 Okay, so last time we were talking about degenerations of polarized hodg structures. So now we'll talk about degenerations of mixed hodg structures, so of mixed hodg structures. So the first thing we need to talk about are relative weight filtrations. So the difference with the pure cases, we now have a weight filtration. So the setup here is going to be V is a finite dimensional vector space, say over a Q or some field of characteristic zero, W dot equals the filtration. So for example, this could be a mixed hodg structure and it's weight filtration. And N takes VW dot into itself as a nilpotent endomorphism. So by this I mean that the nilpotent endomorphism of V in it preserves W. So let's just recall from last time. So if you have a vector space, maybe I'll call it A, and you have a nilpotent endomorphism, say phi, say this is nilpotent, then from this you can construct a filtration which I'll call W phi. This is the weight filtration of phi, and it has the property that phi of Wk is contained inside Wk minus 2, and it has the property that phi to the k induces an isomorphism between graded Wk of A into graded W minus k of A. And you can construct this out of the Jordan form. It always exists. And then if A had had, I'll call it W equals W phi. So it's OK because if V has, say, weight m, so for example it's a hard structure of weight m, what we want to do is make center this weight filtration at m. So shift to get filtration m dot centered at the weight m. So in this case, you would have n to the k would induce an isomorphism, that should be an isomorphism here, graded m m plus k of A into graded m m minus k of A. And again, this will always exist. So it always exists. So there's no restriction at all on the n. But in this case here, what we have is we have a graded, say, Wm of n is going to take graded, maybe I'll use a simpler notation, we're going to have graded a map on the mth graded quotient. So this is just the induced map. And this guy's nilpotent, so that means it has a weight filtration. And we're going to shift it so it's centered here. So we're going to let m sub m dot be equal to the shifted, so the weight filtration of nm shifted, well, let's just say centered at. We want to center it at m because this guy has weight m. And so what's the definition of a weight filtration of n takes v into v relative to W dot is a filtration m dot of v with the property that the induced filtration satisfying. And the first condition we want is that the filtration induced on graded Wm of v is mm dot. So in other words, it cuts out the correct weight filtration on the graded quotients and two, you also require that n of mk is contained inside mk minus 2. So a simple factor, so remarks is that one is relative weight filtration, so not every n takes v W dot into itself has a relative weight filtration. And two, if it exists, it's unique. Relative weight filtrations, this is easy. Relative weight filtrations are unique when they exist. So let me do a little example here. If graded W dot of n is equal to zero, then you can easily check that m dot exists if and only if n of Wk is contained in Wk minus 2, and in this case m dot equals W dot. And so it's easy to see here, maybe I should give an example of where it doesn't exist. So I could take v equals, say, h1 of some curve relative to, say, x0 and x. So c is, we'll take an x0 and here's an x1. And so we have an exact sequence, I guess I want. And so this is isomorphic to, say, q. And now, suppose I looked at the following endomorphism, I just take x and I let it travel around a loop here, say gamma. So that'll, you'll get a corresponding nilpotent endomorphism of this. It's going to be trivial here. It's going to be trivial on the graded quotient, but the endomorphism will be non-trivial. So it can't be, yeah. So give this a weight filtration. So this is going to be, say, this is going to be equal to W0 and this guy here, W minus 1. They're the standard weight filtrations in the situation. This guy does not have, so let x move around, gamma and I'll be sloppy, say this is nh1 not equal to 0 in h1 of c. And then n equals log of the automorphism, the corresponding automorphism, and n is not equal to 0, grwn.is equal to 0, but no, no relative weight filtration. It's a silly thing because for there to be a relative weight filtration, n would have to lower weights by 2, but if it lowered weights by 2, it would be trivial. So and there are better examples. I didn't think of one. Right, so let's talk about degenerations of mixed odd structures. So this is going to be very similar to degenerations of odd structures. So the input is, one, we have a filtered local system over some base. I'm going to take the base to be one-dimensional for simplicity. You can do all of this in higher dimensions. But in fact, you can test to see if something's a variation just by restricting to a curve. But anyway, so this is a filtered local system, q local system. And let's assume that the local monodromy, so my c will be a possibly affine curve. This will be, say, projective. So assume that the local monodromy is unipotent. This is true in our, this will be true in our situation. It's always true after a finite base change. And the second thing we want is a hodge filtration. Yeah, it'll be a weight, at the moment it's just a filtered local system. But yeah, and it will be the weight filtration, a hodge filtration of v. And this is defined to be, so we'll call it f dot. So that extends to a filtration by sub bundles of the deline's canonical extension. So, so what this is giving us, as in the case of variations of hodge structure, if you have, here's your curve over here, and here's some cusp here, p. It's giving you a vector space that you associate with that, that cusp. And now, and we're gonna have a hodge filtration. Here it is here by holomorphic sub bundles. That'll cut out a hodge filtration on each of these fibers. And the weight filtration is going to behave well. And so this