 Thank you, Luke. So firstly I want to thank my colleagues at Orsay and London for this very nice invitation, which I really appreciate. I always feel it's like waking from a dream when one comes from the marshes to East Anglia to here in Orsay especially. In fact, I never was in the time I was at Orsay, Jean-Marc was not there, he only arrived later. But I think the time which I really got to know him best of all was when we suffered what I think was the worst journey in my life together. Because people don't realize today, but this was long ago in about 1980, but the only inexpensive way. So we were both invited to Princeton and the only inexpensive way of getting across the Atlantic from Paris was with this incredible arrangement I think with Icelandic air that we had to first get a train in the middle of the night to some capital, maybe in Luxembourg. Exactly. And then we flew to Iceland and then we spent a few more hours waiting in there for the next flight to take us to New York. And then finally when we got to New York we had to wait hours again for a bus to take us to Princeton. But you could buy a sweater in greater weight. But I think it's the worst journey I've ever encountered in my life and Jean-Marc and I sustained each other throughout this journey. I think if he hadn't been there I would have perished and I really appreciated it. With Jean-Marc, with Jean-Pierre-Van Tain-Bergé, in fact we do have one paper together, one joint paper with Sujata, which I want to say a little bit about before I begin talking about the other subject. So let's suppose I have a finite extension F of Q and I have an Arbelian variety A defined over F and I take any prime P and then I look at the field F infinity which is F adjoining the P power division, all the P power division points, the coordinates. P power division points on the curve and I'm going to write G for the Galois group of this extension F infinity over F. Now around about 1960 I think Seher proved that the cohomology groups H i of G acting on A P infinity are finite for all i greater than or equal to naught. I've forgotten exactly why he wanted this theorem but the proof is not at all. I mean it's an interesting theorem that he proved and curiously when I had been studying the Iwasawa theory of A over this field F infinity and in particular when you came to compute the, well let's assume that now for simplicity that G, well no it's vital now that G has no element of order P and so it has finite P cohomological dimension then if you study the Iwasawa theory of the Selma group, the P infinity Selma group of A over the field F infinity and you compute its Euler characteristic you see that to have it consistent with what you would expect from the conjecture of Bertrand assuming that Seher is finite and there are no points of infinite order that you have to prove that chi G of A P infinity, the Euler characteristic, right? of the cardinality of the H i G A P infinity to the minus one to the i you have to prove that this is equal to one and I asked Seher and he said he had no idea whether this was true or what but then in fact Sujata and I discovered that there was a very simple proof of this fact and we have a little compound unit where we set it out I'm not going to talk about the proof but it's very very simple it uses E My's theorem and some other basic facts but it was actually very easy and so we were very pleased and then naturally well so that where Jean-Pierre Venton-Bergé intervened was that we were this well Sujata and I were at Misery, Misery in California and Jean-Pierre I think was actually down the hill out of Misery in Berkeley but he asked the question what happens if you take instead of a if you now take F the analogous question for F to be a finite extension of Q P what happens so in fact in this case there is to my knowledge there is no application it doesn't come into the Iwasawa theory but you see rather easily that in fact the even the analog of Seher's theorem is not always true that you have to at least assume that A has potential good reduction over F and again you have to assume of course to talk about Euler-Carrie with the streaks that G has no element of order P and it turns out that and this is the paper we wrote with Jean-Pierre but but it was really a great deal of the most of the I because it's very difficult to prove so in fact it's still true the theorem that we prove in this paper that subject to these conditions A has potential good reduction that even locally of course G now is the Galois group of the local field Chi G AP infinity is one provided A has potential good reduction over F but the proof is really quite difficult and there are many side things that came out of it along the way in this paper we wrote and it was a great pleasure I must admit working with Jean-Pierre on that paper so that was my one time in which I really worked closely with him and I had looked at his one or two of his other papers carefully but it was indeed very striking and very nice that the work we did together on that so what I want to talk now is to get back to this Iwasawa theory that in fact as I said suggested to us the global form of this theorem although I'm not going to discuss that but before I do that I want to go back into the history a little bit to remind you of the background so I'm going to be just talking about the case now from now on K well Q will be a prime always which is congruent to 7 modulo 8 and K will be the imaginary quadratic field Q a join the square root of minus Q and of course this condition here this will be vital for the whole theory that I want to talk about this tells us that