3/4 Universal mixed elliptic motives
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Transkript: Englisch(automatisch erzeugt)
00:32
OK, so last time we were talking about degenerations of polarized hot structures. So now we'll talk about degenerations
00:41
of mixed hot structures. So of mixed hot structures. So the first thing we need to talk about are relative weight filtrations.
01:12
So the difference with the pure cases, we now have a weight filtration. So the set up here is going to be V
01:23
as a finite dimensional vector space, say over Q or some field of characteristic 0. W dot equals the filtration. So for example, this could be a mixed hot structure
01:42
and it's weight filtration. And N takes V W dot into itself as a nil-potent endomorphism. So by this I mean that the nil-potent endo,
02:03
N is a nil-potent endomorphism of V and it preserves W. So let's just recall from last time. So if you have a vector space, maybe I'll call it A,
02:26
and you have a nil-potent endomorphism say phi, say this is nil-potent. Then from this you can construct a filtration which I'll call W phi dot. This is the weight filtration of phi.
02:49
And it has the property that phi of W k is contained inside W k minus 2.
03:00
And it has the property that phi to the k induces an isomorphism between graded W k of A into graded W minus k of A. And you can construct this out of the Jordan form. It always exists.
03:21
And then if A had had. So you have a bit of a classification thing on it, because you've got W here for phi? I'll call it W equals W phi. So it's OK, because if phi has, say, weight M.
03:46
So for example, it's a hard structure of weight M. What we want to do is center this weight filtration at M. So shift to get filtration M dot centered at the weight M.
04:14
So in this case, you would have N to the k would induce an isomorphism.
04:21
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.
04:44
So there's no restriction at all on the end. But in this case here, what we have is we have a graded, say, W M of N is going to take graded.
05:05
Maybe I'll use a simpler notation. We're going to have a map on the M-th graded quotient.
05:23
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,
05:47
so the weight filtration of N M shifted.
06:02
Well, let's just say centered at. We want to center it at M because this guy has weight M. Right? And so what's the definition of a relative weight filtration?
06:30
A weight filtration of N takes V into V relative to W dot is a filtration M dot of V with the property
06:50
that the induced filtration is satisfying. The first condition we want is that the filtration induced
07:11
on graded W M of V is M M dot.
07:27
So in other words, it cuts out the correct weight filtration on the graded quotients. And two, you also require that N of M K is contained inside M K minus 2.
07:49
And a simple fact, so remarks, is
08:05
that one is relative weight filtration. So not every N takes V W dot into itself
08:24
has a relative weight filtration. And two, if it exists, it's unique. Relative weight filtrations, this is easy.
08:42
Relative weight filtrations are unique when they exist. And so let me do a little example here.
09:10
If graded W dot of N is equal to 0, then you can easily check that M dot exists if and only
09:29
if N of W K is contained in W K minus 2. And in this case, M dot equals W dot.
09:50
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
10:09
to, say, X0 and X. So C will take an X0.
10:21
And here's an X1. And so we have an exact sequence. I guess I won't.
10:44
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.
11:01
So 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. So give this a weight filtration. So this is going to be, say, equal to W0.
11:22
And this guy here, W minus 1. They're the standard weight filtrations in this situation. This guy does not have. So let X move around gamma.
11:46
And I'll be sloppy. Say this is in H1 not equal to 0 in H1 of C. And then n equals log of the corresponding automorphism.
12:01
And n is not equal to 0. Grwn dot is equal to 0, but no relative weight filtration. It's a silly thing, because for there
12:21
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. And there are better examples. I didn't think of one. So let's talk about degenerations
12:41
of mixed-hodge structures. So this is going to be very similar to degenerations
13:03
of hodge structures. So the input is 1. We have a filtered local system over some base.
13:21
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. Anyway, so this is a filtered local system.
13:46
Assisted Q local system. And let's assume that the local monodromy.
14:07
So my C will be an affine, well, a possibly affine curve. This will be, say, projective. So assume that the local monodromy is unipotent.
14:29
This will be true in our situation. It's always true after a finite base change.
14:44
And the second thing we want is a hodge filtration. At the moment, it's just a filtered local system.
