Introduction to elliptic curves and BSD Conjecture by Sujatha Ramadorai

0:12 okay so i' like to thank all the younger co-organizers who did all the work and I'd like to welcome you all um I want to say a few words about the Genesis of this conference so in 2014 December 2014 there was a preliminary workshop on the birs V and conjecture theoretical and computational aspects of the births and conjecture in Beijing in China at the Beijing International Center for mathematical research and so we largely covered you know the statement of the conjecture the theoretical aspects and we thought we will follow it up with something where more people in India there were very few students from India who participated in that conference there were lots of Chinese students few

0:58 Indian researchers but not many Indian students and so we thought we should have something in India where there are more Indian students exposed to this subjects and there are a few Chinese speakers who will be speaking in the workshop uh part of it and many of the lectures that were covered in China they are available in you know type form introduction uh whatever results were covered and so on it will be distributed along the next few days printed copy or we put it on the website so that people download it look at it whatever but I would strongly urge the younger participants to go through those notes as much as you can I'm sure not many don't feel disheartened you might

1:43 not understand everything there and so on but that's the way mathematics is done you don't understand everything the first time in fact uh in born in Max blank Institute born Professor hook and harda gter harda very famous German mathematicians they used to say like that doing mathematics is like going to church when you go to church first of all in those days in Latin all the Lany lury nothing you wouldn't understand anything but you had to go there continuously so that eventually you'll understand it so mathematics is like that so keep reading it and you will understand you won't understand it the first two times but the third time your brain will ask you some question you know why is this true why is that true and so on and that's the way to pursue a subject and dig deeper into the subject so just don't come here with the

2:29 attitude that I'm here to listen to the lectures take notes please go through the additional material that will be made available to you or we'll give you the website notes where it's available go through that in you know alongside hearing the lectures and interact a lot with the speakers and especially the conference speakers also so what is very beautiful about the births SP and dire conjecture is you know it's all these intricate pieces small pieces and then they fit together perfectly so why they fit together is a mystery why they fit together so well is a mystery and of course that's the theoretical side it predicts something but then the beautiful side is it's also computational you can check and checking that the formula is correct that the

3:16 conjecture is true for various elliptic curves that itself is a task you know it's a project for a student who is learning the subject it's equivalent to a project to understand exactly how you compute and then because you you need a lot of theoretical understanding to actually implement the computation or to understand how the computations work so it is that's what is very nice about number theory on the one hand it has this very abstract theoretical side but on the other hand you can actually see that it's not just some empty dreams or some fancy dreams or fantasy but it's actually something that can be concretely verified and when you verify that's the computational aspect when you verify you will realize that you're actually picking up a lot of the theory you're understanding a lot of the theory

4:02 of why these things are coming and what are the generalizations and so on the other beautiful thing about the births conjecture is that it's just the very base of a vast set of conjectures and this is the simplest form of one such conjecture which already is a million dollar problem as you know so the other generalizations of this conjecture are even deeper and there's a very active area of research both on the theoretical side and on the the computational side on possible generalizations and so on so anybody who you know wishes to understand this deeper and if you work hard at it and it it is very fruitful it is very productive for you in your future career as a researcher especially if you're interested in number Theory okay so with that so this is available

4:49 I'm going to you know this PDF file I'm going to either distribute it or maybe printed versions will be made available this is the introduction to the birth and conjecture so how many of you here have you heard have you come across this conjecture before not just the name but in detail okay so not uh not many so um for the younger students pay close attention because like I say it has two parts the theoretical and the computational part and each of it is very deep you know each part of the puzzle like I said it's a big puzzle which fits together beautifully and each part of the puzzle is a subject by itself okay so um the basic object of study here is what are

5:36 called elliptic curves the births unit entire conjecture is about is a conjecture about elliptic curves and uh um don't know how I go down yeah so what it does so we first need to understand I'm going to introduce three players at a very preliminary level the first is elliptic curves and then um there is a function let okay let's first talk about elliptic curve the simplest way to think of elliptic curves of course you can study eliptic

6:23 curves on the other hand are the most simple of what is called the aan variety but you can think of it algebraically you can think of it topologically you can think of it analytically you can think of it from the point of view of algebraic geometry or whatever but for our purposes for the purposes of studying the birds and conjecture it's best to think of elliptic curve as an equation in three dimensional space so it's a projective equation so it's an equation which is homogeneous three variables so that's the most general form of the equation that's called the VR equation here what's return return here this y^ S Zed you know so this is a homogeneous equation right every homogeneous equation of

7:09 degree 3 so of course I mean then you're interested in the solutions of these equations so these coefficients A1 A2 A3 A4 they can be in any good thing that you like it can be in the rational numbers it can be in the integers it can be in uh complex numbers it can be in the real numbers or it can be in more fancy fields which are called the piic fields and then you want to study the solutions of this equation okay and note that this is projective space so P2 so of course there's one obvious solution you put xal yal Z equals 0 that's an obvious solution 0 equals z but that's not allowed because you're working in projective space in projective space you're working with homogeneous coordinates x y z so in projective space

7:55 you have coordinates let me write x y z and I write this sign because instead of x y z because what this means is if I take a value X not X1 X2 this is the same as multiplying it by any scalar in the corresponding field that your where your coefficients are okay so it's it's projective that's what is meant by projective coordinates whereas when you work with aine coordinates you can convert it into and aine meaning DEH homogenize it that is by just putting Z equals 1 so you're looking only at one part of the equation but that part is good enough and for you to relate to it

