 and he will talk about rigid co-cycles and singular moduli for real quadratic fields. Thank you very much. Can everyone 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 Horvath 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 we'll be mostly motivational and including also some very classical backgrounds 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 I'll call Klein's J invariant which is defined by a very explicit q expansion. So this is n cubed q to the n divided by 1 minus q to the n where n is greater than or equal to 1. You'll recognize the Eisenstein series of weight 4 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 1 over q plus 744 plus this magical number 196884q etc and I can view this as a function on the Poirier upper half plane in the variable q being e to the 2 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 Poirier 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 1 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 0 perhaps a slightly more interesting randomly chosen example is the j function evaluated at the square root of minus 15 and I have to look at my notes which I computed for you is minus 5 squared times 3 cubed times this magnificent number 637 plus 283 square root of 5 divided out by twice the square root of 5 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 well 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 3 to the power 6 5 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 is minus 3 cubed 5 squared times 283 which happens to be a prime number okay 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 joined the square root of minus 15 which is the field of definition of the argument 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 is a genus field it's a bi quadratic field obtained by joining the square root of minus 3 and the square root of 5 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 an including world war people had the feeling that this theory was really satisfactory concluded now a huge renaissance for singular moduli happened much later and this came with the very celebrated work of gross and zagie 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 1 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 yeah so j tau 1 minus j tau 2 where the discriminant of tau 1 is less than zero and the discriminant of tau 2 is also less than zero and in the paper of gross and zagie 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 a 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 the origins of this work of gross and zagie and fortunately zagie had some significant birthday I forget which one a couple of years ago and did gross gave a talk on the occasion of this birthday and he made public also the letter that he received from zagie 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 their 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 and 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 um 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 1 and tau 2 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're gonna 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 to me is 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 want to 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 want to highlight one table so zagiae computes many things in this letter and I'll just 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 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 this is what zagiae 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 let's make another one here of x going this time from 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 x's in this range yeah 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 the table okay who's willing to make a conjecture observation zero every prime factor that arises so that divides the difference of these two singular moduli seems to arise somewhere in this table correct okay that seems to be true now 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 uh because I probably made a horrendous mistake three squared five thank you yeah this is the problem with a a board talk and 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 uh 13 I wrote yes this is a three uh 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 is and he described 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 and he kind of walks you through his thought process he comes up with a formula and then he proves it in uh 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 uh doing that particular exercise so that's what happens in that letter now the rest is kind of history uh because zagir challenges gross in the letter to find the proof that's different from his proof because his proof uses it's a very analytic proof it uses families of eisenstein series uh it revolves around the families of eisenstein series that was written down by hecke about a hundred years ago and he challenges gross to find an algebraic proof which gross does so they give two proofs in their first paper later on they combine the two into the contributions that 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 Birch and Swinnett and Dyer conjecture relating the heights of Hegner points to the first derivative of the central value of the L function of an elliptic curve so this is gross agai 86 and then there's also gross conons agai 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 but piatic analytic means invariance 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 r m points so real quadratic fields instead of imaginary quadratic fields and so this course will aim to discuss the foundations of this theory so real quadratic analogs and this is the subject of rigid co cycles and everything that I'll mention is joint work with Harry Darmore 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 alternative approaches that I should mention and 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 there is stark's conjecture so stark's conjectures they revolve around leading terms of L series and they can be seen as refinements of Dirichlet's class number formula now the original conjectures of stark were entirely Archimedean so complex analysis and it's interesting to note that since we're at a computational conference if you're interested only in the very classical purpose of singular moduli namely generating abelian extensions the conjectures of stark suffice entirely to solve this problem for real quadratic fields even though they remain to this day completely conjectural if you have a computer you can assume they're true get a result verify that that result is correct and that's in fact what many of the computer algebra packages do now what the what star's conjectures come up with the invariance they come up with are units so they're very far removed from these singular moduli which have interesting prime factorizations