15:01
And it will be the hodge filtration of V. And this is defined to be, we'll call it F dot.
15:20
So that extends to a filtration by sub-bundles of Deline's canonical extension.
15:48
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 cusp.
16:03
And now, we're going to 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.
16:22
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.
16:47
And we also require the connection has regular singular points. So nabla takes V bar into V bar tensor omega 1
17:02
C bar log S. So you're allowed to have logarithmic singularities. So to say that that is the canonical extension of V is just to say that these two properties hold.
17:22
And so we want this to satisfy Griffith's transversality. So this means that if you take a section of Fp of V bar
18:01
and you differentiate it, you get something that's at worst in Fp minus 1 of V bar tensor omega 1 C bar log log. For all p.
18:29
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.
18:54
So it gets its rational structure from the q local system, so with induced hodge and weight
19:05
filtrations. And now the next condition we want is actually a non-trivial condition, is that for each p in S, let's just
19:34
let np be equal to the residue at p of the connection.
19:41
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 Deline's construction implies that all the weight bundles extend through as well. So the w dot extends as well.
20:03
But this is not a hypothesis, it's automatic. And so what we want is that each np,
20:23
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.
20:41
And five is that for each v in the tangent space of p of C bar, this guy not equal to 0, so I'll call it vv.
21:06
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.
21:32
Then with the weight filtration and the Hodge filtration, I want to take m dot.
21:42
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.
22:01
So it's a more complicated structure. And if you go back a good exercise to look at the definition of the variation of Hodge structure, in that case, there's w dot, or the w dot is trivial, and then you just have m dot's going to be the monodromy weight filtration shifted. So anyway, these are basically the axioms
22:22
of an admissible variation. Such a thing is called an admissible variation. Is the fact that the residue preserves the weight filtration automatically? Yeah, because the local monodromy preserves w dot. It's unipotent, so it's logarithm well.
22:44
The logarithm is just a polynomial in the monodromy. I mean, the weight filtration would extend across using its w canonical extension, right? Right, so you can look at the,
23:01
because the local system was filtered, so each wm of v is a local system, you can take its canonical extension. Naturality just implies that that just gives you a filtration of the canonical extension. And so the local monodromy is unipotent,
23:23
and so its logarithm is nilpotent. And it will preserve the weight filtration, because the monodromy, because the filtration is by flat sub-bundles, or by sub-local systems. Anyway, such a, whatever, a v, w dot, f dot, et cetera,
23:48
is called an admissible variation.
24:05
And this definition may look very strange, so the question is, is it natural? And the answer is, oh, let's,
24:22
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.
24:50
So one is, so this is due to Steenbrink and Zucker. By the way, I think the definition
25:01
of the relative weight filtration was made by Deline and Veitou. 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 Steenbrink and Zucker, and they showed that local systems of,
25:24
maybe I'll just say it this way. Suppose I have an x over c, and this is a family of smooth, but not necessarily projective, smooth varieties.
25:40
And I want it to be, by family, I mean topologically, locally trivial. And so in this case here, local systems are k f lower star of q. And every fiber here has a weight filtration.
26:01
Deline's construction of a, so here I'm assuming for every t and c, xt is a smooth variety, so its cohomology has a mixed hot structure. You can easily see the weight filtration's locally constant and these guys are admissible variations
26:26
of mixed hot structure. This is also a good source for relative weight filtrations. They wrote out a lot of the details. Then there's Guillen, Navarro, Poeta, I think.
26:51
This is my memory. They did the more general case, x over c here, but this is just a topologically, locally trivial family
27:07
of complex varieties. So not necessarily smooth, not necessarily complete, right? They can be singular, open, whatever.
27:23
And again here, you'll get the same result. Rk f lower star of q is an admissible variation of mixed hot structure. This subject loves abbreviations. And then another one we'll need is three,
27:44
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.
28:08
So again, family just means topologically locally trivial here. And so here you can look at the local system.
28:20
So you can look at the local system. Maybe I'll like to call it p. So over t in here, you would look at pi one of xt, and you would look at it with the base point sigma of t, and you would take its unipotent completion
28:40
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 hot structure. And we'll be interested in this
29:00
when this is the universal elliptic curve minus its zero section, and this is really just a section of tangent vectors along the zero section. Sorry? You said that the Lie of the Fourier perturbations contained in p was p? Well, this is, by p I mean the local system whose fiber over t is the unipotent fundamental group.
