 Can you hear me all right? Fantastic, okay. Thanks very much for the introduction and thanks for the invitation also. It's really a great pleasure to be able to speak here. So this course will be a short course, so there's only three lectures. And the main purpose is to give an introduction to the foundations of the theory of rigid co-cycles. And it's used to give an attempted construction of singular moduli for real quadratic fields. Now, this theory hasn't been around for that long, only a couple of years. And what I'll do here is I'll give a sort of friendly introduction to the foundations of the theory, which can really be made very elementary and very computational as well. Now, this is in sharp contrast with the increasing number of theoretical results that are emerging also in this area, which in contrast have kind of formidable prerequisites to appearing in their proofs, and in particular the proofs of special cases of some of the observations that we're about to make in this course. So instead, we'll take a very kind of elementary and computational approach in this course. The main idea of course being to embody as much as possible the spirit of this summer school, which is number theory informed by computation. And I very much hope to make it sufficiently concrete and computational for you to apply everything that you've learned so far in this summer school, all your skills with your favorite computer algebra system and really engage with this theory in a very concrete and examples-based way. To help you do that, there's a series of exercises which you can find online and we have a problem session, first problem session later today. So the official TA for that is James Rickards. I don't know if he's around, if you can raise your hand, yeah, there he is. He'll be leading those sessions. There's also my student, Horvak Damjohnson, who knows a fair deal about these computations that go into it. Is he here? I don't know. Oh, there he is, yeah. So those two people will be around and you can ask them lots of questions. And I'll be around also and encourage you to do so. Okay, I think that should cover the practical side of things, if there are any questions, please interrupt me. So today will be mostly motivational and including also some very classical background. And on Thursday and Friday, I will start in earnest with the rigid co-cycles. And I would like to begin today by discussing the first two words here in the subtitle, which is singular moduli. These boards really vibrate a lot. Okay, I'll do my best. Okay, so the values of the modulo j function, which has appeared many times already, I think in other courses, which are called clients j invariant, which is defined by a very explicit Q expansion. So this is n cubed Q to the n divided by one minus Q to the n, where n is greater than or equal to one. You'll recognize the Eisenstein's using weight four here and I'll cube it and divide it out by the following infinite product, which defines the Ramanujan delta function. If you expand this out as a Q series, you obtain one over Q plus 744 plus its magical number 196884Q, et cetera. And I can view this as a function on the Poircari upper half-plane in the variable Q in e to the two pi i tau. The sentence starts with the values and I'll be interested in very specific values, namely those values of tau in the upper half-plane, which satisfy a quadratic equation with integer coefficients. These are called CM points. So these are points in the Poircari upper half-plane, which I'll denote by age sub infinity, which is just a set of all complex numbers whose imaginary part is positive. So what about these values? I'll claim that they are arithmetically rich, which is not a mathematically very precise term. So let me try and point out a number of ways in which they are arithmetically rich, perhaps guided by a few basic examples, most of which, I mean the most standard ones. Or if you start by evaluating the j function at i square root of minus one, there's a unique such square root in the upper half-plane, you get the number 1728, which appeared in many talks already, as well as the j invariant of a cube root of unity, which is zero, perhaps a slightly more interesting, randomly chosen example, is the j function evaluated at the square root of minus 15, and I'll have to look at my notes, which I computed for you as minus five squared times three cubed times this magnificent number, 637 plus 283 square root of five, divided out by twice the square root of five, okay? Minus 15, sorry, this is not very clear. Thank you, minus 15, exactly. Now there's two things I'll note about this number. It's an algebraic number. It's an algebraic number that's defined over a different field than the argument that we fed into the j function. What that field is, I'll tell you in a second, and you probably already know, but I would like to point out two things. The first is that we can compute its norm, it's an algebraic number, and we can factorize that norm. What we get is minus three to the power of six, five cubed, 11 cubed, this is a cube. So we obtain an integer, that's an algebraic integer, if we take its norm we get an honest, rational integer, and if you factorize it, we find that it's extremely smooth. It's divisible only by very small prime numbers to some large exponents. I'll also mention, because it'll come back potentially later, that it also has a trace, and that trace, well you can kind of read it off from this, I guess it's minus three cubed, five squared times 283, which happens to be a prime number. I'll get back to these two observations and why I bother telling you this about this particular j-value. Now, classically, the reason people were very interested in these j-variants, oh