 I will start now. So this is lecture three and I will begin with some brief reminders, a brief recap of what we did last week and then the theme today will be to focus on periods. So the main objects in this business are a pair of affine group schemes and a comparison isomorphism between them. So more precisely we had G11B, which is the relative completion of gamma equals SL2Z with respect to the inclusion, the natural map rho into the Q points of the algebraic group SL2. So B stands for Betty and this is just the group completion that I defined in the first lecture but it has an interpretation as local systems as I explained in the second lecture. So this is an affine group scheme. So it's a pro algebraic matrix group or projective limit of algebraic matrix groups over Q and it is equipped with a zarisci dense homomorphism which I didn't give a name but maybe rho tilde from gamma into its rational points. So we think of this group as some sort of algebraic hull or algebraic envelope of SL2Z. Of course an affine group scheme is just given by its affine ring which is simply a commutative Hoth algebra. Okay, so then the other group in the game is the Diram relative completion which I defined in terms of algebraic vector bundles with an integrable connection on M11, satisfying some conditions and it's the Diram relative completion. Again it's an affine group scheme over Q so once again it's given by a commutative Hoth algebra and these two things are related as follows. So there's a canonical isomorphism that I called comp which after extending scalars to C is a canonical isomorphism between these two group schemes. So these two schemes become isomorphic after extending scalars. More we know something about their structure by the definition of relative completion in both cases we have an exact sequence. They are extensions of the algebraic group SL2 by something which is prounipotent in both cases. So this is prounipotent, in other words it's a projective limit of unipotent matrix groups. Unipotent group is always conjugate to, you can always represent it by a subgroup of the group of matrices. I remind you with ones on the diagonal and the rest above the diagonal. So it's a projective limit of these things. So here I've written SL2 B and SL2 DR. So this can be confusing but it actually clarifies things considerably. In both cases it's just SL2, SL2 beti o SL2 Diram is just the group SL2. The index is just to keep track of where we're working and it's because in this comparison which sends SL2 to SL2. SL2 beti o SL2 Diram, the comparison is non-trivial. So it's very useful to keep track of which copy we're working with. So it also says that the comparison respects this decomposition so comp maps U11 isomorphically onto U11 etc. Finally we know something about the structure of the general shape of these groups. So the easiest way to write this down is to define little U to be the Lie algebra of the unipotent radical of the Diram part. Then this is isomorphic to the completion of the a freely algebra on generators coming from modular forms. So last time I called them, there were generators corresponding to Eisenstein series and each of them comes with a copy of a standard representation of SL2 of rank 2n plus 1. For each cusp form there's a two dimensional space of such representation. So if you choose a basis you've got two generators for every cusp form of weight 2n plus 2 and this corresponds to Eisenstein series of weight 2n plus 2. So this is for n bigger than or equal to 1. So these generators are not canonical, that's important to emphasise. So this lecture is going to be entirely about trying to understand this isomorphism comp and that's going to give some interesting numbers. The definition is the follows. The ring of multiple modular values. So I'm not entirely sure how to denote these. Sometimes I think I write mmv gamma, possibly I've written in one place rather z gamma possibly. So this is a ring of numbers, a subring of c. So smallest ring r containing c such that, smallest q algebra such that the comparison map is defined over it. So g, in other words it's all the numbers that turn up in the comparison map. Maybe a more satisfying way to say this is the comparison map induces a map on the affine ring. That goes in the opposite direction. So if you take an element in the affine ring of the DRAM group and you apply the comparison map on the level of the affine ring, then you get something in this q algebra and certain coefficients which appear. And r is the ring, sorry mmv gamma is the ring generated by all those coefficients. So this ring does not have a canonical basis in any way. So for now it's just a ring. And what I want to do is sort of dig down into this and try to make, understand as much as we can about it. So the first thing we can do by the first point is use the fact that gamma is risky dense in the Betty relative completion. So we have a map from SL2z into the q points of the Betty relative completion and then we can apply this comparison map and that sends us into the complex points of the DRAM relative completion. And in fact, in fact we know by definition of r that it necessarily lands in mmv gamma. That's the definition of mmv gamma, that's the smallest ring, so this is true. And this map is risky dense. So that means these numbers are going to be generated by the images of elements in SL2z. Now gamma was just the topological fundamental group of m11 with respect to a tangential base point, which I defined last time, d by dq, which I sometimes also call 1 at infinity. It's a unit tangent vector at the cusp. And as you all know, this is generated, SL2z is generated by two matrices, which go by the name of s and t. And these correspond to the module transformations, tor maps to minus one of a tor inversion and t is translation, tor maps to tor plus one. OK, so these elements s and t should be thought of as paths in m11, in the complex points of m11. So let's look at that. So for t it all takes place in the q disk. So if this is a picture of the q disk, I remind you that q equals e to the 2 pi i tor. Then it's punctured at the origin, and our base point was this tangent vector d by dq. And so the path t can be represented as a path that goes along this tangent vector and loops once in the positive direction around the origin and then comes back along this tangent vector. So that's t. So t is a loop around the cusp. And indeed it must be by this formula, tor goes to tor plus one corresponds to winding around the origin. So how do we think about s? Well if this is a picture of the upper half plane, then we have the cusp up here. And the tangent vector sticking down like this. Then we think of s as a path from the cusp with this initial velocity, the tangent vector, all the way down to zero. So in fact as a representation of s we can just take the imaginary axis. So this picture is slightly misleading so we're working upstairs in the upper half plane. Of course on the overfold m11 we should think of this. This is a picture of m11 with the cusp and some tangent vector sticking out. Then s is indeed a loop because this point zero is equivalent to the cusp zero is equivalent to the cusp infinity. So we should think of s as some sort of path in m11. But when we work on the upper half plane we should think of it as a path between zero and infinity. Ok so because sl2z is the risky dense and is generated by s and t it turns out as we shall see that the ring of multiple modular values is generated as a ring. By the coefficients of the image of t and s. So by that I mean we have these matrices t and s which live in sl2z and the image is some element of this group and the coefficients are certain numbers. So it turns out the coefficients of t are very uninteresting they're just powers of 2pi i. The element t itself is very important. This only produces powers of 2pi i. So all the information then is contained in this single element s. So that's the interesting part. So before proceeding with this let me just give something more familiar because by now it's increasingly well known. And explain the analogue in the case of p1-3 points just so that we get our bearings. This case is very slightly different because we're not looking at the fundamental group but the groupoid. So we're going to look at paths from one point to a different point. We could