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


Everybody don't get an opportunity to do this, man. Every morning you gotta wake up, man, feeling blessed. Another opportunity, every single day. Don't never take it for granted. Never take it more than you wake up for granted, dawg. Let's go out here and take advantage of every opportunity, man.  Let's go out here and ball out, play together. Like I said, all three phases, man. We always say, it ain't never about them. It's always going by us. What we do, how we play, the type of attitude we bring with it. The mentality, man, dawg, man. Let's go out here and eat some dog food today, man. Let's go out here. Hey, we're on a roll, man.   Take over your time, man. Take over your time, man. It's out of luck. Get it. Yes, sir. All right. All right. All right. Woo, woo, woo. Good enough, man. We did a little more than that, man. All right. Woo, woo, woo.  One, two, three.


Okay, good afternoon. Welcome to CMC markets on Friday the 2nd of December and this month non-farm payrolls report for November with me, Michael Houston. Before we get started, the general risk warning, I will be recording the video. So if any of you want to go back and re-listen to any parts that might be of interest to you, you can feel free to do so. I'll post it on YouTube after the event and then we can go on and talk in general terms about obviously what I'm expecting from the report, whether or not the report actually matters in the wider scheme of things in terms of US monetary policy. And also once I've finished going through the key chart points on the various indices as well as currencies have also a quick preview of next week's USPPI numbers, obviously the CPI numbers that come after that, and obviously the Fed meeting, which also coincides with in the same week, the European Central Bank and the Bank of England. So it's going to be very, very busy week, the week beginning the 12th of December. But for the here and now, let's concentrate on the US payrolls report. And it's coming off the back of a very strong month for European stock markets. And the DAX closed 8.6% higher on the month, obviously 100 made a gain of 6.74%. And the March higher in the dollar, US dollar has come to a shuttering halt posted its biggest monthly decline on the dollar index since 2010. So I mean, what's what's caused this sharp turnaround when it comes to the US dollar? Well, quite simply, it's down to the fact that I think, and the market is starting to think that inflation has peaked. And I mean, there's certainly been evidence of that for quite some time. I've been talking about the prospect the US CPI peaked as long as four or five weeks ago, and perhaps even longer. Because if you look at how the headline number has been coming down over the course of the past few months, it's come down every single month on the headline number in the US since June. And those those drops have now started to accelerate. And that's no better illustrated not just in the US numbers, but also in the headline PPI numbers for Europe, particularly in Italy, in Germany, and this morning as well, in the EU PPI numbers that came out at 10am this morning, where we saw a monthly decline in October of 2.9% PPI month on month, which dropped the annual number from 41.9 to 30.8. Now, obviously, that is still very, very high. But it's the extent of the decline that I think is really shaping the narrative, if you like about, you've got central banks, you've got central bankers talking about the fact that they don't think inflation has peaked. You got Madam Lagarde earlier this week, making that very same statement and that inflation expectations, you know, needs to be well anchored. Well, they haven't been well anchored for about the last 12 months. And for the last for a good part of the previous two years, you've had central bankers saying inflation is transitory. And they were saying inflation was transitory as recently as February this year. I mean, up until March this year, the Fed was still growing its balance sheet. And then since March, the upper bound of the Fed funds rate has gone from 0.25% to 4% since March.  Now you've got central bankers articulating concerns about policy lags, about gross slowdowns. You've seen a slowdown rate hikes from the likes of the RBA. In November, they hiked by 25 basis points when they're expected to hike by 50. You've got the Bank of Canada next week and the RBA is next week as well. You've got the Bank of Canada who were expected to hike by 75 basis points last month and hiked by 50. So there is growing evidence that inflation has already peaked. And yet you've got central bankers now saying, well, actually, we don't think that it has. Well, why should we believe you? You were telling us inflation was transitory up until February this year. Now that it looks as if it's coming down, you're saying that it's going to be staying up at current levels for quite some time to come. So sorry, don't believe you. You know, you've already proved that you're not particularly reliable when it comes to transitory. So why should we believe you when you're starting to say it's become more persistent? Now, obviously, maybe I'm being a little bit harsh. I would argue that I'm not. You know, some of what central bankers have done has been the architects of why we're in the situation that we're in at the moment. Nonetheless, we are approaching some very, very key levels on a number of major indices, notwithstanding the S&P 500. If we look at this daily chart here, if I draw a line through the peaks from earlier this year, we come in pretty much where we are at the moment. And yes, we have broken above the 200 