 Okay, so this is the fourth and final lecture where I try to say all the things that I wish I'd said in earlier lectures. Unfortunately, I won't talk about mixed modular forms, I didn't get that far and I apologise for that. But I will say a few things that I believe are important and to illustrate what they are and what they mean, I will first motivate with the case of P1 minus 3 points. Even though I don't want to talk about P1 minus 3 points very much at all, I just will write down the key ingredients as I see it in the theory and then I will replicate them or give generalisations of them in the case of M11. So the zeroth section then is motivation for this lecture from P1 minus 3 points. And I'm going to briefly summarise most of the ingredients in the motivic theory and say why they're important. So what we have here is, as I've mentioned this a few times already, we have Betty and Durram fundamental groups. Oh sorry, not fundamental groups, fundamental tosses of paths because we're going from 0 to 1. So this is the unipotent completion or the Durram pi1 of P1 minus 3 points from a tangent vector at 1 to a negative tangent vector of length 1 at 1. And these are schemes over q and they are connected, they are related to each other by a comparison isomorphism. So this is a morphism of schemes. And then there's an element that plays a very important role which is the drachemar in the topological fundamental group which is the path, the straight line path going from 0 to 1 or rather the tangent vector of length 1 at 0 to the tangent vector of length minus 1 at 1 and it's simply the straight line, no my line is not very straight, do that again. So that's the drachemar and it gives an element in Betty fundamental group, it's rational points and then we push it across into the Durram fundamental group and its image in the complex points of the Durram fundamental group is the Drinfeld associator. So I've explained that a number of times already. So there's a sum over some words in two letters and the coefficients are shuffle regularised, multiple z to values. And last time we discussed what the analogue of this should be in genus 1 and discussed the analogue of the relations satisfied which in the case of the Drinfeld associator, the hexagon and pentagon equations. Right, so then this we've more or less covered. So the next stage in the theory is to make things, is to put in the motivic point of view. So the way that this is done these days in this situation though of course in the early days we didn't have a category of mixed-hate motives so you won't have to do something else. But now we say that these schemes are the realisations of something else, a motivic fundamental group. I'll write with a subscript m, motivic fundamental group. So what that means to say that the scheme is a realisation of a motive, what it means is that rather the affine ring, we have an object which we think of as the affine ring of a scheme which is an end object in a category of mixed-hate motives over the integers. And its betty realisation is the affine ring of the betty scheme and its dirham realisation is the affine ring of the dirham scheme and you have other realisations as well. So this was done by Dylian Gontrov, defined by Dylian Gontrov. Okay, so then what do you get from this? Well the next stage in the story is to interpret this. So a motive or an object in an Abelian category or a Tanakhian category of motives is simply a vector space plus the action of a group. So the category of mixed-hate motives, because it is a Tanakhian category, is the same as the representations of a certain affine group scheme that I'm going to write. So the group schemes if I do it correctly and consistently will have curly G's, so that the the motivic galore groups will have curly G's whereas all other groups will not have a tail, so I hope I do that consistently. So a mixed-hate motive is simply a vector space plus the action of a particular group, which is the Doram motivic galore group of this category. There's the Doram motivic galore group of this category and this function is the Doram realisation function. So what we're getting then is some schemes and the extra data of a group, MTZ, acting on one of them, on the Doram one. So we get some object with a group acting on it and this encodes all the motivic theory. Of course you can also place Doram with Betty if you like, but there's no loss of information and just restricting to a single fibre functor and Doram is by far and away the most convenient in this story. So the point of this is that the structure of OPI1M as a motive is completely equivalent by the Tanakathiam to the action of this group on the scheme OPI1 Doram and this in turn is completely equivalent to the action of this group on, I'd like to say, multiple zeta values, but to make this rigorous I have to put little motivic in brackets. So the first point is that this group action really contains all the information, it really knows everything that there is to know about PI1-3, P1-3 point, or the fundamental group of P1-3 points. No, so I skipped this because there's always a map, PI1 top, I won't repeat the rest, goes into the Betty points, into the rational points of the Betty and it's the risky dense and the same happens in relative completion. So I skip that step when I am. So the way I prefer to think about this is that you can imagine that there's a Galois theory of transcendental numbers like multiple zeta values and the action of this Galois group on these numbers should be, so that Galois action is clearly conjectural but you can make it absolutely rigorous by replacing numbers with something called motivic periods and then this action of this group is completely equivalent to an action on the motivic versions of these numbers. So that's something that's quite concrete and is used a lot and the point is that it's all completely encoded in the date of this action and without wishing to give an entire course on this because I've done it before you can deduce a lot of fun things. For example to prove results here between multiple zeta values but between periods you just compute numbers. So here you can prove them using complex analysis for example and because of this equivalence you can push them back to statements into the actual motive and you can result deduce results about the eladic structure of p1-3 points, it's a fundamental group and you can deduce results about piatic periods as well. So that illustrates the power of this point of view. Okay so all that to say that this technology of having motivic multiple zetas and hopf algebra or co-algebra structure on them is used all the time but it really comes from this group action. So the key point then is to determine this group action and for me it's one of the most important points in the whole theory. So we need to know how this motivic Galois group acts on 0 pi1-deram. In other words we get a map from this group into the group of automorphisms 0 pi1-deram that's what it means for this group to act on on this scheme and we know that this Galois group satisfies certain constraints. It's constrained in some way so it lands in some subgroup of the group of automorphisms that I will just denote with a prime for now since I don't want to go into the whole theory. And the key point here is that this subgroup can be described quite explicitly and it turns out actually to be isomorphic as a scheme but not as a scheme because there is no group structure here. It turns out to be isomorphic to the fundamental group itself and what you get from that then something slightly strange you get an action of this pi1 on itself which exactly reflects the action of the galois group. So this is in some sense this is what's confusing about the theory because in p1-3 points the role of this automorphism group gets confused with the role of the torso of paths and that causes a lot of confusion. In the case of m1-1 we'll see that it's very slightly different. Okay so this was first done by Ihara and it's as I mentioned it's extremely important so we can describe this action the action of this this group on a pi1-deram explicitly. So there's an explicit formula perhaps I'll just give it I didn't prepare this so my conventions may be the wrong way around but essentially if you represent this by group like formal power series in two variables then you get an operation on formal power series in two variables which is something like multiplied by f on the