8:42 concretely another example I can give is all of you in topology or in your MSE would have learned about the stereographic projection right so in real terms we know that the Earth is a sphere but we don't imagine it in three dimensions we project it to the two two dimensional plane and see imagine a map as being flat so this DEH homogenization is something like that you just looking at one part of this whole thing what you're missing out is not it is important but you're not losing too much information so it's okay for you to De homogenize and study the solutions after de homogenizing so here yeah so after you de homogenize you'll get uh this equation here for

9:28 instance just putting Z equal 1 and to remember that it is De homogenized I'm taking changing equation the variables from Capital to small okay and all of these coefficients are like I said in any favorite field of yours K is any field and you can work with that field okay and we are only going to study the aine part this is called aine part of the equation one aine part of the equation when I put Z equals 1 and roughly that's what we are going to be studying and we are going to be studying the solutions and you can make various coordinate changes remember again in coordinate geometry when you was when you studied the hyperbola or the parabola or linear equations and so

10:14 on you were always made to allow you are always allowed to make certain coordinate changes you're not changing information by changing coordinates okay so roughly that's what I'm discussing next you can make certain coordinate changes and then if you're always we'll assume that the characteristic is not two two is invertible that's that makes life simpler if you want two to be not invertible if you want to work with Fields with characteristic two for instance something like ZM 2 if you want to work over such Fields you have to be much more careful so we are not going to consider that equation uh consider that case and we'll be working with characteristic not two Fields then you can make various change of coordinates and ultimately reduce it to something of

11:00 this form y^ 2 so elliptic curves you can ultimately reduce so let me assume K is a field characteristic of K is not equal to 2 then you can actually assume that you can after changing coordinates you can assume that the curve is of this form y^2 equals X cubed + a x + B where this A and B can be explicitly formulated in terms of the original coefficients remember my first equation had in the D homogenized sorry in the homogenized form as a homogeneous equation you had all these coordinates A1 A2 A3 A4 A5 A6 and so on right so

11:46 this capital A and B you can actually give explicit formulas for those in terms of those coefficients so you take the most General homogeneous equation in degree 3 which is of that original form you have a y squ z etc etc de homogenized then you can reduce it to this form and the characteristic is not to so this is a cubic on this side right important invariant associated with this this is called Delta which is the discriminant and that is the formula there minus 16 into a cub + 27 b^2 and for those of you who remember how to solve we all learn how to solve quadratic equations in high school minus B plus orus square root of B Square - 4 a c by 2 a but solving cubic equations you don't learn

12:32 it in high school it's much more difficult it's a BC if you're lucky I don't think you even learn it in BC you learn it only in Masters at least in my days it used to be the case I don't know if it's now taught in BC but when you're are solving cubic equations you come across this invariant for cubic equations called the discriminant so this is very closely related to that discriminant okay and we will be considering only those cases where this Del Dela is not zero this is called the discriminant of the elliptic curve and I'm going to assume that my curve is smooth so I can now you know I can just look at the right hand side and just remember it as y^2 = F ofx where f of x

13:20 is a cubic equation over K yeah yeah yeah no I this is the aine part so it's okay I'm still looking only at the aine equation right I remember I've de homogenized again if you want to relate to something familiar remember you have a square uh sorry you have a sphere and there when you're are taking the stereographic projection this point is omitted because you're projecting from that point right so you're looking at only a fine part but there is something else that point this is called the point at

14:05 Infinity so something similar is happening here you have de homogenized it but in the homogeneous part if you look at the first equation again there you'll see an obvious solution 0 1 0 will be a solution yeah you see if you look at this equation here if you look at this equation here there is one obvious solution I can put xal 0 y = 1 zal 0 this is legitimate because this is not 0 0 0 so this is a legitimate point of the projective plane and this obviously satisfies this equation but when I have DEH homogenized it this point is missed because when I'm DEH homogenizing it Z coordinate has to

14:52 be one so there is still this point okay so it's just like when you're projecting from a sphere to the plane you have that point at Infinity so even here when you have de homogenized you have that point at Infinity okay so this is the elliptic curve so you can remember y² = F ofx this is a cuic equation over K and then I want Delta not equal to zero this will be arranged I can arrange this if f of x has distinct roots so this is for you for in the most simplest of terms you can think that an elliptic curve is the study of solutions

15:39 of equations of this form y^2 = F ofx where f ofx is a degree 3 equation with distinct roots and you're looking for all those roots but along with those Solutions you have this other point 0 1 0 okay so this set we will refer to it as an elliptic curve for where for for the purposes of this Workshop that is enough if you remember that so now of course then you want to ask how big is this set is it we know it's non empty because there is that point at Infinity 0 1 0 is there but beyond that are there other equations now that will depend on the field right where can you find equations can you find it over the field that you're considering how many equations does it

16:26 have so the simplest of field we will consider is q rationals so what about rationals you can go to the real numbers what about real numbers you can go to the complex numbers what about complex numbers and so on so roughly that if you just understand that the study of elliptic curves or all these fields each field is very deep studying the solutions over each field introduces a new aspect in the study of elliptic curves and then how do you put all this information together so roughly you need that is the basis of the bir and dire conjecture okay so now um we are going to study therefore let me so therefore an elliptic curve over K so there are these things you have to remember y^2 = F ofx degree 3 then

17:15 there's a discriminant Delta not equal to zero or you can think of f of x is having three distinct Roots then look at the solutions e of K Solutions in K that is all the values X and Y in this field which satisfies this equation along with the point at Infinity so remember that this constitutes an elliptic for today okay and now the first important theorem is that this has a much finer structure it's not just a set when you take any two points in the solution set you can actually manufacture a third point or if you are given even one point you can manufacture another solution

