 This talk will be an introduction to the Birch-Swinnerson-Dyer conjecture at a roughly graduate student level. So first of all it's named after two people, not three, so Birch is Brian Birch and Swinnerson-Dyer is Peter Swinnerson-Dyer or Professor Sir Peter Swinnerson-Dyer. And the Birch-Swinnerson-Dyer conjecture relates the following two objects. We take the rank of a Mordell group of an elliptic curve and we're going to relate it to the order of zero of the L function at s equals one. So I better start by explaining what all these various things are like what is an elliptic curve, what is the Mordell group and so on. And then I'll say a little bit about Birch and Swinnerson-Dyer's motivation for finding the conjecture and then I'll summarise some of the progress that has been made on it. So let's start by quickly reviewing what an elliptic curve is. So an elliptic curve is roughly an equation of this form, so y squared equals x cubed plus bx plus c. So b and c are fixed constants and we've got a curve like this, so typically an elliptic curve, if you draw it it might look something like this. And the problem here b and c are going to be rational and the problem we want to do, we want to solve is find all rational points. For instance a well-known historical example studied by Fermat is an elliptic curve y squared equals x cubed minus two and this obviously has one rational point five three for example. And if you look at it you see it's really rather a difficult problem to figure out whether or not it has other rational points. One reason for the interest of elliptic curves is that in some sense the simplest curves we don't really understand. So if a curve is degree two, we've got degree two curves pretty much under control since the time of Gauss or something. But degree three, there are still lots of basic things we don't know about them. So this is sort of simplest non-trivial example. Elliptic curves have very little to do with ellipses, they're called elliptic curves for roundabout historical reasons. So the first thing we can do is we can make the rational points into a sort of group. So if we intersect a straight line with this curve, you see it intersects the curve in three points. And we can define a group structure on the points of the elliptic curve by saying a plus b plus c equals naught if and only if a b c lie on a line. And this turns out to make the points on the elliptic curve into an abelian group. That's not at all obvious, by the way. In particular, checking associativity of this addition is quite complicated. I should say that the zero point of the group is a sort of point at infinity. So strictly speaking, it's not quite a solution of this. So that gives us some extra structure to work with. In particular, the points with rational coordinates all form a group because you can check if a and b have rational coordinates, then so does c. And this group is called the Mordell group. So the Mordell group is just a group of rational points on an elliptic curve that I'll call e for short. And the Mordell group of an elliptic curve has been a sort of central topic of study and number theory for about a century. The reason why it's called a Mordell group is the Mordell proved that it's finitely generated. This means that it's a finite abelian group, which are pretty well understood, direct some number of copies of z, where this number here is called the rank of the Mordell v group. And we would like to understand both this finite part and the rank, and we sort of pretty much understand the finite part. So Barry Mazer showed that the finite part has order at most 12, so we've sort of got it under control. The rank is much more mysterious. So for example, one obvious question is, can the rank be arbitrary large, or is there some bound to the ranks of all rational elliptic curves? And opinion about this has varied backwards and forwards quite a lot. At the moment the consensus seems to be that the rank is probably bounded. So Park, Poonen, Voight and Wood have a paper on the archive a few years ago where they suggest there are a finite number of rank greater than 21. So that's still conjectural. You can't do much better than this, because various people in particular L-keys are rather good at finding elliptic curves of large rank, and he's found infinitely many of rank 19 when I last heard, and one of rank 28. Numbers may have gone up since I last checked this, of course. So the rank of an elliptic curve is still a somewhat mysterious object, and it's this rank of the elliptic curve that the Birch-Swintersen-Dyer conjecture tells us or conjectures something about. So we notice first of all that the rank is equal to naught if and only if the number of points is finite. So the first thing Birch-Swintersen-Dyer looked at is to try and tell when is the number of points on an elliptic curve finite. And the key idea is the following. If the number of points on E is infinite, then the number of points mod P should be larger than expected. So this is a very informal sort of statement