left and then you do some non commutative substitution like this. So f and g are functions and our group like formal power series in summary coefficients in summary. So you get a very concrete formula this was discovered by Ihara and then from this it's a very short argument to dualise this this is explained in my lecture at the ICM proceedings. You dualise this and you get a a co-action formula for multiplicative values essentially and we use that all the time. So really that the heart is understanding this group action so this implies formula for co-action on material chemis interviews and you can really use this to compute it's actually absolutely extraordinary that this whole philosophy gives anything at all I feel like saying that the more you understand this the more surprising it becomes that these very general considerations actually give you any information at all but in fact you get an enormous amount of information from this co-action in fact it completely determines all the relations between multiples. So the next stage then is some input from the material theory more precisely Borel's theorems on algebra k theory tell us something about the size of this material Galois group and in fact we know that the the Lie algebra of the unipotent radical of this material Galois group or it's associated graded is isomorphic to the freely algebra on generators sigma 3, sigma 5, sigma 7 and so on where sigma 2n plus 1 is in degree minus 2n minus 1 and these are the very famous zeta elements something's called sulye elements graded you can just put if you don't like it you can put the completion of the free Lie algebra on these elements and so these sigmas correspond in some sense to the odd zeta values which in turn control the structure of the whole structure of the ring of multiple zeta values through this mechanism so this controls this Lie algebra controls the structure of motivic mz v's and hence all mz v's so you get from this I don't want to do the whole course on this but you you get upper bounds from for the the the dimensions of the space of numbers in this way it's very concrete and so the final piece I want to get to is the theorem that I proved a few years ago which was called previously called the Delaney-Hara conjecture which is that this Lie algebra nor was the the sigma 3 sigma 5 act freely so let me write a freely algebra as a blackboard L so then this freely algebra this Lie algebra acts freely on on zero pi 1 the RAM and that implies that in fact that this the pi 1 of p1 minus 3 points in fact generates the whole category of mixtape motives over the integers so now what I want to I really don't want to talk about this but I will repeat all of this for genus 1 and explain how to get a group so we don't have a category of motives in this case but I will explain how to cook up something some category of realizations that will do the job we're going to get a group acting on everything I want to then describe the hot structure underlying the the relative completion then I want to describe the automorphisms of this structure and how the Gallo group acts and then I want to explain what the analogs of these sigmas are going to be and I will conclude with a freeness theorem which is an analog of this result in the case of genus 1 sorry what I'm going to say later this is a theorem oh yeah I'll state a theorem of I'll state a freeness theorem generalizes this that involves zeta elements but also modular elements but the caveat is that that's not the whole story that there is an infinite sequence of infinite sequences of generators they're not just modular and zeta elements they're ranking Selberg elements and etc etc etc so that's the issue but in but it's a story this is a tricky very tricky combinatorial and analytic argument it uses some some difficult identity due to zaggy between multiple zetas proved by fragment Lindelof principle it uses a tricky combinatorial argument in in the genus 1 situation this theorem is going to pop out without without any effort it's just going to pop out of the structure from the description of this group the analog of this group so we'll see that okay so now the case and one one so this is what we're really interested in and the first point is then as a substitute for motives we're going to work with hot structures so the Betty and Duran completions G11B and DR have not a not just a mixed hot structure but a limiting mixed hot structure so this is a new feature that we don't see in genus 0 and it's very important indeed as I will try to explain at the very very end so this was computed by Hain defined by Hain and it's very slightly different context but it's equivalent okay so what does a limiting mixed hot structure have well it has a geometric weight filtration w but it also has another weight filtration m and the it has a hodge filtration so this is called the monodromy or I might just call it the weight filtration without any adjective and f is the hodge filtration right so the the the weight filtration as a motive if you like is m it's not w and we think of this as as a mixed hot structure with an extra with an extra filtration w so how this is going to be encoded so I want to encode this data in the following way that we're going to have some rings og1 betti affine ring og1 1 diram and there's a comparison isomorphism between them and this is going to be encoded as a w filtered end object of a category H of mixed hot structures which I'm going to define now so what's going on here is that we've got some some local systems or variations of variations over m1 1 and on those there's a weight filtration w and so you can you can you can look at the the w n part of that you can stop the filtration at a certain point and you get a variation and when you take the limit you get a genuine mixed hot structure so so the upshot is we get w for each step in the w filtration we get an actual mixed hot structure and there's a lot of extra data that goes into limiting mixed hot structure that I'm going to ignore for now but it will come back come back very shortly so the first thing then is to define a category H so that H be category whose objects are triples consisting of so I think a version of this category was first written down by Deline so vb and vdr are finite dimensional q vector spaces with an increasing filtration m so my habit is to write w in this context like everybody else but I have to we have to remind ourselves that it's m now weight filtration is denoted by m so if I accidentally write w please stop me so vdr also has a decreasing filtration f so these are filtrations of q vector spaces and they are finite and exhaustive then C is an isomorphism from the could between the complexifications of these vector spaces which respects the weight filtration m there's the data of a real Frobenius which is very useful especially for constructing modular forms I mentioned in the first lecture and which I won't have time to do since actually I won't need it I'm just going to drop it and put it in brackets and skip that and then the key condition is that the vector space vb equipped with the filtration m and equipped with the filtration f on its complexification is a graded polarizable q mixed hot structure whose definition I won't give it's very well known and it's just some linear linear algebra of conditions on the filtration and then the morphisms in this category are what you think they are so morphism is given between a triple is is what you think so it's like it's given by linear maps phi b phi b prime phi diram that respects everything so there's a commutative diagram involving C and C prime that needs to commute there's a these maps need to respect the M filtrations the effiltration in this case and so on and so forth so I'll just say that this has to be compatible with the above data right so then this forms a an abelian category there's an obvious notion of direct some there's no notion of dual notion of tensor product so it's a q linear abelian tensor category with duals in other words it's in fact a tenacian category so and it comes equipped with two fiber functors omega Betty or diram which is a functor from this category to vector spaces finite dimensional vector spaces over q and it sends a triple to the corresponding either the Betty vector space or the diram vector space so that's a fiber functor that's a neutral tenacian category over q and so from this we get a group of course so we let G so this is what's going to