18:01 this you can't do in general right if I give you any equation and if you solve it and you find one solution it's not obvious that you can use that solution to produce more solutions so already that's very interesting about the elliptic curves so that in fact is the very it's not a deep theorem it's not difficult uh but um it's very counterintuitive in the following sense so here these are uh this is Delta and I mentioned remember I mentioned that Delta you can actually express it in terms of the other coefficients and so on so this is the formula okay so we can skip all this like I said uh it'll be given to you in the note form so you can just read about it and uh yeah so let's skip this

18:48 uh I'm just going to concentrate on uh this theorem okay now I don't think I've explicitly stated it here but let's write theorem theorem one e over K and elliptic curve K is a field so when I say elliptic curve I mean all this characteristic of K is not two and then I'm looking at the solution Delta of such an equation with Delta not equal to zero and then I remember that

19:34 that's a set and then I add this point at Infinity so that is my e of k e of k e over K and elliptic curve then the set e of K I take all the solutions has the structure of an aelan group so this is why once you given one solution you can get more solutions because it has an ailan group structure so I take one point then I know that if I take there is a multiplicative operation right once it's an aan group you have a binary operation on it plus

20:19 or dot let let me call it dot so if I take two points or okay so aan group what does this mean so this means that e of K comes with a structure plus so I can take any point p in E of K and then it makes sense for me to talk of p plus p and then keep repeating it P plus P plus p and so on and this theorem is telling me that p+ P is again going to be a solution of that equation and this P plus P this is a complicated operation it's not something very naive like I take a solution X Y suppose this is my point p and and of course I'll homo even if I homogenize this x y and then I have some Z

21:05 coordinate so it's really let me put it one in projective coordinates and then I take a point Q which is the form X1 y1 so it's really X1 y1 1 it's not addition in the naive sense it's not that I add x + y X1 y + y1 1 + 1 or multiply X into X1 y into i1 and so on it's a more complicated operation and there you involve actually U what you studied in coordinate geometry so for those of you who don't know how this operation is defined that's an exercise go spend you know it's there in Wikipedia it's there in any uh you can find lots of articles on the web try to understand how this addition operation is defined and why it depends only on

21:51 the field so you see what you will do is once you add this you will get a new Point let me call it X2 Y2 1 now this X2 Y2 will also lie in K whenever X Y X1 y1 lies in k then X2 Y2 is also going to be in K okay and it involves a lot of uh not difficult but deep ideas like bzu theorem you know it's a theorem uh all of you know I hope you know or even if you don't know it remember when you studied coordinate geometry and when you took intersections let's say a line intersecting a circle if I take a circle and if I take a general line how many points can it intersect

22:37 in h at most two right if it's a tangent then it will intersect that point but that point is actually counted twice because it's a tangent why is this happening because the equation for a circle is x² + y sare = z Square what degree is it line has degree so 1 into 2 is 2 so that's why it integr intersects it in two points but now if you look at this this elliptic curve has homogeneous degree three a line has homogeneous degree one so ideally it should intersect it in three points 3 into 1 three points so you really use that so in fact there's a pictorial uh depiction of that let me just show it to you so that you sort of understand why this

23:24 works let's do it over the reals the idea is something similar so over the reals most often the elliptic Curve will look something like this this is the aine part because this is a plane so this is the aine part of the solutions and then remember there's this point at Infinity somewhere up there so if I want to add two points suppose I want to add two points these two points then the idea is the following okay let's take these two points so you first draw a straight line through these points now this is coordinate geometry right you know the general equation for a line that cuts

24:11 two points in a plane so you have that equation for the line H now because this line is of degree 1 and your elliptic curve has homogeneous degree 3 this line is already intersecting the elliptic curve in those two points but there is still a third point which it should intersect and you don't know what the third point is okay but in principle there's a third point so what you do is then take the point at infinity and then join it to that third Point see this this line through these two points it's intersecting the elliptic curve in a third point so now

24:56 start with that third point that's one point and then this point at Infinity is the second point so join those two points by a straight line right I mean all this can be explicitly worked out in equations you know this is simple coordinate geometry you're just taking two points taking a line through them therefore you have the equation for the line then you're intersecting this line with the elliptic curve and you know again in coordinate geometry how to find intersection points so therefore you find the third point then take that third Point join that third point to this point at Infinity so then you have a line two

25:41 points now again that line has to intersect the elliptic curve in a third Point these two points are already on the elliptic curve and you have to that line has to inter intersect the elliptic curve again at a third point that third point is how you define the addition of these two points okay and this is explicit you have to write down the equations do the intersection formula yeah otherwise it won't be smooth yeah because this equation this uh D I mean that it explicitly intersects three points you need it it's easier if it's smooth you need smoothness

26:29 no you in fact you can generalize it you know they need not be distinct so for instance if I want to double this point if I want to add this point to itself then I just take the tangent at this point so that tangent is already intersecting it twice so then I find one other point okay join that point to the point at Infinity then that will intersect it at a third point and that's twice this point okay but but it's a little tiresome to actually write down the explicit equations but it's done in any number of textbooks silver man T Silverman all those textbooks it's done and I would strongly urge you in the evenings most of them are actually available online or you can read it but

27:14 please you know these are the kind of things which you should be doing in the evening or when you are free you should someday when I mean if you have attended a workshop on the births Winer and dire conjecture you should know how to add two points of the elliptic curves that's the least takeaway from this Workshop okay all right so therefore e of K has a structure of an aelan group now what kind of an aelan group is this that depends on K that depends on the field that you're working with so that gives much more structure to the study of elliptic curves you can study the elliptic curves over different fields you can study it over Q you can study it over R you can study it over C