take paths based at a single point and in which case it would be very much closer to this picture to the M11 picture. But I hesitated to do that because I think the path torsor is actually the more well known setting so let me explain that. So here on M04 we have the base points that one takes are the tangent vector 1 at the origin and the negative unit tangent vector at the point 1. Of course these are not drawn to scale of course. So the topological, so not fundamental group but homotopy classes of paths from this tangent vector to the other tangent vector is what I just said. So these are homotopy classes of paths from 0 to 1 essentially. There are a couple of paths that play a key role or there's one path that plays a very crucial role. So we have this picture, make it a bit bigger. Then there is the so-called DCH, o droi semae, the straight path from, it's a straight line from 1, 0 to minus 1, 1. So it's really literally the map from the unit interval into C minus 0, 1, the open interval into this along the real axis. So it just goes along with unit speed here and comes in unit speed here. That's indeed a path between two tangential base points because its initial velocity is 1 and its final velocity is minus 1 coming in. So there's another path that plays a role here. So you could take DCH but you could pre-compose it with a loop round the origin. So you could look at gamma nought. You could go round a loop round the origin and then go to 1. That's another example of a path. In M11 the straight line path is played by S and the role of this little loop here is played by T. So in this setting we work with unipodent completion. So the betty torso of paths from 0 to 1 is the unipodent completion. So this is unipodent completion and the diram version of the unipodent completion, the same thing. So this can be described quite explicitly. I'm going to put them over here. So opi1 betty is a scheme and its r points are a commutative Q algebra. So this is a scheme of a Q. Its r points are given by the set of group-like formal power series. So invertible power series in two non-commuting variables which satisfy some algebraic equations. So I wrote this down in the first lecture I think. Group-like formal power series. So I gave a delta of xi equals 1 plus 1 tensor xi plus xi tensor 1. And the diram, the affine ring of the diram torso of paths is really the Q vector space generated by words in two non-commuting letters, e0 and e1, where e0 corresponds to the differential form dx over x and e1 corresponds to dx over 1 minus x. So I don't want to give an entire course on p1 minus 3 points but this is just to fix ideas and to have a concrete analogy. So in this setting we have a very, very similar setup. The topological fundamental groupoid, the space of homotopy classes of paths, maps into the betty fundamental torso of paths more precisely into its rational points and then we have a comparison isomorphism. So here I forgot to say exactly as in as before we have 0 pi1, we have a comparison isomorphism. So then this goes to 0 pi1 into its complex points. So it's an exact analogy of what I mentioned earlier and what this does is it takes this straight line path which is just a path from 0 to 1 and it sends it to something over here which is very famous which is the Drinfeld Associator. So concretely the Drinfeld Associator can be written down at least formally. So it is again a formal power series indexed by words in e1 and then the coefficient of each word is what's called a shuffle regularised multiple zeta value and this is a formal power series in two letters e0 e1 and we know what it looks like, it starts off 1 plus zeta 2 e0 e1 minus e1 e0 depending on certain sign conventions, et cetera, it goes on forever and it can be viewed as the generating series of multiple zeta values. So after that long digression I hope that motivates exactly what's going on in this picture for m11 we've got exactly the same situation, the small difference that we've got the fundamental group with respect to a base point and not a torsor of paths, in that respect it's simpler. So the element t, or rather the image of the element t over here is really the same element as this gamma zero here, it's just a loop around the origin. So t only produces powers of 2 pi i but it's very important, I'll just say this because I might not have time to explain it it's some kind of non-Abelian analogue of the pizza in a product on modular forms. And in principle we can determine its image completely, so it can be computed completely or the element s on the other hand we should think of as the analogue of the Drinfeld Associator in fact they're related in some way for m11 and it's in the same way that the Drinfeld Associator its coefficients generate multiple zeta values, the coefficients of this gadget generate multiple modular values. But it's much, much, much richer than the Drinfeld Associator, a much more complicated beast. So now I want to explain relations between these numbers, as I progress it will become more and more concrete. So for the Drinfeld Associator it's well known that it satisfies the associated relations, namely the hexagon and pentagon relations. And so I want to write down an analogue of that for m11 and so these are co-cycle relations. So we had this exact sequence, unipodent radical, where dot is either betty or diram relative completion. And in particular the comparison isomorphism restricts to or induces a comparison between these copies of SL2. So let me write that down explicitly and dispense with it. So SL2 betty as I mentioned before is just SL2, it's just a label and same with SL2 diram, it's also just a copy of SL2. But the isomorphism is non-trivial so on the level of points, if I take a matrix A, B, C, D then what this does is it maps it to another matrix, what I'll call gamma bar and it multiplies B by 2 pi i and C by 2 pi i inverse. And so this is a minor detail that can be ignored as a first approximation but it's very irritating if you don't distinguish betty and diram copies of SL2 otherwise you get yourself into a pickle. So it's important to bear this in mind. The reason for this will become clear when I talk about the underlying hodge structure. Fundamentally this SL2 acts on the H1 of the tate elliptic curve and that H1 is a copy of a Q of 0 and a Q of minus 1. And that explains this 2 pi i sneaking in. OK so now to proceed, we have to choose a splitting of one of these short exact sequences. So this is on points, so what I've done is I've written it down on complex points. Here this is a point in SL2 betty brackets C and I've written down its image in SL2 diram brackets. This is a product of schemes but to write this down you can write down what this is on the affine ring. Exactly but I've just written it on points. So now let's choose a splitting of this diram exact sequence. So this is possible on the level of points by Levy's theorem or version of Levy's theorem you can always split such a short exact sequence of algebraic groups schemes on the level of points and on the level of actual group schemes this is a theorem due to Moster. So what we have then is we're looking at this exact sequence and we're going to choose a splitting. So what that means is we're going to write G11 diram is isomorphic to SL2 diram U11 diram. And so this means I'm viewing SL2 acting on the right so right action on on the right of U11 diram. Okay so this splitting is non-canonical it depends on some choices unfortunately and I do not believe that there is a canonical such splitting. Okay so now we do the same thing we rewrite this map so we had gamma SL2z going into G11 the rational points of G11 betty by the comparison this gives a map into the complex points of G diram and then we've chosen our splitting so we land in the complex points of this semi direct product. And what that does it takes a matrix gamma in gamma and it maps it to in the first component it maps it to gamma bar which is essentially the same matrix gamma but with this irritating 2.i's creeping in. And and something else that I'm going to call C gamma. And of course the ring of multiple modular values are generated by the coefficients of the C gammas for all gammas for all elements gamma in the group SL2z. So this map is a this map is a homomorphism and so that's equivalent. It's equivalent to a one line