play the role of a motivic galore groups it's going to get a curly G B or dir there be two such groups is defined to be the automorphisms of the corresponding fiber functor in this category so this is an affine group scheme of a q and it plays the role of the motivic galore group and in fact it's in the case of mixtape motives it's no loss to work in a category realizations the corresponding group acts in an identical way to the motivic galore groups it's literally the same thing okay so okay so now I want to state a theorem about I want to put relative completion view relative completion as an object in this category somehow so let me remind you briefly that G 11 the Betty and diram relative completions that we defined are group schemes of a q affine group schemes so what that means is that the ring of functions in either case are commutative hopf algebras that's what it means for this to be an affine group scheme so in particular there's a lot of data that comes into this but there's a co-product right so we have a co-product and other stuff and the comparison isomorphism then is an isomorphism of hopf algebras right so so now what we want to do then is view the affine bettering the affine deram ring and this comparison is a triple and it's going to be a triple in this category I need I need the the border razor no there is none that's strange yeah okay so the theorem then which is most of the work is contained in the work of decaying this is a corollary of Haynes work that the affine ring of the Betty relative completion has natural filtrations w and m deram has natural filtrations w and f such that such that og on Betty is I'll say it this way then explain a little bit what it means so this is a w filtered hopf algebra object in H rather end object so what that means slightly more concretely so this is how we encode this this geometric weight filtration it's just saying that for every n if we we only consider the weight n part the Wn part of these rings then then this is this is an object in fact an end object it may be infinite dimensional maybe sort of a limit of objects in age but if we if we take any m and filter piece of it it will be finite dimensional and and it is compatible and this structure is compatible with the data that goes into hopf algebra in other words this co these co-products are consistent with all these filtrations and all these structures so compatible with the hopf algebra structure so there's more to come for this theorem a little bit more but I'm just going to postpone it we can be a bit more precise about this let me stop with pause the theorem for now so that's already a very quite a tight constraint on on the structure of this thing but something that seems completely trivial but again is also extremely important is the local monodromy at the cusp so I sometimes call this inertia for reasons that will become clear so the local monodromy at the cusp defines a map from the topological fundamental group of gm the tangent vector 1 at 0 into topological fundamental group of m 1 1 d by dq which is just SL 2 z so I've drawn this picture already this is the this is the the chart given by the the punctured disc so if we draw the punctured disc d star and remove the origin then we have the tangent vector of length 1 which is just the same thing as d by dq and here we have a loop going around the origin in a positive direction and that gives us in m 1 1 it gives us a a a loop around the cusp and it corresponds precisely to the matrix t which we studied last time so essentially we have a copy of the motivic fundamental group of gm which is a very simple thing sitting inside relative completion vessel to z and so this this thing is is geometric if you like and therefore this morphism this homomorphism of fundamental groups actually gives morphisms of the on the level of completions another way to say that is in fact by universal properties of completion relative completion you deduce that the same is true on the level of b and dr we get a map into g 1 1 b slash dr so these are morphisms of group schemes so how was this encoded and we want to say that this a morphism like this is a morphism compatible with with hodge theory and the way to say that is then that somehow this is a morphism in the category h so I will use that sort of language but what what such a statement means is that on on the affine rings you get a genuine map between objects of age so to spell that out this morphism of group schemes translates into a homomorphism of hopf algebras in the opposite direction and hence a map of triples so what this means to be a morphism of group schemes in the category h the definition is that on the affine rings we're getting a morphism of hopf algebras objects in h all right so that's exactly what it means so this encodes so in the theory of limiting mixed-hodge structure there's a very important role played by the the nilpaton nilpaton operator and this is how it comes into this here how it comes into the theory so it's encoded by the data of a map of the metallic fundamental group of GM into our group scheme so to to make that a little bit more concrete and make the connection this object is very simple in fact it's Lie algebra so I think of this group as a group in H then I can take its it's it's pro-unipotent I can take its Lie algebra and its Lie algebra is the Tate object I mean that the it's pro-unipotent so it's completely determined by its Lie algebra what I mean so that and the Lie algebra is just the Tate object Q of 1 where Q of 1 is Q of 1 is an object of H and it's just the object given by the pair of vector spaces Q and Q and the isomorphism between them sends 1 to 2 pi i inverse so that's the the Tate object and so if you're familiar with the theory of limiting mixed-hodge structures what we've got then is just a map on the level of Lie algebra is we've got a map from Q of 1 into here and the image of a generator here therefore gives an an endomorphism on this so this encodes the nilperton operator n which is also the logarithm just the logarithm of this path log t so t viewed as as an element in in bet you'll drum in the theory of limiting mixed-hodge structures and this also explains why when last week we computed the periods of of relative completion along t's in other words we computed iterated integrals of modular forms along t and I explained that they were only involved 2 pi i powers of 2 pi i and that's clear from this picture because they pull back to periods of gm and the only periods of gm are periods of Q of 1 and the period of Q 1 is is essentially a 2 pi i so this remark explained makes it obvious why the periods of t only involve 2 pi i or the powers of 2 pi i okay so in some sense this t thing is trivial it comes from something that's geometrically very trivial but the point is that it sits inside relative completion in quite a complicated way and as I explained last time that's reflected by the fact that Eisenstein series have a have a non trivial zero through a coefficient and that involves Bernoulli numbers so you can you can actually I'm not going to have time to do it but you can write out what n looks like the image of n in this in Betio and Durham and you get a power series involving Eisenstein generators Bernoulli numbers and you also get the pizza in a product between cast forms so that it's actually quite a tricky object so now let me reformulate this this local monodomi in a different way again another way to say it then is that we've got a map of Durham fundamental groups so we've got a homomorphism or morphism of group schemes and since these are Durham realizations of pro-objects in H they get an action of this Galois group G Durham H is going to act on both of them in a compatible way so compatibly with this morphism so that seems very trivial but it's actually gives a huge amount of information on the this this action of this material galo group so before proceeding with the description of this this action I want to write down remind you of the structure of relative completion and explain its hodge explain what these three filtrations look like it's slightly tricky so here's a description of the hodge structure or really rather just the filtrations on the Durham relative completion so it's much simpler to write things in terms of the Lie algebra so we'll call that U11 Durham is the radical pro-unipotent radical of G11 Durham and let lowercase U11 Durham be its Lie algebra so this has a mixed hodge structure