28:01 you can study it over what are called local fields and so on and so forth and then each time you get a different kind of aan group okay so let's take a simple case for instance let's take a field that we all familiar with k equals the complex numbers so there what what is actually happening those of you have done topology you'll recognize a little of this see when you are thinking of an elliptic turb in terms of complex numbers then it's a Genus zero curve sorry genus one curve genus genus just measures the number of holes that you have in your surface so a sphere has no holes so it is genus zero genus one it has one hole so it's like a Taurus how

28:46 do you get a Taurus so what you do so if I want to look at the anelliptic curve Solutions over an elliptic curve then what you really need is a lattice see you have the integral points here right and this gives a lattice what do you mean by a lce I just take these points then over the complex numbers these points give you all of the elliptic curve somehow okay but concretely what do I want I told you that it's a Genus one elliptic curve so if you want to think of it topologically it's a Taurus how do I get a Taurus from the plane Z cross Z that's where the lce comes in

29:31 Z and Z is a lce that's what I have mapped out there right I'm actually using one 1 0 but there's no reason to use one 0 1 0 I can use any other lce right so what is really happening there is what you have probably seen in topology just look at this part so you're left with a rectangle topologically how do you get a Taurus from a rectangle stick these two sides get a cylinder stick those two sides you get a Taurus right in mathematical terms very rigorously if you want to express this especially if you want to see the structure of an aelan group what's happening is you have a latis

30:18 Lambda inside the complex numbers so for instance it can be this lattice or it can be even some smaller lce like this which you span over the entire complex plane okay and then you look at C this is an ailan group your lce is also an aelan group and it's a subgroup of the complex numbers so it makes perfect sense to talk about the quotient C Mod Lambda right and if you come back and look at this what you are doing when you're identifying Lambda see these are the four latice points and going modulo means you're killing it so all these

31:03 four distinct points you're identifying it so that's the first part you're going to identify this point and this point this point and this point so you make a cylinder then all of these so then you only left with one point here one point here that's going to be identified so that's making the taus like that so topologically that's what is happening from this algebraic operation okay so you can think of an elliptic curve over complex numbers e of C you can just think of it as C Mod Lambda so the only important information that you need to remember is this ltis Lambda it might happen you take two different latices but when do they give the same elliptic curve in the geometric

31:51 sense that happens when you can go from one lce to the other by a simple multiplication of a scalar this is what's called an isometry of a lce or homothety of a lce okay so so that's a simplest case and this has a structure of an aent group and you see this is huge this group is huge C Mod Lambda it's an aelan group all right but it's not a finitely generated aelan group at all in no sense and that's why from the number theoretic point of view when you start stud the elliptic curves over what are called number Fields they are very interesting so let me remind you definition a number field

32:39 K is nothing but a finite extension of Q finite extension of the rational numbers so for instance how do you get finite extensions of Q just take any equation over Q solve those equations attach those roots to that Q then you get a field extension that's a number field so for instance Q attach square root of two square root of two is not in Q or Q attach I the field generated by q and I that's a number field and these are all finite extensions of Q because as a vector space over Q it has finite Dimension right so that's an elliptic

33:25 curve over a number field and what the most powerful theorem here theorem 2 is that let K be a number field this is called a model way theorem let K be a number field e over K elliptic curve then the solutions e of K Remember by this I mean the aine solutions and also the point at Infinity okay always e of K has we know that it has a structure of an ailan

34:10 group but more is true it has a structure e of K is finite is a finitely generated aan group so this is what is miraculous and this is what makes the study of elliptic curves very interesting from the number theoretic point of view because once you know that it's a finitely generated aan group you have many more questions that you can ask what is the first question you can ask you can use the structure theory for finitely generated ailan groups so you know that by the structure

34:55 theorem therefore of K is isomorphic to certain free copies of Zed this is the free part because any finitely generated ailan group has a free part which is the rank and then a tsion part right and that tsion part you can finite you can write it further as cyly groups now it's Z mod P1 to the^ N1 Z mod q1 to the power N2 and so on so you have all those invariant so you have understood this structure if you understand the rank and if you understand the tsion part what the tsion part looks like so let me just write e tsion k this is the tsion part this is going to now be a finite group

35:42 so it's a finite ailan group so therefore it's a direct sum of cyclic groups and what kind of cyclic groups occur there that's the next question but already from this you have one very important invariant which is R okay okay so this is called algebraic rank definition the invariant r coming from the structure is called the algebraic or model way rank I'll use either of these terms interchangeably model way rank of the elliptic curve e over the field K it's important that you

36:27 remember that it's over K because you're solving you're looking for Solutions over that ground field right now notice one other simple thing that you can do already so I have an elliptic curve over K and let me take an extension l/ K extension of K finite infinite don't care but some extension now I can then look at the solutions of e over K and of e over L and of course because K is a sub field of L you have e of K is contained in E

37:15 ofl right and anything can happen it might not have it might have very few points here and even if you go to a small extension it might already already have large number of points it might acquire a rank here e of K could be finite but e of L could end up being very large you could actually get R then immediately you ask questions like how big can R be is there a bound on R meaning is there a universal bound such that any elliptica or any number field cannot have rank bigger than that okay so all of these many of these questions are open I think the largest known elliptic also can you explicitly write down an

38:02 elliptic curve of rank something over a given field so you see these are all computational aspects of the question so some answers are known some answers are still wide open you know nobody knows the answer and so on so this is how the computational aspect of the study of elliptic curves automatically enters the picture given an elliptic curve how can you compute its Rank and so on and each of these actually took I won't say centuries but definitely Decades of research to actually arrive at an answer okay all right so this is one story so if you have understood things up to this part this is one half of what you need to understand the birds wi Andre conjecture now in this we have not used anything except