day moving average. So that does suggest that we're starting to see a turnaround in sentiment. The bigger question is whether or not we see a breakout today. And I got to say, I struggle with that. We've basically rallied thus far on the premise that inflation has peaked, but that it won't cause a significant earning slowdown and a significant period of economic recession, even though bomb markets are potentially arguing for that very scenario. They're pricing in a significant recession in 2023. In fact, here in the UK, we're probably already in a recession, given the fact that Q3 GDP contracted and Q4 is unlikely to be any better, given some of the data. And the same applies to the EU as well, certain countries in the EU. It's going to be a significant struggle to really suggest that we're going to see any sort of growth in Q4 and that could well spill over into Q1. So things are going to remain very difficult. So we've seen a fairly decent rebound. The bigger question is, can we sustain it? We're certainly on the basis of dollar weakness. I think we can certainly sustain dollar weakness. Why do I say that?  What we've also seen is a big technical breakout on amongst a number of key currency pairs, euro dollar, being a case in point. Now, I thought that it would go a lot lower than it has. The fact that we're back above parity and are broken above the 200 day moving average, the 50 day moving average is moving to more positive, suggests that we can definitely potentially see 10620. Why 10620? Well, that's basically using FIB levels here. We can project 38.2 Fibonacci retracement from the highs this year to the lows and a pullback from there. So there's certainly potential for us to go quite a bit higher or a little bit higher. Also, we can draw a little trend line through these peaks here and you've got a nice little upward channel forming through these levels here. So certainly I'm expecting to see further dollar weakness in the short term. Nowhere is that better illustrated in the way that dollar yen has behaved over the course of the past few days. In fact, I think we can go quite a bit lower and we can head back towards these lows in August at around about 130 40. That's that was the key breakout level to the upside. We've now tested lower. We've broken below the previous support 140. There's 140 to 250. But 140 here is likely to be a very key barrier to further gains. And we're also below the 200 day moving average. So momentum is clearly against the dollar when it comes to dollar yen. And you've also got Bank of Japan officials certainly arguing the case for further or potential change in monetary policy from the Bank of Japan. And that's likely to be positive for the yen and negative for the dollar. So you'll potentially be doubling down on that as well. So I've got the countdown marker going there. Cable, you asked me about cable, Richard. So big question here for me is cable has, have we got further to go higher? Really what's happened in cable has been a symptom of dollar weakness. We've broken above the 200 day moving average. We're testing the 50% retracement level of the moves from the peaks back in June last year to the record lows that we saw a 10340. We certainly have potential. I'm more bullish on cable now, despite the deteriorating fundamentals than I was, say, for example, a couple of months ago. But everyone was bearish on cable pretty much every man and his dog. So even if you are very negative on the UK economy, that doesn't necessarily mean that the pound can't go higher, because bearish sentiment is so negative. As we head towards year in, you're going to have a certain amount of short covering. So certainly against the dollar, an awful lot of the more hawkish scenarios as far as dollar gains are getting priced out, which is basically why you're seeing the pound ratchet higher, much more aggressively than is the case against the euro. And essentially, that's also why you're seeing a game against the euro. But in the context of the overall range that it's been in over the course of the past year or so, if you look at the pound there, it's not really going anywhere. And it's holding above the 200 day moving average. So I think euro sterling is likely to continue to remain in a range. And while the dollar remains weak, the likelihood is we'll potentially see further sterling gains. Certainly, I think my mindset is that we could potentially see 130 cable within the first six months of next year, obviously depending on how the data plays out. But certainly in the context of these payrolls numbers, let's talk about the actual numbers themselves. I'm just going to quickly look at the DAX. And again, here, we're pushing against key resistance on the DAX, again, from the highs this year, 61.85 level, big, big level around about 14,576. So it's going to take quite something to push the DAX much above this series of peaks through here. So again, pushing against resistance in the on the S&P, pushing against resistance on the DAX, and also pushing against the upper end of the recent ranges on the FTSE 100 as well. Potentially bullish candle there. But if you look at what the FTSE has done this year, it's towards the top end of its recent range. I don't see that changing much between now and the end of the year. So what are we expecting on the headline numbers? We can see the forecasts in red here, 200,000, which is down from last month's 261, which was better than forecast. Unemployment rate 3.7% average hourly earnings expected to come in slightly lower. If we come in pretty much in line with expectations, I don't anticipate there being a significant move one way or the other. It's going to take