guy here will have a natural flat connection. The characterization of the extended connection is that it will follow from what I say here. So we want, so this here implies that the residue at p of nabla is nilpotent, or p in s. And we also require the connection as regular singular point. So nabla takes v bar into v bar, oops, v bar tensor omega 1c bar log s. So you're allowed to have logarithmic singularities. So these, to say that that is the canonical extension of v is to just to say that these two properties hold, oops. And so we want this to satisfy Griffith's transversality. So this means that if you take a section of fp of v bar and you differentiate it, you get something that's at worst in fp minus 1 of v bar tensor omega 1c bar, mm-hmm, for all p. OK. And so the next thing is the obvious condition here. We need that fiber by fiber we get a mixed hodge structure. So each fiber of, say, v over c is a mixed hodge structure. So it gets its rational structure from the q-local system. So with induced hodge and weight filtrations. And now the next condition we want is actually a non-trivial condition is that for each p in s, let's just let np be equal to the residue at p of the connection. So this is nilpotent. So we assume that the residues were all nilpotent. And by the way, I should say here, too, the naturality of Delene's construction implies that all the weight bundles extend through as well. So the w dot extends as well. But this is not a hypothesis. It's automatic. And so what we want is that each np, so it takes the fiber of the canonical extension over here with its weight filtration into itself, has a relative weight filtration. So you want the relative weight filtration to exist. And 5 is that for hv in the tangent space of p of c bar, this guy not equal to 0. So I'll call it vv. So I explained how to construct out of a tangent vector a q-structure on the fiber of the canonical extension. So this is the q-structure on vp. Then with the weight filtration and the hodge filtration, sorry, I want to take m dot. This is the relative weight filtration. And f dot is a mixed hodge structure filtered by w dot. Filtered in the category of mixed hodge structures. So it's a more complicated structure. And if you go back a good exercise is to look at the definition of the variation of hodge structure, in that case, there's no w dot, or the w dot is trivial. And then you just have m dot is going to be the monadromi weight filtration shifted. So anyway, so these are basically the axioms of an admissible variation. Such a thing is called an admissible variation. Is that the fact? I mean, the residue preserves the weight filtration that's automatic, is it automatic? Yeah, because the local monadromi preserves w dot. It's unipotent. So it's a logarithm well. The logarithm is just a polynomial in the monadromi. I mean, the weight filtration will extend across using its delinear canonical extension. Right, so you can look at the, because the local system was filtered, right? So each Wm of v is a local system. You can take its canonical extension. Then you could, naturality just implies that that just gives you a filtration of the canonical extension. And so the local monadromi is unipotent. And so its logarithm is nipotent. And it will preserve the weight filtration because the monadromi, because the filtration is by flat sub bundles or by sub-local systems. Anyway, such a whatever, a Vw dot f dot, etc., is called an admissible variation. And this definition may look very strange. So the question is, is it natural? The answer is, let's, so when you look at this condition, it looks very strange. But when you try to construct these things, if you're in the business of constructing them, it's usually not a problem to construct the m dot, it just comes out. So, and this will become clearer when I discuss these examples. So, one is, so this is due to Steinbrink and Zucker. By the way, I think the definition of the relative weight filtration was made by Deline in V2. Now, I'm sure he guessed it from, I haven't looked at it, but I'm sure he guessed that from the elatic setup. The first Hodge theory case where this was considered was by Steinbrink and Zucker, and they showed that local systems of, yeah, maybe I'll just say it this way. Suppose I have an x over c, and this is family of smooth, but not necessarily projective, smooth varieties. And I want it to be, by family I mean topologically locally trivial. And so in this case here, local systems, k, f lower star of q. And every fiber here has a weight filtration. The Lienz construction of a, so here I'm assuming for every t and c, x, t is a smooth variety. So its cohomology has a mixed hard structure. You can easily see the weight filtration is locally constant. And these guys are admissible variations of mixed hard structure. This is also a good source for relative weight filtrations. They wrote out a lot of the details. Then there's Guillain, Navarro, Poeta, I think, this is my memory. They did the case, the more general case, x over c here. But this is just a topologically locally trivial family of complex varieties. So not necessarily smooth, not necessarily complete, right? They can be singular, open, whatever. And again here, you'll get the same result. Rk, f lower star of q is an admissible variation of mixed hard structure. This subject loves abbreviations. And then another one we'll need is three, this is myself, is that if you can look at the following situation. So we'll take a section here, say sigma. So this is a pointed family of smooth varieties. So again, family just means topologically locally trivial here. And so here you can look at the local system. So you can look at the local system. Maybe I like to call it p. So over t in here, you would look at pi 1 of xt and you would look at it with the base point sigma of t and you would take its unipotent completion and maybe take the coordinate ring of that or the Lie algebra of that. So we'll need