 Mae ymddydd ymddydd yn ystod i'r newydd, y newydd homofig yn ymddwyr sy'n cyd-1. A'r ystyried ymddydd ymddydd ymddydd yn teimlo. Mae ychwanegwch, mae'n rhywbeth a'r oedd yn gwybod yw'n ffordd am ychydig ymddydd ymddwyr, That's the right object and how it's connected to geometry and mixed motifs. So, if you're interested in that part of the theory, then you can skip straight to lecture four, and hopefully there will be a future, my future self will be explaining that. Failing that, there's some pre-prints on the archive that are very elementary and very easy to read. So, in these lectures, I want to start at the very beginning. ac mae'n rhaid i'w ddweud yn ystod o'r hynny. Felly, rwy'n meddwl yng Nghyrch Gwtendig yn 1984, yn ysgisdynt ym Mhrogyrn. A Nghyrch Gwtendig yn ystod o'r gweithio'r ffordd o'r gweithio'r galwau ar y system o'r ffordd ymgyrch. Fy ffordd ymgyrch gyda'r ffordd o'r ffordd. So pi1 hat of MGN with respect to some system of base points of fundamental groups or rather fundamental group oids of MGN, the modular space of curves of genus G with N marked points. So he laid out this huge program. The impetus for this was, according to him, inspired by Belly's theorem, I think I hope I got the accent in the right place. So this is 1980 I believe, and a corollary of Belly's theorem, which we'll see again in a minute, is that the absolute Galois group Q bar over Q already acts faithfully pi1 hat already for genus zero with four marked points. I'm going to leave the choice of base point blank for now. So Goddindig's hope was that you could try to understand the absolute Galois group geometrically, and that's inspired a lot of work. But the point of view changed significantly in around 1989 when Deline Drenfeld and Iharra independently changed perspective somewhat. And here they only focused on the case M04, which is isomorphic to the projective line minus three points. And that's certainly unfair. Drenfeld also considers the case M05 in a significant way. And instead of looking at the profanac completion, they switched focus to prounipotent, or in Iharra's case pro-unipotent pro-L completion of the fundamental group. P1 minus zero on affinity. This is back to some base point. So this is very simple. This is just a free group on two generators. So nothing could be more concrete. And so later than in 2005, Deline and Gontroff, as laid out in Deline's original paper along those lines, proved that the unipotent completion, which I'll write with a superscript un, of P1 minus zero on infinity, and now I better specify the base point. So for now it's just a symbol. It's a tangent vector of unit length emanating from the point zero. So it's called a tangential base point. We'll come to that later. So this object is the betty realisation of a motivic fundamental group, pi1-mot, which is a pro-object in a category, or the category of mixed-state motives over z. So concretely, that leads to some very classical objects. So in the Aladdin setting, which was especially studied by Iharra and his school, these are iterated extensions of cyclotomic characters. And the betty and deram realisations, or rather their comparison, leads to periods, and the periods of this object are given by certain integrals, which are multiple zeta values. So here we had the absolute Galois group acting on the profinite completion. So what's the analogue of that statement? Well instead of the absolute Galois group we have something called the betty motivic Galois group, the fundamental group of this category, which acts on this unipotent completion. So there will be a lot of words that you may not be familiar with, but this is just the motivational part of the talk, which will last for about an hour before actually starting with some definitions. So this is an extension of the multiplicative group by a unipotent, a pro-unipotent group, whose graded Lie algebra is free on generators sigma 3, sigma 5 in degree minus 2n minus 1. And so these elements are very well known, they're called zeta elements or Sule elements. Okay, so in this course what I want to do is to take this theory, this modification of Gottendieck's original theory, and try to extend it to the whole case of MGN in particular. And the most important first step in that is the case M11. So in this course I want to study the motivic version, in other words this sort of analogue of Gottendieck's original programme, in the case of M11. So M11, the modular stack of elliptic curves. And the orbital fundamental group, so the fundamental group that plays the key role here is the orbital fundamental group. Again there's some base point which I'll explain in the next lecture. It's a tangent vector at the cusp, so it's d by dq, a unit tangent vector in the q-disk. But this is equal to the group SL2z. So this whole course is just the study of SL2z. And it's quite surprising how much mathematics is going to come out of this situation, already very simple. So the first problem when we want to copy or generalise this motivic story for M11 is we run into a problem. We can, the naive idea is to look at the unipotent completion of SL2z. So what's the unipotent completion and we find that it is trivial. So that's not very interesting. Another thing you might try to do is look at the algebraic completion. And this is absolutely huge. It somehow looks at you're studying all possible algebraic vector bundles on M11. Already the algebraic completion for P1 minus 3 points is enormous and unstudiable at this point. But there's something in between called relative completion which I will define today. I'm not going to talk about it. It's the algebraic hull. I will define relative completion. You can write it as a limit of all possible relative completion. It's some sort of algebraic envelope of the fundamental. There's a Tanachian definition which I can give it to you when I give the Tanachian definition of this. So the relative completion is something that sits in between that's just the right size. And it is just right. I mean the short answer is you look at all SL2z representations. That's a Tanachian category. Take its Tanachian fundamental group. So it's huge. It's very, very large. And this will be something exactly in between the two. So this theory here, relative completion, contains, that's closely related to the theory of universal mixed elliptic motives of Heine and Matsumoto. So their theory is something which is very much mixed-tate. So it's very closely related to P1 minus 3 points. But the difference here, this object as a motive is going to be very much not mixed-tate. It's much, much bigger. It's what you could call mixed-modular. So now some motivation. Why do this? So the question that motivates me is how can we classify or how can we find in nature all mixed motives? So this is clearly a very only ambitious question. But let me illustrate with two situations which are known. So two examples where the answer is more or less known. So here are some examples of pure motives which we know how to construct. And essentially a rank two motive, this is slightly vague, but this has motivation. Rank two motive, so over Q let's say, by the work of Wiles and Company comes from the cohomology of a modular curve. Of course