a sizable miss towards the downside to the downside for markets to come off quite significantly, because at the moment, there's a bit of a balance. Markets are pricing in inflation coming down and a soft landing. If job growth collapses, which is unlikely at the moment because of 10 million vacancies in the US economy, if job growth collapses, then obviously that predicates a much harder landing than perhaps markets are pricing in. So anything over and above 150, 180, 200,000 is probably going to be in line with expectations. But certainly in terms of the data that we've seen this week, particularly in manufacturing, the ADP report saw the loss of 86,000 manufacturing jobs. So it'll be interesting to see whether or not the non-farm payrolls report manufacturing payrolls numbers there show a similar decline. Certainly there was weak growth in the October numbers of only 32,000 manufacturing jobs were added in October, expecting a rise of 18,000 manufacturing jobs in the November ones. So it'll be interesting to see whether or not that continues. The bigger thing here, I think with respect to the actual report itself is wage growth. So at the moment, we're expecting 4.6%. That's going to be a 12 month low. So again, that will keep the Fed sweet in terms of, yeah, we're going to go 50 basis points. We already know we're going to go 50 basis points. It's really about what comes after that. And what will shape what comes after that will be not only today's payrolls report, but it will also be USPPI next week, which is due to come out on the 9th of December and USCPI the following week. And USPPI next week is expected to fall by quite a significant amount. 8% was the headline rate for PPI in October. Market projections for that from a drop to 7.1. And in October core PPI was 6.7. That's expected to fall to 5.8. So again, that feeds into the weaker dollar narrative and the potential that January rate hike could be as little as 25 basis points. And that's essentially what we're talking about here. The terminal rate after the 50 basis point rate hike that we're going to get this month will be could be as low as 5% or even lower than that because the Fed funds rate will be four, four and a half after the payrolls number, not the payrolls number the Fed meets in December. So counting down in terms of Euro dollar, I'm still very much a seller of dollar strength. So sell dollar on rallies. Now that we've broken out on these key chart points on the on the on the upside in terms of Euro dollar and cable, dollar yen selling into weakness around about 137.50 overall, I think given the progress that we've seen this week, we're on I would be surprised if we see a break to new highs on the S&P and the DAX and that's probably going to come back and bite me in about five seconds time. So here we go over now. Keep quiet. 263. That's a strong number. A positive revision higher 284. So that's very positive five. Wow. That's dollar stress. That's a really dollar strength story there wage growth. The Fed will not want to see that 5.1% and revised up in previous months to 4.9. So and unchanged at 2.9. Let's have a look at the other numbers. Manufacturing payrolls rose by 14,000. So yeah, I mean, all of this makes is a complete contrast to what we've seen earlier this week. Week data, there is nothing wrong with the US labor market on the basis of those numbers. Even the Canadian jobs report is stronger than was anticipated 10.10,100 jobs added in November. So we're going to let it get a little bit of a pause here. And obviously equity markets are lower on the back of the dollar rebounding. Not really a surprise there. Dollar going a little bit bid. Let's have a look at the 10 year yield. That's up 10 basis points from 3.5 to 3.6% nine basis points now. So certainly no soft landing. No hard landing either. Probably a US economy that's an awful lot more resilient than perhaps an awful lot of us gave it credit for. But certainly I think the tenure or the tone of Powell's comments earlier this week do suggest that the Fed is starting to worry about policy lags and the policy lag effects. So very strong number, very good for the dollar. That particular payrolls report also got a price in the fact that generally as we head towards Christmas, you get an awful lot of what I would call temporary hires. As a result, retailers hiring for Thanksgiving and Christmas. So it'll be very interesting to see whether or not the big jumps that we've seen not only in the October revision, but the November report carry themselves over into the new year. But certainly that number is a surprise. And what's even more surprising is the significant jump in wages, which the Fed will probably not welcome and could bring could could prompt a few analysts say, well, maybe the Fed should go by another 75 basis points. Personally, I think that is unlikely. It shifts the balance slightly towards 75 in December, but given Powell's comments earlier this week, it would be unusual for the Fed to continue to go down the route, tightening aggressively from the 75 they've seen thus far. So let me just double check to make sure I've answered all of your questions because it's just going down. So I've done sterling dollar, dolly and dollar CAD. Let's have a quick look at dollar CAD for you, Richard. I missed that out. My apologies for that. So this is what I did earlier. Let me just get out of that. Of course, in doing that, I've forgotten to send I've forgotten to send out the the price alert for it, which is a little bit disappointing. Never mind. I will do that now. Well, it's three minutes late. A little bit disappointing. There we go. So where was I? Okay, so we're going