as well and I'm going to describe it so as I mentioned a few times already this is isomorphic to the completion of a freely algebra on certain generators and they were given by Eisenstein series and for each cusp form there were a pair of generators EF prime and EF double prime so we're here here F cusp form so last time we chose a basis of cusp forms with rational coefficients rational Fourier coefficients okay and so these generators are non canonical now so briefly we have so these X's and Y's were elements were basis of a vector space so everything basically everything can be promoted to the category H so since the beginning of these lectures we've had a Betty thing going on a dram thing going on and the bottom line is that everything can just take place in H so the in particular this vector space has been playing a role VN we can now view it as an object in H and I remind you that VN Durham was this vector space of these these bold generators X and Y but now we can put a mix what we gain now is a mix hot structure so these X and Y aren't just variables now they're going to have M and W and F filtrations so VN H was defined to be way back in the first lecture the symmetric and symmetric power of V1 so I just need to describe the Hodge theory of V1 and in fact V1 of H is as an object in H it's simply a direct sum of two Tate objects and why is that the case so this is a well-known fact that the if you take the limit mix hot structure on the cohomology of the universal elliptic curve so this was the universal elliptic curve at the fiber at the point at the tangential base point d by dq this has a limiting mix hot structure and it's exactly this so I think I want I want homology so what is it plus plus one so I want homology so put another way that tells us that X the meaning of the variable X then is that it's a copy of Q of 0 so and the meaning of Y is that it's going to span a copy of Q of 1 so in terms of the M and F filtrations the Hodge numbers with respect to M and F are here 0 comma 0 I shouldn't put that so the Hodge numbers with respect to M and F and here minus 1 minus 1 and they're both going to sit we're going to view these since we're working with W filtered objects in H we're going to put stick these in W 0 and then the yes yes X and Y form a basis of V1 and the powers of X form a basis of its symmetric power so exactly X X is a generator here and Y X is a generator of the DRAM component of this vector space and Y is a generator of the DRAM component of this vector space so it's just saying that that these X and Ys carry weights essentially it's not a big deal but now the truck crux of the matter then is that the Eisenstein generators also have a mixed Hodge structure so they correspond to Q of 1 so that's the M and F Hodge numbers they are again minus 1 and minus 1 but so this photos from the work of Steinbrink and Zucker on limiting mixed Hodge structures of curves this is going to sit in weight in geometric weight minus 2 and minus 2 and then the cusp forms and that's really because it has a corresponding differential form has a pole at the cusp pushes the W weight up or down in this case and then EF prime EF double prime is going to be a copy of VF 1 where VF was the Hodge structure of a cusp form so I defined in an early lecture the motive of I mentioned the motive of cusp forms they have a Hodge structure and they will both take twisted by 1 so the Hodge numbers of the motive of a cusp form twisted by 1 are 2n minus 1 and minus 1 2n and these are going to sit in W equals minus 1 so so this is pretty tricky and I have to admit that we don't really know how to extract all the information from these filtrations at present you go to sort of three-dimensional picture with these three filtrations it's quite hard to visualize it gives a lot of constraints certainly the M and W are going to play a very important role and give a lot of constraints but I have the feeling that there's more to be extracted from this so what I'm going to do now is maybe we have a brief break and I will draw a picture if I can of this Lie algebra with its Hodge structure which is a moment I've been dreading because it's quite hard to get it right on the board so I can do that once you have a coffee and then you'll when you come back you'll see a beautiful picture will make all these filtrations abundantly clear okay so this is a this is a drawing of of the Lie algebra of G or more precisely of SL2 semi-direct U11 Duran so SL2 is is up in weight zero it's generated by these two differential operators xd by dy and yd by dx and their commutator is H which is the degree in y minus the degree in x or the other way around so then we have so that's SL2 and the rest are the generators of U11 Duran so we ignore the Hodge filtration F for now and we just look at the the M weight filtration and the W going down the blackboard so this is the geometric weight filtration and M is the monodromy weight filtration now the first oh yeah the first thing to say is that negative numbers go to the right that's that's a bad habit that we've got got into but it it's stuck it's convenient to do it this way it's harder to draw the other way so negative numbers go down here but they go to the right and positive numbers the left and the first surprising thing which is that all the cusp forms are floating very high at the top so the cusp forms are all sitting in W equals minus one so for every cuspital generator you have an EF prime and an EF double prime so EF here stands for both copies it means EF prime and EF double prime and we get some elements sitting way up at the top in the W filtration then the Eisenstein generators go way down in the W filtration then the other thing to say is that the this this SL2 acts in the obvious way on on these blocks so xd by dy moves you left two blocks yd by dx moves you to the right by two blocks and indeed this generates a standard representation of SL2 this generates a representation of SL2 of dimension 3 and so on and so forth so here on the left we have highest weight vectors and the extremity we have lowest weight vectors which are annihilated by by by this operator here yd by dx so yeah so what I've drawn what I explained so you out of the room so what I've drawn is the semi-direct product of SL2 on u11 Durand so it's a Lie algebra u1 it's a Lie algebra so that freely algebra in the category of mix-hot structures with an action of SL2 so the SL2 I'll say to get SL2 is up here at the top in orange it's xd by dy and yd by dx and the commutator is an H which is the degree in x minus degree in y and the h invariance is this red line here and the red line sits so the red line is what should I put it so the red line so the red line is m equals w and it contains all the SL2 invariance that's going to be important so already we don't we don't have any SL2 invariance in in the generators you have to take Lie brackets of at least two of these things to get an SL2 invariant piece so the the key point is that the Eisenstein series go down in the W filtration very fast and they're all lined up to begin at the m equals minus two column next end to the right but all the cusp forms are very much up at the top and that's some very important fact that that that is just true and it's useful but we I don't feel we've really fully exploited this very particular structure I think there's a lot more information that can be obtained out of this so this is m and w then I've ignored the the hodge filtration f which I've written a little table there for convenience so if you want to think of the hodge filtration f you can imagine another filtration coming out of the blackboard in three dimensions and things are sitting these things are sitting at strange diagonals coming in and out of the blackboard if you have sort of a three-dimensional picture with f as well and I just don't know how to draw that on a piece of paper but if someone has an idea that would be useful okay so this is what the the algebra of Durham the Durham rate of completion looks like in all its glory or rather than most of its glory since we've ignored the hodge filtration for this picture just just as a remark by comparison you know if we were to draw the same picture for p1 minus three points the Durham fundamental group is just it's just the freely algebra on two generators e0 and e1 so the corresponding picture for for this we'll only have one filtration we only have m and we just have e0 and e1 sitting in one slot