 Today, I have to deal with a certain amount of technical stuff, but in order for it to make sense, I'm going to give an overview of, you know, where I'm going and what I'm doing, and then I've got to explain what all the pieces are, and it's going to take me longer than I thought to explain all the pieces, I think. So, oh, and the first thing, too, is that I put up the notes from the previous lecture and I'll put up these notes and some other references, you know, I've written loads of notes for myself to try to get all this stuff straight, both in terms of notation and other things. All right, so the first point here is what's, right, so I'm going to give this overview, but in order to define it precisely, I'm going to need certain things, say, from theory of limits of mixed-todd structures, somewhat technical, and I'm going to go through that in the context of certain local systems over moduli of curves, moduli of elliptic curves, so that I can state the definition. I don't think I'll get there today, but so the first is that M11, this is the moduli stack, whoops, of smooth elliptic curves, and so this is a stack over z, it's got good reduction at every prime, so this is important for us. The second thing is that, so this is the universal elliptic curve, and we need the most basic, essentially elliptic motive in this case, in this relative case, and that's going to be the local system H, which will be R1F, lower star of Q, suitably interpreted, it'll have Betty incarnations, Q Dharam, which I'll mention briefly, Hodge-Ellatic et al., and Chrislin, about which I know essentially nothing, but, so this is a local system, it's fiber over E, the fiber over the moduli point of E is just H1 of EQ, and so we have, so one way to think of M11, if we just think of M11 analytic, well let me just, so I'll take D star to denote some puncture disc in whatever theory you're thinking about, there's an inclusion of this into M11, and this would be the Q disc. So Q is a local, if you look inside M11 bar, it has a unique cusp that corresponds to the nodal cubic, that's the unique cusp here, and just think of it as being a neighborhood in whatever theory, and right, so it's the upper half plane mod this guy, this is in sort of the Betty type situation, or the analytic situation, and Q equals E to the 2 pi i tau, and then in the more arithmetic situation we take it to be this guy here, so, sorry it's the puncture disc, so I need to, and then we have a tangential base point, which is DDQ, which is an element of the tangent space at say the cusp of M bar 11, and this tangent vector is non-zero at every prime, so it all primes P, and so it's essentially this is corresponding to a more or less to the Tate curve, and so one thing I'll talk about later today in both the Hodgson also in the et al situation is that we can talk about the fiber of this local system over this tangent vector, maybe here I'll call it, for the time being I'll call it say T, over T, so if you calculate this in these various theories, this is going to be Q plus Q of minus one, so however you interpret these guys here, and it's actually split it to direct some, and the key point here is the key observation is that this is the, well I'll say this is a mixed Tate motive over Z, it's a very trivial one in a way, but it's, it is one, so in fact in this category there are no extensions of Q by Q of minus one, so what does this mean, so in the Hodgson situation this is going to be a limit mixed Hodgson structure, and so what does it mean in the et al situation, it's the GQ module, it's the GQ action on the Tate module of the Tate curve, a lot of Tates here over, so I think it's Q bar, so that's an algebraically closed field, so you look at the Tate curve over this and you compute the Galois action and it decomposes in this way, so and you have to twist it and tensor it with, okay, so the next thing is that fundamental groups, so we have that perfectly standard pi one of M one one analytic, say with this base point, this is naturally isomorphic to SL2Z, and so over Q bar is isomorphic to the profinite completion, which is, which is definitely bigger than the SL2 of the profinite completion of Z, so this is profinite completion, so I wrote these down because we're interested in local systems, so here's the approximate definition universal mixed elliptic motive, so it's a universal mixed elliptic motive, say over Z, you can put lots of adjectives in here, which I'll omit, is a motivic local system V over M one one, so here partly because of ignorance and partly I think because of lack of technology is these are going to be compatible realizations, so variations of mixed hard structure, least sheaves and so on that have various properties, satisfying one is that each, so such a motivic local system