their theorem is much stronger, you need to look at a single prime. And the cohomology is something which is Abelian. And in the mixed case there are very few examples which are known. The one that's going to be self-motivation is we can look at iterated extensions of some rank one motives, particular tape motives, over Z. And these are all generated by not the cohomology, but the fundamental group, unipodent completion of P1-3 points. And the fundamental group is something non-Abelian. So the idea is to take the product of these two situations and try to construct extensions of motives of modular forms by looking at not the cohomology, but the fundamental group of modular curves. So this theorem is something I proved in 2012, I think, that the motivic fundamental group generates all mixed tape motives over Z. Right, so what is the program then? The program is to take gamma, a subgroup of SL2Z, of finite index. So for now let me not necessarily assume that it's a congruent subgroup and take any subgroup of finite index. And put Y gamma to be the upper half plane quotiented by gamma. I put double slash to denote an all before quotient in the case that gamma has torsion, for example. And you want to study the system of relative completions. So they will be some objects, pi1 rel. Again, I'm not going to specify base points for now. And they are going to have plus the action of a group on them, a motivic galore group. So the question looking in the background, which is whether the analog of such a theorem holds in this case. So the question that I would love to know the answer to is do these objects. So let's just restrict the question and consider the case of congruent subgroups, some modular curves, generate all possible mixed modular motives. All possible extensions of motives of modular forms. And immediately there's a problem because there's no eraser, but there is now. So the problem here is that there is no appropriate motivic theory that's going to help us with this question, unfortunately. And so we have to have a workaround. And the workaround, which is already the case in Deline's original paper on p1-3 points. The workaround is to work in a Tanachian category of realizations. This will be something very concrete. So I would call such a category curly H. So what we're getting then, the objects that we're studying then in fact, for each realization we're going to get a different copy of this completion. So what the data is then is we get a collection of groups or other affine group schemes. If you prefer to think in terms of that affine ring, these are hopf algebras. So we get affine group schemes, pi1 rel omega for every realization omega, where omega will be betty doram, and you can add more realizations. Plus, there will be the action of a pro algebraic G omega H, which plays the role of a galore group upon it. So that's what we're studying. It will be, in fact, I'll only focus on the betty and the doram. So we're going to have two group schemes, two hopf algebras, that are comparable in some way. And there's going to be a group acting on each one. In fact, I'll only focus on one. There'll be the doram group acting on the doram group scheme. So we've got a pair of groups that isomorphic in some way, and a group acting on them. It's very concrete. Sorry, I didn't understand this part. Is this action of a motivic galore group acting on this space Y gamma? It's acting on the group. It's acting on the group. It's actually on the group. So we get a group with a group acting on it. So this is the game we're going to play. Is it important that Y gamma is your curve? Well, yes, it is. Well, in some sense, yes, because you really want the, so if you want to go in hard dimensions, so if there's no H1, if there are no modular forms, so the pi1, unipodent pi1 relative, is built out of the H1 of the space with curvature and some vector bundle. So if the H1 vanishes, then this thing is trivial. And that's the reason why this is trivial, because there are no modular forms of weight two on level one. So you can fix that. You can do something else, but it doesn't have a nice name relative to fundamental groups and some higher thing. So it's not here if you can replace a part of the thing by zigo? Yeah, I think you can. I think you can. But if you take pi1 rel, you're not going to get anything very interesting. But I think you can do something else. But that's a bit futuristic. Already we have so much to, already we'll see that the case of SL2Z on its own, wherever it is here, is incredibly rich. And I'd be surprised if we understand everything in my lifetime. So definition then, a mixed modular motive, maybe not a motive in the strict sense, but it's certainly a system of realizations that come from algebraic geometry, is a sub quotient of this fundamental group, this relative completion. I prefer to think in terms of the affine ring. So this is a group scheme, its affine ring is a hopf algebra. So dot is the various realizations omega and some base point in a category of realizations h. So this enables us to define some working definition of what a mixed modular motive should be. And if we look back at this theorem, then the analog would be that a mixed tape motive over z is a sub quotient of the affine ring of that. But by this theorem, that's exactly the definition of it, that's exactly mixed tape motives. So inspired by this, it seems a reasonable definition in the absence of the material theory to define a mixed modular motive to be a sub quotient of something that comes from a modular curve. So then we can define, this is the continuation of definition, the category of mixed modular motives for gamma will be the full Tanachian subcategory of the category of realizations h generated by this thing. I'll write it up again. And it has certain periods, and I want to call the periods multiple modular values in analogy with the fact that the periods of the fundamental, the unipodent fundamental group of p1 minus 2 points are multiple zeta values. And so what we've got then is we've got a group, this sort of metallic gamma group, is going to act on this other affine group scheme if I fix some realisation omega, say d by dq. And this action factors, so it has a huge kernel, but it factors through a quotient g omega mm gamma. And this is, the connection with this category is that the category of mmm gamma is equivalent to the representations of this group. So the reason why this group action is important is because it precisely controls how big this is as an object, as a mixed motive in this sense, as an object of h. So studying the structure of this group and studying this group action is equivalent to seeing how rich this category is. So if this action were completely trivial, so if this acted in a completely trivial way, then that would mean that this category mmm gamma is trivial, uninteresting. So we want this action to be as rich as possible and that means that we've got a very rich category of mixed modular motives. And that means you've got a very good chance of generating all possible such objects. G omega doesn't depend on gamma. G, this is a motivic gallery. It just depends on h, this thing. This doesn't depend on gamma in any way, no. Yeah, it's