so this is the analogous picture for p1 minus three points is that it's not very interesting but we see this incredible richness in on m11 on SL2z so what we've got then is this this object that just exists it's a motive what's a drama realization of a motive and the Galois group is going to act on this this whole thing and clearly it's very rich indeed so I now want to try to describe if I've got to say so of course these are just the generators and then you have Lee brackets you have commutators between these elements so the Lee bracket of e4x squared and e6x to the four will be somewhere in w minus 10 and it'll be over here in in this column so for example you have e4x squared Lee bracket bracket and so on and so forth so when you take Lee brackets of Eisenstein elements they're going to move to the right so you say that the action of SL2z on v1 is a natural one no so to get SL2 to act I need to choose a splitting so we had yeah so it's like slightly cheated here or I hadn't until I wrote down a commutator but we have G Durand 11 and it sits in an exact sequence SL2 which I put a Durand for bookkeeping purposes and it has a unipotent radical but then what we can do is then to write things to compute it's always useful to split this so we choose a splitting and I mentioned this last time but the fact what's new here so SL2 acts on the right and you have SL2 what's new now is that we have hod structures so the fact of the matter is that you can always choose a splitting compatibly with all the MW and F filtrations so you can split compatibly with WM and F and then what that once you've chosen a splitting that's the same thing as choosing an action of SL2 on G or on on U and on the level of the Lie algebra you get this guy and that's what I've drawn no but the SL2 Z sorry no there's no action of SL2 Z per se no it's really an action of SL2 the Lie algebra yeah so SL2 Z will appear but in a different and a slightly different way so that's correct so that's really the Betty so SL2 Z is really the Betty side because SL2 Z is the fundamental group and you always get a map from the fundamental group into the Betty into the Q rational points of the Betty relative conclusion but this is I've drawn Durham here so it is not the best way to think about it in the Durham picture so the way that SL2 acts so this is what I explained last time so how does SL2 Z act here when in fact you're right but maybe I'll recap so I did this last time SL2 Z acts here via these co-cycles so we had SL2 Z going to G Betty 1 1 Q and then by the comparison that gave us something in G 11 Durham C isomorphic to SL2 Durham semi-direct Q 11 C and so every element every matrix gives us gamma I called this gamma barks there's some irritating 2 pi eyes but in this basis you won't see them and then a co-cycle C gamma so this co-cycle somehow it if you think about the path gamma ends up being spread throughout the whole Durham picture so the path S which we had last time is going to do what do you think it does up here on the SL2 part and then it's going to have some the co-cycle CS is going to spread it out with all these periods all through the Durham that's that's gamma goes to gamma bar so I explained I explained that last time it's just essentially the same matrix but with 2 pi eyes in there and and then I switched I've switched bases from these Durham and Betty bases and that eats up the 2 pi eyes exactly exactly it's just bookkeeping so that's sometimes so here I'm writing the bold X's and last time when I did the co-cycle I wrote it using the the other X and Y but these these are essentially the same and these differ by multiplying or dividing by 2 pi eyes it's just exactly that's the origin of the 2 pi eye which is very irritating but it's it's very important right so now let me describe the Galois action then so Galon inverted comm is it's not a classical Galois group in any sense so G1 run Durham is Durham component of a pro-object in H therefore it has an action of the automorphisms of the category H so this action knows everything this is the be this is the the holy grail if we can understand this action we would know everything there is to know about mixed model emotives so this action I'll just write that quickly knows everything I completely determines the structure of G1 1 as an object of H and hence the category which I mentioned a few lectures back so define a category of mixed modular motives MMM gamma I'll mind you the definition here it's just the full subcategory so full subcategory of H generated by just full ten a full ten I can subcategory of H generated by the affron ring og 1 1 that means it's all subobjects and quotient objects duels tensor products thereof etc so that's a cat that's what I define to be the category of mixed modular motives and understanding this category is equivalent to understanding this action of course we can enhance this category H I'm just working with Betty and Durham because it's the way the quickest way to get information about this action you can throw in other realizations if you like so then the question is how on earth could we possibly compute this action and it seems hopeless but in fact surprisingly you can get very far so I'm going to restrict the action not of the full Gallo group but just look at the unipodent radical and I feel like saying that that the the semi-simple part the action of the semi-simple part we sort of know that there's not much to say it's really encoded by the action of heck operators and it's it's understanding the pure objects in this category and they're the motives of of modular forms which we know so all the interesting stuff is in the unipodent radical and its action on so it the unipodent radical acts on everything in sight and it's going to determine all the extensions in this category and that's what we really want to know right okay so how do what do we know about this action well we know that it respects it has to respect the local monodromy so we know a few other things as well that I haven't had time to discuss but most of the content is in this little picture I'm going to draw so we have the local monodromy so this is the fundamental group of GM just given by single loop and we had this map local monodromy into G11 and I'm going to call this local monodromy Kappa for want of a better better name and then G11 is a an extension of SL2 by its prunipodent radical and how what what does this local monodromy look like well it takes a little loop in the Q disk which I called gamma zero and it sends it as I explained to T here but it also has a component up here which is which is interesting and complicated so this this morphism from G1 to SL2 is a morphism in H and this is a subgroup in the category H so all this everything here every morphism in this diagram is compatible with the hodge theory and come from morphisms in the category H so that means the metallic Gallo group Gallo Group of H respects all the maps in this diagram and in fact we can be much more precise that in fact SL2 which is just the it's our fine ring is is given by endomorphisms of this vector space V all right so it's it's just a tate motive it's very simple and as I explained earlier this fundamental group of the punctured disk they are both tate well first of all they're pure objects and even simpler they're pure tate objects in H so they're incredibly simple and the action of this metallic Gallo Group on them factors through a very very small quotient and in particular since for simplicity I'm only going to look at the unipodent radical the unipodent radical of this category acts trivially by definition on all pure objects so it's going to act trivially on both SL2 and this pi 1 rather their Duran components because it acts trivially on all pure objects in H by definite that's the definition of you it's the subgroup which which acts trivially on on pure objects or direct sums of pure objects okay so now definition if I have an exact sequence of affine group schemes over K where K has characteristic zero and S is reductive or pro reductive and U is pro unipotent so this is the situation we've got with G11 over there and what we've got then is this automorph this group of symmetries acts trivially on this acts trivially on this and therefore in particular it preserves it preserves this this subgroup so we want to understand when you have an affine group scheme acting on a short exact sequence of affine group schemes what does it look like so if we take