two splits in K two okay will be the ring of integers of K and two okay will always be a product of two primes P and P two distinct primes because of this condition and I suppose I've fixed an embedding of K into C now in 1838 Dirichlet of course was a Frenchman I think he grew up in Germany and his parents were moved there by Napoleon I believe and but he spent much of his life in Berlin because he was married to the daughter of Mendelssohn and it was only very late in life he was Gauss's successor at Goettingen but this is theorem this marvelous theorem I want to prove he actually proved much well I think he died about 1850 so it's he proved the following theorem so let me write H I'll need it in what follows H will be the class number and he proved the theorem which is the model for all that follows that I'm going to talk about is that H is he gives gave this marvelous formula for H it's R minus N where you look at this so what are these two integers R and N so you look at the set from one up to Q minus one over two and R is the number of quadratic residues of squares mod Q in this set and N is the number of non squares modulo Q so you see this is an exact formula for the order test it out yourselves Q equals seven and so on and in fact let me point out so it's proven in fact this right hand side here is essentially L chi zero where chi is the character of the imaginary quadratic field and I want to point out that not only has no one ever proven this formula without using combining L functions and arithmetic but no one of course it implies in particular that R is always strictly greater than N because the class number is a positive integer and no one has ever proven has found an elementary proof of this fact so this is a very remarkable formula What does it mean elementary in the sense of not using L functions? Well elementary without you minding without using L functions or some equivalent limiting process it's completely unknown after all it should be a purely arithmetic statement that R should be greater than N but there should be some arithmetic proof but we don't never know and curiously it seems that well with the possible exception of Eisenstein none of Dirichlet himself died soon after he was in Goettingen his wife first died then he died but of his students I think only Eisenstein perhaps came close to guessing that there was some analog of this formula for elliptic curves so that's what I want to talk about now in the rest of my lecture so of course and this was the great discovery of Bertrand's when it's in dire that there is such a formula but with many different features that I'll be talking about in the rest of my lecture so of course their formula works for any Arbelian variety over a number field but I'm just going to talk to what is perhaps the simplest case and where certainly we know by far the most about the conjecture namely that I want to now so K will have a H will capital H will be the Hilbert class field of K of course amongst these fields that I'm looking at only Q equals 7 is the only one with class number one all the rest have class number bigger one than one and this degree here of course is H so I'm going to be interested in elliptic curves E defined over H with complex multiplication by the whole maximal order of K of K that's what I want to talk about I mean it's also a very interesting question to talk about well I'll say perhaps a little bit about that later when for other imaginary quadratic fields in these two but just to give you some idea of before I I mean later on I will write down one analog of this formula but let me just tell you the most elementary thing which comes out of a very special case which I so it's a curious consequence of this conjecture which just involves these imaginary quadratic fields so let me define now the field L which is going to be K adjoin the square root of minus the square root of minus Q perfectly honest quadratic extension of K and I'm going to so what is what primes ramify in there well of course Q does but in addition let me suppose you see just one I have I have to pick a sign of square root of minus Q and so I'm going to let the P be the one with odd P is one minus the square root of minus Q over two being positive once you fix the sign of square root of minus Q this determines one of these five primes completely so P will be here and now it's a little exercise in algebraic number theory to see that it's precisely I mean apart from Q this prime P ramifies in here and of course this is a degree too so it's with ramification index two sorry P is over two the guppy P is a prime over two yeah yeah right as usual I mean here two to two okay is PB bar yeah yeah yeah so now let's so this is a totally imaginary quadratic extension of Q and so it's unit group it's it's classical unit group will have ranked one and I'm going to write eta for a fundamental unit of the field L and now I can look at eta it's a global element but I can look it in the completion eta lies in Lv the completion of L at V at this ramified prime and therefore I can there I can also take its logarithm logarithm of V of eta is of course the sum from n greater than or equal to one of minus one to the n minus one eta minus one to the n over n so we have this element always of the completion at V now I want to tell you the theorem which comes out of part of the Bert's-Winnett and Dyer conjecture it states that if the prime Q is congruent to seven modulo 16 then the order at V of the logarithm of eta is always exactly two of course we don't we don't have any really nice formula for log for eta or anything like that but whatever happens the order