29:25
And I will eventually give a quick definition of this. I'm gonna say something about Tanaki and business in a little bit, just in the interest of efficiency. It's basically the best unipotent approximation to the fundamental group of the curve or whatever.
29:45
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 hot structure and relative weight filtration and so on.
30:02
And I thought that would be a bad, that was a bad place to start the first lecture. Or it would have been, so.
30:38
So now I'll remind you that this is joint
30:43
with Makoto Matsumoto. Okay, so I'll very briefly recall the rough definition.
31:07
The words will probably make a little more sense. So what is a universal,
31:27
so is a compatible set of filtered local systems.
31:43
And I'll be more precise about this later, but for example, you would have Betty, Q Durham, this together gives you Hodge, and you would have L Addict. So here you would get a Le Chiffre.
32:08
And there are two basic conditions with GRW dot, yeah, maybe I'll call it,
32:23
I'm basically gonna call it VW dot. And with GRW dot of V, isomorphic to a direct sum of SNHR, so H is our basic local system,
32:44
and you'd want something, and so when I say Hodge, implicit in this,
33:02
I mean that I get an admissible variation of mixed Hodge structure. And then also have a filtered mixed tape motive
33:20
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,
33:40
W dot in MTM. Right, so it's gonna be a filtered mixed tape motive, and the mixed tape motive weight filtration's called M. That's because it's gonna be a relative weight filtration such that the fiber of V over the integral base point,
34:01
D DQ, is V M dot W dot. So I'm gonna explain what this all means, but just, this is the overview.
34:29
Okay, so let's look at the official definition. So, a universal mixed elliptic motive,
34:52
it consists of, so it's best to start with the mixed tape motive, a filtered object,
35:09
V W dot of mixed tape motives over Z,
35:20
and denote the weight filtration of V by M dot. So I did that to drive Francis nuts, everybody nuts.
35:41
Two, a representation rho which takes SL two Z into the automorphisms of the Betty realization of V and it preserves this guy here.
36:04
And this guy I should point out, as I hope I explained clearly before, this is naturally isomorphic to pi one of M one one analytic with the base point D DQ. You can lift the base point, the tangent vector D DQ
36:21
to the upper half plane by, as the imaginary axis. So, sorry, yeah, well, you only need the germ
36:43
of it up near plus infinity, you know, sorry? Well, it's part of the, it's part of the definition.
37:07
Ah, okay, okay, let me make, well, no, but you could have all sorts of other representations of SL, SL two Z is virtually free. So take a finite index subgroup, it's free,
37:21
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
37:41
things like SNH tensed with some simple hard structure that's not a Q of R, right? This is a different category, it's going to have a different Tanakhian fundamental group. And I'll try to explain that later.
38:01
I mean, this was a bit of a surprise to me. So I want, huh? If you assume that those associated gradients are the direct sum of SNH R, doesn't that automatically imply that the hard structure at that tangential vector has to be mixed tape?
38:22
No, it doesn't, I think. Ah, 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
38:42
that's not in, that has periods that aren't multisadors. So you just take any mixed Hodge tape structure whose periods aren't multisadors. And now look at the constant variation over M one one with that as fiber. That won't be in this category, because the limit will be that constant hard structure,
39:01
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.
39:22
Yeah. Right, so, sorry?
39:45
When you say mixed tape motive, you actually mean like an object in the green contour. Yeah, I said somewhere here, I think I said it's an object of MTMZ, not just some hard structure that's of type, these graded quotients are of type PP.
40:02
You know, it's a more stringent condition, so you can then say, well, are there any interesting examples? So we'll get to that in a minute. Oh yeah, so,
40:25
where each GRWM of, I just lost track of where I am here.
40:52
Where did I go? Well, let me just write it down. So, this will give us,
41:04
this will give us here a local system, VW dot over M one one analytic, right? And we want each
41:21
is a sum. This is just as a local system of S, N, H. There can be various N where N is congruent to M mod two, right?