calculation that this is equivalent to the equation Cgh equals Cj slash h slash is my notation of the standard notation for a right action times Ch where the multiplication takes place in U. So this is holds for all gh in gamma. So this is an equation which defines a non-Abelian group co-cycle and as a result it gives quadratic relations between the coefficients of these of these elements. Right, so let me briefly remind remind you some notation concerning non-Abelian group co-homology. So this is a digression. So if we have a group G, any old group G acting on a non-Abelian group. So my notes are called a stupidly, but it's not necessarily a belian on a non-Abelian group. So group scheme, for example. Then this the set of co-cycles Z1 GA set of non-Abelian co-cycles is the set of maps from G into a satisfying precisely this equation. So here my actions on the right, for left action the formulae are very slightly different. So such that Cgh equals Cg slash h Ch. So that's the definition of a non-Abelian group co-cycle and there's an equivalence relation on these. So this is a set, it's not like co-homology of, sorry, it's not like group co-cycles on a non-Abelian group. There's no group structure here, this is just a set and it contains a distinguished element which is the trivial co-cycle where G goes to the identity, every element of G goes to 1. So it's just a pointed set, but on that pointed set there's an equivalence relation. So we say that two non-Abelian co-cycles are equivalent if they differ by co-boundary. So that means if there exists a B in A in element B such that C prime G equals B inverse slash G Cg B for all G and G. And if I didn't mess up the formulae you can check that C prime defined in this manner, that any C prime defined in this manner by twisting one element B indeed satisfies this co-cycle equation. So that's an equivalence relation and we say that something is a co-boundary if it's equivalent to the trivial co-cycle. In that case it would be the co-cycle given by G maps to B inverse slash G B for some B. It's indeed a group action of A on the right. Exactly, yes. Absolutely. And then you're right. So H1 is the quotient, so I won't use this so much this time. Model the equivalence relation and you can think of this as a base and Z1 as a total space over H1 and the fibres as you say admit an A action. And that's a good way to think about it. And then the final mark I want to say is, which is very well known, is that this space of co-cycles can be interpreted as just a hom. So if you, in fact I've already used it up here. So Z1 GA is canonically in bijection with the group homomorphisms from G into, group homomorphisms from G into the semi direct product of G with A, which are the identity on G. So this is maps, G maps to G something. And this something defines a co-cycle. So we've already used this to define the co-cycle that we're interested in. So that's a very easy thing to prove. Okay, so by this final remark, the non-Abelian co-cycles are simply the homomorphisms of a group. And since we have a presentation for SL2Z, was it gone, here, then we can explicitly write down all these, all the equations, all the necessary and sufficient equations for C to be a group co-cycle. So the first remark is that minus one in gamma acts trivially and that's going to imply that, so it's very easy to show that the co-cycle evaluated on the element minus the identity is going to be trivial. And this is also related to the fact that the local chart on M11 was the stack quotient of the punctured disk by plus or minus one. And it's also related to the fact that there are no modular forms of odd weights for SL2Z. So that's different ways you can think about this fact, but C of minus one is one in this business. And now that means we're really interested, we've really got a co-cycle of gamma modulo plus or minus one, which is also known as PSL2Z. And this has a presentation, so the image of S in this quotient satisfies S squared equals one. Maybe I'll put congruent to one just to emphasise that this identity is in the quotient, so the matrix S itself literally satisfies S squared equals minus one. And it satisfies, and likewise U cubed equals minus one, so U cubed is congruent to one, where U is the element T dot S, which is one minus one, one, zero. So from this presentation, it means that we get an immediate corollary that this co-cycle equation, which is up here, so let me call it something, dagger, I can leave this down here. So the co-cycle equation dagger is equivalent to, well, first of all, by definition C, it's literally saying that C is an element, a non-Abenian co-cycle of SL2Z in U duram 11C. And that's in turn equivalent because we have this explicit presentation, and by this remark that to define a homomorphism on the group, it suffices to define it on generators, satisfying equations given by the relations. You immediately get the SSS bar times CS equals one, and I've run out of room. Let me squeeze it in here. And we get C U bar squared C U U bar C U equals one. So we get these two equations coming from this presentation, and where C U equals CT slash S bar CS. So there we see very explicitly that this co-cycle, the value of the co-cycle on any group element is completely determined by its values just on S and T, S and T generators. And so you have to just specify two elements in U11 duram, CS and CT, and you've got a co-cycle if and only if these three equations are satisfied. And so these are a complete set of equations, and they imply relations between multiple modular values. And the important thing to remark is that, again I've said this before, that CT we essentially know it completely. It just involves 2 pi i's, so you think of CT as being known somehow, you can view this as a system of equations satisfied by CS. Now we want to actually compute something, so we have some relations satisfied by these numbers. So it's as if we've written down formally what Z is, and we know some equations satisfied by its coefficients. But we now want to actually try to compute some integrals and get a grip on some of the coefficients explicitly. Sorry? So the co-cycles give relations, but they're not the only relations. So there are some other relations that we know that essentially of the form a certain combination of MMVs are multiple zeta values. And the reason for that is because there's a geometric reason, which I haven't talked about, which is that approximately these group schemes act on the fundamental group of the punctual elliptic curve, which is a mixtape motive. So that mixes in multiple zeta values into this picture in some complicated way and gives some other relations. But we don't know, but still, even throwing that into the mix, we still don't know all the relations. And there seems to be a lot more, we don't know really where they come from. This is already a very strong constraint though, it's very powerful as it is. So I'll say some more about that actually later on. But even more fundamentally, the difference here is that every period, every coefficient here can be written down as an iterated integral of some differential forms along this path dCh from 0 to 1. And unfortunately in the case of SL2z, you can't do that explicitly in all cases. So you can only do that for a certain piece of the relative completion. And so let me explain what that is and how that works. So periods and what I call the totally holomorphic or sometimes just the holomorphic quotient. OK, so the issue here is that there is no explicit description of the affine ring of OU11 diram. So that these would correspond to differential forms, or iterated integrals in differential forms, that we would then integrate along s from 0 to infinity and they would give numbers. And that's the analog, so I just raised it, but in the case of p1-3 points this was very explicit, it was just a tensor algebra on two generators corresponding to dx over x and dx over 1-x. The reason why there's no explicit description, so we can write down descriptions of this, but it's not explicit because there are essentially non-trivial massy products in this business and the non-trivial cut products. And you have to choose a sort of system of massy product, you have to construct some kind of minimal model to do something explicit here. So that's a bit of a pain. There's some recent progress by