because we don't quite know what larger than expected means. So the point is if you've got lots and lots of points on the elliptic curve, which you would have if they're an infinite number of rational points, then you can reduce all these rational points mod P and get points mod P, and that sort of means you're slightly more likely to have a point modulo P. So what we do is we let Np be the number of points on the elliptic curve mod P. In other words, we're trying to solve y squared equals x cubed plus bx plus c mod P, and that number is not quite equal to Np because we really ought to counterpoint at infinity at the time being a bit sloppy about. So if you've got a rational point, then most of the time you can reduce it mod P as long as P doesn't divide the denominator. So lots of rational points should make Np a little bit bigger than usual. You can see that Np is going to be approximately P because there are P choices for x, and then there are either 0 or 2 values of y, and there's a sort of 50% chance of there being 0 and 50% chance of there being 2 values of y because it depends on whether this is a quadratic residue or not. So we expect about P points mod P. So Np over P should be approximately equal to 1 in general. So this suggests you might want to look at the product Np over P, where this is a product over all primes. And maybe if this is equal to infinity, this might correspond to infinitely many points on E, and if it's finite, this might correspond to a finite number of points. Well, one problem is there's no reason why this should actually converge even if it's finite. I mean, these numbers might just sort of oscillate up and down a little bit. It's really not clear. But what you can do instead of looking at this product, you can look at the product for P less than equal to N of Np over P. And look at this as N tends to infinity. And try and estimate how much this is growing. And this is what Berkson, Swinnerson-Dyer did with a very early computer. And this was back in the 1960s, so computers were these huge things that filled an entire room and probably less powerful than what you can get these days as a $50 pocket calculator. But whatever, they managed to use this computer to calculate this sum. And what they found is that indeed the product of P less than equal to N Np over P seems sort of finite or bounded if the rank is equal to 0 and seems to tend to infinity if the rank is greater than 0. Well, obviously you can look into that a bit more closely and try and compare the rate of growth of the product over P less than equal to N of Np over P with the order of the 0. So they looked at this too. They tried to see how this number grew with N and compared it with the order of an elliptic curve. And what they found is that this expression here is, grows roughly like log of N to the G, where G is the, sorry, not the order of 0, the rank of the elliptic curve. So I'm going to say the rank turns out to be the order of the 0 of an L function, but that's coming up in the next sheet. So they did some numerical calculations and this seemed to grow very roughly like that. It doesn't mean, this isn't asymptotic to that, but the calculation suggested it sort of lay between some constant times this and some other constant times this. So the number of zeros modulo P does seem to be related to the rank of the Mordell group. Well, you can make this a little bit more precise as follows. So there's quite a lot known about the number of points on an elliptic curve modulo P. So if we have an elliptic curve modulo P, then the number of points is equal to P minus alpha P minus alpha P prime plus 1, where alpha P is a complex number of absolute value, the square root of P. This is the famous Riemann hypothesis. Well, it's the Riemann hypothesis for elliptic curves, which was first proved by Hasse. It's quite a lot easier than the Riemann hypothesis for the Riemann zeta function. And the elliptic curve has a zeta function associated with it. So first of all, we can write down a local zeta function of an elliptic curve. So this local zeta function zeta of P is, for various complicated reasons, defined to be X with sum of N P to the M over M times Q to the MS, where this is the number of points over the field with P to the M elements. And this is, turns out, to be equal to 1 minus alpha P times P to the minus S times 1 minus alpha bar P P to the minus S divided by 1 minus P to the minus S, 1 minus P to the 1 minus S. So that's a local zeta function. And you define the zeta function of an elliptic curve E to be the product overall primes of the local zeta function, except that's not quite true because there are a certain number of bad primes where you need to think a little bit harder about it. So this is a quality is not quite true, but I'm going to sort of ignore that. Now, you notice this product is, looks a bit like the product for the Riemann zeta function. So the ordinary Riemann zeta function looks like the product over P of 1 over 1 minus P to the minus S. And if you notice, this 1 over 1 minus P to