41:40
So, just as a topological local system, let me see if I can get these pages. Oh yeah.
42:06
And now the next thing is three. A filtered vector bundle.
42:22
V, W dot over M one one bar over Q. So this is going to be the Q to ROM story. With connection,
42:44
nabla which takes V into V tensor omega one M one one over Q log P. And this guy here is the cusp.
43:00
The cusp Q equals zero. And a filtration, you need a hodge filtration such that the connection satisfies Griffith's transversality, right?
43:27
So, you only lose one degree. And the residue at P of the connection is nilpotent.
43:44
Actually, I needn't have added that. That's gonna be automatic. And so now, we can combine this with a flat connection.
44:04
I forgot to say this, we want a flat connection. And so, then we want the obvious isomorphism
44:27
of the Q to ROM setup.
44:40
So, what we want is if we took V, W dot, nabla and we tensor it with C, this and we restrict it to M one one analytic or just M one one, say C, this should be isomorphic to
45:04
and it's natural flat connection. Right, so this just giving a Q to ROM version of the vector bundle underlying this guy here. And together, they form
45:25
an admissible variation of mixed hard structure over M one one analytic, right? So, and then the last part's the elastic part.
45:50
In fact, one can probably leave out some of these pieces but it's still the way I think about it. I think some, the existence of some are consequences of the other parts.
46:02
Five, so the representation rho L, which is gonna take pi one of M one one over Q bar,
46:21
DDQ into ort VL. So, VL is going to be the elastic realization of the mixed state motive induced by rho. So, this guy here is naturally isomorphic to the profinite completion of SL two Z.
46:43
So, this is SL two Z 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.
47:02
So, the Galois group acts on this because this base point here is Q rational and it also acts here because the L-adic realization of V is a Galois module. Yep, all right, so that's the end of the definition.
47:24
All right, so, a couple of quick remarks. Sorry? To provide the VL in definition of what can be brought. No, we start out with a mixed state motive and a mixed state motive has an L-adic realization.
47:42
This is the L-adic realization. So, that comes equipped with the Galois action. What's the origin of this parity thing? Sorry, I'm wrong with the atom up here. Well, because this is a variation of weight one and I mean, this condition is forced by
48:01
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.
48:21
The way it occurs as a variation is unique up to tape twist. So, you can't put an even power in an odd weight and 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 similarly can define
48:47
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.
49:00
Here, there's only basically one choice. And you can also do, and you can also do mixed elliptic motives over things like M1, N plus R and you can take a level, I don't know what, lambda, you know, level structure.
49:22
And maybe here you have to walk over some ring of integers and so, this guy here, I will denote this category by M, E, M, N plus R. The main examples are going to be very simple.
49:44
And so, the category I'm calling M, E, M is really equal to M, E, M1. That's the one I've been talking about, but you can do it with a tangent vector.
50:22
So, let's look at some examples. So, the first one is, these are the simple.
50:41
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. They occur, I've already shown you that these guys degenerate to a direct sum of tape motives.
51:00
So, these are, we've already verified this. The second example, what I'll call geometrically constant. So, these are just,
51:22
you just take, i.e., you just take a V in MTM, and you just look at, and pull back.
51:45
So, think of the mixed tape motive as being a local system here, you just pull it back here, and it'll be a, as a variation of hard structure, it'll just be a constant variation of mixed hard structure. And as a Le Chiffre, it'll have trivial monodromy on the geometric part of the fundamental group.
52:03
It'll just be a Galois representation. Okay, so these guys,
52:29
and third example, and I, actually I'll talk about a slightly more general situation.
52:41
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.
53:02
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, you can pull back along the zero section to get something over here.
53:21
So, their restriction to the zero section give, you get these simple extensions, which are of the form S two N H. And we'll see in a bit that these correspond
53:44
to Eisenstein series. Non-trivial extension that corresponds to the norm G two N plus two, the Eisenstein series.
54:04
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. You know, if you think what we're trying to do is construct an extension for every elliptic curve, these are the generalizations of the values
54:22
of the Riemann zeta function. You know, that you get in the theory of mixed state motives. So, these are analogs of zeta values.