just some category of realizations to set up some kind of group action on this. So this is some realizations. So it could be, oh, this is omega. For some omega. Sorry, it's the system for all omega in realizations. So I don't know the definition of periods here. Are they dependent on this omega? The periods will, so this is just motivational. I'll give proper definitions later. But the periods will be, so what will happen is you'll have, there'll be two situations, omega-deram and omega-betty, and they will be connected by a comparison isomorphism. So what we'll have is p, p1, rel, deram, sorry, p1, rel, betty is going to be isomorphic to p1, rel, deram. And the periods are going to be the numbers that come out of this isomorphism. In other words, you can look at the smallest ring, subring of C, such that this is true. And that will be a ring of numbers, and those are periods. They are the periods of the relative completion. I will give definitions as I go along. Here I just want to explain where we're headed and why it's interesting and what the tools are. So the tools, where can I put this? The tools are the following. So what we've got then at the centre of the picture is this group GMMM Gamma. It's very mysterious. It's precisely what we'd like to understand and it's going to act on this relative completion, which, as you'll see, is something very concrete. Another way to say this is that, so this is at the centre of what we would like to understand. Another way to say this is that there's a homomorphism from this particular Galar group. It's going to embed into a group of automorphisms, this relative completion. And we want to get some sort of control on this. And a priori it looks impossible, but there are three tools that we can use. First is to compute integrals, so periods, and they will enable us to show that this action is non-trivial in many cases. So in other words, it's going to give a lower bound for the image of this Metific Galar group in this automorphism group. And on the other hand, we have Baylinson's conjectures. We've picked an upper bound for this group. So we get an upper bound from here, and we also get an upper bound from relations. So there are relations, if you like relations between periods, for example, control the image. And these things give an upper bound. So this is very similar to the case of P1-3 points where we have all the Metific Themes that we could possibly want. The difference being only that in the case of the Mixtate case, this is a theorem. The upper bound on the size of the Metific Galar group comes from Borell's theorem. But in this case, we don't have that. We have to appeal to Baylinson's conjecture. And if you like, this is a testing ground for Baylinson's conjecture. And this will not be fixed by defining some Abelian category of motives. It's really something much stronger. OK, so those are the tools. And now I want to draw another sort of motivational picture of where, how this theory fits together and how it connects to other subjects. It controls the image. So here, this is some abstract group. And relations will say that this lands in some subgroup defined by some equations. So we have a group sitting inside another group. We have no idea what this looks like, a priori. And it seems completely impossible because this is extremely mysterious. But we will see that we can describe this quite explicitly and these relations will cut out some very explicit subgroup of this. And then we can find lower bound and try to compare the two and try to get them to squeeze it from both ends and actually nail down this part. So I will prove some real theorems about this group and hence about this category. So the landscape is we have sort of, at the centre of everything, we have the case M11. So this is SL2. So SL2 clearly plays a very important role in all of this. And one direction of generalisation following Grotendiggs' esquistin programme is to replace M11 with MgN. Look at the relative completions of mapping class groups. Now from this, this is connected to, because you get a, the fibres of MgN of MgN minus one are curves with N sections. Out of this, let me cut a long story short, you're getting an action on unipodent fundamental groups of curves and you're getting symplectic derasions on freely algebras. And by a theorem of concebitures this is in turn related to the cohomology of the outer automorphisms of the free group on N generators. And a different direction of generalisation, which I hinted at earlier, is to look at relative completions of fundamental groups of all quotions of the upper half plane by gamma, so not necessarily congruence, not necessarily congruence subgroups. And these receive all, so we know that this has to be extremely rich because it already receives all pure motifs of curves, of algebraic curves of a Q bar. This is because of Belly's theorem, which implies that every smooth connected projective algebraic curve over Q bar can be uniformised as a quotient of the upper half plane by some finite index subgroup of SL2 there. Take it to be torsion free. So this is going to be extremely rich. It's some huge mixed motive and it's certainly going to contain the motives of all curves, all possible algebraic curves. So this phenomenon Gordon Dick refers to in his Esquistam programme and he calls SL2Z a machine-a-motif, but he's really thinking about the case of pure motifs there, not mixed motifs. No, it's absolutely not a congruence subgroup here, so non-conguence. So that's the picture and the two different directions of generalisation and we see that the key thing to understand is just the case of SL2Z of M11. So that's what I'm going to exclusively focus on in this course. The key example is therefore M11 and gamma is just SL2Z, so nothing could be more concrete. There's another reason I'd like to say why this particular object is extremely important for this generalisation and that's due to something called the two-tower principle which is first stated by Gordon Dick to my knowledge in an extremely vague comment in his Esquistam. Since then it's been refined somewhat. And that's basically that in order to understand this system of fundamental groups plus that action H, then you only need to look at the first two modular spaces in the modular tower. So there are four spaces that play an important role. There's unipotent completion of M05, there's unipotent completion of M04 which I remind you is just P1-3 points. So this is classical. It's been studied a lot by these papers by Deline Drinfeld and Ihar and it's entirely mixed-tate. So in terms of motors it doesn't generate very much. And then we also have P1-Rail M11 and we have P1-Rail M12. And the two-tower principle is that it's a kind of van Kampens theorem that if you look at the compactification of MGN then its one skeleton is generated by these spaces. So the fundamental groups already generate all the objects and the relation, the two skeleton is generated by these spaces. So these are going to give relations. So what this means is that it really suffices only to understand everything, MGN you only need to look at these four spaces. And that's why this space is particularly important. So then you could ask