Marle-Woe in his thesis on how to write down, how to explain this in terms of certain complexes. But it's not going to be as nice as p1-3 points, we can just write down a basis completely explicitly. And I don't know if there's some natural choice of a system of higher massy products in this setting which would solve this issue, we don't know that yet. But what we can do for now, we can define, we can describe a piece of this, which I like to call the totally holomorphic quotient. So what that is is, so we have u11-deram and it has some quotient u11-deram whole. Now sort of informally u11-deram or its Lie algebra was generated by some Eisenstein classes and then two generators for every cost form, eF prime and eF double prime. And here informally this is generated by throwing away the eF double prime and just keeping the Eisenstein classes and the holomorphic part of the modular generators. So this, I have to say some words, this is not motivic. So this u11-deram does come from algebraic geometry and it has a mixed hot structure, but this thing does not. It's just a, it's a diram thing, it only exists in the diram realisation. So it is not motivic in whichever sense you wish to interpret motivic, any reasonable sense. So for example it does not have a mixed hot structure or it does not have a Betty analogue or a Betty analogue. So it's defined in terms of the Hodge filtration, in fact you can define it as the, I haven't spoken about the Hodge structure on this yet, but you can just define it to be the quotient by the normaliser of f0, where f is the Hodge filtration on u-deram. Okay, but it turns out that this thing can be written down explicitly and its periods can be written down explicitly. So let's do that now. So what that means, I forgot to write a line, so what that means is that this is some subring, o-u-1-1-deram whole is some subring inside u-1-1-deram and it has a completely explicit description. And this is going to give us differential forms that we can then integrate along s and t. And I mean what is the path goes 1 into s? The path from 0 to infinity. In that behalf plane it's this, it's the imaginary axis. You can't sit on a disk because it leaves the disk and goes on a big detour and comes back into the disk. The disk is just sort of something local up here. So now let me describe, so we're going to get more and more concrete as the lecture progresses. So recall some notation. We had, I had a vector space v-n-deram which was a certain algebraic vector bundle which was the nth symmetric power of the H1 of the universe vector code. We had a curve at the fiber d by dq. Concretely it's just a vector space and we can choose, we have a set of generators, x, the i, that I'm going to denote by blackboard, bold, x and y. And this has a right action of gamma. And in fact we already saw this, the Betty version which was the fiber of this canonical local system at the same tangential base point was, sorry, this is a direct sum q. So it's just a vector space of homogeneous polynomials in two variables of degree n. And this we definitely saw was this vector space with the right action of gamma. And the point of having these different notations is because SL2, Betty and SL2-deram are not quite the same when the comparison isomorphism has this factor of 2 pi i. So this is just to keep track of that. And y corresponds to 2 pi i y and x corresponds to x. So as a first approximation you can just ignore this distinction between x's and y's and just take the standard Betty, x and y in the module of forms literature. But if you actually want to work here you have to be very careful with these different normalisations depending on where you're working. This is due to the comparison, this is exactly the comparison morphism between SL2, Betty and SL2-deram. It's completely equivalent. I don't want to dwell too long on this, it's really trivial but it's just important to do things carefully otherwise things become incredibly confusing and none of the hots theory works out if the weight's all wrong. So now let curly bn be a cube basis for the space of cusp forms of weight n with rational Fourier coefficients. So I call this Sn in an earlier lecture. And so then we can write down this thing explicitly, it's just a tensor algebra. So then the affine ring O U11 dr whole is nothing other than the tensor co-algebra generated on a q vector space generated by certain symbols. The definition is it's just the tensor algebra on a space of modular forms but it's nice to write down a basis to do computations. So having chosen a basis this can be made very concrete. We have symbols E2n plus 2x to the Iy to the j where I plus j equals 2n and these correspond to all n beginning with a 1. So these correspond to Eisenstein series and so E4, E6, etc. And then we have the cusp forms before we had two generators EF prime and EF double prime, one corresponding to holomorphic modular forms, the other weakly holomorphic modular forms. But here we throw out the weakly holomorphic and we just keep the holomorphic part. So we just have a single generator for every cusp form now. So I plus j equals 2n and F a basis element and 2n plus 2. So these are cuspital generators. So tensor co-algebra is just the tensor algebra. So if you have a V a vector space it's just a TV is just the direct sum V to the tensor n. Co-algebra means that there's a co-product on it. So precisely we have a basis of, this is a Q vector space, the basis is given by words in the symbols, in these symbols. So we take words in these symbols, the non-commuting letters and the co-product is given by deconcatenation. So if you have a word w1 up to wn in certain symbols then the co-product sends it to the linear combination i equals 0 to n. So here the notation gets a little bit confusing. So what I like to do is if you had a word for example e4 x squared and then e6 y cubed or something, then it's convenient because these x and ys are the same letter, it's convenient to put a little subscript. So the leftmost one gets a subscript one and so the i-thletter gets a subscript i. So that's just a bookkeeping notation. You don't have to do that but it's just, I like to do this in my papers and I might do it later on. So if you see the subscript x and y it's just keeping track of which in which slot, which x belongs to which letter. So we had this co-cycle gamma going to the full unipodent radical of the DRAM relative completion and it's dependent on a choice of splitting. And then now we can look at something smaller and look at its image in the holomorphic quotient. And so now we get a new co-cycle, so we had this huge co-cycle, a full co-cycle C and now we have a smaller co-cycle where we've thrown out a lot of the periods. You've thrown out all the iterated integrals which aren't totally holomorphic. But already this is going to be very interesting. And so what we get from this then is automatically, since this is a group homomorphism, this holomorphic co-cycle is whole, yeah sorry, it looks like HD, it's whole, my writing is bad. Yeah thank you, whole, standing for totally holomorphic. And this gives us a co-cycle in this quotient. And now the point is that, so here's a key point that I can't explain yet without doing some hodge theory. And that's that this co-cycle depended on a choice of splitting of an exact sequence at the very beginning. Now if you choose that splitting to be compatible with the hodge theory and you can do that, then it turns out that this bit that the part on DR whole actually splits canonically. So this co-cycle is actually canonical, it does not depend on any choices. And there's a very good hodge theoretic reason for that. So it does not depend on the choice, on any choices of splitting. So provided your initial choice of splitting respected the hodge and weight filtrations, then this is some part in the hodge filtration and it separates things out and the SL2 is just can be split off very easily. Okay so this thing is canonical so we can try to hope to write it down. And its coefficients are regularised iterated integrals, also known as iterated Eichler or Schimura integrals. So what I meant to say was that the coefficients are regularised iterated Eichler Schimura integrals. So these were defined by Manning a few years ago where there isn't the word regularised. So in the convergent case. So this is something