54:54
So, four, and this example is important for us. So, let's look at M one one.
55:01
So, I'm gonna look at M one vector one. 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 gonna look at the local system. Again, I'm gonna call it P. And the fiber of this guy over here
55:21
is what I'm gonna call P E V. So, this is a pronial potently algebra. So, P E V is defined to be the Lie algebra of pi one unipotent of the elliptic curve
55:42
minus its identity. So, this is E minus zero V. So, this is a freely algebra. So, this guy here, just to give you some idea, is isomorphic to the freely algebra
56:02
on H one of E completed. This is not natural. And the weight filtration will be its lower central series.
56:21
So, this is a pro. And here, this is one reason I need some decorations
56:40
because I need base points. So, do you want a coffee break?
57:08
I can keep going. Maybe I, why don't I throw something up on the board and then we can take? Three is a certain quotient of four. Sorry? Three examples of three. Yes. Well, these things are gonna occur everywhere.
57:24
And so, yeah, they certainly occur inside this guy up here. So, the next thing is,
57:40
so, the Tanachian fundamental group of M E M. And you can also do with some decorations here.
58:02
They're very, 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 M E M,
58:28
N plus R, are Tanachian over Q. And this means they're the category of representations
58:40
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 one of M E M.
59:05
So, let me stop here. I'll start up after the break. I'll do a very, very quick resume of some things about Tanachian categories, just so. And in fact, then I'll rigorously construct unipotent completion in a way that Francis thinks is illuminating,
59:21
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
59:41
that we can't write down explicitly. So, I'll do that after the break. So, this will be.
01:00:00
be very brief. So f equals a field of characteristic 0. And so a neutral category over f
01:00:24
is a rigid abelian tensor category, C.
01:00:46
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.
01:01:01
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,
01:01:38
and they're usually denoted by omega,
01:01:40
from C into vector spaces into finite dimensional. This is going to be finite dimensional vector spaces over f. That preserves tensor products.
01:02:03
So the first example, one, g is any group.
01:02:52
And you can look at the f representations of g. So this is just representations of g on a finite dimensional vector space.
01:03:03
Category of finite dimensional g modules, finite dimensional over f g modules.
01:03:25
And two, mixed tape motives and mixed elliptic motives, n plus r. And I should say here the trivial object here
01:03:40
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 hard structures. And here the trivial example is q of 0 again.
01:04:06
And four, you can look at mixed hard structures x. So it's equal to the category of admissible variations.
01:04:24
And so x here is a smooth variety. So the basic fact we need is that I
01:04:46
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 vec f
01:05:04
is a fiber functor, is any fiber functor. So that's one of these exact and faithful functors. Then c is equivalent to rep f g, where
01:05:27
g is the tensor automorphisms of omega. So what's an automorphism here?
01:05:40
These are natural transformations. Actually, I should say natural isomorphisms from omega to omega that preserve that respect tensor product.
01:06:07
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 affine. It's also pro-algebraic, pro-linear algebraic actually.
01:06:30
And so our goal is to write down this group when the category is just MEM, mixed elliptic motives.
01:06:43
And so we're going to do notation is that g is equal to pi 1 of c with base point, this fiber functor.
01:07:16
So let's do an example, a more interesting example,
01:07:23
unipotent completion. So how do you define the unipotent completion of a discrete group? Well, one way is to use Tanaki in categories.
01:07:41
Sorry, discrete group. And so f equals a field of characteristic 0. And in this case, we're going to take
01:08:01
c to be the category of finite dimensional unipotent representations of gamma.
01:08:22
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.
01:08:41
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,
01:09:04
by gamma submodules 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 submodules,
01:09:21
and all the associated gradients are just trivial modules. And now this category is Tanakian. So omega takes c into vec f. It just takes a representation, goes to its underlying vector
01:09:41
space. So it's clearly faithful. It's clearly exact. And so the definition of unipotent completion is, definition is that gamma unipotent over f
01:10:03
is defined to be pi 1 of this category with respect to this fiber functor.
01:10:22
So 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. I'll just remark. So this is an older way of thinking about it, but one way is if gamma is finitely generated.
01:10:44
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.
01:11:03
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.