why am I just focusing on that? Surely I need to be studying M12 as well as M11 but in fact the passage from here to here is very small. In fact you can show that the passage, so the study of the relative completion of M12 is equivalent to essentially the study of the relative completion of M11 plus a little bit more plus a so-called monodromy representation on the fibre, which is the unipotent completion. So unipotent completion of the fundamental group of the tate elliptic curve. And again this is again a mixed-tate object and it's very closely related to this space here. All the new information then is coming from this single object. That's precisely why I'm going to focus only on this. It really contains, it knows everything. This monodromy representation of the fibre is this one. There's a representation of this. I have to be very slightly careful here but essentially there's a representation of this on that. So you've got a huge very rich object which is not at all mixed-tate, it's a mixed-modular, a very big thing which we'll describe and it acts on a very small mixed-tate thing. And once you know that action, that's enough to recover all the information from here. So all the new information is really contained in this object. OK, so applications, the clock's an hour, an hour out. So briefly some applications of this. Some of which have been worked out, others haven't. One sort of very low-brow sort of accessible application is that already when we study p1 minus 3 points, there's all sorts of strange modular phenomena that occur. And this has been known for a long time that if you really want to understand the finer structure of this situation, it's being controlled somehow by modular stuff. So let me just give you one example of this. Modular phenomena in mixed-tate motives. So there's nothing modular at all about mixed-tate motives a priori but the modular world casts a shadow in some strange way. So to illustrate this, I'm just going to give a relation between multiple zito values that's due to, found by Gangle, Canico and Zaggy. And they looked at double zito values, first studied by Euler in the 18th century, and they found this relation. So they found an exotic relation that occurs for the first time in weight 12, and they showed that these relations are in fact in one-to-one correspondence with cusp forms for SL2Z. And in fact this relation comes in a direct way from the Ramanujan function so that this one corresponds to the Ramanujan function delta in an explicit way. So already if you're just studying multiple zito values you see that modular forms are coming in and doing something very subtle. So that's one first motivation for passing to M11 after having studied, after having understood the case P1 minus 2 points. A second application is about which I'll say very little, well nothing in fact, amplitudes in high energy physics. There are increasing examples of numbers that are required for amplitude calculations in high energy physics that are of mixed modular type. And you need a theory to construct these, study these, understand these. And that's exactly what we're doing here. Ffiamon amplitudes, there are amplitudes in superyang mills in string theory, super string theory, strong connections with. Another application which is part of this question that if it's the case that as we hope that we can generate all possible mixed modular motives out of these fundamental groups, we should be able to find some very, very simple examples. We should find what are called Rankin-Selberg extensions. So these are classes. We should find extensions which generate whatever these symbols mean. We should be able to construct all possible extensions of a trivial object by a tensor product of two modular forms. So Vf is the demotive, if you like, of cusp form. So already this is not known how to do this and it has applications to the Bertzwinnerton dark injection. So one of the first things we hope to be able to find is these guys, and I think I know. I haven't particularly studied this question, but I think we know where to find them. OK, so the plan then, if I don't run out of time, is to have sort of two parts. The first part, which I'll do this afternoon after the break, is to study and define relative completions just of groups. So this is purely group theoretic and we want to focus on the case sl2z. Then I want to explain that there is a mix-hot structure or to describe the mix-hot structure on this relative completion. Then I want to explain what the periods are. So these are given by certain integrals, generalising Eichler integrals. These are very classical integrals of modular forms from 0 to infinity. So I'll explain what the periods are and they're related to Manin's iterated Shimura integrals. And then, finally, study the action, define and then study the action of a Galois, or Tanaka group, G on the H on this situation and then to do some theorems about in the direction of this question. If I have time, I will talk about what I had originally attended to talk about, which are mixed modular forms. So all this is some algebraic geometry, if you like. And it's not too hard, but it's complicated. And my hope was that you could realise this entire theory as modular forms. And it's quite surprising, but it's possible to do this. So this is, in some sense, an elementary approach to one. It's completely elementary in the sense that it's on the graduate sort of level. I've written up sort of introductory notes to this for Zaggy's 65th birthday conference. So this is called a class of non-holomorphic modular forms. One, there's a series one, two and three. In fact, it's so elementary, one's going to talk about it because you wouldn't believe me necessarily that it was in any way connected to anything interesting. That's why I'm going to all the trouble to do this first. But in case I don't get to this point, let me just give you an idea of what's going on here and how it's related to one. And so what do you have? We have a motive or some big object, Pi 1 rail, which I apologise, I still haven't defined. And we can, instead of looking at loops, the fundamental group of loops from the cusp to itself, we can look at paths from the cusp to a point Z and allow the point Z to move in the complex points of M11. And that defines some sort of local system or variation of hot structure, if you like. And it has periods. So the periods will give us functions and some of these, but not all of them, are iterated integrals of modular forms. So iterated integrals of cusp forms were studied by Manning a few years ago. And in the sort of the abelian part of this, the abelian quotient will give back exactly the classical Eichler-Schimmer theory. And so these are integrals of modular forms. They're absolutely not modular. They are multi-valued, well, they're functions on the upper half plane, and hence they're multi-valued functions on H mod gamma. So they're not gamma invariant in any way. But you can modify this. There's a different notion of period, which I like to call the single-valued period. And another way to think of it is as a periodic period for the infinite prime. So you could call this an infinity-adic period. And now it gives real analytic