to be doing dollar CAD. So the strength of the dollar could well take it back to this trend line here. We got the 50 day moving average, which thus far over the course of the past few weeks has managed to cap any dollar gains. And I would expect that to continue to be the case. So we could see further Canadian dollar weakness, dollar strength, particularly if the Bank of Canada goes by 25 basis points next week. Because we do have a central bank meeting next week. And the Bank of Canada is likely to go by 25 and not 50, given the fact that it dialed back from 75 previous month, which should see the Canada weaken potentially further the dollar gain run into resistance through there. But certainly in terms of this particular chart, we would look to see a little bit of a dollar gain up to around 135, 60, 70, and then for it to drift back down again on the back of that. So certainly still nothing changes in terms of my selling dollar strength scenario in terms of these payrolls numbers. But what if these payroll numbers will do is will keep the dollar bid heading into the end of the week. So we could we'll see Euro dollar slip back down towards the bottom of yesterday's range there, which is around about 104, 103, 80, be a little bit of support there. What we won't want to see is a close on the lows here because that would then signal a bearish reversal and signal a little bit of weakness back down towards 102. And perhaps that's what we need in terms of a shakeout of some of those dollar positions that are probably a little bit stale, where people have got long of euros and a little bit of short dollars. But certainly it does tie in quite nicely with the FIB level on the cable chart here, which held really nicely at 123 in the move higher yesterday and was unable to break through today. So we could see a little bit of a pullback in cable and Euro over the course of the next few days on the back of these numbers, but much will depend on PPI US PPI, which is due on the 9th of December, which is next Thursday. And if that comes in particularly weak, we could say three or four days of a little bit of dollar strength and then we'll get a little bit of dollar weakness on the back of the US PPI number as the oversold nature of the recent dollar losses starts to unwind itself. So hopefully that helps you out with the dollar CAD. Can I talk about Sentinel-1? What if I knew what it was? Let me just see if I can find it. I'm guessing it's a, wow, okay. Let's have a look at that. From a technical basis, I'm assuming you mean and nothing else.  Well that's interesting. I mean potentially we could be near a short-term base with gap lower. I mean this sort pattern on a technical level usually tends to be positive. It looks like a hammer if that helps. So I think if we get a break back above 16 here, we could well see a little bit of a pullback in the short to medium term. Let's also have a look a little bit from the technical point of view. There we go. I mean it's definitely bearish but I would be certainly interested to see what happens and whether or not this move off the lows. We filled the gap. What we want to do is now get back above the gap. So there's a little bit of gapology in terms of this particular chart here. We gap lower. We've got a hammer. We've now come higher. We filled the gap. The big question is do we basically re-surface up here or do we come back down here? At the moment it looks as if a base could be in but you know it's a fairly high risk trade based on the chart and the price action that I'm seeing at the moment on this particular chart there. So hopefully that helps. Euro yen. Euro yen. Let's have a look at that. Right well nice little trend there. Let's have a look at see if we can find something discernible. Well I think I've just answered that particular question. Taken the lows from March. So big trend line currently which we rebounded from this morning on this particular daily chart. So as long as we hold above today's lows which are currently at 140 76 then we could see a fairly decent rebound. But what I would say is the fact that we're back in the cloud. Any rallies are likely to be limited to around about 144 on that basis before we drift back down again. So should see a limited rebound from current levels. Test back into the cloud. Draw a line through these peaks here for a retest of the line the uptrend line with a potential break towards the downside. So that's what my Euro yen chart looks like there. Okay so have I got have I missed anybody in terms of questions? So at the moment looking looking at the current state of play as we head into Christmas and the end of the year. I'm not convinced that there's an awful lot of further upside in terms of equity markets and the reason for that is if if you've made any money this year you're going to want to protect it. If you've done your boots this year you won't want to do any more and ultimately everything is waiting C mode for the Fed, the ECB and the Bank of England. So that suggests to me that we're probably seeing the short-term peaks will chop around in the S&P, the NASDAQ and the DAX and will retreat a little bit from the highs that we've seen in the past few days. That's not to say that we can't go higher but I think that for the here and now will probably chop around within the current range that we've been in. So with respect to the DAX we could slip back to around about 14-250, 14-500 between now and the central bank meetings barring anything black, swan-ified, black swan event, anything out of China, talk of a China reopening. I got to say I still think that's highly unlikely. I think what could happen or what's most likely to happen