the minus S looks just like this factor here. So you find this, the zeta function of elliptic curve is the ordinary Riemann zeta function times the ordinary Riemann zeta function of S minus 1. Now that comes from this term, of course, times, sorry, divided by a certain L series. This is the so-called L series of an elliptic curve. And this is this L series that's going to appear in the Berks-Swinterson-Dyer conjecture. So the L series of the elliptic curve is just 1 over the product of 1 minus alpha P to the minus S times 1 minus alpha bar P to the minus S. So it's a product of that over all primes. And what you notice is that N P over P, which is P minus alpha P minus alpha bar P plus 1, is equal to 1 over L P of 1. So the product over all these numbers N P over P is essentially 1 over L of 1, where this is the L function of the elliptic curve. So this is the product that Berks and Swinterson-Dyer were investigating, and this is the elliptic curve. Well, the trouble is this product does not converge. So you've got to be a bit careful. We're just sort of arguing informally. But you see that if you pretend that it does converge, you see that this would be finite if the L function doesn't vanish at 1. And if it diverges to infinity, then you would expect this number here to be 0. There's a big problem, by the way, at the time Berks and Swinterson-Dyer were working, it wasn't at all clear that this L series was defined for S equals 1. So this equality shouldn't be taken too literally. Neither side is really properly defined. Anyway, this sort of suggests that the number of points of E is finite should be equivalent to saying that the elliptic curve is, the L function of the elliptic curve is nonzero at 1. So this is the first form of the Berks, Swinterson-Dyer conjecture. That it gives a criterion for an elliptic curve to have an infinite number of points. It has an infinite number of points if and only if its L series vanishes at 1. Rather, I should say this has to be an analytic continuation of the L series, because as I said, the L series doesn't actually converge at S equals 1. So that's a bit of a problem. Well, you can extend this a bit. So you remember I said earlier that the product over P less than or equal to N of NP over P sort of grows a bit like log of N to the G times some constant or whatever. And this turns out to correspond very roughly to saying that the L series, L of S, has 0 of order G at S equals 1. It doesn't correspond to precisely that because as I said, this isn't really clearly a well-defined function, but informally you might guess that this growth rate sort of formally corresponds to having a 0 of order G. So this gives the next form of the Berks, Swinterson-Dyer conjecture, which says the rank of the Maudelvet group of elliptic curve is equal to the order of 0 of the elliptic curve at S equals 1. So this is one form of the Berks, Swinterson-Dyer conjecture. Berks and Swinterson-Dyer actually refined this a bit because you can say L E of S is equal to S minus 1 to the G, where you conjecture this G is the same as the rank G here, times some constant. And Berks and Swinterson-Dyer also pinned down what they felt this constant ought to be, and it turns out to be rather interesting and rather complicated. So it depends on things like the Tamagawa numbers of the elliptic curve and it depends also on the Tate Shafarovic group and on a few other bits and pieces. And so there was a sort of joke about the Berks, Swinterson-Dyer conjecture at the time which said it was conjecturing the equality of two things, neither of which were known to be well defined because, as I said, first of all it wasn't known that this is defined at S equals 1 and secondly it wasn't known at the time that the Tate Shafarovic group was finite. In fact there wasn't then a single example of any rational elliptic curve whose Tate Shafarovic group was known to be finite. So this formula involving the order of the Tate Shafarovic group was again a little bit dubious. So that's what the Berks, Swinterson-Dyer conjecture says, roughly speaking. What I'm going to do now is to discuss some of the progress that has been made on proving it and we've actually come quite a long way since the conjecture has originally stayed. So I guess the first substantial progress was made by Cotes and Wiles, I guess this would be in the late 70s. It said that suppose elliptic curve has complex multiplication. Well, I need to explain what that means. Then if the L-series doesn't vanish at S equals 0, then the Mordell group is finite. So this is one implication of the Berks, Swinterson-Dyer conjecture for some very special elliptic curves and they have to have complex multiplication and they have to have rank 0. So let me explain what complex multiplication is. So most elliptic curves have only the obvious endomorphism. So we can ask for endomorphisms of an elliptic curve E and they all have endomorphisms which take a point A to N times A for N in the integers. And a few of them have extra endomorphisms. So let me give some