has to be exactly two it turns out this is equivalent to a statement about a certain component of a Tate Shepherd yes you actually say that on eta the log series converges and if so why well because I mean if it's congruent to one eta is certainly congruent to one modulo modulo V there's no problem about it converging what is not obvious is what is its order but now let me make a conjecture so this is just half the primes congruent to seven mod eight that if Q is congruent to 15 mod 16 which is the other half of the primes then odd V log V eta is always strictly greater than two but I can't prove this statement but I I think it's true I mean it turns out to be equivalent to a certain component of the Tate Shepherd-Avich group of a certain elliptic curve that I'll be writing down is non-trivial and anyway but I have no idea how to to prove this statement in an elementary way you know you can fool around with it as much as you like I can't find an elementary proof okay so now that's so that's to give you just one down to a thing which by the way you can think of this is another way I think this is an exact form of the viatic Brouwer-Zegel theorem because this ramification index is always two and so you get that the order it's an exact form of the Brouwer-Zegel theorem I don't know any other viatic exact statement of it like that okay so now I want to my heavens when did I start quarter yeah well anyway let me get now hurry on so the first question is so I'm interested I've told you in these curves E up here which I defined over the Hilbert class field of course there are none so let me make a remark so fact classical theory of complex multiplication that H is in fact K join J of okay well now here you see I'm thinking of of okay as a lattice it is of course we've picked an embedding in the complex plane and so you can take its it's the modular function J is a in the upper half plane so this is a class a very classical theory of complex multiplication of course I think Kronecker was also a student of Gauss in fact I believe it's Kronecker's you're gonna try not quite sure who whether he proved it now so we want to write down the all the curves which he defined over H with this property here and the first thing we want to do is to write down the simplest one right I want to write an equation so far I don't we've had a single equation in the lectures so far but I want to write one and curiously it's not as far as I know in the literature so definition although the sort of background to it is classical so this was discovered by young Chinese mathematician of myself Yongxiang Li and it's in a paper we have on the archive so D is going to I'm going to call it D is the elliptic curve y squared equals x cubed minus now you'll forgive me for writing these negative powers when I say something about it J of okay so this is the J function to the one third now here I want there is a real root of J of okay J of okay is real there's a real roots I'm taking the real root cube root there plus 2 to the minus 5 3 to the minus 3 J of okay minus 12 cubed to the one half and it doesn't matter really which which square root I take here now it's it's not obvious that in fact this is defined over H by classical theorems from the theory of complex multiplication by classical results of Weber and Songen do you have an x in the equation I'm sorry I've left out there of course there's an x here thank you so what's interesting about this curve I mean it's I believe it's the simplest elliptic curve with complex multiplication with these probably one why is it well for one thing it certainly has complex multiplication because the J invariant of the curve J of D it's a simple calculation I leave it to you to check that it's J of okay so that tells me there is complex multiplication secondly what's interesting about it is that the discriminant of this equation delta of D I leave it to you it's a little exercise is one so the only primes of bad reduction are the primes dividing two and three and it's even not quite obvious a priori that there is such a such a curve even exists with more sophisticated machinery so the J the coefficients that you have written like this so the cube root and the square root are in C they are not in H understand it no no the coefficients here this is this theorem of Weber and Songun but these there is a cube root of J of okay which belongs to H and a square root of this which belongs to H this is the theorem of Weber in fact this theorem is true certainly for all K of discriminant prime to H it is sick sorry but that it's still defined over H for all such discriminant doesn't matter all you need is it it's prime to six and now it's a simple general fact because of course we only have plus there are no cubicle cortic twists here so the fact is that every e defined over H with N h e equals okay is of the form is a quadratic twist of D by some alpha in H cross which is not a square right otherwise the equation of it is the same as as if you just put the alpha here it's a quadratic twist so that's the complete description of all these curves and well we're interested in their arithmetic so the first thing is it's the important theorem of derring well and one should say derring and HECA because HECA proved I mean the actual L functions are L functions of grossing characters that they haven't that the L so the complex L series L e over H s is entire and satisfies its good functional equation so there's no problem at all with the complex L series of these curves and now push that there and bring this down sorry I miss up something so it's so classically the Galois group of H over K permutes