38:47 algebra right you're solving equations you're using the structure of finitely generated aent groups therefore you get an invariant which is algebraic rank now what is amazing about the births un and dire conjecture is it's going to define something on the other side which is completely analytic in nature and the way you cook this up is you know you don't even understand why somebody should be studying this but there is a motivation it comes from the reman zeta function and so on so just believe me when I say that it's not just ad hoc but there's a method in the madness so you manufacture something else which is completely analytic in nature and then from there you'll manufacture an invariant and then the first part of the bir SP and dire conjecture predicts that

39:35 this algebraic invariant you have defined and that analytic invariant that you have cooked up using some very strange things they are actually equal so let's come to the analytic part now how much time do I have 20 minutes okay good so we can look at the analytic side so is this clear so far the algebraic side any questions so remember yeah sorry then they'll have to lie in a straight line you see those that point at Infinity p and Q then p+ Q will uh be that point at

40:24 infinity speaker please yeah yeah yes that's in fact another important thing that will be inbuilt later so the question is the following which we will talk about later that will come when we actually so the birs wi and dire like I said at the first level you have the weak birs wi and dire conjecture which predicts equality of two different invariants an algebraic invariant and an analytic invariant and then there's an exact formula this is what is even more uh amazing about the birs entire conjecture you see this analytic function that I'm going to Define is actually going to be a power Series in one variable over a complex variable okay so it's a holomorphic function analytic function so you can think of it as a power Series in one

41:11 variable so then you look at the leading term of that power series every Power series you has an order right the first non-zero leading term so the birs V and D conjecture the exact formula predicts the exact value of this leading term then it will act the coordinates because each coordinate three is an element of K bar so I can take a sigma which is in the Galva group I can add Sigma on each of those coordinates of the point then I get Sigma X Sigma X1 Sigma X2 or Sigma X Sigma y Sigma Zed KN that's again going to be a point on the elliptic curve so that'll again give you a point in E of K Bar okay so that means that this Galva group is acting on this set

41:58 set and the points which it leaves invariant you can actually it's easy to see that e of K Bar fixed so I'm going to put GK this just means the fixed points of of e of kar the gala group fixes that point so take all those points which is fixed by every element of the Galva group then that is nothing but the solution defined overk e of K Bar the Galva group acts on it take the fixed part of it that's nothing but the elliptic curve overk and this will play an important role in the exact formula you know you'll Jazz this up further you'll study this action you'll Define something and then you

42:44 that will come up in the exact formula and that's in fact one of the mysterious things in the bird's conjecture exact formula all right so now to come to the analytic side I'm not going to write it I'm just because I'm going to send this to you or you'll get it um okay so in order to manufacture the analytic side what you need to remember you know again for those of you who haven't seen this deeper I would urge you to these are the things that you should be doing in the evening so let's take the simple case of k equals Q then we all know I mean that's one of the first things we study in algebraic number Theory okay what do we know about Q so we know that Q sits inside the real

43:31 numbers as a dense subset right Q is a metric space and the metric that comes is actually the metric that's coming from the complex absolute value restricted to R restricted to Q you get a metric so Q is a metric space with respect to the absolute value and then once it's a metric space you can complete it what do you get when you complete it you get the real numbers so therefore Q is dense in the real numbers with respect to that metric but there are more metrics that you can Define on Q the simplest one being so you have the absolute metric I'm going to put Infinity for that so when you complete with respect to that metric let me put absolute value Infinity so when you complete you get the real numbers but what you can also do is you can use a another metric which actually

44:19 for every Prime p i can define a metric which I'll denote by VP absolute value with respect to P I'm not going to be very precise because I want you to go and dig out this metric for yourself understand how it works and so on but what does a metric do in any topological space what is the use of a metric distance now here I'm going to use the distance what does the distance say how close are two objects in that set here I'm going to be simplistic and say in order to define a metric you only need to understand how close something is to zero right you need to distance you know need to know the distance between two points but I say it's enough to know the distance from zero because that's

45:04 inbuilt in the metric formula distance between X and Zed you understand the distance between X and Zed if you understand the distance between X and Y and Y and Z right so I'm going to use that principle here so with respect to this metric a point in Q is close to zero if it is divisible by a very very high power of P so that's the distance function and I can do this for every p and then you actually have to check that Define defines a metric okay so again for those of you who haven't seen it there's an exercise do these things in the evenings and then therefore you can complete it when you once you have a metric you complete it with respect to

45:50 the metric then you get a field complete you get something called QP which is called the pic local field and again Q is dense because this is a completion Q sits as a dense Subspace inside QP and just as inside Q you have the special latice Zed right I just have to take the field of quotients of Zed to get Q so similarly inside each of this QP I have something called the pic integers zp which is an integral domain and I just take have to take the field of quotients of zp to get QP so all this is

46:39 automatically available once you study Q so then you see your whole paradigm shift now you want to study an elliptic curve over Q you know that e of Q is finitely generated and you know that it has a rank and so on but the only way to understand e of Q is by studying I mean not only way it gives you more Le leway to understand e of Q I can study e of QP QP is huge by the way it's by no way a finite extension or anything like that it's really huge as a field I study e of QP for every Prime p and then understand how can I get more information about e of Q from these

47:25 larger fields so that's one of the side stories in the theory of elliptic curves and why is this interesting because QP is a much even though it's large it has more structure and it's easier to manipulate for instance zp is a local ring and a local ring is very simple local ring only has a maximal ideal and in this case the maximal ideal is even principal is generated by a single element so there's a maximal maximal ideal and then the units so local rings are in between rings and Fields they are not as beautiful as Fields but they have much nicer structure okay so zp is a local ring so