01:11:23
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.
01:11:48
And it defines a topology on this by the powers of this ideal. So this is just, so we can give this the iatic topology. And so this is just also the direct limit
01:12:00
of the Hom f gamma mod i to the n f. And this is a Hopf algebra. You can easily check.
01:12:21
And this more or less comes out of Quillen. A long time ago, Quillen wrote a paper on rational homotopy theory. And buried in the appendix is a section on what he called Malteff completion, but nowadays is called unipotent completion.
01:12:45
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.
01:13:04
And this is just the inverse limit. I should say the completed enveloping algebra of f gamma mod i to the n.
01:13:29
And so our problem is to compute pi 1 of M e m.
01:13:46
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.
01:14:01
If you use different fiber functors, the fundamental groups are isomorphic to an isomorphism unique up to conjugation. So let me start with generalities.
01:14:22
So most of this is very soft, with one exception. The first statement is, well, we have a functor from mixed tait motives into mixed elliptic motives. This just takes a mixed tait motive
01:14:40
to the geometrically constant things. So we've got M 1 1, and we map down to spec z. And you can take a mixed tait motive over here and pull it back here. So these are the geometrically constant guys. And so this gives us a map from pi 1 of M e m
01:15:12
to pi 1 of M t m. And this guy, you can argue directly that it's subjective,
01:15:20
but it's easier to see that you also have a functor from mixed elliptic motives into mixed tait motives. This is just the fiber over d dq. Or if you think of the definition,
01:15:42
a mixed elliptic motive consisted of a mixed tait motive plus a whole bunch of other stuff. So you just forget everything else and just take the mixed tait motive. And this is going to give us a splitting of this guy here. So this will give us pi 1 of mixed tait motives
01:16:06
into pi 1 of M e m. And 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
01:16:21
is the same as its fiber everywhere.
01:16:47
So what we can do is define pi 1 geometric of mixed elliptic motives to be
01:17:00
equal to the kernel of pi 1 mixed elliptic motives mapping to pi 1 of mixed tait 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.
01:17:24
It maps to pi 1 of mixed tait motives. And it's got to be subjective. The kernel is pi 1 geometric of mixed elliptic motives.
01:17:42
And we have a section here. This section is given by the base point, DDQ. In fact, we can compare this with, for example, the et al. fundamental group of M 1, 1. And it's compatible with this here. There will be maps.
01:18:01
But I'll discuss that later. I'll just recall here. Pi 1. So if we look at pi 1 of mixed tait motives, it maps to GM.
01:18:21
So this is basically the fundamental group of the category of split mixed tait 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.
01:18:46
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.
01:19:08
Odd things here. I better put z9. Completed. Completed means take power series, Lie power series.
01:19:42
So proposition is the first statement is that there is a natural homomorphism from SL
01:20:06
to z into the geometric fundamental group of mixed elliptic motives. And here I'll be more precise. I should be the fiber functor is the Betty fiber functor.
01:20:24
And I'm going to take the Q rational points of this group. And so the Betty fiber functor takes the fiber. So omega Betty takes V, a mixed elliptic motive.
01:20:41
And it takes it to the fiber of this over d dQ. And then it takes the Betty realization of that. And two, it is the risky dense.
01:21:08
So this guy here is the only. This is not that hard, but everything else I'm saying
01:21:20
is soft. This requires some work. So let me sketch the proof. The proof of one is easy. So we have functors from mixed tait motives. We can take it to mixed elliptic motives.
01:21:41
So this takes a mixed tait motive to the constant, geometrically constant, mixed elliptic motive. And now here we can map into representations of SL to z, say Q representations.
01:22:02
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. I'm just taking it to the rho. And now if you look at what happens here,
01:22:21
so the fundamental group of this won't be SL to Q 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,
01:22:41
you take vec Q. And then you just take the forgetful functor here, this commute. So you get a map here in omega beta. And you'll get the map into here. And why does it land in the geometric part? Well, if you continue on to pi 1 of mixed tait motives,
01:23:04
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
01:23:22
that you landed in the geometric fundamental group. So this part is completely soft, the first assertion. Two, so whoops.
01:23:55
I've got an old version and a new version of this page here, so I have to go delete that.