any short exact sequence of affine group schemes of pro unipotent kernel and pro reductive quotient and in fact this is the general picture for any such group scheme then we can define the automorphisms which respect Pi of G and they are the automorphisms of the group scheme G so these are group group homomorphisms such that Pi alpha equals Pi now when I write this what I mean is is for every ring I'm looking at the we can take that the ring R we can take the R points of this group scheme and we're looking at automorphisms on the level of points so what this is is a functor but I write it without reference to the the ring of points you're taking in so it's a functor from commutative k-algebras to groups and one has to be careful here it's not the case in general if you take a group scheme it's automorphisms is not a group scheme but there are conditions under which it's true in which it is a representable functor in this case it's going to be representable that's not a problem but I don't want to go into that so I'll just say that this is ought for me as a functor from commutative k-algebras so given a commutative k-algebra R the automorphisms of R are the isomorphisms of the R points of G to itself which commute with this projection it is in I have to think yeah I think I think it is in this case yeah certainly in the in the application it's definitely representable so somewhere I wrote down some conditions so if you take you can prove that so if you have a pro-unipotent group scheme it's automorphisms are representable and there's a general question if you have some filtrations on the group schemes then there's there's a condition for representability that I wrote in some paper and that's on the archive and in in this case I think it's fine I didn't actually check but for the case of SL2Z the case we're going to apply this to it's definitely it's definitely representable I'm just saying that one has to be a little bit careful and when you think of automorphisms this is automorphisms on the level of points you could also ask well anyway I don't want to go into it there's there's some subtle questions related to this which would just don't arise here so then the theorem is that for every splitting sigma of this I make s act on the right I think for every splitting in this sort of exact sequence there is a canonical isomorphism of this automorphism group with you semi-direct s invariance of you or to you s invariance so what this means is that the automorphisms of you s invariance is the isomorphism the automorphisms of you to itself such that they commute with with phi so the s s-equivariant so that the s s-equivariant and so what what are elements in this thing so first of all I'll write down elements in you semi-direct or us and I'll explain what this superscript means so what they are on the level of points they're given by pairs b, phi where b is in you and phi is an equivariant automorphism and this is an equivalence relation so we say that a pair b, phi is equivalent to b a a inverse phi a so this is conjugation by an element a for any a s invariant and interview so this is pairs b, phi they form a group with the obvious semi-direct product the usual semi-direct product law and modular equivalence relation which is multiplying the right by an element of an s invariant element of you and conjugate by an s invariant element of you here and we denote the equivalence class by square brackets b, phi so what we get then because the action of this group here preserves respects pure objects and this is pure and this is pure it's going to act on g in such a way that it commutes with this projection pi and it's going to fix the image of kappa so to say that in equations all we get is a map from the Durand Galois group to a certain automorphism group a Durand which is a subgroup of the pi respecting automorphisms of g11 and it's the subgroup of autos automorphisms which do which first of all they've got to respect the the hodge structure and more precisely the weight filtrations w and m they don't have to respect the hodge filtration f but they have to respect w and m and they have to preserve so what I mean is that they they preserve kappa and they respect so they respect w and m what I meant to say is and they preserve kappa they leave it in variant so in my paper there's actually there's an extra condition that we know that these automorphism satisfy that I'm not talking about so in my paper a actually means something else plus an extra condition and last there's not time to explain that so then we can make this concrete then so if we pick a splitting g11 Durand SL2 Durand u11 Durand then we can write down this group in quite a concrete way using using the theorem over there so let so here we have t and kappa plus b image of a generator gamma of pi 1 Durand and this kappa plus is really another word kappa plus really equals the co-cycle ct which we partly computed last time just a slightly different notation the context is very slightly different but you can think of this literally as as this this power series ct which we computed and it's just involves some complicated expressions in Bernoulli numbers and and well we computed part of it I mean it also involves it also incodes pizza and inner products so concretely then what a Durand then via the previous them can be written as being the W and M preserving elements in u11 Durand semi-direct u11 Durand SL2 ORT u11 Durand SL2 i.e. in other words it's the elements every mativic automorphism can be represented by an equivalence class b comma phi where b is here and phi is here such that it preserves kappa in other words it satisfies equation b slash t phi kappa plus b inverse equals kappa plus and that's the condition this is the condition that the Galois group preserves the image of of this under the morphism kappa and so this defines a group a subgroup of automorphisms of relative completion and it really is the analog it's a genus one analog of the gottendic type in a group okay so to explain this definition a bit more carefully let me explain how it acts on co-cycles and it should should enlighten the discussion so the point is that this is actually a massive constraint on what these mativic element these mativic automorphisms can possibly look like and this this condition that looks innocuous is in fact extremely restrictive so as a remark let's explain how this automorphism group acts on co-cycles so we get an action of this automorphism group and in particular these equivalence classes on co-cycles non-Abelian co-cycles Z gamma 1 u1 1 Durand so I explained last time how a splitting gives you a splitting of G as a semi-direct product gives us co-cycles and therefore the automorphism group is going to act on the space of co-cycles so how does it if we start off with a co-cycle C non-Abelian co-cycle then B5 is going to change C and give us a new co-cycle so we take a non-Abelian co-cycle and we transform it to get a new one and the new co-cycle so if the old co-cycle is the map G goes to CG then the new co-cycle is G goes to B slash G Phi CG B inverse so this is C prime G so this is how this group of automorphism acts on co-cycles and it's a very easy but quite nice exercise to check that this operation indeed preserves the co-cycle equations so one way to think about this which I think is the most enlightening is that the space of non-Abelian co-cycles is you can think of this as a total space over a base which is the non-Abelian co-homology classes equivalence classes of co-cycles and and the way I think of this action or one way to think of this action is useful is that the element Phi is giving us an automorphism of the co-homology class and the element B is twisting the representative of the co-cycle within that co-homology class so that explains this dual nature of this automorphism that Phi changes the point in the base and then B selects a point in the fiber so we see immediately that this group respects so we think of this group as a group action on multiple modular values and it's clear from this that it respects the relations between multiple modular values at least those that come from the co-cycle equations so perhaps since I have a tiny bit of time let me just sneak in how this acts on on an example of a co-cycle because I think it's very enlightening so we had last time I we computed the co-cycle of an Eisenstein series and I hope I got my normalizations the same as last time if not I apologize so what it