what do you do c bar z bar plus d just multiply the complex conjugate and what do you get here. Now, let me make a simplification here just to make life simple I just use this interior because this is what I am going to be interested in. So, A B C D are going to be in T s. So, no c bar no d bar. So, this comes as A C mod z squared plus plus A D z plus B C z bar. Now, if you look at the imaginary part of this is real this is real. So, this is this is a part that contributes the imaginary part then this take out the real part the imaginary part of z bar is negative of the imaginary part of this. So, you get A D minus B C times imaginary part of z divided by C z plus d whole square. Now, A D minus B C by the definition is 1. So, this is imaginary part of z divided by C. So, it is sign is exactly same as imaginary sign of z. So, that is where I am going to stop because I do not have time, but tomorrow I am going to finish this off. So, this tau is going it is a very interesting transformation it looks somewhat funny that you may mapping this in this fashion, but it is a the most general transformation that reserves for example, circles. If you take a circle and apply tau on it and look at the curve that you get it will be a circle. If you take line and apply tau on it what happens to a line a line also goes to circle. So, circles and lines together go to circles and lines. So, it is basically general like this if you take two lines with a certain angle and look at the corresponding curves under tau and the point where they intersect you look at the corresponding point where the corresponding tau curves intersect look at the angle of intersection there that we preserved. And this is this how we can actually characterize all the class of entire all transformation we preserved this. So, it is a very interesting subclass of transformation which preserve a lot of properties. And these are going to be useful for us also because the property that we want from this function f is essentially invariance under tau. So, f of z or f of tau z we would want to be roughly equal to f of z not completely, but roughly.