etc so their units they're very very different invariance there's also a proposal of man in using non-commutative geometry which is similar in spirit to what happened in stark but that's even more speculative there is also a non-archimedean version of these conjectures of stark that's the periodic growth stark conjecture which is a theorem there's a recent work also of Das Gupte and Karte spectacular breakthrough in this area and it produces also invariance that are algebraic and generate abelian extensions of real quadratic fields and even more general totally real fields but again the invariance are very very different it's a periodic construction and they construct p units so again there's no rich factorizations like there are in the differences of singular moduli which is what led to all these other applications so for the classical point of view there's also these other approaches that are worth mentioning another very interesting one to mention is to look at the j invariance still but the j invariant you can't evaluate at a real quadratic argument because it lies just outside of the domain of the j function but what you can do is you can take a real quadratic singularity at its conjugate and that defines for your geodesic in the upper half plane you can consider cycle integrals of the j function along such geodesics and these numbers they're very interesting but they never appear to be algebraic that didn't stop people I mean they can still be interesting if they're not algebraic and so you can read for instance very interesting things about them in the work of kaneco and also duke imamoglian toth so I just like to mention also these alternative approaches the invariance reconstructor go in a very different direction they're orthogonal to these in many ways and I'll focus of course on the rich factorizations that they exhibit and I'll try and illustrate that as much as possible with explicit computations and then we can try and guess some conjectures in the spirit of what zagir does in his letter okay any questions while I check also how I'm doing on time yes quite a disaster yes thank you okay now to explain and to start explaining rigid co-cycles I have to tell you a little bit about reduction theory for quadratic forms which is a very classical subject and that's what I want to do for the remainder of today's lecture and everything here is very classical and goes back essentially to the work of gauss but in order to do explicit computations later it's important that we get all of these definitions straight now a quadratic form for the purposes of this course at least will mean an integral binary quadratic form in other words it's an element a x squared plus b x y plus c y squared it's a homogeneous polynomial in two variables with integer coefficients which I'll abbreviate quite often with this angle brackets notation a comma b comma c I'll say that such a quadratic form is primitive if the greatest common divisor of a b and c is equal to one now the set of quadratic forms is endowed with an action of s l 2 z this is a right action and it's an action by ring automorphism on this polynomial ring z x y whereby a matrix p q r s in s l 2 z meaning p s minus r q is equal to one sends x to p x plus q y and it sends y to r x plus s y now this action preserves the sets which held the note by curly f sub d and what these are they're the primitive forms whose discriminant defined by b squared minus four ac is fixed so these sets curly f sub d are respected or preserved or acted upon by this action of s l 2 z okay now the equivalence classes of this action if I take primitive forms modular this action of s l 2 z of a fixed discriminant there's a number of equivalence classes and whenever this discriminant is not a square which I might as well assume because we won't have much use at least in these lectures for the square case when it is not a square there is a bijection between the s l 2 z equivalence classes on f d for this action so modular s l 2 z with a group that nowadays we know as the narrow Picard group of z d plus square root d divided by two so the quadratic order of discriminant d we can look at its narrow Picard groups it's a class group a d need not be fundamental so this is more traditionally denoted by Picard group in the narrow sense so I quotient out modulo ideal equivalence by principle ideals that are generated by a totally positive element what is this bijection this bijection will take such a form abc and it'll send it to the class of the avertible ideal generated by 2 a comma minus b plus square d when this is a bijection it was Gauss really who for the first time discovered that if you took equivalence of quadratic forms with respect to this group s l 2 z as opposed to the group gl 2 z which is more common at the time because people were thinking about other questions then you actually get a group structure and it's precisely this sorry oh sorry oh d d is a discriminant sorry I didn't say that is that what you mean yeah sorry I didn't say this but when these are discriminant thank you d is a discriminant that it's and it's not a square yes so whenever I write d I implicitly mean that it's a discriminant in case I forget but I'll try and remember thank you okay now an important notion for us is to find within these equivalence classes distinguished representatives and this is the classical topic of reduction theory we will define very special elements in these equivalence classes that are called reduced oh yes thank you in fact let me redo this thank you so we define reduced forms in two separate cases so we'll do this according to the sign of the discriminant d yes the first case is when the discriminant d is less than zero so these what we call the definite forms now we say that a form abc is reduced if the following condition the following inequalities hold on the integers a b and c so I want that b in absolute value is at most a so this is less than or equal to a and a in turn has to be less than or equal to c and at the same time if equality happens in either of these two inequalities then b must be positive sorry is so if equality holds yes thank you that's true I've I may have been a little bit not careful with the negative discriminance because we'll be talking about positive discriminance and I think you are probably correct you're completely correct so how do I fix this in the easiest way so when d is negative I will want to restrict to positive definite forms okay yeah maybe the easiest way to do now because I'm on the board and I have to fix it somehow quickly to get to the rest of my material is to just make d positive would that be a very offensive solution to you this is the only case we will actually talk about and I apologize for this oversight but you don't seem happy with this with this fix okay okay yeah I don't I don't want to