actual modular forms on M11. So this huge object, this huge mixed motive, has some kind of realisation as some non-holomorphic modular forms. So it's a real infinity. Infinite prime is real. Yeah, that's a real place, exactly. Yep, it's a real period, a single-valued period. So to explain what the single-valued period is in case I don't get that far, let me just give a simple example of the fundamental group of GM, so P1 minus 0 infinity, at the point 1. So the unipodent completion is something very simple. We'll see that later. But let me just tell you that the periods of this thing are completely described by a single integral, namely the integral from 1 to z. So we've got GM, we've got sort of 0, we've got the point 1 and the point z. We've got some path from 1 to z. And everything is determined by a single integral, dx over x, which is log z. So dx over x, well, spans h1 around. And this is, of course, a multi-valued function. And what would be the single-valued period in this case? So there's a very general recipe. In this case it gives just the real part of this integral. And it's log of the absolute value of z. And now this is a well-defined function on C star. So a priori, the periods of this system of fundamental groups is a multi-valued function. It's a function on a universal covering space. But the single-valued period means that the function descends and it's actually a well-defined function on the space. So if you play the same game here, iterated integrals and modular forms are functions on the upper half plane. But when we take the single-valued period, they actually become modular. They transform in a good way with respect to the modular group. If you have this complex private infinity, then do you get this complex analytic model? Yeah, you could think of it that way. But it's not really the same. The analogy is not quite right. You can think of it that way if you want. Who will give you back this period? So the complex period is a comparison between Duran co-modul theory and Betty co-modul theory. You're integrating differential forms over cycles. And this is integrating a differential form against the dual of a differential form. It's comparing Duran with Duran. That's completely different. So in previous years, do you have these multi-valued functions back? No, you can't get them back from this. There's some loss of information when you go to this. There should be, yeah. So the real Frobenius gives you a period, which is the single-value period, and you could replace it with periodic Frobenius. And that should give periodic periods. But that's not been worked out at all in this setting. So in the case of p1-3 points over here, that's been worked out and these are periodic multiple z values. But in the modular case, we haven't looked at that yet. So yes, there should be some sort of periodic modular forms looking around. So finally, let me compare and contrast what's going on. Say briefly what these mixed modular forms look like and why they deserve the name. So a classical modular form, as you know, is a function that transforms in a certain way. And it has a Fourier expansion. It's a Fourier expansion where the an are algebraic integers. Maybe I should just write down the transformation law. So we have something like f, I'm sure everybody knows this. Faz plus b over cz plus d equals cz plus d to the n f of z. So on this side, we have functions on the upper half plane, which are real analytic and they have a Fourier expansion that's a bit different. So they have a Fourier expansion. Well, it's not really Fourier expansion, but they have an expansion in terms of q and q bar where q equals e to the 2 pi i z. And I'm not going to say what the sum's over. It's just, anyway. And then in this case, the a ns aren't algebraic numbers anymore, but these numbers are periods. So they're conjecturally transcendental numbers like values of zeta functions. Here, the transformation law now is a bit different because we have two modular weights, r and s, for r and s integers. That's how it transforms. So here, we have an action of Gal q bar over q on modular forms, which acts on their coefficients. On this side, we should have some action of a motivic Galaw group. So this group g certainly acts on everything in sight, but I don't entirely know how precise it acts for now. What's the interpretation of a modular form? Well, it's a section. You don't consider it to be mass forms. You consider they are not mass forms. No, absolutely not mass forms. So this is the whole point. There's a total change of direction from the mass, the series of mass forms. So they are not eigenvalues of the Laplassian. So it turns out that the filtration at the first level do happen to be eigenfunctions of the Laplassian, but thereafter, it's absolutely not the case. So there is some overlap with the theory of mass forms, but very little. It goes in a very different generalisation of classical modular forms from mass forms, but there is a small overlap. And these are interpreted in a very different way. There is a small overlap. These are interpreted as sections of the Hodge bundle, so L tends to N, which we view as a sub bundle of the nth symmetric power of the cohomology of the universal elliptic curve. You can think of this as a pure variation of Hodge structure. So it's a sim N, H1E. So these are sections of some variation of pure Hodge structures. It's a local section of something global. No, they're the global sections. These are global sections. And so what are these guys? So why is this a natural generalisation? Because we're looking not at holomorphic sections, but now real analytic sections, global sections. Of not the cohomology of the universal elliptic curve, but the entire fundamental group of the elliptic curve. And we should put unipotent completion here. So we take the punctured universal elliptic curve and there's this huge non-Abelian generalisation of its cohomology, and the real analytic sections of that give precisely these objects. So this thing is not pure anymore. It's an iterated extension of these guys VN. And we can replace this with the full relative completion of M11, if we like it. So this run out of space. So the left-hand side, again, is somehow Abelian because we're only looking at cohomology, which is Abelian. And the right-hand side is non-Abelian. So here we have, this is Abelian. And it relates to modular forms related to pure motives. And on this side, this is Pi 1 is non-Abelian. The punctured curve is non-Abelian. And they're related to mixed motives. And we can keep going. So here we can define L functions of modular forms, which are classical, due to Hecker. And very interestingly here we get some new types of L functions. I don't know what to call them, sort of new mixed L functions, which are very interesting and haven't been studied at all yet. OK, so I hope that this motivates why this class of modular forms that can be presented as a really elementary definition actually comes from something that's very deep. And this theory has several applications, which I may or may not get to, depending on time. One application comes from physics, which is the theory of modular graph functions. So this is not completely