you can write down very explicitly and that's what we're going to do. We could take a break here or we could just press on and finish early depending on what you prefer. Okay so we'll take a very brief five minute break and then we'll write down some iterated integrals. Okay so I'll explain now how to calculate these totally holomorphic periods. So first a notation. Given a modular form let me write f underline tool to be 2 pi i f tool x minus tool y to the 2n d tool. If f modular of weight 2n plus 2. So I've been a bit inconsistent over this. Sometimes it's convenient to normalise this by 2 pi i to the power 2n plus 1. Here it's convenient to normalise it just with a 2 pi i. It doesn't make much difference but sometimes I've used this notation to mean something very slightly different. The first remark which won't play a role is that if we write this in terms of the parameter q and rewrite these betty generators as diram generators then this is x minus log q y to the 2n d log q. So what happens is that the y eats up well all the 2 pi i's get eaten up and you see that this is actually rational. So what do we think of f tool is thought of as a section of v 2n tensor omega 1 on the upper half plane. Every holomorphic model of forms gives you some section of this trivial bundle of the upper half plane. We define a one form, a formal one form omega whole to be sum over n. For all coss forms of weight n we take a basis of coss forms and we take some symbol for every coss form and the symbol keeps track of this particular one form. You can do this in a basis free manner but this is convenient for computations. So that's the cuspital part and then the Eisenstein series 2n plus 2, so it should be n beginning with a 1, underline, tour and then we have this Eisenstein symbol e2n plus 2. This is the Eisenstein series that's normalised to have completely rational. So the first coefficient of q is just 1 and it has rational q coefficients. Ok, so now let me keep that there. Now we can take iterated integrals in that and this was first done by Manin in some two very short but very nice papers a few years ago. So for all two points tour 1, tour 2 in the upper half plane define, so this was done by Manin, i whole, so the transport if you like from tour 1 to tour 2 is a formal power series 1 plus this plus integral from tour 1 to tour 2 i whole i whole plus our dot dot. Now this is a formal power series in es prime xi yj e2n plus 2 xi yj. I'm using, I'm mixing metaphors here. I'm taking the Duran generators, but the Betty X and Y, which is not a very hygienic thing to do, but it simplifies the formulae considerably. So that's why, and it's actually what's done in the modular form literature. It's confusing at first. You've got a Duran generator with the Betty guys here. It's just important that's said once. This is one, it's this formal differential form, yep. It's a formal, it's like a, yeah. So it's a formal power series connection. And this is its transport. So this is a non commutative formal power series. So I don't know if this is literally a special case of Chen's work on iterated integrals. In the case it's a very slight variant because you've got differential forms with values in a vector model, in a trivial vector model. So it's a tiny modification of Chen's general theory in this case. So let me, if you're not familiar with this, let me briefly remind you what an iterated integral is. So this was developed extensively by Chen in a long, over many years. And I think in the physics literature it was also considered, and bears the name of, bears Dyson's name, though the theory was really worked out. The vast majority of the foundational work on this was done by Chen. You have the dates. It's 1970. I thought it was earlier than that, but okay. Okay, I take your word for it. And so the base code, I'll be very brief because this is very well known. If we take some differential one forms, some smooth one forms, and I can put, you know, if we've got, they could be vector valued in a trivial vector bundle, for example, in this case, but it's sections of a vector bundle. It makes hardly any difference. Smooth one forms on a smooth manifold M, and we give ourselves a path, gamma, and we take an open path. So if you have a smooth path on M, then the iterated integral of this, the sequence of one forms along gamma is defined to be the, in physics called the time-ordered integral, the integral over simplex, f, n, t, n, d, t, n, where f, i, t is defined, sorry, f, i, t, d, t is defined to be gamma up a star omega i for i equals 1 to n. So what's going on here is that when you've got a smooth map from the interval to M, if you have a one form on M, we can put it back to the interval 0, 1, on which we have the coordinate t. And so any one form on the interval can be written some function times dt. And so what you do then is you take, so this is just omega 1 written parametrically. The idea is that you take a primitive of it, take an indefinite integral from 0 to t, then you multiply it by the next, so that's a function, and now you multiply it by omega 2 and that gives you a one form. You take a primitive of that to get another function. You multiply it by the next differential form, integrate, multiply, integrate, multiply, integrate. So that's the notion of an iterated integral. Physics is the time-order integral. Exactly, time-order integral. So the way I like to think of it is if you have a path integral, you imagine a point travelling along a path. And what's going on here is that you're sort of firing endpoints along a path one after the other. And you're sampling the path as you're sort of firing endpoints in sequential order. Okay, so there's a huge theory here that I'm not going to go into. Exactly, yes. Absolutely, yeah. So let me skip that theory because you could be an entire course, and just write down the properties of this particular instance of a formal power series of iterated integrals. So I should just say briefly that iterated integrals satisfy lots of properties. If you take the product of two iterated integrals along the same path, that can be written as a linear combination of iterated integrals along that path via the so-called shuffle product formula. There are formally for what happens when you compose paths, when you reverse paths, and so on and so forth. So in this case, some of these formally give the following... The point I should make is that, in fact, because this formula goes closed, this thing does not depend on the choice of path. So it's independent, and because the upper half plane is simply connected, it's independent of the choice of smooth path between tool one and tool two. So it's really a function of the endpoints tool one and tool two. And that's because this differential form is integrable. In other words, it's closed and it's worked product with itself. Vanishes, which is clear because we're on a complex one-dimensional space. Okay, so having said that, this is a function of two variables. So the integral from tool zero to tool two is simply the product of non-commutative form of power series. Then it satisfies a differential equation. So this is i whole omega whole tool one minus omega tool zero i whole t zero t one. So when I write omega whole on the side, we think of these symbols EF prime and E2002 as acting via left and right multiplication respectively on formal power series. There are shuffle product identities, which means that if you take any two coefficients of this power series and multiply them together, they can be written as a linear combination of other coefficients. In other words, the vector space generated by the coefficients forms an algebra. And finally something which is some sense new in this situation because it's not a general feature of Chen's theory, whereas everything else is. What we gain is the action of SL2z and these integrals are equivariant. So this is the only feature which is a novelty. And the reason for this is because the differential form we're integrating in the first place is itself SL2 equivariant. That's going to be very important. Okay, so now these are some integrals between two points on the upper half plane. Now I want to take one of these points to be the cusp and things are going to diverge. So we need to do regularisation with respect to the tangent vector at the cusp, which is playing