is that they'll be more flexible about lockdown restrictions. They won't be locking everyone up 24-7, padlocking them behind the front doors and throwing away the key. I think if anything what the recent unrest has taught us is that the natives are becoming restless and they need to be, there needs to be a much more nuanced approach to Chinese lockdowns. Infections are still going to continue to rise. There won't be a full relaxation of COVID restrictions much before the end of Q1 beginning of Q2 next year but I think you could see a certain degree of flexibility on the part of Chinese authorities that you probably haven't seen before and with China trade numbers due out next week it might lead to be awful. We're expecting to see declines in both imports and exports in excess of 4% for exports, a 4% decline and a 6.7% decline in imports. I think the Chinese authorities will want to at least try and keep the economy ticking over until the early part of next year when they can look to fully relax restrictions then. Let's just quickly have a quick look at gold as well because I know some people like to talk about that and that's run out of steam at the 200 day moving average. We did close above it but it also happened to coincide with, it did it a couple of times through here as well. The fact that we dropped back on the back of the jump higher in yields, the US 10 year is now 3.62% so it's now 11 and a half basis points higher. The inflation slowdown or the wage growth slowdown could take a little bit while to play out but certainly in terms of wages, the wages numbers welcome news for workers as narrow as the gap between headline inflation and wages but the Fed will be concerned about a wage price spiral so not that there is one it's just something that central bankers like to bang on about. Okay so it's now quarter two unless I'm assuming that was an attempt at humor, is England's play over the 50 day moving average? Yeah well it depends whether you're talking cricket or football Alan because if you're talking cricket yeah it's well above the 50 day moving average. If it's England's football then it's middling because they were quite good against Iran and rubbish against the US and they were pretty good against Wales so we'll have to see what they do against Senegal next week. Very hit and miss indeed. Does anyone else have any questions ladies and gents because if not I'm going to wind this up and I'm going to wish you all well obviously I won't speak to an awful lot of you before Christmas so I'd like to wish you all a very early merry Christmas and a happy new year and I'll speak to you all same time similar place in January otherwise thanks very much for tuning in over the last 12 months it's been a pleasure and I'll see you all next year as I say a very Christmas and happy new year and profitable trading to all of you thanks very much for listening


on everyone Sam here with late in sports cards and I am doing 2022 tops definitive baseball hobby 3 box case break 37 random teams here are the results to the random again you can find these anytime on the website under the results tab as well as the link in chat currently let's go ahead and get started good luck everyone really pumped for this haven't broken definitive basically since like release day so ovens looking so forward to this one right box uno if you know it's minus 26 and back to back tournaments holy cow wow man empty all right we've got patch go to the Padres Manny Machado nice one there three of 30 on the machado patch pretty card we've got what is this game use memorabilia helmet definitive helmet relic card twins josh donaldson 14 of 35 there you go twins that's cool that's really cool we've got uh definitive bat auto giants eight of ten will clark green there you go giants back to back got giants again this time it is brandon crawford one out of 50 book and one of 50 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congrats on that that was sick congrats d-backs combo spot hit and to end it off go into the braves four of 25 dale murphy five fielding awards rob g said the cardinal suck matt refuses to be insulted by you rob last box of the case let's end it with a banger here return to sender on that return to sender we've got eight of 30 blue patch ronald akunya jr there you go braves nice akunya 33 of 35 as a filly andrew mccutchen jumbo jersey nice cutch we've got definitive bat auto or bat relic auto 37 of 40 world series winning bernie williams congrats yanks nice one there josh h that's cool nice bernie got 18 of 50 jumbo patch auto salvador perez going to the royals there you go royals jake w all i have to say about the brewers is the cardinals never had to install a slide to entertain their fans we've got dual auto the other a's dual auto ricky henderson mark maguire 16 of 25 congrats kred g thank you lord i appreciate that congrats craig nice hit there we've got gold framed auto ryan sandberg 28 of 30 go into the cubs there you go uh jimmy c on the combo hit with the rhino we've got purple rookie autograph six of 10 jaren diran for the red sox tray h picking up a nice jd rookie auto and to end it off go into the new york yankies dj lamehue gold framed patch autograph four out of five nice one josh h sick patch on the dj part of the ny logo that will do it for the break right there gonna do a quick recap yes yeah we'll do a quick recap that is just uh top loading a few more cards so we've got salvador perez patch auto to 50 royals uh bernie williams bat auto to 40 yankies andrew mccutchen jersey philly's 35 uh ronald acunia jr patch eight of 30 uh braves seth beer rookie auto green 10 of 25 diamond backs dale murphy four out