proven, but I strongly suspect that this class of objects that were discovered by physicists are particular examples of this class of modular forms. So these are studied by Green, Van Hover, so I should be alphabetical, Green, Russell, Van Hover, and more recently by Zabini. It also overlaps, so as I mentioned earlier, these mixed modular forms are a huge family of new objects. And sort of the very first level actually corresponds with something classical. They correspond with mass, sorry, weak harmonic mass forms, and hence the theory of molecular forms. In some sense, this is a huge non-Abelian generalisation of this particular limit. So that's it for motivation, and maybe we have a coffee break now, and then I actually start doing some mathematics. OK, so in this second half, I'm just going to do some group theory. So I'm going to define relative completion, and this is a notion due to what was studied by Richard Hain. In fact, he's one of the only people who's really studied this. So we start off with gamma, a group, and we have R, a reductive algebraic group over field K of characteristic zero. And we give ourselves a representation, rho, from gamma into the K points of R, and we assume it is the risky dense. So we're really only interested in one example, which is when gamma is SL2Z, R is the algebraic group SL2, and K is Q, and the representation is the inclusion of SL2Z into SL2Q. So that's the example to bear in mind throughout this. OK, so the definition is that there exists, so I'll give the definition in a minute, but here first come the properties, there exists a pro algebraic, so that means a projective limit of algebraic matrix groups, affine group scheme, G gamma relative to rho, defined over K, and this is called the relative completion. And what it looks like is that it's a group and it has a map to R. Sometimes I might accidentally call this group S, forgive me if I do. So a reductive map to R, and its kernel is that it's unipodent radical, so it's an extension of R by a pro unipodent group. Furthermore, it's equipped with a map rho tilde from the group into the K rational points of the relative completion, such that if you then project back to R, you get back the original representation rho. So it will be a group with these properties and it's essentially the largest possible group that satisfies this. So more precisely, it satisfies a universal property, which is given any homomorphism, gamma into the K rational points of G, a pro algebraic group scheme, over K, which is extension of R by pro unipodent group, so of R by pro unipodent, then affine group scheme, always affine group schemes. Maybe I forgot to write it in the hands. It's always been affine group scheme. Then there exists a unique morphism, G gamma rho to G of group schemes, affine group schemes of a K, such that pi factors through the relative completion. So we have this universal map from our original group into the K points of our relative completion and given any representation into any algebraic group, affine algebraic group of this kind, then it factors through the relative completion. Oh sorry, pi here. So you can think of this group as an algebraic group that's some sort of envelope of gamma. So one stupid definition is to notice that the set of groups of this type, extensions of R by pro unipodent group scheme, form a category, and you can take the projective limit of all possible such objects and you can take that. That obviously satisfies the universal property and you can take that as a definition. That's not a very enlightening definition so I'm going to take a more high-brow definition. It's a categorical definition. So let's see gamma rho be the category whose objects are representations of gamma. So gamma representations v. So this is, v is a finite dimensional K vector space with a gamma action. Plus, so equipped with a filtration. So you take a representation of your group gamma and we suppose that it has an increasing filtration by subrepresentations such that where the successive quotients are gamma R modules. So a representation of the affine group scheme R. In other words, it's an OR co-module. And the action of gamma on this quotient factors through the representation rho. So what this is is the category of all gamma representations with a filtration and we're saying that the successive quotients are going to be of a very fixed specific type. We're going to control what type of, what the associated gradient looks like but then we're going to allow, then that's the only constraint. So it's the category of representations of gamma with a filtration such that the successive quotients are of a specified type. This is a very natural problem. R, yes, R was part of the data. So we start off with gamma and we start off with R and we start off with this data. How does it appear in the name of your category? No, it's gamma rho. It's intrinsic to rho that rho is a map from gamma to R. So yeah, the notation is not great. The associated gradient that you have is, is this ingredient under gamma? Yeah, of course, yeah. So each VI I've written here is a gamma subrepresentation and so gamma preserves the filtration before it acts on the quotient. So you're saying that we've got, we've got, yeah, so gamma's acting on this filtration, it acts on the associated graded and then we're saying well on the associated graded let's put a very strong constraint on the associated graded. You can make it trivial for example and then you're saying gamma act looks like upper triangular matrices or something. That would be unabodent completion. So the morphisms in this category are obvious and some morphisms which respect the filtration and they're not compatible. So then this C gamma rho is a neutral tanachyn category over K and it has a fibre functor omega from C gamma rho to rectus bases over K. So there's a functor from every gamma representation that just forgets all the group actions. So there's an obvious functor to rectus bases, it's an exact tensor functor. So this category has a tensor product, it has duels, has all caches of representations, you can take the dual of a representation, the contagregent, take tensor products, direct sums, that's what's called a tanachyn category and there's this functor that respects those structures. And then from this the sort of abstract definition then is g, the relative completion gamma rho then is defined to be the automorphisms of omega in this category. So what that means if you're not familiar with the tanachyn theory, every time you have a tanachyn category you can construct a big algebraic group which essentially all symmetries, all possible symmetries and the theorem is that they respect the structures on C gamma rho and the theorem is that when you have a tanachyn category it's equivalent, the functor omega gives an equivalence to the representations of this, I find group scheme. So we start off with a discrete group, we define a category and say that's actually the representations of an algebraic group and the way to think of this is essentially all the algebraic constraints, so if you were to try to write down such a gamma representation which I'll do later, you see that there are algebraic equations that need