this local system here. So this is actually in general infinite dimensional, but it'll be a pro thing, but this is an admissible variation of mixed hard structure. And we'll be interested in this when this is the universal elliptic curve, minus zero section, and this is really just a section of tangent factors along the zero section. Sorry, except that the Lie algebra of completion is contained in p, what's p? Well, this is by p, I mean the local system, whose fiber over t is the unipotent fundamental group. And I will eventually give a quick definition of this. I'm gonna say something about Tanakian business in a little bit. Just for in the interest of efficiency, it's basically the best unipotent approximation to the fundamental group of the curve or whatever. All right, so now I can define universal mixed elliptic motives, finally. And to do that, I needed the notion of a variation of mixed hard structure, and relative weight filtration, and so on. And I thought that would be a bad, that was a bad place to start the first lecture. I would have been, so, and I'll remind you that this is joint with Makoto Matsumoto. Okay, so I'll very briefly recall the rough definition. The words will probably make a little more sense. So, what is a universal, so is a compatible set of filtered local systems? And I'll be more precise about this later, but for example, you would have Betty, Kudoram, this together gives you Hodge, and you would have L-addict. So here you would get a least sheaf. And there are two basic conditions with grw. Yeah, maybe I'll call it, I'm basically going to call it vw. And with grw.ofv, isomorphic to a direct sum of SNHr. So H is our basic local system, and you'd want something, right? And so when I say Hodge implicit in this, I mean that I get an admissible variation of mixed hard structure. And then also have a filtered mixed TateMotive over z. And so let's call it v. And it's got a weight filtration. This is the weight filtration in MTM. And it's going to be filtered by another filtration, w.inMTM, right? So it's going to be a filtered mixed TateMotive. And the mixed TateMotive weight filtration is called m. That's because it's going to be a relative weight filtration, such that the fiber of v over the integral base point, ddq, is vm.w. So I'm going to explain what this all means, but just this is the overview. OK, so let's look at the official definition. So a universal mixed elliptic motive consists of, so it's best to start with the mixed TateMotive, a filtered object vw. of mixed TateMotives over z. And denote the weight filtration of v by m dot. So I did that to drive Francis nuts, everybody nuts, to a representation rho, which takes sl2z into the automorphisms of the Betty realization of v. And it preserves this guy here. And this guy I should point out, as I hope I explained clearly before, this is naturally isomorphic to pi1 of m11 analytic with the base point ddq. You can lift the base point, the tangent vector ddq, to the upper half plane by, as the imaginary axis. So sorry? The box of this guy. Yeah. Well, you only need the germ of it up near plus infinity. Sorry? This condition is not our automatic. Which one? The last one. If I just take some invisible variation of cross structure. Well, it's part of the definition. But the fact that the associative gradient is just given by correct sums of something. OK. Let me make, well, no, you could have all sorts of other representations of sl2z is virtually free. So take a finite index subgroup. It's free. You can map it just about anywhere. And that'll give you a local system on a finite cover, and then you can push it down. But which variations come from geometry? Aren't these all the ones that come from geometry? No, I can construct ones that come from geometry. Sorry? Yeah. And I want to stress, note that we're not allowing things like snh, tensed with some simple hod structure that's not a q of r. This is a different category. It's going to have a different Tanakian fundamental group. And I'll try to explain that later. I mean, this was a bit of a surprise to me. So I want to. If you assume that the associative gradient r is the direct sum of snhr, doesn't that automatically imply that the hod structure, that tangential vector, has to be mixed-tape? No. It doesn't, I think. Sorry. It implies it has to be mixed-tape, but it doesn't imply that it's an object of the, you know. But that's what I mean. But you do get all the realizations. No, you can get just take any mixed-tape motive that's not in, that has periods that aren't multisetas. So you just take any mixed-hodged-tape structures, periods aren't multisetas. And now look at the constant variation over m11 with that as fiber. That won't be in this category, because the limit will be that constant hod structure. But it's not the hodge, say, the hodge realization of something that's in MTMZ. So where are we? I've lost my blackboard. Which board am I on? Oh, I just did this, and I'm about to erase this. Right. So sorry. When you say mixed-tape motive, do you actually mean like an object in the beam construct? Yeah. I said somewhere here, I think I said it's an object of MTMZ, not just some hodge structure that's of type, this graded quotient sort of type Pp. You know, it's a more stringent condition. So you can then say, well, are there any interesting examples? So I'll get to that in a minute. Oh, yeah. So where each grwm of, I just lost track of where I am here. Where'd I go? Well, let me just write it down. So this will give us here a local system vw dot over m11 analytic. And we want each is a sum. This is just as a local system of SNH. There can be various n where n is congruent to m mod 2. So just as a topological local system, let me see if I can get these pages. And now the next thing is 3, a filtered vector bundle, vw dot over m11 bar over q. So this is going to be the q to rump story with connection. Navla, which takes v into v tensor omega 1 m1 1 over q log p. And