looked like so this co-cycle was obtained by integrating an Eisenstein series from zero to infinity and it was it was 2 pi i times some rational co-cycle which I defined explicitly plus some some some constant times an odd zeta value over 2 pi i to the 2n times a co-boundary and I explained last time that this was this is in some sense rash this is rational so this is 2 pi i rational and this is a co-boundary and it's transcendental and I also explained that that the cohomology class of this co-cycle is just this piece this is zero and cohomology and that's consistent with the Mann and Drinfeld theorem this should be a rational cohomology class and the way it said that this this comes from a tate motive so its period should just be 2 pi i rational but this it has a non-trivial transcendental part which is a co-boundary so if we unravel this formula the material gallery group is going to act on this on these co-cycles and what it does is it it it it scales all the well it does something to all the generators e and f and in this case it's going to do nothing because to lowest order phi is always the identity so we're not going to see this at all in this example and then it it multiplies by by it it modifies by co-boundary so if you take the coefficient of e2n plus 2 in this what you find is that the co-cycle stays put it's the same thing that comes in here then you get plus b slash g minus b which is the addition of a co-boundary so under so what b phi does to this it modifies this co-cycle by a co-boundary so what it does it adds i it takes this co-cycle and when you compute c prime it's going to be the same thing plus so c prime equals c plus some constant times 2 pi i boundary of y to the 2n okay and so duly you can interpret that as an action of the motivic gallery group on the number on the odd zeta value and so that's equivalent to saying that the motivic gallery group transforms the odd zeta value and modifies its value by adding some multiple of 2 pi i to the 2n plus 1 and as we know or as you might know from my other lectures that this is indeed how the betty motivic gallery group acts on motivic multiple zeta values so this is so in some sense that the fact that the co-cycle of an Einstein series has this this transcendental term in it is really fundamental because it reflects the first non-trivial piece of the action of the motivic gallery group right so now let me retranslate these so these are all groups of automorphism acting on groups it's simpler to think in terms of Liege algebras Liege algebra reformulation so now let me take g 11 to be Liege 11 so I'm going to drop the dirams because it gets a bit tedious oh no okay I'll be I'll keep the dirams and why not u 11 is the Liege algebra of the unipotent radical of the relative completion and u h is the Liege algebra of this unipotent radical of this this gallery group then if we translate all the above stuff into Liege algebras then we get an action of this Liege algebra u drum h on the Liege algebra of relative completion via in the following extremely specific way so it having chosen a splitting as before it's going to map to this semi-direct product and then it goes to the SL2 equivalent derivations obviously algebra and so given a an element sigma it will always manifest itself in a very specific form of B delta where the equivalence relation on these derivations is that B delta is equivalent to B plus a delta minus adjoint of a for any a in u 11 SL2 invariant now this this equivalence relation shouldn't frighten you because as we see from this picture here there are actually not very many SL2 equivalent elements in this Liege algebra you have to go quite far you have to take quite complicated commutators before you even see the first invariant element so we can really sort of ignore this as a first approximation and as I'll explain that we can also think of deltas on approximation as well and so the the the material elements can really be thought of to first order in terms of just an element in this Liege algebra which which we understand fairly well hmm sorry oh this is equivalence relation oh a sigma sigma is some element in here so sigma maps to along this map sigma so sigma is an element of the Liege algebra of automorphons this category I don't see sigma no it should depend on sigma I suppose B sigma delta sigma yeah yes you're right it this the B and delta depend on sigma but I'm just saying if you give a sigma let me call the image B delta and the image of this but there was this inertial condition the image lands in the subspace of derivations satisfying the Liege algebra version of this condition which is B comma n plus delta n equals 0 this is the inertial condition where n is the logarithm of t kappa plus and I claim that we can we can write this down more or less explicitly it's it's it's something like I she didn't prepare again but it's involves a sum of e2n plus 2x to the 2n with some Bernoulli number factor here and some some coefficients so n to lowest order is a power series that involves all the Eisenstein series with some Bernoulli number coefficient coming in here so this is something that's this element and it's something quite concrete and we know a lot about it so we have a Liege algebra of derivations that respect this condition this is something very concrete you could put this on a computer if you like and and unexplore it so now define you mmm respectively it's Liege algebra ummm to be the image of UH so this is so UH we've got so what is UMMM another way to think about it is that you UMMM is the Liege algebra of the unipotent radical of aught tensor mmm of omega Duran so the the category of mixed modular motives as I defined it earlier is a Tanakhian category it's completely determined by its its its Galois group and this is the this is the unipotent radical so this beast completely describes all the extensions where it describes everything about this category but it in particular it gives us all the information about extensions and iterated extensions in the category mmm gamma so the holy grail is to to get a presentation right down generators and relations of this Liege algebra if we knew that then we would know exactly which exactly the the structure of the category of mixed modulo motives so that's exactly what I'm going to try to do now first I want to explain on this geometric picture what this constraint is so every the upshot of this is that every I'll call these natively elements UMMM can always be represented by a pair B Delta so as Cathy pointed out I should have the dependence on Sigma but I'm lazy and what we think of we call B Sigma is the geometric part and Delta Sigma we call the arithmetic part no reasons for this so let's draw a picture we can in the same spirit we can draw a picture of the Liege algebra of the derivation algebra of this Liege algebra I'll put it up here so if the Liege algebra has a mixed hot structure then its derivation algebra also has a mixed hot structure in the words the derivation algebra of this is an object of the category H and if we draw a picture of it with the M and W filtrations then we have this line M equals W that's the line in red on this picture and given any if I have some material element here if I want to ask what are the possible what are the elements of a given type of a given weight so they'll be in a certain M column I've been a fixed M column and they will look like this so we have the delta part so the delta part is SL2 aquavariant and as I mentioned earlier if you're SL2 aquavariant you've got to lie on the diagonal so the delta part entirely sits the arithmetic part sits in this slot here and then we have all this stuff here which is add B and we call the so what I should say that when I write an element like this I'm really choosing a splitting of the W filtration but the intuition is perfect the top part which is canonical is called the geometric head and then so there's all this stuff given by B this is just given by an element of the Liege algebra U and then there's an arithmetic part which is mysterious slightly mysterious and then there's all the rest is an infinite tail that goes down in the W filtration so that's the picture of of a motivic Liege algebra element and I emphasize you know to draw this picture we what it implicitly means that we split the W and M filtration but that's fine as I explained earlier so now it gets fun because we have all these filtrations and all these constraints and we can do