48:12 QP is actually the field of quotients of a local ring you only have to invert one element which is p to go from zp to QP so QP in fact you can write this as zp Inver P okay so I'm going to use this information to explain what happens on the analytical side so each of this is called a local for every Prime what you get correspondingly that is local and I have a bunch of local information and I want to get Global Information that is Global Information meaning information up at the Q level from QP and that again is a very important principle in amb theary if I'm

48:59 given local information can I piece them together this local information you should think of them as pieces of a puzzle can I somehow assemble them to see some part of the larger picture that is information over q that is the global information and that principle is called local to Global principle and the first place where it is very um beautifully operative is in the study of quad quadratic equations H in Kowski principle you take a quadratic equation you want to understand if that quadratic equation has a solution over Q then the has in Kowski principal says that it has a solution over Q if and only if it has a solution over the reals and over all the QPS and and you see that's the other

49:47 thing about studying QP so this I said is a local ring it has a principal ideal and so therefore I can study the residue field the residue field is just ZM p z which is actually a finite field and finite fields are much easier to manipulate do computations with and so on so all the important information about QP even though QP is huge you can hope that much of that useful information is stored in zp and the information in zp you can manipulate by just studying information over ZM P which is somehow very imaginable ZM p is finite you can program your computer to see if it has Solutions over ZM p and then understand when it has Solutions over zp and therefore over QP and so on so roughly that's that's the idea okay

50:33 and so therefore you define the local factors which is defined as follows um all right so here there are some terms which you don't understand but they are all there in um the notes that I'm going to give you but roughly what I want you to understand is if I take let me take K is a number field as usual uh from now on K is a number field so for for Simplicity if you want to keep track of what's Happening you can take example k equals q but just like for K you have these Bunch for Q you have these bunch of primes for K if I take any finite extension each of these primes there's an analogous uh extension of these primes these are called places or valuations on

51:22 K on k okay so there are these local objects even when you go to a number field which is just a fancy generalization of the prime numbers okay all these again are deeper and I would strongly urge you to spend more time understanding those and so on but for now just remember that there are these analoges of P when you go to finite extensions and then corresponding to each of those analoges I'm going to give give an equation LV e over K of T so for instance when you're working over Q this is going to be LP of e/ Q so what form this equation takes it's going to be at most a quadratic equation what form that quadratic

52:08 equation takes will depend on the behavior of e at the prime P what does it mean behavior of e at the prime P so remember your e is y^2 = f ofx in the affine form and f of x is actually an equation in Q of X it's an equation with coeffient I in k q in this y STW form it's not difficult to imagine you can clear denominators and then make this transform this into an equation over Zed right I just clear the denominators and I can get an integral form of the equation that's called an integral form of the equation so in fact you can even work with an integral vras form but all of these I warn you I'm making it

52:53 simplistic to actually see that it's rigorous and it work Works each of this is a chapter in Silverman's book but it's very very um tractable it's not difficult you know you can read it at your leisure you can read Tate's book or Tate Silverman and then so roughly therefore what I'm telling you is that you can think of this elliptic curve as actually so from this you can actually go to y^2 = F ofx where the coefficients are actually in Z of X cubic equation with coefficients in Zed and once you have integral coefficients I can reduce the coefficients modular Prime so I can look at Y Bar squ I just look at its reduction F Bar X xar which is in Z mod P of X for every

53:42 Prime P I can do this for every Prime P right I'm just looking at the coefficients reducing them mod P so for instance to give you a concrete example suppose we have y^2 = X Cub minus 43x + 24 so when I look at it mod 2 this is going to be y^2 = X Cub - x over z m 2 I'll put a bar so that you understand that this is reduction over z m 3 it's going to be Y bar^ 2 = xar cubed - 1 - x again it has the same form Z mod 3 over Z mod 5 it has X bar cubed - 3x + 4 3x bar +

54:31 4/ z m 5 but each of this is an elliptic curve over this field in its own right but you don't know the discriminant you see we Define those invariance discriminant etc etc so if I study the solutions for these equations over the finite Fields I know that it has a structure of an aelan group but remember it won't have a rank because you're working with finite Fields so it'll be a finite set so you count the solutions over each of these finite sets so that defines some other invariant called AP which is very important from the point of view of modular forms we'll come to that later but anyway so what you're doing is once I give you an elliptic curve over Q you

55:16 can assume change coefficients assume that they are over Zed you can reduce it at every Prime count the solutions of that elliptic cve over each of those primes now different things can happen you see over this for instance if I take you your elliptic curve over some prime could be becoming something like Y Bar squal xar cubed this is far from smooth you know when you plot it it'll look something like this or your reduced curve could be something like this these are not smooth there are singularities here so depending on what happens when you reduce the curve depending on on whether it is still smooth or it looks like this or looks like that your equation is defined

56:01 so if it is still an elliptic curve even after you reduce suppose you get Y Bar squ equals F barx over the reduced field over ZM p and this still has three distinct Solutions over that field then that means it is smooth so then we say that it has good reduction so it is being good at that prime or it could have split multiplicative reduction meaning it could have a point like this and then two tangents and so on depends on that local Behavior at the tangent so these are all technical terms split multiplicative reduction not split multiplicative reduction additive reduction it just depends on what happens to the equation when you look at it modular or prime these are all technical terms but again you see this

56:48 is where the computational site comes in today you have programs where you can feed in an elliptic curve feed the elliptic curve equation give the prime and it will tell you whether it has split multiplicative reduction or additive reduction and so on okay but underlying it is a lot of theory which needs algebraic geometry a little bit of algebra but depending on that reduction I'm going to involve local factors LV and then I'm going to assemble all these local factors together that's product of V LV e over K I'm changing the variable QV to the minus s minus one okay I'm just introducing uh a new variable and