the role of our base point. So now let me explain to you very concretely how to do that. It's very straightforward. So we need to define, and the problem is the cusp forms pose no problem at all because they go to zero very fast in the name of the cusp. The problem of the Eisenstein series which have a zero free coefficient and that gives you divergence as you go to infinity. So the thing to do is to isolate out this divergence and take what's essentially the residue of this one form in the Q-disc. And as I just said, all the cusp forms drop out, all we get is the zeroth free coefficient of the Eisenstein series. So this is some kind of residue of omega whole at Q equal zero. And so this notation here for any modular form f0 underline tour will be defined to be 2 pi i, and then we just take the zeroth free coefficient x minus tall y to n d tall where f is the a and f are the free coefficients of f. So for cusp form it gives zero, it vanishes all together. And then now for any points tall zero, tall one in c. So here we think of c, the meaning of c is that it's really the tangent space at the origin of the disc. That's the geometric meaning of this c. It's the tangent space to m11 at the cusp, m11 compactified at the cusp. But we can take iterated integrals with these and let's call that i whole infinity tall zero, tall one. And it's the same thing, so it's the iterated integrals but now of this differential form here. So this is something very explicit. We know that the constant terms of the Eisenstein series are just Bernoulli numbers. So omega whole is some formal power series, some completely explicit formal power series whose coefficients are Bernoulli numbers. And so we can put the two pieces together and define the regularized iterated integrals. So define i whole just of tall, so you can write, if you prefer, you can write this tall comma infinity. But I just write it as tall. And the definition is you take the limit as epsilon goes to the cusp of i whole tall epsilon times i whole. Infinity epsilon zero. So there's a very nice geometric reason for why this formula should be what it is. I've given lectures about this before and I don't have time here. So let's just take this as a definition. And you can believe me that this converges very nicely as epsilon goes to infinity. Essentially this part sort of cancels out all the divergences in this iterated integral. And it converges extremely fast. So another convenient notation is to write this, to define this to be the iterated integral from tall to the tangent vector at the cusp of omega whole. So we think of this as a formal power series where we've just taken the upper limit of integration before here tall two to be the tangent vector at the cusp. And this is the definition. OK, that's what this notation means now. And from that you can extract very concrete explicit formulae for computing these iterated integrals. So these are the regularised with respect to the tangent vector, the unit tangent vector at the cusp. So this thing now turns out to be the unique solution to a slightly different, essentially the same differential equation. But now it's just a function of one variable. So d of this equals minus omega whole tall, high whole tall. And the constant of integration of initial conditions are fixed by the fact that its value at the tangent vector at the cusp is just one. That uniquely determines this power series of iterated integrals. So this is the same differential equation as before. What this condition means, another way to think of it is that there is some sort of regularised limit as tall goes to the cusp at unit speeds or along this tangent vector. But it tends to, if you regularise it in some way, its limit is just one. So this is very explicit and this convergence is exponentially fast. So it's very convenient to compute with just two or three free coefficients of a model of form. You can get these numbers for any value of tall to hundreds of digits very quickly. So then how does a group SL2Z act? Well for all gamma we have an immediate property of this uniqueness in this differential equation is this equation c whole gamma. So you could take this as the definition of c whole gamma if you're analytically minded. I define it a different way but this is completely equivalent. So here's the c whole gamma is a formal power series in these symbols and that's by how I wrote down the definition of Duram Hall. You can also think of it as a point in a complex point of this group scheme. So this equation you can check immediately. So if you take this equation and apply it with gamma equals gh and apply it with gamma equals g and gamma with h then you immediately deduce the co-cycle equations. So that's a very simple exercise. The co-cycle equations for c whole follow from this formula. That's a definition of c gamma. The first line is a definition. So I define it so I'm a top down approach and this is a formula for c gamma whole. But you could take it as a definition if you like. This is a perfectly good definition. And so this is Z1 gamma u. So these iterative integral satisfy shuffle product relations and these co-cycle relations. So in particular they're determined by the value on t and s. So from this formalism you can show that the value on t is in fact it only sees what's happening in the neighbourhood of the cusp and it only depends therefore on this infinite part of i-hole. So it only depends on this power series here which is obtained by integrating omega-hole. So it only sees the Eisenstein series and it's essentially some completely elementary integral involving Bernoulli numbers. So here you're going to get Bernoulli numbers and here you're going to get products of Bernoulli numbers and then you get products of three Bernoulli numbers and so on. But it's completely elementary and you can compute it explicitly to all orders. So it only involves the residues of the Eisenstein forms and therefore only involves Bernoulli numbers and of course powers of 2 pi i. So we can write this down explicitly and then sort of the meat is s and again this is mysterious. By messing around with this formula you can write it down like this. So you can look at just iterated integrals from i which is in the upper half plane to the cusp. So s i equals i it's a fixed point of the involution s and these are superfast conversions. Ok so now having written down these holomorphic iterated integrals I now want to explain and give some examples. The simplest examples in the case of length one in other words is a single iterated integral and in that case we retrieve the classical theory of periods of modular forms. So I'll dispense with that and then I'll say something about length two examples. So the first example is where we just look at a single iterated integral of a cusp form. So it's just a piece of this so the first non-trivial term in this expression. So we take an element in R, this is our basis of cusp forms with rational free coefficients. It doesn't have to be on our basis but it's just convenient. Way two m plus two. And then we can take the coefficient of, so we have this formal power series c whole gamma. So this is some formal power series in all these non-commuting variables. And we take the coefficient corresponding to E f prime. So the coefficient of E f prime in this power series is c whole gamma E f prime. That's the definition that means that take the coefficient of this single letter in this power series. And that's going to define a co-cycle in v two m. So it's going to be an abelian co-cycle. And the reason for this is if you take my non-abelian co-cycle equation, which I've raised, but it's c whole gh equals c whole g slash h c whole h. And just read off and think about what it means to take the coefficients of a letter of length one. Then what that gives you is exactly the equation c whole gh E f prime equals c whole g E f prime slash h plus c whole h E f prime. So taking off just the linear part, so the terms of length one in this expression essentially abelianises it. And you get this classical abelian co-cycle equation. So this is very standard in the third model of forms. So there's a from the general formula we get c whole on s. So the value on t is zero, it's not very interesting. So I should maybe say c t whole e prime f is zero that follows from this expression here. So it's a