examples and an example of a curve with extra endomorphisms. We might look at the curve y squared equals x cubed minus x. And this actually has some extra automorphisms. You see here the only automorphisms would be when N is plus or minus 1. And it certainly has these automorphisms because we can change y to minus y and map x to x and this will be an automorphism. But there are some other automorphisms to this because we can change y to I times y and x to minus x. So this is some extra unexpected symmetries which are things called complex multiplications. They turn out to be multiplications by complex numbers if you represent the elliptic curve as a torus. So that's what complex multiplication is. Coates and Wiles also assumed it was complex multiplication by something of class number 1 but that's the minor technical detail. So very roughly the idea of their proof is that if the Mordell group is infinite then they showed that L1, which could be written as some sort of algebraic number times some transcendental constant, the algebraic number would have to be divisible by an infinite number of primes which are sort of actually vaguely related to irregular primes but never mind. So you could say roughly speaking that if the Mordell group is infinite then L1 is divisible by an infinite number of primes. And this shouldn't be taken too seriously because this is actually a real number so saying it's divisible by prime is kind of meaningless but you can make sense of that. And so the next piece of work that I want to talk about was by Gross and Zagier and they showed that if the elliptic curve E is modular, whatever that means and I'll explain that in a moment. And the L function has an order 1, 0 at s equals 1 then this implies that the Mordell group is infinite and that the proof of this used something called Heidener points. So let me explain very roughly what a modular curve is and what a Heidener point is. So modular means roughly that the elliptic curve can be parameterised by modular functions. This means there's a map from the upper half plane to the elliptic curve given by some sort of modular function. And a modular function is one that, well the similar sort of modular function being variant under sl2z acting on the upper half plane by a b c d tau equals a tau plus b over c tau plus d. It turns out that you can't actually use these sorts of modular functions to parameterise elliptic curves but there are some slightly more complicated sorts of modular functions which will map the upper half plane to elliptic curves. And it turns out that if you can do this then you have really good control over the elliptic curve. So modular elliptic curves as they're called turn out to be particularly nice. And now what is a Heidener point? Well a Heidener point tau in the upper half plane is one that satisfies a quadratic equation a tau squared plus b tau plus c equals 0. Well actually that's not quite true. If you take one of these points and map it to the elliptic curve it turns out that it's not quite a rational point but it's a point with algebraic coordinates. But then by fiddling around with that and summing over Galois groups or something you can get a rational point out of it. So by looking at these quadratic irrationals in the upper half plane we can get points on the elliptic curve. And the idea of Gross and Zagier's proof was they showed that the height of a Heidener point on the elliptic curve e is related to the derivative of the L series at s equals 1. It's some complicated constant times this derivative. So the height of a Heidener point is some way of measuring its size that I won't go into in at least not in this lecture. And the proof is very simple in principle. You just calculate the height of a Heidener point and calculate the derivative of the L series and check they're equal. In practice however the calculation is horrendously complicated because each of these things is incredibly difficult to calculate and when you calculate both of these you get a complicated sum of many different terms and you have to check that each of these terms match up and so on. And the calculation is well over 100 pages long and is really terrifying to look at. Incidentally there's another interesting consequence of modular elliptic curves. Let me give an example of a modular elliptic curve. So suppose you take the elliptic curve y squared minus y equals x cubed minus x squared. Then it turns out to be related to the following modular form. A modular form is a slight variation of a modular function. And it's right, sum of cnq to the n is given by the following series. It's q times product 1 minus q to the n times 1 minus q to the 11n. So if you've done modular forms you might recognize that this is a modular form of level 11. And it turns out to be very closely related to this elliptic curve because if the number of solutions modulo p is written as p minus cp then the numbers cp are given by the coefficients of this modular form. And this turns out to be closely related to the fact this elliptic curve is actually