this guy here is the cusp, the cusp q equals 0. And a filtration, you need a hodge filtration such that the connection satisfies Griffith's transversality. So you only lose one degree. And the residue at p of the connection is nilpotent. Actually, I needn't have added that. That's going to be automatic. And so now we can combine this with a flat connection. I forgot to say this. We want a flat connection. And so then we want the obvious isomorphism of the q to rump setup. So what we want is if we took vw dot, nabla, and we tensor it with c, and we restrict it to m11 analytic, or just m11, say c, this should be isomorphic to, and it's natural flat connection. So this is just giving a q to rump version of the vector bundle underlying this guy here. And together, they form an admissible variation of mixed hodge structure over m11. So then the last part's the analytic part. In fact, one can probably leave out some of these pieces still the way I think about it. But I think some, the existence of some are consequences of the other parts. 4, 5. So the representation rho l, which is going to take pi1 of m11 over q bar ddq into ought vl. So vl is going to be the ellatic realization of the mixed-hate motive induced by rho. So this guy here is naturally isomorphic to the profinite completion of sl2z. So this is sl2z hat. So the representation from rho on the Betty version of this will induce a representation here. And we want this to be gq-equivariant. So the Galois group acts on this because this base point here is q-rational. And it also acts here because the ellatic realization of v is a Galois module. Yep. All right, so that's the end of the definition. All right, so a couple of quick remarks. Sorry? You have to provide the vl independently of what can be rho. No, we start out with a mixed-hate motive. And a mixed-hate motive has an ellatic realization. And this is the ellatic realization. So that comes equipped with the Galois action. What was the origin of this parity thing? And sorry, I'm in the wrong way with the atom. Well, because this is a variation of weight 1. And I mean, this condition is forced by saying that this guy is a variation. So snh is a variation of weight n. And then when you take twist it, the parity of the weight doesn't change. And you can easily prove that because this is a simple local system, it can occur as a variation of hot structure. The way it occurs is a variation is unique up to take twist. So you can't put an even power in an odd weight, so on. So the remark here, I'll just make an obvious remark. You can do something similar. There are a few technical difficulties, but there are. So one is similarly can define mixed elliptic motives over, say, m1 n plus r. You have to choose a base point, and then you have to show it's independent of the base point. Here, there's only basically one choice. And you can also do mixed elliptic motives over things like m1 n plus r. And you can take a level structure. And maybe here you have to walk over some ring of integers. And so this guy here, I will denote this category by mem n plus r. The main examples are going to be very simple. So the category I'm calling mem is really equal to mem1. That's the one I've been talking about. But you can do it with a tangent vector. So let's look at some examples. So the first one is these are the simple. In any of these categories, these are the simple guys, the only simple guys. These are just, you can just take s n h r. I've already shown you that these guys degenerate to a direct sum of tate motives. So we've already verified this. The second example, what I'll call geometrically constant. So these are just, you just take, i.e. You just take a v in mtm. And you just look at and pull back. So think of the mixed-tate motive as being a local system here. You just pull it back here. And it'll be, as a variation of hard structure, it'll just be a constant variation of mixed hard structure. And as a least sheaf, it'll have trivial monodromy on the geometric part of the fundamental group. It'll just be a Gal-Y representation. So these guys, third example, and actually, I'll talk about a slightly more general situation, the elliptic polylogarithms of Baylinson and Levin. So they write it down as one thing. I think they call it the elliptic polylogarithm. But I like to split it into pieces. So this is some variation. This is some variation, say, p over the universal elliptic curve. So we could have also defined mixed-elliptic motives here. You can pull back along the zero section to get something over here. So their restriction to the zero section give you get these simple extensions, which are of the form S2NH. We'll see in a bit that these correspond to Eisenstein series. A non-trivial extension that corresponds to the norm G2N plus to the Eisenstein series. And these are going to be the most basic. These are all the simple extensions. So I like to think of these are the analogs. If you think what we're trying to construct an extension for every elliptic curve, these are the generalizations of the values of the Riemann zeta function. That you get in the theory of mixed-tate motives. So these are analogs, obzative values. So four, and this example is important for us. So let's look at m11. So I'm going to look at m1 vector 1. So here I have to look at that. So a typical element of this guy here would be an elliptic curve plus a non-zero tangent vector. And I'm going to look at the local system. Again, I'm going to call it p. And the fiber of this guy over here is what I'm going to call pEv. So this is a pronial potent, le algebra. So pEv is defined to be the le algebra of pi1 unipotent of the elliptic curve minus its identity. So this is E minus 0V. So this is a freely algebra. So this guy here, just to give you some idea, is isomorphic to the freely algebra on H1 of E completed. This is not natural. And the weight filtration will be its lower central series. OK? So this is a pro. And here, this is one reason