of 25 autograph for the braves matt olson matt Chapman dual to 10 a's nice hit their a's jim tome 12 of 40 jersey auto white sox mike ishtremsky blue patch to 30 giants sick patch there we've got freddy freeman 33 of 50 braves patch jackson koar rookie auto royals to 50 mike schmitt definitive uh awards one of 25 philly's auto got gold schmitt ronado dual auto to 35 cardinals got uh mani machado to 30 patch padres will clark eight of 10 green patch auto giants and brandon crawford won a 50 jumbo patch auto giants a few more here we've got josh donaldson on the uh what is that uh helmet relic twins jaren diran rookie auto to 10 red sox dj le mayhew four or five patch auto yankies ryan sandberg 28 to 30 auto cubs a couple of mag cards we've got one of one dylan carlson button autograph relic absolutely sick congrats lauren that is so cool we've got again for the a's dual bat jersey autograph ricky henderson mark maguire 16 out of 25 got vlad guerrero junior one of 30 gold framed auto blue jays and four out of five tom glavin gold framed patch auto mets congrats everyone on the break thanks again for joining up next we've got some derrick jeter members only stadium club gonna be a fun time see you soon


Seamus, is he fit and Amadu anana, is he fit as well? Seamus and Amadu have been on the grass, we'll make a decision on them. Amdras is still a bit away, but has been on the grass with the sports science team, just easing back into a kind of return to training program. You also get back Abdulai Dukhore after his suspension as well as all the attributes that he brings to human field. Are you also getting someone back who in some ways feels he has to make amends for what happened against birds because obviously you've been without him for most of the second half against him and the subsequent game? I certainly don't ask players to make amends for moments in a game. You know, I think the moment was clear, it was decided he got sent off, but you want the passion, the pride, the belief in that player and he's been delivering that. So I certainly don't ask players for any reasons and you know things can happen in a football match, but you want them to look after yourselves while they're not playing and be ready to come back in and he certainly looked after himself and he'll be ready to come back in. What difference does a player like that make to your midfield? There's no guarantees, but you know, when we came here and he came into the side and he did make a difference, you know, whether he's playing in sort of more like a three in there or more like the sort of pushed on of the three with the sort of number 10 role, he's made a difference with his energy, with his application, scored, set up moments, you know, just a driving force to be on a team unit, but we want the whole team to be like that and he was the catalyst in certain games. And other players followed that. Now we want the other players to lead it and then play his part in it. Obviously his availability is a positive. A point and a clean sheet at Palace is another positive. Dominic getting 90 minutes is a positive, as well at the same time you've also slipped back into the bottom three. So how do you balance the two negatives and the positives? Just realities always work on realities. Good to see Don back out there, that's for sure. Got more minutes in than what we thought. The tempo of the game was slightly lower than some so that we could allow that to happen. He felt good, obviously we monitored it during the game and he's recovered from that well from the physical idea of it and the recovery period. He looked sharp against day in training. Yeah, the idea of a clean sheet away is always welcome of course. I thought we could have took the game on a little bit more actually on the day. But you know, work in progress particularly away, a lot of noise about their way form for a couple of seasons there. Yeah, and then the idea of the position in the league, we've had a million times, happy to bore you with it again. The league table is the one at the end of the season accounts. You know, you have to monitor what's going on but we can only monitor what we're doing. You know, results outside of us, as I've said a million times too, if they come our way brilliant, if they help you out brilliant. But the way that we're trying to plan it is to help ourselves out and take our business ourselves. Taking the game on, take it comes into the thinking again this week because Eddie spoke and he said the last couple of away games generally did not start the way they did at home to Spurs. I'd imagine that your aim is to ensure they don't get that opportunity to start in that way. I go to St Park as well. Well, they've turned into, you know, a good very strong unit. You've seen that this season, you know, from signing a lot of players in January again in the summer but also the good coaching, the good methods Eddie uses. You know, you're always respected Eddie and what he's done and what he's continuing to do. So we know they're a good outfit and that's been seen not only by what your perception is but the league table and some of their performances and if he suggests and they haven't started as well then that's something for them to worry about. We know we've got to start well in every game. Every game, sorry, and a lot so far since opening we have done and we've got to continue that but also add to it and what I talk about the consistency throughout a game because we've had strong periods in games when we haven't got our nos in front, we haven't taken chance, et cetera, et cetera. You've got to keep that consistent performance throughout the 90 odd minutes, 95 minutes or