will come with a weight filtration, so each grade, each weight graded quotient is a direct sum of these simple elliptic motives, sum of local systems of the form symmetric powers of H twisted by R, and just in case there's any confusion, it's SNH tensed with Q of R, so I take the symmetric power then the twist, and to the fiber of V, the fiber which maybe I'll just call with a plain V of V over T, so this is going to be in the realizations this will be some sort of limit mixed hard structure in the etel situation, you will restrict to this guy up here, wherever I put it, up there is an object of MTM, and when I write MTM I really mean mixed eight motives over spec Z, so by this I mean, so these are going to be variations of mixed hard structure and so on, and here I mean that there is a mixed eight motive and the fiber of the variation of mixed hard structure will be the hodge realization of the mixed eight motive and so on, same with the ellatic story, the fiber of the least sheaf over T will be the ellatic realization of the mixed eight motive, but in order for this to be comprehensible I have to talk about variations of mixed hard structure and so on, and I have to talk, get into a little bit about moduli of elliptic curves, so that's what I'm going to do today, just explain what all the words mean, so Can I ask a quick question? The V lives over M11 or M11 bar? Well you'll have the local system will lie over M11, I mean M11 bar is simply connected So what does it mean to take a fiber of this? Well, I understand the low realizations, but Well for example, because key hat is, you said key hat is a conventional vector at the paper, right? One goal today is to explain that, but the point is simply this, that the local systems themselves don't extend, but you look at the corresponding flat vector bundles, they will extend and you'll get a connection with the logarithmic singularity, and then you have to define the limit mixed hard structure, and that I'll do in detail. In the etal situation, you have a map of the punctured disk, well, so you have a map of the punctured disk into M11, and then if I take the, what's the, so geometric point that lies above this, it's going to be Q bar, and then I guess we have to take roots of this, yep, right? So this is, this is then a geometric point of this guy, and we'll be the fiber over that, so in fact that's, that's, this is actually T, alright, so let's, yeah, but I'm going to explain, I would, I'm going to explain the hard story, not, maybe in excruciating detail, I don't know, it's not going to be excruciating I hope, but, but there's some really important point here because you might say, see this local system, H doesn't look very T, you know, but when it degenerates it becomes T as I explained there before, it's the same thing with the mixed hard structure, and what gets confusing here is, or what's potentially confusing is that there are two weight filtrations, there's one coming from the monodroming, there's one coming from this global local system, alright, so that's the best thing to do is just get on with it, so I think I've already messed up my numbering, moduli of elliptic curves, so I realize a lot of this is well known, but I'm going to repeat some of it because first of all just to establish notation, and then we need to be, it's going to help us because we're going to need to do certain calculations, so the first thing is, I, this is a notation I like and I'm trying to propagate is that this, this is equal to the moduli stack of smooth projective curves of genus G with, yeah, so everywhere here 2G minus 2 plus r plus n is positive, so with n marked points, and I should stress they're labeled, so it's an ordered set of n points, and r non-zero tangent vectors, and all the anchor points of the tangent vectors have to be distinct and distinct from the other points, so a curve like this might look like, and now here we might put our n marked points, x1, x2 up to say xn, and then here we might have these tangent vectors say v1, v2 up to vr, whoops, yeah, all of these guys have to be anchored at different points, and each of these, and they have to be, these points are all distinct, and these points are distinct, and distinct from these, so for example if you forgot the tangent vector you, you could turn a tangent vector into another point, so maybe I should say, okay there are points say y1, yr here, and y1 to yr, x1 to xn are n plus r distinct points, and if you're a topologist, and we'll discuss this later, if you took the real oriented blow up at each of these points you would get a surface with boundary, so this is the algebraic way of doing boundary components that topologists like, so actually I denote the moduli point of such a curve by I'd write it as something like cx1, xn, v1, vr, alright, so