to be satisfied for that to be a representation and this encodes all those possible algebraic relations. So this is the best way to define relative completion and from this it's immediate that all the properties hold, the universal property and so on. So for example you get immediately that G gamma rho K is a risky dense and universal property. So that's an exercise in tanachyn categories. Okay, so let's do an example. The simplest example is when R is just a trivial group and then we get something quite classical. So example R is spec K just a trivial group. So that means this category looks at extensions where the successive quotients are trivial gamma representations. Gamma acts trivially on the successive quotients. And so rho gamma to K trivial, then the relative completion, no spec K is such a trivial group to one. Then the relative completion is what's called the unipotent completion and it's pro-unipotent. So I should have said this earlier, pro-unipotent group is a subgroup. So a pro-unipotent algebraic group, it's a finite type, then you can view it as a subgroup, you can always write it as a subgroup of a group of upper triangular matrices. That's what unipotent means and pro-unipotent is a project with ones on the diagonal. So this unipotent completion means that you're mapping your original discrete group into a group of upper triangular matrices in some way. And so this is called the maths of completion. So a brief comment about the structure of relative completion. We have a functor from this category C gamma rho to finite dimensional representations of gamma, which is the obvious one. Given such an object V, it is by definition a representation of gamma. So there's an obvious functor from this category to representations of gamma. And it induces for all objects, for all V objects of the finite dimensional k representations of r. So given a representation of r, then that's certainly by rho a representation of V as well. So we get a map from the n extensions of the trivial object by V in this category, and we forget all the structures except, we forget all the filtrations and everything, and we get an extension in the category of representations of gamma, and this implies a map from the cohomology of V to hn gamma V. So we get a natural map, and this is the main tool to understand the structure of this relative completion. So we get a map from here, and it's very easy to show that this map on cohomology is an isomorphism for n equals 1, and it's injective for n equals 2. And this is the tool we'll use to get a real grip on this relative completion. This map is for the fibromath. Did they forget about the subcet? So here we forget everything. We forget the filtration, and we forget the gamma representation. In this one it's not the same one. Here we forget the filtration, but we keep the gamma action. So now let me maybe draw some pictures of what such an object in this category looks like, and then you'll see what is the question to which it is the answer. So it solves a certain problem in representation. So suppose we want to construct a very simple object in this category, just a simple extension. So a simple extension would be... So we take Vr, a favourite representation of R, and we want to construct an extension E of K, the trivial representation. So K is the trivial representation. Maybe I should write 1, the trivial representation by it. And so what this means is that we can think of this as the object in this category E with the filtration, with just one step in the filtration, 0. So this is an object of this in C gamma row. Okay, so that's what the simplest possible type of such thing would be, and let's say we want to construct all possible such gadgets. So what we would do then is, if we forget the group action for now, as vector spaces, these are K vector spaces, so they're split. So to construct it, what you could do is take E to be V direct sum K as a K vector space, right? And then we want to put, and then put a gamma action on it. So now let me... The gamma will act then on V direct sum K. So it will be a map from every element in the group. We'll give us a matrix, if you like, in some appropriate basis of V direct sum K. And it will look like this with respect to this decomposition of V plus K. Here it's going to be the identity. It's going to act trivially on this component. And on V it has to act by row G because that's the condition in this category. The action of gamma on V factors through row, which is part of the data. And then we have the liberty of putting something here. A sub G. So A sub G, so row G acts on V, and A G is what? It is a homomorphism from K to V. OK, so that... So for every element of the group, we're going to write down... We're going to write down by hand this representation. And we want this to be a representation. So that means that it should be a group homomorphism. And that means that we want to check that MGH equals MGMH. And if you work it out... So this is very familiar. If you multiply these matrices together, we get... because row is a representation, it's a homomorphism. And here we get row GAH plus AG. And so the condition is that AGH equals... Let me write this row G... Well, let me write it like this. Row GAH plus AG. And that's... So recall that Z1, the co-cycles of gamma in a group... So let me write this GAH plus AG to simplify notations. And the one co-chains of a group are the maps from gamma into V, which satisfies the co-cycle condition AGH equals AG plus GAH. And the co-boundaries are such maps of the form A equals V minus GV for some fixed vector. So they're co-boundaries. You can check that these satisfy the co-cycle condition. And the co-homorrhagy is, of course, the quotient. Right. They mean the connection between extension and the... Exactly. So this is very well known. So extensions... So, in other words, splittings... Splittings of this exact sequence are... So this star corresponds to group co-cycles. And this explains why it's natural that the structure of this category, the commodity of the category, is very closely related to the co-homorrhagy of the group we started off with. So then if you want to go beyond this, what is the problem we're trying to solve? Well, beyond simple extensions, in general, you want to construct iterated extensions. So with the same sort of notation, you're going to get, I don't know, some matrices like this. And on the diagonal, you're going to get row one, row two, row three, row four. So these are representations of R. And then you want to put some co-chains here. So the Aij are going to be co-chains. And in order for this to be a representation, they satisfy some system of equations involving cut products and co-values. And so the way this is formalised is by using some kind of bar construction on group co-cycles. But you have to be very careful with this because a cut product in group co-homorrhagy is non-commutative. So any homotopy commutative, and so you've got to be very careful, you've got to take into account these homotopies between cut products when you construct a bar construction. So that's how to think of this category. It's really iterated extensions. So you're having two different categories, one C, Cama, and