whatever and I think that's still a work in progress but we are looking to start well ourselves, of course, and enablely supported by our fans. Eddie's also spoken about Anthony Gordon shutting out the noise at goodness and park. I mean he left pretty much the day before you were announced as Everton manager but you've been able to glee in exactly what went on. What reaction do you expect Anthony Gordon and do you also have to try and ensure that your plays don't get caught up in any kind of noise around that as well? I don't see why they get caught up in any noise about it. I certainly am not. You know it was before my time there so I don't really know to know too much what's happened in the past because we're trying to build a future. So you know that player left, other players get a chance because of that player leaving, et cetera, et cetera. But I don't see why players would be too worried about that than just focusing on the game. In terms of the pre-match routine, I believe there's a change as well. You are going to be arriving by the team bus. What's the thinking behind that change because I'd imagine it's something to do with the welcome that the fans are planning for the players? It was well documented last season about it. A couple of fan groups mentioned about it this season. I've got a massive respect for them because purely from what I've seen, what I've heard from the fans, I've been away since I've been here. So you know they wanted to play their part if that can make a difference. Beyond that we want the team to play their part. Obviously we're looking for the team to make a difference. But if the fans can continue the support they have then which has been fantastic then we respect that totally. Have you seen pictures of that welcome at Goodison Park? What do you make of it? Well it wasn't here but I think the atmosphere or it creates helps to create an atmosphere, sorry. But like I said the focus remains on the whistle blowing because that alone can't win you a game. You've got to make sure that you're in the right frame of mind anyway regardless. But if the fans can help play their part then that's part of it. Every little bit I suppose. I just wonder as well if it's something that plays a part in a recognition as to how important this game is. I know every game is important but obviously I'm talking from a perspective that time is in essence. I don't think so. I think all of our fans, all Evertonians, not just the ones that come in the stadium I think everyone knows how important this is where the games are. Well they're all important since we got it but you know we're beginning to get a fit to look to the squad with injuries and suspensions. They're all important games but of course this last clutch of games are bound to be important because of what it means. Thank you Sean. Thank you. Obviously only three home games left talking about the fans. What would your message be to the fans going into these games? The same as it has been since we got it. We can only ask everyone to support us. That's all we can do and play our part in giving something back which we've been trying to do. A couple of drop-offs but mainly the team are putting a lot onto performances put a lot on the pitch to go and turn things around so generally I've been happy with that but we have a couple of drop-offs which need corrected. But it's that dual responsibility like said I'd like to think fans support the team anyway but we've got to give them something to support. Sometimes they play their part in getting us going sometimes we play a part in getting them going with the way that we're performing. But they've been terrific so I've got absolutely no question mark over the fans support at all. It can be obviously pretty intense under the lights at Goodison as well. Is the point to try and make it as uncomfortable as possible for the opposition? The team have to make it uncomfortable. The fans play their part with the noise in the stadium and the atmosphere has been terrific since I've been here. So I've been on the other side of that so I know what it means but the team have to bring that as well. That's the important focus. The team have to provide a performance that can win games. Just a word on Newcastle as well. Six out of seven I think they've won going into this. Big chance of Champions League qualification. What have you made of their improvement overall? I think they've found that the good mixture of recruiting well, building an energy and a belief in the team unit, good coaching and management. It looks like they've found a nice mixture there. Recruitment is tough. I don't know how much you spend, lots or little. It's still tough. They've invested a lot of money but they've been quite obviously been coached well. They've got a good feel about it. And they've kind of bought a good feel to what they're trying to do. And then they've definitely changed the rhetoric. They've changed the story. We all know the noise at Newcastle was not positive for a long, long time. And that's partly what we're trying to do here. We've got to keep changing the perception of what's going on and keep changing the thought of it. The only way of doing that inevitably is by performances and taking games on and winning games. Sean, promising return from Dominic Calvert-Lewin to the Crystal Palace to get the best out of him that the service from wide areas needs to improve. Does it? Thanks. Do you think it needs to improve? Those, doesn't it? Statistically, we have improved that. The quality of chance is what I'm always focusing on. I've said it a million times. You read them stats and it says they have this many chances. I saw them. It's