the different possibilities of I'm sorry the Galois group of the Galois group of H over K which is which is order is H so this this sense in E so J of E so it is sense J okay to J of some ideal and so you get another curve over H well look don't let me waste I mean this is an elementary exercise don't don't let me how do you twist by an element of the field you take out well if it's a twist if you like by the quadratic extension H cross H join the square root of alpha yeah and and well explicitly you would just put the alpha on the left hand it's alpha its equation is alpha y squared equals the right hand side yeah okay so now what you'd expect the conjecture one important case of the conjecture BSD to tell one would be the following two statements firstly that L E over H one is non-zero if and only if EH and Shah E over H are both finite so of course the teacher of a ravage group is the most infamous let me remind you what it is even though it is the most mysterious and infamous group in the whole of mathematics Shah of E over H is the kernel of H one H E to the product overall V of H one H V E it's always conjecture to be finite but the only time we can prove it would be in some situation like this where the L function gets involved and I mean there's conjecture here secondly there's the the Birch there's an exact formula for I'm going to write down an important case a bit later for the order of Shah E over H in terms of very roughly of L E over H one so this is what Birch and Swinney and I would be telling one one in the first important example of Birch Swinney and I in this case would be these two statements now the first comment I want to make to you is that we have no idea how to prove this in this generality the the reason is that we have to introduce to to tackle this conjecture we have to introduce what I'm going to call the the Arbelian hypothesis namely that if you look if you are joined to H the coordinates of all the torsion points on the curve well of course because it's complex multiplication you get an Arbelian extension of H but what we're going to insist on is that actually this is an Arbelian extension of K and it's well known that not all curves satisfy this property in fact the in our well in our case the the ones which certainly which will be interested in will be which do well firstly it's a lemma if you like the D this curve I wrote down here which I said I believe is the simplest such curve complex multiplication D satisfies the Arbelian hypothesis and so and thus so does D twisted by alpha the quadratic twist of D obviously if you take D twisted by alpha but now you have to assume that with alpha not an arbitrary element of H but in K cross modulo take away but not a square so you have to take quadratic twists by by elements of K so these in the end will be the curves that I'm interested in and now I want to write down what I believe is a I'm going to call it a probable theorem I want to say a little bit about the proof and some reasons why so I'm being brutally honest it's not the complete proof is not written down we're working on it all the time but this is joint work it includes quite a lot of classical work I'll say what extra work you have to do with Kazuka Lee Tian and myself that if D satisfies the Arbelian hypothesis then this is true sorry and so do sorry I mean if D alpha that's yeah I mean if if he satisfies the Arbelian hypothesis let's just put it like that then this is true yeah so I mean we're we're gradually writing down the comprehensible put the words you see you to prove a theorem like this you have the only way we can do it is to use the Wasawa theory so it means you have to do it for every prime P and there are these difficult primes which are the most interesting so they occur so difficult most difficult primes are when the prime P equals to the primes P which divide H and the primes of bad reduction and so you there are there are many different aspects which come in once you look at these problems these primes for example the Tamagawa factors which are there lurking there in the Bertz von Netendier formula which Gauss never had to worry about of course there are no Tamagawa factors in his class number formulae but and but it's important for us to we life is a bit easier for us because for all these curves too is a potentially ordinary prime because it splits and there's a lot of interesting work going on at the moment by other people handling the situation of these difficult primes when two is inert or ramified in K and that they have very interesting applications but I won't be going into that well so what I want to do in the for much of the remainder of my time is to rather it's like if I can use this one yes but are you think one can reach it's like trying to reach heaven yeah one can though right so I mean the I don't want to go into the it's to elaborate to talk about the you've got to prove a main conjecture and so on but at least I want to talk about how this theorem applies to firstly let's look at the the curves D so the theorem so of course to apply this theorem you have to have the non vanishing theorem that and the first theorem which was proven by Lee and myself is that in fact it's true L D over H one is non-zero for all primes Q Cormorant to 7 mod 8 that again in fact we use it was our theory to prove this I don't know how to prove it in any other way but it turns out to fall out rather nicely out of the it was our theory in fact let me mention here I told you that the curve D there you can see that there makes sense for any it's defined over H for any imaginary quadratic field of discriminant prime to 6 and I really