cospital co-cycle and value on s is essentially the integral from s inverse the tangent vector infinity to tangent vector infinity of f tau d tau. But in fact because this modular form vanishes at the cosp, we don't need to take a tangential base point at all. We can just ignore that fact and take a classical integral from zero to i infinity and that's going to be perfectly convergent. So up to some normalisation of two pi this is exactly the Eichler integral or Eichler-Schimmer integral. So this is a big subject and it's been studied in great detail. So you can expand this as a polynomial in x and y and you get integrals of f tau times a path tau, which is nothing other than a melin transform of f and the melin transform of a modular form is its l function. So to cut a long story short you can write this in the form some k equals one to the two n minus one of some coefficient which is elementary and I can't be bothered to write down times the value of the l function of f at a point k x to the k minus one y to the two n minus k minus one. So this is, let's call this pf, which is also called the period polynomial of the cosp form f. So up to some even power of two pi, there's some normalisation. So the l function here was defined by Hecker, let me remind you, lfs is defined to be the sum of the Fourier coefficients of f over n to the s and this converges for all real part of s sufficiently big and you can be precise about this. So there's something funny about this, the definition of the l function which always seems very strange and that it's that you don't, this definition also works for Eisenstein series, but you never take the first Fourier coefficient you take, you start with a one and not a zero. And that's always sort of struck me as being slightly strange because the functional equation for this l function comes from the modular behaviour of f with respect to the inversion s. And if you remove the first coefficient, you break that invariance completely. So it's kind of strange that this l function defined in this manner where the sum starts at one and not zero still satisfies the functional equation. But actually if you think about it, that troncating at zero is exactly this tangential base point. So if you work out this formula and I regret that I don't have time to do it. If you work out this formula in length one, what it does is that you take the holomorphic integral and you subtract something and that exactly has the effect of it exactly explains this trick of Hecker's to get the l function in all cases. So that's a very nice little lemmer, it's a nice remark. And I regret that I don't have time to explain that. It's interesting only for the Eisenstein series, but it becomes absolutely crucial in higher length iterated integrals. You really have to regularise in a careful way and the formula is more complicated. But when you do this Hecker business, it's always strange, you've got an integral from, you've got essentially the Mennon transform from zero to infinity and you've always got this weird term of integrating from zero to one of the residue. So essentially you integrate from zero to infinity of f minus its constant Fourier coefficient. And then there's an extra term which is an integral from zero to one of the constant term. And the meaning of that extra term integral from zero to one is an integral in the time, it's the integral along the tangent vector of length one in the tangent space. And that's very satisfying because it gives a very clear geometric meaning to these classical formulae. Anyway, so I digress. This is very well understood. And then let me just spell out the co-cycle equations in case you haven't seen it before. The co-cycle equations for s and t amount to functional equations on pf, which are called the period polynomial relations. So that's this first equation here, corresponds to this. And then a second equation corresponds to a three-term equation. So these are called the period polynomial relations. And an example to have time quickly. So an example if we take f is the Ramanujan cusp form of weight 12, then we can write this polynomial explicitly as a holomorphic period, which is real omega plus 36 over 691. So I hope these coefficients are correct. I didn't copy them down incorrectly. Plus, so it has an even part and it has an odd part. So the odd part is given by the other holomorphic period. So these polynomials are very famous and they sharpen all sorts of places. So they're the smallest non-trivial solutions to these sets of equations. Okay, so then a remark here. So here we're only looking at the holomorphic iterative integrals. That means we're only integrating f. If we want to capture the quasi-periods, as I mentioned in the last lecture, if we want to get the quasi-periods, so again this is something that's not at all been considered classically, then to capture them we need to consider the full co-cycle Cs, not just its totally holomorphic part, just C whole s. So here these iterative integrals are given by holomorphic moduloforms. We're studying this at the moment. But I claim that in the length one case you can do these same calculations and that will involve integrating a weakly holomorphic moduloform with poles at the cusp. So you have to be careful how to do that. But then you can capture the quasi-periods of the motive of f. So there are some numbers that I call eta plus and i eta minus. This is called cycle, we are not too isolated. This is Eichlerchymru, yeah. So this is very classical and it's been studied in great detail by Zaggyr and Manin studying in the 60s I think and there's a huge literature on this by now. So Manin did a lot more. I regret I don't have time to talk about it. So Manin proved something very beautiful, something called the universal coefficients theorem that he showed that there's an action of the Hecker operators on these polynomials. So once you have an action of the Hecker operators you get the eigenvalues and you can regenerate your moduloform. So the period polynomial completely determines the moduloform uniquely and I regret I don't have time to talk more about it but he explained how to get the Hecker operators to act on these polynomials. So it's a very beautiful theory and we don't really know how to generalize this in any reasonable way in higher lengths. So these give some extra relations over and above these co-cycle relations satisfied by these period polynomials. Okay, so now the second example is the case of an Eisenstein series and this is less classical. I mean this is equivalent to stuff that's been known for a long time and in particular I'm sure that Romanujan knew the formula I'm going to write down but what was known was this whole notion of tangential base point. So these formulae were sort of ad hoc but now equipped with the notion of tangential base point we get a completely canonical, we can define a canonical Eisenstein co-cycle and that's by integrating all the way to the tangent vector of the cusp. So in the past you had to truncate somewhere or regularize it with some choice but this is completely canonical and the formulae equivalent to formulae in the literature. So what we get then is the coefficient of E2n plus 2 is an abelian co-cycle which is canonical. Now let me write it down. So first we define a rational co-cycle. So there's a lot of confusion on this in the literature so I think it's good to spend a little bit of time on it. So first let me define a rational co-cycle that I call E0 and there's a slick way to write this down as a generating series but I think it's more instructive just to give the formula. So there's a Bernoulli number B2n over 2n factorial so this looks a bit strange but there's a reason for it. Then you take x plus y 2n minus 1 minus x to the 2n minus 1 over y. So this is a polynomial, the y factors out and this is obtained as an easy integral around the cusp. I gave the formulae earlier, it's an exercise to do that. The value on S, the quick way to get this is through the L function of the Eisenstein series which is a product of Riemann's zeta functions and if you take the value of a product of Riemann's zeta functions what you find is this expression with products of Bernoulli numbers. So again I hope that I didn't miscopy this formula but it's in my paper somewhere. So this expression with product of Bernoulli numbers