I need some decorations, because I need base points. So do you want a coffee break? I can keep going. Maybe why don't I throw something up on the board, and then we can take? Three is a certain question to four. Sorry? Three examples. Yes. Well, these things are going to occur everywhere. And so, yeah, they certainly occur inside this guy up here. So the next thing is the Tanakian fundamental group of MEM. And you can also do with some decorations here. They don't change. When you change decorations, this guy, the fundamental group of this guy changes in some predictable way. So yeah, so I'll just start with a remark. Is that the categories MEM and plus R are Tanakian over Q. And this means that the category of representations of some affine group scheme, or the same, it's also a pro-algebraic group. And this leads us to the following problem. Compute pi 1 of MEM. So let me stop here. I'll start up after the break. I'll do a very, very quick resume of some things about Tanakian categories, just so. And in fact, then I'll rigorously construct unipotent completion in a way that Francis thinks is illuminating, but I think other ways are more illuminating. And then I will go on to discuss this problem. And in fact, we know a lot. We don't know it. We can't completely compute it. You run up against some standard conjectures in number theory at some point. But we have very good evidence that it's got a certain presentation that we can't write down explicitly. So I'll do that after the break. So this will be very brief. So f equals a field of characteristic 0. And so a neutral category over f is a rigid, abelian tensor category. C, I'll say a bit about this, where there's a trivial object, which we'll denote by 1, is isomorphic to f. So a tensor category has a tensor product. And here it's got to be commutative, associative, and so on, all sorts of axioms have to be satisfied. These are exactly the axioms that are satisfied by the category of representations of a group. You have an associative tensor product and so on. And it has to admit an exact and faithful from C, and they're usually denoted by omega, from C into vector spaces into finite dimensional. This is going to be finite dimensional vector spaces over f that preserves tensor products. So the first example, 1, g is any group. And you can look at the f representations of g. So this is just representations of g on a finite dimensional vector space. Category of finite dimensional g modules, finite dimensional over f g modules. And 2, mixtape motives and mixed elliptic motives, n plus r. And I should say here the trivial object here is just equal to q, the trivial representation. I should say f, the trivial representation. Here the trivial object, 1, is just q of 0. Another good example is mixed hodg structures. And here the trivial example is q of 0 again. And 4, you can look at mixed hodg structures, x. So it's equal to the category of admissible variations. And so x here is a smooth variety. And so the basic fact we need is that I should say such a functor is called a fiber functor. And there can be many of them. The basic fact is that if omega takes c into veck f is a fiber functor, is any fiber functor. So that's one of these exact and faithful functors. Then c is equivalent to rep fg, where g is the tensor automorphisms of omega. So what's an automorphism here? These are natural transformations. Actually, I should say natural isomorphisms from omega to omega that preserve that respect tensor product. And so we'll denote, so basically it's saying that c is the category of representations of this group. This group here is going to be, it's affine. It's also pro-algebraic, pro-linear algebraic actually. And so our goal is to write down this group when the category is just MEM, mixed elliptic motives. And so we're going to do notation is that g is equal to pi 1 of c with base point, this fiber functor. So let's do an example, a more interesting example, unipotent completion. So how do you define the unipotent completion of a discrete group? Well, one way is to use Tanakian categories. Sorry, discrete group. And so f equals a field of characteristic 0. And in this case, we're going to take c to be the category of finite dimensional unipotent represented representations of gamma. And these are all defined over f. So there's two ways to say what a unipotent representation is. It's a homomorphism of gamma. One way to say it is, so this will be contained in glnf, it's a representation that can be conjugated into unipotent matrices. Or you can also say just a representation that admits a filtration v equals v0 contains v1 contains v2 and so on, where by gamma sub-modules where each vj mod vj plus 1 is a trivial gamma module. So that's a more coordinate free way to say it. It's just that you can filter the modules by sub-modules and all the associated graded to just trivial modules. And now this category is Tanakian. So omega takes c into veck f. It just takes a representation, goes to its underlying vector space. So it's clearly faithful. Clearly exact. And so the definition of unipotent completion is definition is that gamma unipotent over f is defined to be pi1 of this category with respect to this fiber functor. If you're as good at Tanakian categories as Francis is, you can easily work with this definition. There are others. Maybe I should say something just to remark. So this is an older way of thinking about it, but one way is if gamma is finitely generated. So that definition there does not assume the group's finitely generated. You can look at f gamma, which equals the group algebra. And you can take, there's two ways you can go here. One is you can say what is O of gamma unipotent over f. So what's the coordinate ring of this? And it's actually, it's equal to the HOM continuous. So what does HOM continuous here? So you've got an augmentation from f gamma into f, the standard augmentation. And you've got an ideal, which is the kernel of epsilon, which