yeah, absolutely. I have a tendency to write very small, so please don't hesitate to remind me if I exhibit some recidivism. So classically, and I'm talking now sort of late 19th century, they were notable because of their ability to generate ring class fields of imaginary quadratic fields. And class field theory was in full development in those days. Putting the singular module, I really center stage around the turn of that century. In this particular example, the relevant ring class field is just the Hilbert class field for the imaginary quadratic field q join the square root of minus 15, which is a field of definition of the arguments that we plugged into the j function and out came this number, which is contained in the Hilbert class field, which in this case is a genus field, it's a bicodratic field obtained by joining the square root of minus three and the square root of five. Indeed, it generates that field over the imaginary quadratic field. This singular modulus does. So this is classically why people were very interested in the singular moduli. And for a long time, I think, certainly up to and including World War II, people had the feeling that this theory was really satisfactory concluded. Now, huge renaissance for singular moduli happened much later. And this came with the very celebrated work of Gross and Zagier. Their first paper together dates from 1985. And what they did is something that looks very strange at first sight. They say, take one singular modulus, J tau one, that gives you such an interesting algebraic number that is a generator for a ring class field. And now let's take another one of a discriminant that has nothing to do with the first one. So J tau one minus J tau two, where the discriminant of tau one is less than zero and the discriminant of tau two is also less than zero. And in the paper of Gross and Zagier, the original paper, they required that these were co-prime and fundamental. What they did is this is an algebraic number, an algebraic integer. So you can take its norm and that gives you an honest integer. And what they do in their paper is they give an explicit formula for what this integer is. Looks a little bit bizarre at first sight, but what is so fantastic about this discovery is that it led to really deep and very important developments and the foundations of these discoveries. I mean, if there's anything that screams the theme of this conference, I think it's very much the origins of this work of Gross and Zagier. And fortunately, Zagier had some significant birthday, I forget which one, a couple of years ago. And Dick Gross gave a talk on the occasion of this birthday and he made public also the letter that he received from Zagier in 1983, announcing some of these first discoveries. And this is such a wonderful document. So Gross says the following. He says, singular moduli were studied intensively by the leading number theorist of the 19th century, as we remarked. They're algebraic integers, which generate certain abelian extensions of the imaginary quadratic fields. The theory was believed to have been brought to a very satisfying completion in the early 20th century. That was before Don got his hands on it. In early 1983, Don sent me an amazing letter from Japan containing a proof of a factorization formula for the integer, which is the norm of the difference of two singular moduli of relatively prime discriminants. This was a completely new aspect of the theory which Don had discovered by extensive numerical experimentation. So that's very much in the spirit of this conference. And here you can see this letter. It's very nice. I recommend anyone who hasn't seen it to look at it. It's really fun to read. And I'll read sort of the first page of it. And you'll see in the exercises, the first exercise really is to try and recreate for yourself this moment of doing these numerical calculations and to actually spot some of the patterns without looking at the letter first and try and relive this moment, which seems like a very frivolous exercise. It's very ill-defined, trying to find patterns and see if they match up with patterns that someone else found 40 years ago. But it's very instructive because later on in the third lecture, we'll do precisely that, but in the setting of real quadratic fields. We'll try and mimic as much as possible that initial process of being guided by computations. So this is what Zagir writes. Dick, he says. I've been in Japan for two weeks now and I'm enjoying it tremendously, both for sightseeing and mathematics. However, telling you about the trip can wait till you get to Germany. I'm writing now for mathematical reasons only. Yes, as you may remember, I had asked you whether our results might be generalized to results on the norm of the difference of singular moduli for arbitrary CM points, tau one and tau two, with unrelated discriminants. You poo-pooed the idea, explaining why your method only applies to elliptic curves having CM by the same order. Not daunted, actually I was, I didn't do the calculations till now, I calculated the difference of singular moduli for many different examples of class number one. A somewhat tricky business since my HP has only 10 places. And I found the values and then here's some big table. Now already, I am just mesmerized by this letter because I don't know, I wasn't born, but in 1983, if you were on a trip to Japan, I don't think people had laptops, so this HP with the 10 places he's talking about, I could only surmise he's talking about a calculator with sort of 10, I don't know, but this three years, this is amazing, that he had a calculator with 10 places and worked out all of these tables. You have one? Fantastic, okay. So if you really wanna be hardcore, you should do the first exercise