01:24:05
So this follows from A is the theorem of the fixed part.
01:24:25
And what this theorem says in this case is if you have a variation of mixed tait structure which has trivial monodromy, it has to be constant,
01:24:44
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 tait structures. But you can use that to prove the following statement implies
01:25:02
that an variation of mixed tait structure with trivial monodromy is constant.
01:25:30
And so this implies, so if you think about it, if you map this into here, it's telling you
01:25:42
that the Zariski closure of this has got to generate the normal closure. If you look at the normal closure of SL2z in here and take it to the Zariski closure, it's got to be the whole thing.
01:26:00
So this part's also relatively soft. This implies that the Zariski closure of the normal closure of the image of SL2z
01:26:32
is that we have to see that the image of SL2z,
01:26:44
the closure of it's normal. And the way we do that is you have to use the fact that, so the existence, so this is the non-trivial part. The existence of the limit mixed tait structure
01:27:06
on the relative completion, which I'm about to define,
01:27:20
implies, so you have to use DDQ as your base point, implies that the Zariski closure is normal in, right?
01:27:59
And so this implies, you put these two things together,
01:28:02
you see that the, so you'll see that the Zariski closure of this inside pi 1 of MEM is pi 1 geom. If you don't like that, I've got a version
01:28:30
of this written up somewhere. Anyway, this is important because it's going to allow us to band on the size of the kernel.
01:28:49
OK, so we're going to call a mixed elliptic motive a split if it is a direct sum of SNHRs of simple.
01:29:11
These are the simple objects of mixed elliptic motives. And so split.
01:29:22
It means semi-simple because you say it's an item. Yeah, OK. So these are the semi-simple mixed elliptic motives. Yeah, better. The semi-simple.
01:29:48
So semi-simple mixed elliptic motives is a Tanakian category.
01:30:03
It's a subcategory of the whole thing, which is clear. And then pi 1 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.
01:30:27
This is, you may wonder about the Tate twist, but if you look at, for example, GLH, yes, it's actually the inverse of the determinant. Determinant, if you look at this representation into GM,
01:30:45
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. Because you're always putting H in weight. Yeah.
01:31:00
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
01:31:26
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.
01:31:42
And so what this gives us, so this gives us, we're going to have pi 1 of MEM, and we can use any fiber functor, mapping into GLH. And it's got to be subjective, because we have a splitting.
01:32:01
And the splitting is GRW dot. And so there'll be a kernel. And what we'll see soon is that that kernel
01:32:20
is pro-unipotent. And 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.
01:32:42
And in the theory of motives, you can only extend by stuff of lower weight. But we'll see it another more direct way, that the kernel of pi 1 of mixed elliptic motives into GLH is pro-unipotent.
01:33:19
OK, so let me, I brought up the issue of relative completion.
01:33:22
So let's look at, maybe I'll call it relative unipotent.
01:33:40
And this idea, I might have been the first person to write about it, the idea came to me in a letter from Delene, 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.
01:34:05
r is equal to a reductive f group. You don't have to take it to be reductive, but there's no real difference.
01:34:21
And I'm going to put an example here, r equal to the trivial group. This is going to give us a unipotent completion. So this is just going to generalize what I did earlier.
01:34:42
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.
01:35:03
It's going to be the category of finite dimensional representations, v, of gamma.
01:35:30
Whoops, I left out gamma here as a discrete group. So I'll put it up here. Gamma equals discrete group.
01:35:40
In all of these, there's also a profinite analog. So I'm in this splitting here, but 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, there's an outer factor through the gradient point.
01:36:02
It's true for any fiber function. Yeah, but OK. But I can take, what's the fiber function? I can take it to be graded. Sure, sure, sure. You have to take one, too. Yeah, you're right. I have to take graded and say, Betty, or?
01:36:23
You've chosen the fiber function, didn't you? 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.
01:36:41
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, we've got a base point here, but we may choose a section to get a different base point.
01:37:05
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.
01:37:24
0 equals v0 contained in v1, contained in, say, vn equals v by gamma sub-modules,
01:37:43
such that the associated graded v is in our module. So you've got an action of the algebraic group, the reductive group on the associated graded,
01:38:03
and gamma acts via on gr.v via, you'll have gamma, we'll map into, and this acts on grw.