is the augmentation. And it defines a topology on this by the powers of this ideal. So this is just so we can give this the i-addict topology. And so this is just also the direct limit of the HOM f gamma mod i to the n f. And this is a hopf algebra. You can easily check. So and this more or less comes out of Quillen. Long time ago, Quillen wrote a paper on rational homotopy theory and buried in the appendixes a section on what he called Maltsef completion, but nowadays it's called unipotent completion. Yeah, another way to say it, too, is if you take the universal enveloping algebra of the Lie algebra of pi of gamma unipotent over f, this is just equal to f gamma completed. And this is just the inverse limit. I should say the completed enveloping algebra of f gamma mod i to the n. And so our problem is to compute pi 1 of MEM. So sometimes I'll be precise about the fiber functor because it matters other times. It doesn't matter because you always get the same group. You just get different inner forms of the group. If you use different fiber functors, you get the fundamental groups of isomorphic to an isomorphism unique up to conjugation. So let me start with generalities. So most of this is very soft, with one exception. The first statement is, well, we have a functor from mixed-tate motives into mixed elliptic motives. This just takes a mixed-tate motive to the constant, the geometrically constant things. So this is, we've got M1 1, and we map down to spec z. And you can take a mixed-tate motive over here and pull it back here. So these are the geometrically constant guys. And so this gives us, we've got geometrically constant. So this gives us a map from pi 1 of MEM to pi 1 of MTM. And this guy is, you can argue directly that it's subjective, but it's easier to see that you also have a functor from mixed elliptic motives into mixed-tate motives. This is just the fiber over DDQ. Or if you think of the definition, a mixed elliptic motive consisted of a mixed-tate motive plus a whole bunch of other stuff. So you just take the, forget everything else and just take the mixed-tate motive. And this is going to give us a splitting of this guy here. So this is going to give us, this will give us pi 1 of mixed-tate motives into pi 1 of MEM. And this is, if you think about it, it's clearly a splitting of this. Because if you take a geometrically constant mixed elliptic motive, its fiber over the base point is the same as its fiber everywhere. So what we can do is define pi 1 geometric of mixed elliptic motives to be equal to the kernel of pi 1 mixed elliptic motives mapping to pi 1 of mixed-tate motives. I should remind you of what this guy is here. So what we do is we have an exact sequence, pi 1 of mixed elliptic motives. It maps to pi 1 of mixed-tate motives. And it's got to be subjective. The kernel is pi 1 geometric of mixed elliptic motives. And we have a section here. This section is given by the base point, ddq. In fact, we can compare this with, so for example, the et al fundamental group of M11. And it's compatible with this here. There'll be maps. But I'll discuss that later. I'll just recall here. Pi 1, so if we look at pi 1 of mixed-tate motives, it maps to gm. So this tells, this is basically the fundamental group of the category of split mixed-tate motives. Things that are just direct sums of q of n's. And the kernel, I like to call it k. And k is, so this guy here is pro-unipotent. And every pro-unipotent group is isomorphic to its Lie algebra via the exponential map. So k, little k, will be Lie of k. And this guy is isomorphic to a freely algebra on z3, z5, z7. Odd things here. I better put z9 completed. Completed means take power series, Lie power series. So proposition is the first statement is that there is a natural homomorphism from sl to z into the geometric fundamental group of mixed-aliptic motives. And here, I'll be more precise. I should be the fiber functor is the beddy fiber functor. And I'm going to take the curational points of this group. And so the beddy fiber functor takes the fiber, so a megabeddy takes v, a mixed-aliptic motive, and it takes it to the fiber of this over ddq. And then it takes the beddy realization of that. And two, it is a risky dense. So this guy here is the only, this is not that hard, but it means everything else I'm saying is soft. This requires some work. So let me sketch the proof. The proof of one is easy. So we have functors from mixed-tate motives. We can take it to mixed-aliptic motives. So this takes a mixed-tate motive to the constant, geometrically constant, mixed-aliptic motive. And now here, we can map into representations of sl2z, say, q representations. Sorry? Yes, well, I mean, if you remember the way I wrote down the definition, there was a v, and there was a rho, and a whole bunch of other things, so I'm just taking it to the rho. And now if you look at what happens here, so the fundamental group of this won't be sl2q or something. It'll be much bigger, but there is a natural map. This picture will give you a natural map. And you've got, if you took omega beta here, you take veqq, and then just take the forgetful functor here, this commute. So you get a map here in omega beta, and you will get the map into here. And why does it land in the geometric part? Well, if you continue on to pi1 of mixed-tate motives, say, q, what we know here is that this map here, this is trivial. It takes every one of these guys, goes to the trivial representation. And that's telling you that this here is trivial, which is telling you that you landed in the geometric fundamental group. So this part is completely soft, the first insertion 2. So I've got an old version and a new version of this page here, so I have to go delete that. So this follows from A is the theorem of the fixed part. And what this theorem says in this case is if you have a variation of mixed-tart structure, which has trivial monodromy, it has to be constant, which implies the theorem of