for this course with only that HP. That's for the really hardcore people. Other people can use a laptop and that also makes the exercise, I think, quite a bit more palatable because it doesn't sound like the most fun obstruction to put yourself under. Okay, now I wanna highlight one table. So Zagyyei computes many things in this letter and I'll just pick out sort of one example that we can stare at and try and explore a little bit together. It's the J value at one plus squared of minus seven over two and I'll subtract the J invariant of one plus the squared of minus 43 over two. Now, see, M theory tells us that these are both integers. So the norm is not necessary, it's already an integer and that integer, when you factorize it, is three to the six plus five cubed, so times five cubed rather, times seven times 19 times 73, okay. Again, very smooth number, which is quite remarkable because the J invariants themselves, as we already noted here, maybe this norm, they tend to be very smooth themselves so we have the difference of two smooth integers which is again very smooth. The ABC conjecture tells us that that sort of stuff shouldn't happen very often. Luckily, there's only a finite number of examples here so we can still believe in ABC if we want to. Now, this is what Zagyyei does, is he makes a little table and here, I mean, of course, he had some foresight so this is the part that will seem a little bit strange but he had good reasons to be interested in those particular expressions. So what he's going to do is he's going to take X to be one, three, five, seven, nine, that's probably enough and then let's make another one here of X going this time 11, 13, 15 and 17, okay. Now the expressions that he's going to compute for each of those X is the following. So it's seven times 43 which is the product of the two discriminants of the CM points that we plug in, minus seven and minus 43 but that's the same, you subtract X squared and you divide it by four and he's going to list all of the positive integers that are of that form, right. So this can only happen for odd Xs in this range so those are the only examples where this expression is a positive integer. So he computes those. So for instance here we get seven times 43 which is 301 minus one squared which is 300 divided by four which is 75, okay. So 75 is three times five squared. I'll just fill in the table and you look at the numbers and see if you notice anything suspicious. So this is a table, okay. Who's willing to make a conjecture? Observation zero. Every prime factor that arises that divides the difference of these two singular moduli seems to arise somewhere in this table, correct? That seems to be true. But conversely that's not such a hot statement it seems. Some of the prime factors that appear in this table do not appear on the left. Oh, there's something wrong because I probably made a horrendous mistake. Three squared five, thank you. Yeah, this is a problem with a board token for this reason I want to show you a huge amount of numbers later on in the third lecture. There'll be slides. There's something else wrong. 13, I wrote, yes, this is a three. I can claim that I meant to write a three but this is just a mistake. This is a three. Thank you. Let me scan again while you observe. Now what Zagir goes on to do in his letter and he describes his thought process which is very interesting. He's trying to figure out which primes in this table actually do arise and to what multiplicity they arise. He kind of walks you through his thought process. He comes up with a formula and then he proves it in this letter. He proves the formula exactly and this first exercise is to try and do the same because by doing it I think you'll have really the right reflexes when we get to the real quadratic, okay? So I highly recommend doing that particular exercise. So that's what happens in that letter. Now the rest is kind of history because Zagir challenges Gross in the letter to find a proof that's different from his proof because his proof uses, it's a very analytic proof. It uses families of Eisenstein series. It revolves around the families of Eisenstein series that was written down by Hecker about 100 years ago and he challenges Gross to find an algebraic proof which Gross does. So they give two proofs in their first paper and later on they combine the two into the contributions of the Archimedean and the non-Archimedean primes to a height pairing of Hegner points and these results they led. Let me just say, and I'll say more about this in the final lecture, to progress on the Bach and Swinnett of Diaconjecture. Relating the heights of Hegner points to the first derivative of the central value of the L function of an elliptic. So this is Gross Zagir, 86, and then there's also Gross Kronen Zagir, 87. Yes, some question. Bigger, yes, thank you. Okay, so the goal, this course, will try to do a very similar kind of experiment and to try and construct also by analytic means for the periodic analytic means. In variants that look very similar to the differences of singular moduli at two CM points where we replace these two CM points by a pair of RM points. So real quadratic fields instead of imaginary quadratic fields. And so this course will aim to discuss the foundations of this theory. The real quadratic analogs. And this is the subject of rigid co-cycles and everything that I'll mention is joint work with Harry Darmel. All right, before we do that, it's important to also mention that we're certainly not the first ones to try this. And there's very, there's alternative approaches that I should mention that you may have heard of, but they yield a very different set of invariance. So other approaches have been explored. Of course, very famously,