make Hendrik unhappy that's this this is one of the things I would like to avoid doing today but I also don't want to lose too much valuable time on this what's important right now that you are happy well okay no but sure we should both be happy there should be a nice that's that's that yes okay um sorry Hendrik I I'm very sorry I'll make it up to you in other ways afterwards so I hope that wasn't too confusing for people it's true that it there's an oversight in that if d is negative you can look at either positive uh valued forms so they're definite so they always have the same sign and if the values are always positive or always negative that creates another ambiguity of a sign which I was not careful with here so this is a bit of a cop out but in the notes I will make sure there's a more elegant way that hopefully makes also Hendrik happy okay now the most interesting case for us will be the positive case and this positive case those are the indefinite forms we will say that a quadratic form a b c is nearly reduced if a times c is less than zero so if the integers a and c have opposite signs this is a slightly weaker notion than the notion of reduced but it is the notion that is most important for what is to follow we will say that it is reduced if well the same condition holds and in addition another condition holds which is that b is greater than the absolute value of a plus c now this is a bit of a bizarre condition because you don't often find this condition in textbooks the general notion of reducedness is defined in a slightly more complicated way but it's entirely equivalent to this statement this is not really so well known but it is the definition that I'll take because it's for computational purposes very pleasant and very interesting because it's so simple and much simpler than the notion that you typically see okay okay very good now another definition that will need frequently is when we have a quadratic form whose discriminant is non-square that it has what we'll call the first and a second root and the first root which I'll denote by r in fact let me just define it on the primitive ones which is all that will need will be a complex number attached to a b c and it's the root b plus square root of d divided by 2 a where the convention is of course that when d is positive this square root of d is picked to be a positive real number the second root will have to be the conjugate of that which is this root now these definitions that I've made of reducedness they can be phrased and perhaps they're a bit more palatable when they are phrased in terms of the roots and this is one of the exercises you will find in the notes as well as the check that that an orthodox definition of reducedness in the indefinite case agrees with the ones that you find in other textbooks so I'll just mention here maybe that reducedness can be rephrased in terms of roots c exercises all right a couple of final remarks let's see how we're doing on time okay I'm going very quickly through all of this more classical material one thing that you may not have encountered before but which I'll mention because it can be useful if you're somewhat more visually inclined is the topograph which is a certain tree that was introduced by Conway in his book the sensual quadratic form and you don't need the topograph but if you know the topograph and if you're a little bit handy with visualizing things in terms of the topograph many of the algorithms and ideas and even lemmas and proofs that appear also in the literature a little bit easier to find if you know about the topograph so I'll just put it out there I probably won't mention it again but if you contemplate the topograph a little bit you might get certain ideas more easily certainly it has helped me a lot in my life so Conway considers a tree and I'll denote this tree by curly t for topograph and what it is is it's the inverse image of the j invariant of the segment on the real line between 0 and 17 28 okay so we already saw the j invariant 0 and 17 28 were very special what I'm going to do now is I'm going to take the real interval I'm going to look at all of the elements in the upper half plane that map to that closed interval within the real numbers between 0 and 17 28 yes okay so what does that look like so this is the upper half plane up here this is the real axis which is just not included in the upper half plane and we know already that I and the six roots of unity that lie in the upper half plane will lie in this inverse image and in fact if you look at the whole inverse image you will see that you get a very nice arc this is not a very accurate drawing it's a circle arc which of course is the j invariant is invariant on the translation repeats like so and then it descends down via all the sl2z images of that same arc or that same hyperbolic geodesic and let me attempt to draw part of this topograph for you in a way that's uh that looks somewhat uh appetizing just keeps on repeating where the ends get closer and closer and closer to the real axis I'll stop there at this depth but you can keep going and you can imagine what the picture looks like so this is what the tree T is this is what the topograph is now what can I do I can take a quadratic form f and in fact let me illustrate it here on the following quadratic form f which is one one minus one whose discriminant is five and what I can do is I can take the connected components of the upper half plane minus this tree so the complement of this tree is infinitely many connected components each of which is adjacent to unique cusp the unique element of p1q what I'm going to do is I'm going to label fr comma s where I think of my quadratic form which has two variables x and y and I evaluate them at r comma s for x and y respectively where this r comma s is the cusp in p1q which is uniquely defined by the fact that it's adjacent to this connected component and of course it's an element of p1q so it's well defined up to scaling and I will take r and s to be co-prime integers so in lowest terms with the usual conventions now if I do this on this particular one the unique adjacent cusp to this region up here is infinity which in lowest terms is represented by one comma zero sign is fixed to be positive in the first coordinate so if I evaluate this at one comma zero I just get one I don't know if I colors actually no there's anyway it doesn't matter so the number one appears in this particular region down here so let me center it so that the cusp zero corresponds to this region down here that's going to correspond to zero one and so I get a minus one filled in here and if I keep doing that I get the following sequence of numbers up to the depth that I have drawn so there's minus one minus five minus one 11 here there is one minus five minus 11 minus 11 minus five one 1931 and let me stop there