the fixed part just says that if you have any variation over some x, it says h0 of xv maps to, say, the fiber over x, this is a morphism of mixed-tart structures. But you can use that to prove the following statement. It implies that a variation of mixed-tart structure with trivial monodromy is constant. And so this implies, so if you think about it, saying if you map this into here, it's telling you that the Zariski closure of this has got to generate, sorry, the normal closure. If you look at the normal closure of sl2z in here and take its Zariski closure, it's got to be the whole thing. So this part's also relatively soft. This implies that the Zariski closure, the normal closure of the image of sl2z is, but we have to see that the image of sl2z, the closure of its normal. And the way we do that is you have to use the fact that the existence of the limit mixed-tart structure on the relative completion, which I'm about to define, implies, so you have to use ddq as your base point, implies that the Zariski closure is normal in, right? And so this implies that you put these two things together, you see that the, so you'll see that the Zariski closure of this inside pi1 of MEM is pi1 gm. If you don't like that, I've got a version of this written up somewhere. But anyway, this is important because it's going to allow us to ban on the size of the kernel. OK, so we're going to call a mixed elliptic motive split if it is a direct sum of SNHRs of simple. These are the simple objects of mixed elliptic motives. And so split. It means semi-simple in here. Sorry? It means semi-simple when you say it's an item. Yeah, OK, so these are the semi-simple mixed elliptic motives. Yeah, better. These semi-simple. So semi-simple mixed elliptic motives is a Tanakian category. It's a subcategory of the whole thing, which is clear. And then pi1 of semi-simple mixed elliptic motives, if you think about it, is just isomorphic to GLH. Or it's actually equal to GLH, and it's isomorphic to GL2. This is, you may wonder about the Tate twist, but if you look at, for example, GLH, yeah, it's actually the inverse of the determinant. If you look at this representation into GM, this representation corresponds to q of minus 1. That's because we put h in weight plus 1. If I put h in weight minus 1, it would correspond to q of 1. It equals your other point of information. Yeah, so all right. So now we have semi-simple mixed elliptic motives. This includes into mixed elliptic motives. So this is going to give us a map on fundamental groups the other way. We also have a map back because we can take grw dot. If you have a mixed elliptic motive, it's associated weight graded is semi-simple. And so what this gives us, so this gives us, we're going to have pi 1 of m e m. And we can use any fiber functor mapping into glh. And it's got to be subjective because we have a splitting. And the splitting is grw dot. And so there'll be a kernel. And what we'll see soon is that that kernel is pro-unipotent. We'll see some remark soon. We'll see that the kernel, it stands to reason because the kernel should be telling us about extensions. And in theory of motives, you can only extend by stuff of lower weight. But we'll see it another more direct way. The kernel of pi 1 of mixed elliptic motives into glh is pro-unipotent. OK, so let me, I brought up the issue of relative completion. So let's look at, maybe I'll call it relative unipotent. And this idea, I might have been the first person to write about it. The idea came to me in a letter from Deline. Maybe sent from this very place. So this is a generalization of unipotent completion. And f here is equal to a field of characteristic 0. r is equal to a reductive f group. You don't have to take it to be reductive, but there's no real difference. And I'm going to put an example here. r equal to the trivial group. This is going to give us unipotent completion. So this is just going to generalize what I did earlier. Here, we're going to have rho takes gamma into the f points of r. This is a risky dense representation. And now I'm going to take c as going to be the category of finite dimensional representations, v of gamma. Whoops, I left out gamma here as a discrete group. So I'll put up here. Gamma equals discrete group. In all of these, there's also a profinite analog. I'm assuming that the fiber function on MEM. Sorry? You're assuming something about the fiber function on MEM. It's splitting. Here? Yeah, doesn't have to factor through the gradient one. It's true for any fiber function. Yeah, but I can take what's the fiber function. I can take it to be graded. Sure, sure, sure. You have to set that too. Yeah, you're right. I have to take graded and say, Betty, or? I've got a sequence here. Oh, sorry, I thought it was very easy. Oh, you've chosen the fiber function. No, I think Francis is right. I think I need to say what the fiber function is, but I can take the weight graded fiber function. I can have any fiber function to go this way. And to get one back this way, it's the analog of this picture in topology. We've got a base point here. I've got a base point here, but we may choose a section. We get a different base point. So these are F representations that admit a filtration. So this filtration, the way I'm setting this up, this doesn't have to be unique to some filtration. 