representation of Cama. So you have two distinct homology theories. So I understand this part, but how do you infer this hng, gamma, e, this left hand side? What is the co-homorrhagy here? So there's a general definition of the co-homorrhagy of an algebraic group, and its definition is the group of xn in the category. This is the definition. So that induces this isomorphism. So the definition of the group co-homorrhagy of an affine group scheme can be taken to be the xn in that category of the trivial object by a given. But this also into the same list. But you have this map for the xariskedance map. So how does it induce the isomorphism? Is it necessary to use the xariskedance test? So you're asking what's the proof of this, but n equals one is obvious, because given an extension in the category C, it's such a guy. Now suppose, now view it as a gamma representation, and suppose it splits, then it splits in the category C. There's nothing to do. And therefore, it splits if and only, it splits in gamma if and only if it splits in C. Therefore it's immediate that it's an isomorphism on H1, and on H2 it's not much more difficult. So I'm really far behind. So I'll just explain the unipodent completion of a free group. Unipodent completion is the case where r is trivial. So let's take gamma fn free group on n generators. So this is one of the rare cases where you can write down a completion completely explicitly. So let's call the generators x1, xn. Then the unipodent completion can be written down. So u gamma, so if you take a ring a commutative k algebra, then the a-valued points of this group scheme are given by series s, a formal power series, and some variables y1 up to yn, such that s is invertible and satisfies a Sernaldsberg equation, delta s equals s tensor s. So here delta is a continuous co-product. So this is a completed tensor product, don't worry about it. That sends delta of yi equals 1 tensor yi, that's why I answer 1. And what's the map? So there's a map gamma. So these are non-commutative formal power series, or formal power series in non-commuting variables, in non-commuting formal variables. And there's a map from gamma, the original group, which is a free group, into the k-rational points of u. And it's given by every generator xi simply maps to the exponential of yi, and that's just the power series 1 plus yi plus yi squared over 2 dot dot dot. So every element in the original group is viewed as a formal power series in some new variables, and that's it. So you can think of the yi as logarithms of elements in the original group. So maybe, let me just abbreviate the next thing I wanted to say. So, okay, so from this, you can do a laurais spectral sequence, it's not very difficult, and you can describe g gamma quite explicitly. So the case we're interested in, so we shall only consider cases where h2, so let's, we have gamma, the risky dance in some algebraic group, and let, we let v lambda be a fundamental sort of isomorphism, a system of isomorphism classes of representations of r. So those are the building blocks, and we only consider cases where h1 gamma, or first of all h2 gamma v lambda is 0 always, so there's no h2, and the case where h1 gamma v lambda is finite dimensional. Now in that situation because, so this is injective on h2, that means that this h2 vanishes, and it means, in fact, that all the higher x groups vanish in this situation. So that's a particularly nice situation, and sort of to jump to the conclusion, in that case, in this situation, we get, we have, let u gamma rho be the unipodent radical, use the unipodent radical, then in this case we have all the higher chromology vanishes, and the h1 is given explicitly by the direct sum of all lambda h1 gamma v lambda tensor v lambda dual. So here I'm making another assumption actually that we assume also for simplicity that v lambda are absolutely irreducible. So if you extend to an algebraic closure for all lambda. So this is certainly true for SL2. So okay, then what's the upshot then? Let me just conclude by writing down what the relative completion of SL2 looks like. So our favourite example, gamma equals SL2z, r equals SL2, then let v1 be the standard two-dimensional representation of the algebraic group SL2. It's two-dimensional vector space essentially. So SL2 acts on a two-dimensional, so v1 is just two-dimensional vector space, if you like, and the representation theory of SL2 is very well known. It's just generated by vn, which is the nth symmetric powers of v1. This is an n plus one-dimensional representation. So we can plug that into this theorem. We know that h2 gamma vn is always zero. What we get then is that u gamma rho is the pro-unipotent radical. Then it's Lie algebra. So it's Lie algebra. Lie of u gamma rho is free. It's a free Lie algebra because that's because the h2 here vanishes and therefore, by this result here, well, the h2 vanishes by this result are right here. So it's a free Lie algebra. So in fact it's the completion, often lazy about writing the completions. If you get a free Lie algebra on generators, direct sum h1 gamma vn for every vn symmetric power of the standard representation, tensor vn dual. So that's what the unipotent completion looks like. Let me just conclude, let me not write anymore, just conclude by saying that what is this thing, what is very closely connected to modular forms. So we know by Eichel and Schmurr about which, more next time, in particular the dimension of the space of group cohomology, the dimension of this vector space is exactly the dimension of the space of modular forms of weight n plus 2 of full level plus the dimension of the space of cusp forms. So you get one Eisenstein Cormodgy class and you get two Cormodgy classes for every cusp form. So this is modular forms. So classical holomorphic modular forms for SL2z and these are cusp forms. Yeah, k is equal to q. So, well, let's put c here too. The whole gamma is SL2z. No, I'm just looking at the case SL2z itself. So I'm only going to, in this course, I'm only going to look at the case gamma equal to SL2z. Full level 1. This is just level 1. It's the simplest, most classical case. So what the relative completion then is just an extension of SL2 by a group of upper triangular matrices and its Lie algebra is a free Lie algebra with one generator for every Eisenstein series essentially and two generators for every cusp form. And each such modular form comes with a copy of a standard representation of SL2. And so that's the basic object that we're going to study. And you can see immediately the abelianisation of this just gives back just the Lie algebra but just this vector space itself. And that's just the classical theory of Eichler-Schimmera. I'll finish that. So next week there's a clash with the Paris-Pecain-Tochews. So I suggest next week if, well, one option is to cancel next week's lecture. The other option is to move it. And so I think the most sensible suggestion is to have it from 11.30 to 13 o'clock tomorrow, next week. Yeah, and I think that's a minimal disruption. And then that enables whoever wants to go to the Paris-Pecain-Tochews seminar to do so because it finishes at 11.30. But I'll start at 11.30 sharp and finish at one.