some detective work on what possible durations that can be and the first point is the inertia condition implies a whole bunch of stuff but two things that are very important and the first one is that the geometric head so a bit up there is always a lowest weight vector so that already poses a strong constraint on on on what durations are possible so here we see we could have we could have an element sigma whose whose geometric head is this gadget here and in fact that there is such a one it exists and it corresponds to the z to three extension we saw earlier then you could ask whether there's a derivation whose geometric head is is this element here turns out there can be no such element and so on and so forth the other fact that the inertia condition gives is that this infinite Ted is in fact uniquely determined so once you know what what it looks like it's sort of above the the diagonal then you know all the rest so that's the term and by this inertia condition b comma n and delta n equals zero all right so at last then I can state a theorem about what we know about these durations oh sorry yeah no I missed a key point I'm slightly out of time I'll just get to the punch line so the point is that an element sigma in the abelianization of MMM is equivalent to extensions in the category MMM of SL2z so that we really desperately want to know exactly what derivations are possible because it tells us how to construct extensions which is an important problem and so to exhibit I won't explain that the the machine but to construct such a non-trivial element you the key point is that you have to compute a period you compute a period which is essentially a regulator and that's an analytic argument and that enables you to deduce that some some some element actually exists as non-zero that's images non-zero okay so there's a whole machine to do that the conclusion then is theorem one there exists so these are non-canonical it's that the abelianization is a canonical so there exists first of all we have zeta elements so I call these sigma 2n plus 1 in UMMM for all n and they have a hard structure they are of type q 2n plus 1 and what do they look like well they have the geometric head minus 2 over 2n factorial Eisenstein y to the 2n plus dot dot dot there's there's a geometric part we know the next we know the next term I think I think I know the next time in this essentially nearly all of it in any case and then there's an arithmetic part and then there's an infinite tail and the arithmetic part I actually know this to first order so I know how acts on honestly algebra and it's extremely interesting and it involves some very bizarre quotient of two different Bernoulli numbers somewhat bizarrely and these correspond to to z to 2n plus 1 another way to set is that in the category MMM they correspond to extensions of tape motive by 2n plus 1 so it shows that these extensions exist in the category MMM equivalently that the period the odd zeta values appear as multiple modular values and the proof is we nearly did it the proof is obtained by computing an integral of an Eisenstein series and that produced in the co-boundary part it produced this odd zeta value and that shows that these derivations the machine that shows that they exist and then non-trivial then there's something completely new which are modular elements so they come in in sort of pairs for all f cusp form of weight sorry for every cusp we had a basis of cusp forms with rational coefficient for coefficients and the integer d is any integer bigger than equal to the weight of f and what do these look like let me just write down the one with a single prime the double primes identical so we get some the geometric part is some coefficients that are perfectly computable and then we get a commutator e f prime y to some power an Eisenstein part y to some other power times x1 y2 so this is the notation I mentioned in an earlier lecture to some power plus dot dot dot dot so the geometric head then it is is a commutator of cusp form in Eisenstein series and you get a whole bunch of them and the coefficients that occur are the coefficients that turn up in the period polynomial of f so these are non non-trivial and they correspond to the L values L F D so these derivations are they are of type they are of modular type and they prove that there are non-trivial extensions in MMM of Q by V F D for sufficiently large D bigger than the weight so and the proof of that is Rankin-Selberg or variant of Rankin-Selberg so there are no elements there are no dare possible dare motivic elements that have just a cusp form as their geometric head they do not exist and so there do not exist elements with a geometric head of the form E F prime or double prime y to the 2n and that has a lot of important consequences but I'll skip that okay then theorem 2 which is an analog of the Delaney-Harris conjecture in P1 minus 3 points is that we can deduce Freeness the zeta elements and the modular elements generate a free a free lease of algebra so this is this is great because it means that the corollary is that there exists a huge supply of mixed modular motives so essentially these are mixed modular these are motives of mixed-tate modular type and it's saying that if you if you specify extension data arbitrary extension data then you can construct at least one example of a mixed modular motive with an iterated multi-extension with whatever extension data you like so yeah so it basically says we have the there's some caveats but it it it says that the the category of mixed modular motives is doing what it should it's generating every single example that we can hope to find which needs me on to my next question is it can we expect to find all the extensions as predicted by Baylinson predicted by Baylinson's conjecture so the point is that the story doesn't just stop here there shouldn't be zeta and modular elements but there should be a whole infinite sequence of more and more generators in fact we should expect to find extensions in mmm of the trivial object by symmetric powers and I'll just write down the final theorem and stop so we should get extensions of this type or suddenly they should exist somewhere in nature and the question is do they occur in relative completion and these should correspond to derivations if they occur if they can be exhibited as mixed modular motives sigma v in you mmm and here comes the spanner theorem 3 which is very surprising and the answer is no those absolutely astonished to discover that if we write the at the weight of each modular form fi call it ni sorry 2 ni plus 2 then define a quantity al of v to be 2d which is the take twist here minus sum 2 ik and k plus 1 then if al of v is less than minus 3 it might be less than or equal to but I have a doubt then these extensions cannot occur so this is extremely surprising it means that if Baylinson is right and that these extensions exist as motives then in fact we cannot find them in this geometric setting and I have no idea where in nature where to find such things and the funny thing is that this condition there's a formula due to Carlton that it that tells you what the rank of this X X group in the category real mix of structures is so we know what the dimension of the space should be and when this condition is satisfied it's almost always zero so it's very unusual to that you expect to see some extension that is ruled out by this theorem they're very rare so there's a tiny fraction of motivic extensions that should be out there that we cannot capture using relative completion in this way and this is a big mystery I think the first time such an extension happens so the way that this can happen is when you have many modular forms whose weights are very close to each other and and D is very small and then occasionally you can be in this in this no-go zone of this theorem but in the generic case when D is very large there's absolutely enough space in this derivation algebra for them to exist and I expect them to exist and I expect them to generate a freely algebra so the questions I raised at the beginning of the lecture do those relative completion generate all mixed modular motives well the answer is yes and no yes in the sense that we have these types of freeness theorems that show that you get pretty much everything once you have the once you have the the simple extensions but not all the simple extensions seem to be there in the first place and that's an absolute mystery and