0 equals v0 contained in v1 contained in, say, vn equals v by gamma sub-modules such that the associated graded of v is an r-module. So you've got an action of the algebraic group, a reductive group on the associated graded. And gamma acts on gr.v via you'll have gamma will map into, and this acts on, so grw. So all the graded quotients really come from representations of r, and the way gamma acts on them is via this rational representation of r. So this is Tanakian. So there's an obvious fiber function to just take the underlying vector space. And so the relative completion is, I'll call it, g rel is defined to be pi1 of this category. And here, this is the obvious fiber function. So this important information about this group is proposition is one is g rel is an extension 1 into what I'll call u rel into g rel into r into 1, where u rel is pro-unipotent. Actually, in case I haven't said it, pro-unipotent just means an inverse limit of unipotent groups. And if you like us to explore the queue, do what? I just write rel here for relative completion. I mean, again, I adapt the notation to the situation. Here we have got one group that we care about, and that's sl2z. So sometimes you've got to put the dependence of the group in there. There is a natural homomorphism from gamma into g rel. And to be accurate, I should put our rational points. It is a risky dance. And now, if you have a, maybe here I'll just say some things, just leave that blank there. So some remarks is that suppose if u is pro-unipotent, well, maybe I'll start out with if u is unipotent and u is equal to the Lie algebra of u, then you can easily check that u into u. The exponential maps polynomial. It's a polynomial bijection, and then you've got a well-defined logarithm. That's just because all the unipotent group can be moved inside this guy here, and then it becomes clear. And so if now if you have a unipotent group, and this guy's going to be a nil-potent, if u is pro-unipotent, still true. Every pro-unipotent group is isomorphic to its Lie algebra by the exponential and logarithm maps. That means to know this, you just have to know that. And three, and I can explain this in more detail later on, if u is pro-nipotent, u has presentation, has a minimal presentation. u is isomorphic to the freely algebra on its h1 of u. You have to complete this, and then there will be a map. There will be an injective map from h2 of u into, let's call it, say, phi c into the commutator subalgebra. And you look at the image of c, and you take the closed ideal that generates isomorphic. So this is, again, I write versions of this in various papers, but this is just some generalization of a result of stallings. But it's not canonical. But basically, in a regular group, h2 can sort of be bigger than the relations somehow. There's a tight relationship between the minimal set of generators for the relation ideal and h2. So up here, let me put that up there. So we care about what h1 and h2 are of this guy here. And so it's for all r modules v. By that I mean all rational representations of r. h1 of u tensored with v take r invariance is isomorphic to h1 gamma v. And h2 of u, tens of v, r invariance injects into h2 of gamma v. Let's put this here. So now, so this is telling us something about the presentation. And so maybe a good place to stop today will be, I'll do the example of it, sl2z. So we'll have a lot more to say about it, but let me just start with the basics. So we know that the relative completion of sl2z is going to be. Homology of homology, again, last statement. Homology of homology. This is homology. So in singular homology, the homology is the dual of homology. In this theory, you have to take continuous duals. So the h1 is some sort of in thing. And the h lower one will be a pro thing. And so the big thing is the homology. So let's just look at. That's where you use that r. Sorry? This form that we use as r is relative. Then I use r is a reductable. Yes. It's really important. Yeah. So example, we'll take that f equals q, gamma is equal to sl2z, r is equal to sl2 over q, and rho is just the inclusion sl2z into sl2q. And you might think the completion is just sl2q. It's not because this is a group of rank one, real rank one. So it's not rigid. And so what we get here is that h1 of u, I'll just write u, this result says that this is isomorphic to the direct sum n greater than or equal to 1, h1 of sl2z s2nh tensored with. And I'd like to write it this way, s2nh dual. I know that s2nh dual is the same as s2n. But I want to write it this way. And the next thing I know is that sl2z is virtually free. That's easy to see, because it's absolutely standard, because you can choose some torsion free subgroup of sl2z. The upper half plane divided by that is a non-compact Riemann surface, therefore has free fundamental group. And this implies that h2 of sl2z with any coefficients, if this is divisible, this is a q-module, is equal to 0. And then when you assemble that with the statement here, that tells you that h2 of u is 0. So this is telling us that h2 of u is equal to 0. But this discussion I had over here, of h2 of u is 0, that's saying that u is free. So therefore, u is free. So what we know unnaturally is that, so this is saying that u is isomorphic to the freely algebra on the direct sum of h1 sl2z, s2nh. So I left out the odd powers because the cohomology with odd powers vanishes because the center of sl2z acts non-trivially. There's a standard argument. Maybe I'll explain that in more detail next time. It's a one-line argument. But anyway, we're going to take dual of that, tensor with s2nh. So we're taking, and something I'll explain in a little bit more detail next time, is sl2 acts on this. And the claim is that we have to complete it. And then so we complete by taking the inverse limit of all finite dimensional quotients of this that have an sl2 action. And then g rel is unnaturally isomorphic to sl2 semi-direct product u. And then the final comment is that I'll pursue next time is Eichler-Chimura tells us that these cohomology groups here are basically just modular forms. So you're getting a copy of s2nh for every normalized eigenform. And you're also getting another copy for the complex conjugate of each normalized cusp form. But I'll discuss. I'll start next time with Eichler-Chimura. And I'll explain in more detail about this. And then there's a theorem that says the coordinate ring of this for any base point, including ddq, any base point of m1,1 has a natural mixed-hide structure. And the hodge representations of that are our admissible variations. So I'll stop you.