01:38:30
So the action on the, all the graded quotients really come from representations of r. And the way gamma acts on them is via this rational representation of r.
01:38:48
So this is Tanakian, and this is an obvious fiber function, just take the underlying vector space.
01:39:03
And so the relative completion is, I'll call it, grl is defined to be pi 1 of this category. And here, this is the obvious fiber function.
01:39:52
So some important information about this group is proposition is, one is grl is an extension,
01:40:15
r1 into what I'll call url into grl into r into 1,
01:40:30
where url is pro-unipotent. Actually, in case I haven't said it, pro-unipotent just means an inverse limit of unipotent groups.
01:40:44
And if you like us exponent, u of, do what? I just write rl here for relative completion. I mean, again, I adapt the notation to the situation. Here we've got one group that we care about,
01:41:02
and that's SL2z. So sometimes you've got to put the dependence of the group in there. There's a natural homomorphism from gamma into grl.
01:41:22
And to be accurate, I should put f rational points. It is a risky dense.
01:41:44
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.
01:42:08
Well, maybe I'll start out with if u is unipotent and u is equal to the Lie algebra of u,
01:42:21
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.
01:42:40
That's just because every unipotent group can be moved inside this guy here, and then it becomes clear. And so now if you have a unipotent group, and this guy's going to be an ill potent, if u is pro-unipotent, still true.
01:43:07
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,
01:43:26
if u is pro-nilpotent, u has presentation, has a minimal presentation.
01:43:45
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,
01:44:06
let's call it say phi c, into the commutative subalgebra. And you look at the image of c, and you take the closed ID, all that generates isomorphic.
01:44:31
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.
01:44:42
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.
01:45:06
So we care about what H1 and H2 are of this guy here. And so it's for all r modules v. By that,
01:45:20
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 tensor v,
01:45:50
r invariance, injects into H2 of gamma v. Let's put this here.
01:46:01
So now, this is telling us something about the presentation. So maybe a good place to stop today will be, I'll do the example of SL2z. So we'll have a lot more to say about it, but let me just start with the basics.
01:46:20
So we know that the relative completion of SL2z is going to be a homology of cohomology, and again, that statement, homology of cohomology. This is cohomology. So what's in singular homology, H1, the cohomology is the dual of homology.
01:46:42
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.
01:47:04
So let's just look at it. Where do you use that? Sorry? At this point that you use that, R is relative. R is a relative point. Yes, it's really important. So example, we'll take that F equals Q. Gamma is equal to SL2z.
01:47:36
R is equal to SL2 over Q. And rho takes gamma.
01:47:48
Well, 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,
01:48:02
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
01:48:22
than or equal to 1 H1 of SL2z S2nH tensed with, and I'd like to write it this way, S2nH dual.
01:48:41
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,
01:49:01
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,
01:49:23
if this is divisible, this is a Q module, is equal to 0. And then when you assemble that with this statement here, that tells you that H2 of U is 0.
01:49:42
So this is telling us that H2 of U is equal to 0. But this discussion I had over here, if H2 of U is 0, that's saying that U is free.
01:50:02
So therefore, U is free. OK, so what we know unnaturally is that, so this is saying that U is isomorphic to the freely
01:50:20
algebra on the direct sum of H1 SL2z S2nH. So I left out the odd powers because the cohomology
01:50:41
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.
01:51:03
So we're taking 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. So we complete by taking the inverse limit of all finite dimensional quotients of this
01:51:21
that have an SL2 action. And then Grel is unnaturally isomorphic to SL2 semi-direct product U. And then the final comment is that I'll pursue next time is Eichler-Shimura tells us that these cohomology groups
01:51:49
here are basically just modular forms. So you're getting a copy of S2nH for every normalized eigenform.
01:52:04
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-Shimura. And I'll explain in more detail about this. And then there's a theorem that says the coordinate ring
01:52:20
of this for any base point, including DDQ, any base point of M1,1 has a natural mixed hodge structure. And the hodge representations of that are our admissible variations. So I'll stop you.
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