modular in a way related to this modular function. In particular it turns out the L series of the elliptic curve is a melon transform of a modular form. In particular so the L series of this curve would be the melon transform of this modular form here. In particular this means the L series can be continued to all complex s. So for modular elliptic curves we know the L series does actually make sense at s equals 1. There's another interesting point about it. These modular forms satisfy a function equation and a certain function equation. And this implies that the L series also satisfies a certain function equation. It says that L star of 2 minus s is plus or minus L star of s where L star is L times some junk that I'm not very interested in at the moment. And we have a sign here and the sign is plus or minus depending on whether the order of the zero at 1 is even or odd. So although it can be quite hard to tell what the order of the zero is it's easy to tell whether the order is even or odd. So what Gross and Zagia did was they showed that if the order is 1 then Hegner points give you points on the elliptic curve of infinite order. If the zero has ordered greater than 1 then all the Hegner points turn out to be zero and are much less useful. So the next piece of progress I'm going to talk about was by Collier Vargin Collier Vargin, sorry. Again we assume that he is modular so it can be parameterized by modular functions and he showed that if the L series has a zero of order 0 or 1 then the Mordel Vey group or the Mordel group has rank 0 or 1 respectively. So this is a big chunk of the Birch-Swintersen-Dyer conjecture for ranks 0 or 1. His proof used these things called Euler systems which I don't know everybody says that Euler systems are named because of their relation to the Euler product for the zeta function but I must admit I've always had trouble seeing the connection. One of the interesting things about this was that he was able to actually show that some Tate Shafarovich groups had finite order which until then people had actually known they were finite order. So it actually became possible to do the part of the Birch-Swintersen-Dyer conjecture where you look at the coefficient C where the L series is about some constant times S minus 1 to the G. So far all these results only applied to modular elliptic curves so the next very big breakthrough was by Wiles who showed that many elliptic curves are modular and his proof was soon after extended by Broil and Conrad and Diamond and Taylor who showed not just many elliptic curves are modular but all elliptic curves are modular they sort of extended Wiles's results to cover some more technical cases and this was a really astonishing result when it came out I think most people before Wiles's work was announced thought that the easy part of the Birch-Swintersen-Dyer conjecture would be proving it for modular elliptic curves and it would then be really difficult to show that all elliptic curves are modular this is the famous Tanyama-Shimuro-Vey conjecture so Wiles sort of turned this upside down he showed that he managed to solve what everybody thought was the hard part of the Birch-Swintersen-Dyer conjecture in particular this shows that the L series is always defined at S equals 1 so in particular the Birch-Swintersen-Dyer conjecture actually makes sense before it talked about the L function at S equals 1 and it wasn't even known if this was a well-defined number so since then we've had a result by Baagavar and Shankar who showed that in some sense lots of elliptic curves have ranks 0 and 1 lots means at least a positive proportion and possibly even most of them I'm not quite sure what the current state of the art of this is so in some sense we've made considerable progress on the Birch-Swintersen-Dyer conjecture due to Colovargan and Gross-Sagian coats Wiles we've got pretty good control over the rank 0,1 case we know that probably most curves have ranked 0 or 1 and of course we know they're all modular which was thought to be the hard part of it so what's left to do in the Birch-Swintersen-Dyer conjecture? well what's left, what is still open is the case of ranks greater than 1 and what's the main problem with doing this? well I think one of the main problem is that points on an elliptic curve E with rank greater than 1 seem to be really hard to get hold of so if the rank is 0 then there's no problem because there are only a finite number of points so you're only a finite number of points away from having them all if E has rank 1 then Hagnar points give you a pretty good hold of all the rational points because the Hagnar points generate a subgroup of finite index but if the rank is greater than 1 I don't think anybody knows a sort of general way of writing down something that say generates a subgroup of finite index in the Mordell group so my feeling is that the main obstruction to finishing off the proof of the Birch-Swintersen-Dyer conjecture is finding a good way to generate almost all points on an elliptic curve of rank greater than 1