key chances. It's key quality moments. That's what we're trying to provide. Although I know the stats and the facts, it's more about the quality. How many quality crosses are you getting? How many quality passes? How many quality chances are you creating? So that's what we're looking to do, is always add in to the quality levels, not just the amount. And get it plays in and around him as well? Yeah, of course. Again, someone will look to effect when we came in. There's been really strong moments of that. There's been other games when you haven't done it as well, depending on the score line, depending on how we've pressed in the game, depending on how we've attacked in the game. A lot more goes into it than literally just finding their moments. But Dominic's history suggests he'll play his part in that. And he looked for something to have been out a while. I thought he looked pretty good physically down at Palace. A couple of down days and some recovery days. And I think you'll see another step on from Dom again. With six games to go, it sort of sees the daytime, isn't it? Have you had to sort of re-emphasise that to the players? Because I know, particularly in the games of Gullison, the majority of the time you've had that, but there have been, and you've mentioned it, there have been drop-offs, second half against Fulham, where Fulham scored. That just can't happen anymore, can it? I don't think it should happen anyway, but it does. No, I think that's probably the only glaring one since I've been here. So I overthink one-half, but it was a costly half, of course. No, I don't overthink one-half of football because there's been a lot of good signs, a lot of positive signs. Dealing with injuries, dealing with the suspension to do, who suddenly materialises a very strong player for us. And now we've got them players back in and amongst it. So I'm looking for more, of course. You're always looking for more as a manager. And the onus is on the players to make things happen and not wait for them to happen, not wait for another teammate to make something happen. You're right, yeah. I mean, I speak to him about it all the time, because football doesn't work like that. Now and again, as I say, you get a lucky one when maybe it's not your day or you're quiet and you still find a result, but generally speaking, you make it up and that is one of the keys to being a good football side. How have they responded when you've said that to them because obviously clearly they know that, but poor performances and a lack of confidence can inhibit things like that? It can, but it's only part of the challenge. You know, it's rare to look at the noise that's just changing about Arsenal. Different kind of challenge. You have to find a way through it. That's part of being a professional footballer, that's part of being a manager, that's part of being a coach, to find a way through it. And usually the best way of finding a way through it is through performances, through work on the training ground and building belief. Belief just doesn't suddenly materialise. You build it. And I think we're in a process of building it. We've got more work to do and we've got to do it quickly. We know that. And have the players realised that they can draw on that template from Arsenal and Brentford and those are the games where it has been, you know, a real up and at and performance? Yeah, of course. You know, despite the noise around there about the negative side of it, we work wholeheartedly with the players about the positives because there's been a lot of something since we got here. That was a great marker straight away, of course, against a topside in Arsenal. And re-enhances that, the ideas and the principles and how we work and the group's ownership of that. And there's been a lot of work on that. And like I say, we've stepped away a couple of times. You can argue the fact of injury, suspensions, but I want the whole group mentality to be like that. You know, if there is an injury, it should be a chance for someone else to come in. It should be a chance for them to lay down a marker. So we want that mentality. Somewhat getting better, somewhat getting stronger.


What's up everybody, Jason here for JapaneseCaseRace.com. We're giving away five 18 box baseball mixture spots away with this 2022 Chrome Update Hobby Box Break number seven. And again, guys, first we'll do the break right about it's because of our names, teams distribute the hits out, then at the end we'll take the 30 customer names that I bought in and the top five, get into that 18 box mixer. Let's do the break first. One or three or four times, list of names and teams, one, two, three, and four. Four times, one or three, four, four, Mike, down a mark. Three, one, four, four, one, two, three, four. Phillies down to the Minnesota Twins. Alright guys, so again, stick to what you got. Of course guys, just a quick little filler. I'll let you see your teams really quick. Alright, here we go. Julio, Calvys Auto. Leonard Braves with that one, Mike Freshwater. Warner Franco Purple. Let's see who has the rays. Rays is Raymond. Warner Franco base. Suzuki insert. Torque. Warner Franco Purple. Couple nice rookies. Not too bad. Sweets. Now let's switch teams and do the giveaway now. So again, top five get in guys into the 18 box mixer. Roll, four and four, eight the hard way, top five are in. One, two, three, four, five, six, seven, eight. Boom, eight, eight, top five are in. Tristan, Mike, Mike, Mike, and Tristan. Oh my god, it was just you guys. Geez, what a randomizer. Eight times, top five are in. There's pastry guys' names in there. Another one's in the store guys, jaspeascadesbrakes.com.