comes up an awful lot in this theory and the theory of multiple zeta values, it's very interesting. So then the holomorphic Eisenstein co-cycle then of an Eisenstein series is 2 pi i times this rational co-cycle plus a co-boundary term 2n factorial over 2 and it involves a Riemann's zeta function times a co-boundary term. So this is a co-boundary, I gave the formulae earlier in the first lecture I think I'll give it again. So the co-boundary is on a vector so delta naught of v on gamma is v minus v slash gamma. So for when this is evaluated on T, the element T this doesn't appear at all, you just get the value of E naught on T. On evaluated on S you get this formula evaluated on S plus some zeta value times y to the 2n minus x to the 2n. And this is very interesting because it means as a co-homology class the zeta part drops out completely so this transcendental part disappears and you just get a rational, so the co-homology class associated with Eisenstein series is rational and that's in accordance with the Mann in Drinfeld Theorem but as a co-cycle it's not rational, it's got some conjecturally transcendental coefficient and that's going to be extremely important later on. Okay so this is everything in length one and it's completely classical. So that was the length one story and we've completely described it pretty much and it corresponds to the billionisation of relative completion and it's classical. So things start to get interesting in length two and here we don't know so much. I'll just say briefly what some things I do know. The first interesting case is to look at two Eisenstein series. So the advantage of Eisenstein series is that they are totally holomorphic by nature. There's no non-holomorphic, weekly holomorphic counterpart. So we see everything, we can compute everything with these regularised iterated iclinicals. So we want the coefficient of a word in two Eisenstein series in this power series and this defines some gamma co-chain in V2M tensor V2N. So this thing you can in turn break up into SL2 representations, it'll have many copies. So this is quite a rich object, it's got many different bits to it and the very compact way to write the co-cycle equations on this thing are, if you like, the relations between the coefficients of this co-chain can be encoded, I claim, by the following co-cycle equation. So this is a gamma co-chain, it's in C1 V2M tensor V2N and the most slick way of writing down these equations is in terms of a two-co-cycle, C whole E2M E2N plus 2 equals the co-cycle E2M plus 2 the whole is redundant here. C whole E2M plus 2. So this is a cup product. So the nice thing about this formula it's a recursive expression that gives you a formula for all of these coefficients in terms of the co-cycles of Eisenstein series which you've completely written down explicitly. We see straight away we're going to get some odd zeta values, two odd zeta values and a product of two odd zeta values for starters. And then that's going to completely determine this up to an actual co-cycle which will satisfy the classical co-cycle equations. And those numbers, those coefficients, they're going to appear, for example, at weight 12 you'll get the exact these polynomials but times some new coefficients and they can be computed to very high orders and the question is what are they? So let me just summarise what the techniques are. So I regularise the trade index group of two Eisenstein series is a very interesting object. What do we get? Well we get multiple zeta values, get some non-trivial multiple zeta values occurring and that's essentially in the same way here we get a sort of co-boundary term and the coefficient is an MZV and the depth here can be up to four. Then the main technique for computing this is using the rank and sell bug method. So it's slightly involved so I won't say anything about that. But using the rank and sell bug method which a priori relates to something different but you can use it to do part of this calculation what it produces for you is some coefficients proportional to the L function of every cusp form at non-critical values. So for all values K bigger than or equal to the weight of F. So these are non-critical values of all cusp forms. So again it's perhaps slightly surprising because the Eisenstein series so out of Eisenstein series you're getting cusp forms and when we see the hot structure it'll be even more surprising because the Eisenstein series are somehow tate but there are good reasons why it produces periods of motives of modular forms. So then plus we get something else which is kind of curious we get a different another period which I didn't see in the literature but it's a period of an extension of the motive of a cusp form sorry of the trivial motive if you like by a motive of a cusp form so this is a simple extension so in a category of hot structures or something by a pure modular motive and this L function is essentially the regulator so when you have an extension there's a theory of regulators it gives you an invariant and an invariant of the extension class is the regulator and Baylinson's conjecture predicts that it should be a special value of the L function but there's another period such an extension has another period which is not canonically defined it's defined up to some rational multiple and I don't know what to call it maybe at some point I call it Cfk but we get this number as well showing up as a double Eisenstein integral so that's basically it we get multiple zeta values two different types occurring in two places and only two places single zeta values products of single zeta values two pi i's L values of non critical L values and some sort of companion period that goes with this and before stopping I'll just mention some other sort of slightly surprising consequence of the co-cycle equations is something that I call transference and then I'll stop which is that the co-cycle you don't see this well the co-cycle relations in length n plus one imply relations in length n so what I mean by that is imagine we knew what the co-cycle were for length one here and then we want to solve for this in terms of those and the question is is that obstructed or not so having solved a length having done anzats at length n can you then find a solution at length n plus one the answer is yes for co-cycles in general but the answer is no if you fix the value of your co-cycle on t which we know to all orders so if you ask the problem can we determine at length n plus one such that the value of the co-cycle on t is what it should be then you find it is obstructed and that obstruction is precisely the pizza in a product and it gives something very interesting indeed and then you find that the liftability of these equations to the next length actually provides a constraint at the previous order and for length one length one iterated integrals it gives a well-known fact that so-called the Conan-Zagya so-called extra relation satisfied by these period polynomials which is the period polynomial pf here of a cusp form is orthogonal to the period polynomial of an Eisenstein series an extra relation and in length two we get something even more interesting we find that the iterated integral of two Eisenstein series can be related to or a piece of it can be related to a piece of the iterated integral of an Eisenstein series and a cusp form and this in turn a piece of this can be related to a period of an iterated integral of two cusp forms so I call this transference because coefficients from apparently very different parts of this these iterated integrals are getting transferred from one piece an apparently very different piece and as a final comment so I expect I expect that this could generalise I don't know how to do it that I expect that we find the special values of all rank and sell bug L functions non-critical L values I expect so for small values of K you can use the classical rank and sell bug method but I expect them to occur for all values of K as triple iterated integrals and unfortunately I don't know what's missing here is that there isn't a good analytic technique for computing and there should be some higher rank and sell bug method that enables you to prove that some higher iterated integrals give spit out these numbers and that's very important because it's not known in this case I will stop there