so I can keep going deeper and deeper and you see more and more interesting numbers but let me let me stop there now this can very often guide your intuition especially when it comes to reduction theory for indefinite forms again if you're algebraically inclined you don't need it but for those of you that find this useful it can be useful Conway introduces this notion of the river which for an indefinite quadratic form consists of all of those edges of the topograph that lie between a positive region and a negative region and positive and negative of course referring to the label so here you see so now I'll try and draw it in big kind of I don't know if this is very visible but imagine this is blue or something it's a river so that seems appropriate to color for it here also I'm dividing between one and minus one minus one and one so here I keep going to the left etc I don't know if this is very visible but you can see here that some some infinite path in this tree is being determined by this quadratic form and this is what Conway calls the river so you you can think of quadratic forms so here the quadratic form of course the A and the C appear as the regions labeled here and if I let SL2Z act on this form all of these other numbers are going to show up coming from the fact that SL2Z is going to move one of these edges to the standard edge here in the middle and so it's a way of visualizing really the entire SL2Z orbit of one quadratic form and it's often convenient when we deal with reduced forms and nearly reduced forms in this language so if it's helpful to you try and think in terms of the topograph about all of this reduction theory especially the reduction algorithm which I'm about to state and see if it's if it's for you and if it can mean anything for your intuition okay now the final thing I want to mention and this is really the the only case that we'll look at in this course so it is going to be positive and non-square so define for some form F in the set curly F sub D so this is again a discriminant I didn't say this let me write it as I promised it's a primitive quadratic form of discriminant D which is positive and non-square I'll call it F and I define the following set sigma sub F to be the set of all nearly reduced forms in the SL2Z orbit of this F so it's all of the ABC that are equivalent under this action of SL2Z which I'll denote by a tilde here to the original form F for which A times C is less than zero so the nearly reduced forms that we introduced earlier now this set will play a large role in and what's going to happen later on with rigid co-cycles and one of the exercises is to try and compute it for a couple of forms F now if you're familiar with the topograph for instance it's it's quite easy to see that every SL2Z orbit contains at least one nearly reduced form and it's also very easy to see that the number of nearly reduced forms is finite because if A times C is negative and B squared minus 4ac is a fixed integer D there's finally many possibilities for the A, Bs and Cs and so therefore this set is clearly finite it's also non-empty you can think about why this is true alternatively you could try and explicitly construct an element in this set for a given F and so I'll just mention hope I have a little bit of time okay so I have three minutes to do this that it can be computed efficiently so note that SL2Z is generated by two matrices S and T, T is the name I'll give to this matrix 1 1 0 1 well I might as well write the 0 and S is the matrix minus 1 1 0 0 these two matrices together generate SL2Z and how do they act on quadratic forms ABC by the matrix T which is translation is going to be sent to A well the A doesn't change at all the B is going to change by twice A so it's B plus 2A and the C is just going to be the sum of A, B and C which follows straight from the definition likewise if I apply S to ABC I end up getting the quadratic form C minus B, A now given some quadratic form F the F that I started off with there you can now find a reduced form in its orbit by doing the following procedure so the first step is you could apply a power of T to make sure that B you see you can change B with a multiple of 2A so you can put it in any fixed interval of size 2A with maybe one of the endpoints excluded and the intervals that we will choose here slightly different from the intervals that the Gauss chose we're going to apply unique power of T to ensure that the coefficient B is going to be at most the absolute value of the coefficient A and it's going to be strictly bigger than minus the absolute value of the coefficient A so that's the interval I'm going to put it in at least in those cases where A is greater than or equal to the square root of D if not if A is less than the square root of D I'm going to put it in a different interval I'm going to put it in this interval square root of D and here I'm going to take as my lower bound the square root of D minus 2A to get the length of the interval correct okay this is the first step the second step is to check well if my form is reduced then I stop if not apply S and return to 1 and you can keep repeating this until you hit a reduced form now you can show entirely algebraic here if you like that this procedure stops it even stops after a very small number of steps roughly it's big O of the logarithm of A over the square root of D it doesn't really matter it finishes after a fixed number of steps that is easy to control and once we have a reduced form we can find all the nearly reduced forms in a very similar way and one of the exercises is to work this out and to apply it to a number of examples okay so compute the set sigma f with a similar procedure see exercises now for indefinite forms which is the case that we'll be interested in for the work on real quadratic singular moduli this set sigma f shows up time and time again so I recommend you to compute it in a couple of examples to really familiarize yourself with it and next time I will show you that starting from these sets sigma f you can construct these really bizarre object called the Knopp co-cycles which were constructed by Knopp in the late 70s and which will construct a periodic analog of that will produce invariance for us that seemed to experimentally mimic very closely all of the properties that were observed by Grossensagier for differences of singular moduli before so next time on Thursday I will pick it up and I'll use this reduction theory to define these co-cycles for you and then on Friday I'll show you lots and lots and lots of data and I'll try and recreate this moment where we come up with some conjectures based on numerical computations very similar to what we saw in the letter of Zagier so with that said I think I'll stop I am sorry for going a couple minutes over time and thanks very much I've time for a quick question