57:33 then okay we can skip all this because you can read them in the notes but what I want you to remember is I have now given you a function L of e over k s starting with an elliptic curve studying it over studying the reduced elliptic curve over each of those fi times I get an Oiler product I take that Oiler product and that's called the L function h l function of elliptic curve over that field okay now there's no reason why you should expect anything about this elliptic curve it's a complex variable s is a complex variable but it is conjectured remember the reman zeta function also has an

58:21 Oiler product right it has an Oiler product expansion and it has a power series expansion and the same thing happens here the elliptic curve but it's a deep deep theorem and that goes back you know it's ultimately related to very deep work in langland program and whiles results on modularity and so on what you believe that's again a conjecture you believe that this L function which is actually a par series you know it's analytic Beyond a certain part of the plane and it can be analytically continued to the end entire complex plane that is a conjecture it is proved when your number field is Q so when k equals Q you know that L of

59:07 e over Q has a power series expansion it is analytic over the entire complex plane it's not an easy theorem it's probably the biggest theorem proved in the last century and it involves work of WS diamond tayor on of them okay so here is this very complicated thing where you're but the right way to think about it remember in the reman zeta function also you have a Oiler product and that depends on the prime it's Prime by Prime you multiply the oil factors which are defined Prime by Prime so this is the analog of that so that is a conjecture for elliptic curve the h l function has an analytic continuation and like I said it's a very deep theorem so now you have this is the

59:55 analytic L function the L function is the other side of the bird wi and dire conjecture that I was referring to so you have a function in a complex variable power Series so therefore you can talk of the order of Vanishing of that power series that's called the analytic rank okay so I'll close with the weak form of the births inti conjecture so you see this is the theorem let e be an elliptic curve defined over Q in fact you know it not just for Q but even if you take a solvable extension of Q field extension whose Galva group is solvable so even over those you know that the h l function has analytic continuation to the entire complex plane so therefore once it's an analytic function you can just think of it as a

1:00:40 power series and once it's a power series you have the leading term or the order of the power Series right okay now ignore all this I'll just come to the weak form of the bird and dire conjecture yeah this is the first part which I already mentioned uh the there is a rank and so on birth unit and dire no I'm already going to the strong part okay so let me just write it on the board I can't find it on the oh yeah okay so look at only the first part ignore the second part that's the deeper part of the birs in entire conjecture so this part here conjecture birs dire

1:01:26 conjecture for a number field K elliptic terve defined over K remember this part L of e over k s this is a power series conjecturally you can extend this function to analytic function on the entire complex plane so therefore you can just think of it as a power Series so therefore it has an order the order of this power series at the point s equals 1 is actually equal to the algebraic rank of the elliptic curve that's the weak form of the bir's unit and dire conjecture the second part you know we'll discuss this later B and C Part B and C actually gives once you have a power series you know the leading term so the second part gives an exact formula for the leading term in terms of other very deep invariance Associated to

1:02:13 the elliptic curve and you see already what we have cooked up is you know we have just thrown various things the algebraic side is quite easy it's a finitely generated ailan group you have the rank but for the analytic side you have these Oiler factors which depends on the study of the elliptic curves over the reduced field ZM P there are a lot of things that go into there depending on the kind of reduction that it has at that Prime and so on so there's already a lot of ingredients you have thrown in and then you have manufactured something called the L function which conjecturally is an analytic function so therefore you can think of it as a power series and you know that this conjecture is true in certain number fields for instance k equals Q already for k equals Q the computations are a different story

1:02:59 so anyway so you have this power series therefore you can talk of the order and the weak form of the births unit and conjecture predicts that this order of Vanishing of the L function is equal to the algebraic rank to the number of copies of Zed that occur in E of Q that's the weak form we'll discuss the strong form the exact formula tomorrow okay any questions yeah uh actually it's there in the notes I didn't want to dwell on it too

1:03:49 much yeah so AV and QV so what is this I mean I I'm going to keep this simple you know uh understand it for k equal Q that's the first step if you want to understand it for higher uh because remember this V are generalizations of the primes so let's understand what it is for k equal Q so there a is going to be an AP right and AP minus one is nothing but the number of points of the elliptic curve over Z mod P that's the invariant AP so AP is closely related to the number of points that the elliptic curve has when you reduce it mod P and study it over Z mod P QP is nothing but P it's

1:04:36 just the number of points in the residue field so what happens when you go to an arbitrary number field is the following so let's look at this picture here so you have k/ q so remember this V is always you should think of v as lying over a prime P there might be many V's lying over a single Prime but a v always Lies Over a prime p and then remember I can complete it just like here I said there is something called QP here I can complete it there's something called KV that's a field and K is dense in KV just as Q is dense in QP and inside QP you have zp such that QP is equal to

1:05:23 zp invert P this is a local ring integral domain and this is the qution field which you get only by inverting the generator of the maximal ideal so exactly analogously here you have something called OV this is a local ring and the quotient of OV is KV in fact exactly like this you need to invert only one element or invert 1 / Pi V Pi V is the generator of the maximal ideal here okay so once you have a local ring you all when remember when you have a local domain it comes with a bonus it comes with two Fields take the qution field take the residue field right so here

1:06:10 when I take the residue field when I take the residue field I get ZM P so here I have OV when I take the residue field I get K KV now this KV is going to be a field extension of ZM P it's a finite field it's a finite field extension it's a finite field QV is the number of elements in that finite field okay so those are the invariant that come come there so you see the formula in defining each of those Oiler factors involves deep arithmetic deep algebra you're using local Rings you're using discrete valuation Rings you're using field Theory