wonder whether or not the same statement holds for all of these so they in other words that would be that all of these these the strange family of minimal curves would have in particular finite more del V group but I don't know the situation we haven't really tried hard to prove it but it's interesting so now what I want to do is to at least tell you the exact Bertz Winnett and I a formula in this case and talk about some numerical examples so what is the exact so we know that by our theorem that Shah our probable theorem Shah D over H is finite what is this exact order and remember there's the order of a God given group so even one prime of two being one power of two being out destroys everything but here's the answer so to remember two splits in K into P P bar and now remember odd P of one minus the square root of minus Q over two is strictly positive that's how we we pick the sign of square root of minus or pick Gothic P if you like and in fact now it's a little exercise I leave it to you to check that well let let W one through W R be the primes of H above Gothic P so of course our depends on the the order of the ideal class group of Gothic P and now it turns out it's again a little exercise that D that these are the bad primes these are the bad primes for D over H there and at the other the primes above be Gothic P bar and the primes above three in fact it's a little exercise to check there is good reduction and the the Tamagawa factor at each of these primes is in fact of order four to this is not quite obvious but it's it's well known and not too difficult well it's a little bit difficult to prove but it is for every Q so now let me introduce the period term omega Q so this is going to be essentially the the real period of D over H but I'm going to use the Chowler Selberg formula in it so here and I've got to give you an exact formula so it turns out to be 2 pi to the minus M Q to the minus H over 2 and then the product over the C's between nought and Q which are squares module Q of gamma gamma C over Q and what is the M here M here is in fact an integer as you think about it for a moment since H is odd Q minus 1 is over 4 minus H over 2 so that's the relevant period and now these the theorem well let me again because I've only called the other a probable theorem is that for all primes I mean this is primes Q of course mod 8 we have the exact order of char of D over H is L D over H evaluated at 1 over omega Q squared square root of Q turn all of these if you look at the I mean these are very wrecking now these are the discriminant of H over Q and so on here the square root Q but and now finally we have here 1 over 4 to the R that's the Tamagawa factors that are bad primes and then we have this extra power of 2 2 to the H plus 6 so this is the exact formula and happily some colleagues Dabrowski and others in Poland were able to compute it and what we find out which is not at all trivial of course in fact they they computed this L value and the conjectural order of char for all primes Q congruent to 7 mod 8 and Q less than I think it's about roughly 4,000 I've forgotten exact figure I'm sorry I should have dug it out of the paper but I haven't got it here and so they compute the order of char D over H for all these primes you see it's not at all obvious how I mean you can only compute these out because remember H now is one of them is of the class numbers getting large so in fact the H one of these H's of a degree I think 170 over Q so it's really remarkable this is work of Dabrowski and two other his colleagues with unpronounce unspellable names which I won't write down but I can tell anyone who wants who's interested and of course we know that the one theorem out that I'm known about the Tejchaferevic group apart from this conjectural exact formula and is that it the theorem of Castles and Tate that its order must always be a square of an integer and happily they find that that's true and in fact they find the first few examples this is 0 for Q equals 7 23 31 and sorry ah sorry the order yeah char is okay you're absolutely right excuse me synology and but the order of char D over H for the next prime is 3 squared for Q equals 47 and H equals 5 so these are actually that then the orders of show they tend to grow very rapidly but but these again sort of give evidence intuitive evidence for being this being the simplest such curve with complex modification because as soon as you take others I'm going to say a word about some others in a in a moment like the ones considered by Dick Gross in his thesis let me say a word about them now to finish so definition let's take a now to be the curve D twisted by minus the square root of minus Q always this mysterious twisting coming in so these are these are now actually defined a is defined over even over H but even Q a join j of okay and well if you the orders of this for small Q I mean the smallest Q namely well for seven it is one but once you get beyond that the the order of char have been conjectural calculated by various people and it's enormous it goes off very rapidly so the last theorem I want to mention is the following that we would like to have so by the way relic prove that L a over H one the analog long ago by complex methods whereas we're using two addict there that this is non-zero for all Q congruent to seven mod 8 so we do know that these L a over H are non-zero but in fact Lee and I discovered that there is a very big family of quadratic twists of these for which we can prove the L function of the a's namely just let me write it down let let me call curly M to be the set of all positive rational M which integers