1:06:56 and so on so each of those pieces in the puzzle is a very intricate complex object and that is what is so beautiful about the birth unit and dire conjecture you're doing this arbitrary seemingly arbitrary thing manufacturing something and then those pieces of the puzzle fit so beautifully together to actually enable you to make a very concrete conjecture which is verified or verifiable in a large number of cases yeah any other questions sorry yeah so you see uh we'll see some

1:07:46 parts of it hopefully in the lectures the only known way to understand anything about the BSD or to prove anything deep about the births and dire conjecture see there are two things you can do now I'm going to give you the exact formula in the next case you can compute suppose I give you an elliptic curve and there is a way to compute the rank of elliptic curve and there's a way to compute the L function of the elliptic curve you can actually check the birth unit and the weak form for that elliptic curve you can compute the rank algebraic rank you can write down the L function look at the order of Vanishing at s equals 1 and then see if the two are equal that's one computational side then the next part which we haven't discussed today which we will discuss tomorrow is

1:08:31 giving you an exact formula for the leading term of the L function and suppose you can compute each of those terms in that exact formula then you can check whether that exact formula is true or not so these things you can do and that's routine today you have programs where you just hit a button and then it'll tell you this is the value this is the value and then you check that the formula is true or not true but conceptually or theoretically if you want to prove the bir's wi and dire you don't want to check it elliptic curve by elliptic curve right that's not the way to do it you want a general proof the only way you know how to attack this problem is through something called iasa Theory this is very mysterious you have the complex cell function and you have the rank and you don't understand why

1:09:18 the two of them should be true so iasa Theory takes the following approach let's look at the exact formula if you look at the exact formula it's some quantity number let's say it's some number you have two numbers it's an exact formula with uh two numbers with an left hand side and a right hand side okay this is and the exact formula is predicting that these two are equal now what you will do one way to check that these two numbers are equal is to check if the same power of prime of a given Prime p divides both of this if the P factor is the same how do you check two numbers are the same use the fundamental theorem of arithmetic check

1:10:04 that they have the same factorization so and that's the only way you know how to approach the births wiet Andre conjecture in the exact formula case what you will do is you'll say okay the P part of both sides of the equation is the same and already to do that you need a whole Grand Theory called iawa Theory which is very deep in it self but the beauty of that theory is it tells you why you should expect something like this why is something like this even believable you have two random things one algebraic in nature one analytic in nature why should they be equal so iava theory is telling you that these two are equal because they are both related to a third common thing that's why these two are related

1:10:50 so coming back to your question on the pr K function there is no way you can directly relate the complex cell function and the model way rank so what you do is you manufacture analoges P analoges of these and then say that for each prime P I understand why these two are related there's an analog of the model way rank for a prime P pic rank a and then there is an analog of the complex n function which is the pical function and then those two you can conceptually understand why they are related the main conjecture says that why these two should be related so if you like the births wi and dire conjecture there are two ways to see it of course historically first this

1:11:37 was formulated this is a complex conjecture in the sense it's over the complex numbers and then to attack it you went back to a piic theory where you understand the picture so now with the benefit of hindsight you can say Okay periodically there is this Theory which we understand I can assemble this perodic Theory to generate something over the complex numbers but I don't know how why the assembly should work I know the components work on both sides for piic Theory I know why the components should work okay so the main conjecture in some sense is a combination of all this putting together understanding the piic picture and assembling the piic pictures

1:12:23 to get to the complex picture that is still a mystery you still don't understand the full story of that sorry why did I call what it's see when you think of ailan groups what do you need to know by the structure theorem of aan groups your group is determined by its rank up to isomorphism it's determined by its Rank and and by uh I forget what is the technical word for that anupam I mean the the tsion part when you look at the factors um the technical word for that when you look at the finitely finite ailan groups

1:13:09 you know you can actually write it as Z mod P to the power R1 Z mod P to the^ R2 those increase that covers the P part then you go to another Prime Q there a technical word I've forgotten for that I've forgotten that but invariant factors correct yes and that very name invariant factors is because these factors these are the invariants which determine your group there's only one group you know which has the structure Z direct from Z mod 2 Z mod 4 there are there might be many groups which are isomorphic to that but if your group has invariance rank one

1:13:55 and finite part 2 4 then that says your group has to look like Zed direct some z m 2 direct and Z M 4 if your group has rank two and invariant factors 2 2^ squ or 2 Cub 2 2 S 2 cubed then you know that it has to look like Z the^ 2 plus Z mod 2 plus Z mod 4 plus Z mod 8 right that's why they are called invariant factors these invariants completely determine the structure of your group so here this rank is an invariant it's giving you the rank of your any finitely generated ailan group has two important parts at a first level one is how big is the free part what is the rank then how big is the tsion part at a second level you will describe the

1:14:42 torsion part right I mean these are all MC exercises what are the ailan groups of rank of uh cardinality 36 how many groups can exist which have 36 six elements that depends on the invariant factors so that's why it's called an invariant sorry how do you know that eliptic cures yes it helps to know that an elliptic curve comes from a model form because it's much simpler to write down the L function why does it help remember I mentioned the conjecture that the elliptic curve L function conjecture to be an analytic function on the entire

1:15:29 complex plane if you know modularity if you know that it comes from a model form then you know that L of e over Qs is nothing but L function of that model form and you know that the L function of the model of form is analytic which is a deep result by itself therefore you know that the L function is analytic sorry uh not really I mean no in fact it's no because it's a big challenge you know it only now after no no no look at it historically you knew how to write the L function of an elliptic curve even before you know modularity but after you know modularity

1:16:16 you it makes it simpler to write down the function because