which are square free and whose prime factors congruent to one mod four and inert in K so let's take M to be the set of all these integers and then the theorem that that Lee and I have proven is that for all M generalization well assume Q is congruent to seven modulo 16 so it's only half the Q's then for all M in M L AM over H one is non-zero we have all these quadratic twists and I think what's interesting about the proof is that you really it's the first time I've seen it but you really do have to study the Iwasawa theory not just of a but of the Arbelian variety B which is the restriction of this is vase restriction of scalars from H to K of a so B has dimension H it's a CM Arbelian variety of course and it has a large ring of endomorphisms of but rather I don't have time to go into the proof of what why but you have to use the Iwasawa theory of B I do not know how to prove this if you just work with the Iwasawa theory of A and I don't have time to go into it but let me end with a numerical example you might say well does this theorem hold for such twists of D so in fact I claim that when Q is 23 and M is 901 which is 17 times 53 we which are primes so this is in M that LDM over H one is zero in fact numeric I mean you can prove it's zero but numerically it looks as though it has a zero of order six now you might say why six well you see after all 23 has class number three and it's a little exercise using the restriction of scalars to see that the the rank of any curve in this with complex multiplication say by Q Q square minus 20 the integers of that that it must be divisible by six at least twice divisible by 2h you see and here it seems so since the H is going off to infinity very rapidly you are getting I mean I believe that there are infinitely many M like this say for this case so so and similarly as you go up to larger ones but but of course we can't we can't prove that and even in this case we can't the well I haven't been but the the numerical people can't find the point of infinite order there should be six independent ones but they can't find it but that the L function appears to have a zero of order six is absolutely no doubt so by the way our models here my will of course apply to these curves here when it's finally the proof is finally written down let me finish here thank you how do you call this this which one well it is it turns out that that the the the it was sour there's a relevant it was our module over the field that you get by joining the gothic p infinity division points and to H and it turns out we can prove that that module is zero by a Nakayama's lemma type argument it all boils down to that so that amounts to by Bertrand's when it and diary if you like that that that the even this certain component of the Tate Shaffer-Avich group of either the restriction of scale of the a over they have the same Tate Shaffer-Avich group of course over this large field is actually zero functions of course there are paedical functions but their units and then that's vital in this I mean there the values are units the units in the it was our algebra that's what you prove yeah for out of the main conjecture and other question what could the Dirichlet see of the birds in the entire conjecture you said something to this effect no I'm sorry you said you said that yes Dirichlet no Eisenstein had an inkling of the BSD conjecture well no he didn't have to be fair to Bertrand's when it there's no evidence what Eisenstein did as Vaid points out in his book is that he he did calculate for the curve just one curve I think y-square it was x cubed minus x the value at s equals 1 where it is non-zero so there's absolutely no evidence of course he died very soon there's no evidence that you see for Dirichlet Dirichlet proved his L his L values are never zero I don't think it probably even occurred to Eisenstein or anyone else that these values could be zero you see this is a different world of why Bertrand and I are infinitely more subtle but we shouldn't forget the enormous step Dirichlet made in proving these theorems in getting us going yeah I have two really good questions so the first one is about the last very last week you said that numerical people cannot find a point yeah so usually this indicates a large shaft right shaft lotto if you can't find a point you very good right but I mean here you said as well that you were expecting the last shaft so so my my the common question is why do you have any contractual or actual idea of what contributes to the size of shaft no but but I mean all the numerical evidence is that the order as you look in the family of quadratic twists of any elliptic curve over a number field the order of Shah goes off to infinity with the it's not like real quadratic fields for example where you you know you presumably there are probably infinitely many with class number one not the situation it seems which are but we know nothing except numerical data so you had a second question no it's any more questions yes totally different question what is the family connection between the regret and dr. D. Mendelssohn ah Dirichlet married the daughter of the daughter of her yes I'm sorry it's the daughter yes well I know it's Mendelssohn's sister and they were both children of the Bartoli Mendelssohn family I think it's Mendelssohn's sister look you look it up on the internet but and it's very tragic you see she after he was finally appointed to Goettingen she died very rapidly at very early age under 50 and he appears to and he died very soon afterwards so very sad but but I think they live together long before that in for many years in Goettingen in Berlin sorry