we're interested in, so what we need for the current situation is m11, and here a typical point is that it's a genus one curve and a point, so that's an elliptic curve, but we'll always understand that's an elliptic curve, so I'll just write e, alright, and so, and one, a nice moduli space that's good to work with, it's in fact a scheme, is m1 vector 1, so this is going to be e, and then a vector, and this vector will, will be in the tangent space of 0v, so this topological picture, it's a non-zero tangent vector at the identity, and then m12 is, these are the only three we'll need, it's e0x, so x is not equal to 0, so it's an elliptic curve together with an additional non-zero point, alright, so let's look at these as, first of all, as analytic overfolds, so let's take h to be the upper half plane, alright, so it's just, it's the tau in c, such that m tau is positive, and so I like to think of this as the moduli space of framed elliptic curves, alright, so whenever you solve a moduli problem, or at least the ones I've solved, you sort of rigidify the problem, and if you're doing it analytically, what you can do is you, so what do I mean by this? This is an elliptic curve, so smooth elliptic curve, together with, and I apologize for the abuse of notation, sort of ab, where this here is a basis, it's a symplectic basis of h1 of e with z coefficients, and so why is this the case? Well, if you have, so if you have such an elliptic curve with a framing of h1, you can, it corresponds to, well, it gives you a point in the upper half plane, namely you just take tau to be the integral of omega over b divided by the integral of omega over a, where omega is not equal to 0, it's just a holomorphic, it's just non-zero, b-alien differential, and if you go the other way, if you have a tau in the upper half plane, you just take it up to the curve I like to call e tau, and you take the differential dz, what, sorry, you take, you take the basis 1 tau, so let me explain this, so the notation is that I'll set lambda tau equal to, if this notation is too heavy to yell out, I'll always write this, and e tau will just be c mod lambda tau, so just the obvious elliptic curve you get out of tau, and the basis here, so h1 of e tau is naturally isomorphic to lambda tau, and this has basis 1 and tau, right, that's our basis a and b, so, so SL2z acts on this, we all know that it acts on the upper half plane, and it just acts on the natural way on the framing, so, so we have the SL2z, it acts on h, and, well, the usual action is tau goes to a tau plus b over c tau plus d, but you can also think of it as taking, you can also think of it as taking elliptic curve with a framing a, b, maybe I'll write the framing this way, b, a, and it takes it to the elliptic curve with the framing a, b, c, d, b, a, and then m1, 1 analytic, well, what's an elliptic curve, it's just a framed elliptic curve where you forget the framing, the orbits, any two framings differ by SL2z, so it's going to be the quotient of the upper half plane by SL2z, but I'm going to take the Orbefold quotient, and so very briefly what's the Orbefold quotient of the upper half plane, it means you just work on the upper half plane, but everything has an action of SL2z, and you act SL2z equivalently, so it's just a fancy way of saying work here, but work SL2z equivalently, and a good example of what that's like is just the theory of classical modular forms. All right, so let's look at, actually it's becoming one of my favorite modular spaces, let's look at m1 vector 1 analytic, and first of all a dumb remark is that if you have an elliptic curve and you have a tangent vector, this corresponds exactly to an elliptic curve and an omega, where omega is a non-zero holomorphic differential on E, and why, because the tangent bundles trivial, so you just take it to be, right, if you've got such a differential as a unique tangent vector, which pairs to 1 with your holomorphic differential, right, so this is the same as elliptic curves plus the holomorphic differential, but I want to often think of it this way. All right, so now one way to, there's various ways to do this, but let's take c star across h, and I'm going to take it into framed elliptic curves plus a non-zero abelian differential, and so how am I going to do this, I'm going to take c tau, I'm going to take it to e tau, that's the elliptic curve associated with tau, and then the framing of my lattice is 1 tau, that's no surprise, and then I just which differential do I take, I just take c times dz, so z is the coordinate in the universal cover of e tau, and you can check here that you can think about how sl2zx and ab, so sl2zx, and it will take e tau to e gamma of tau, but that's the same elliptic curve, and then you have to compare the differentials, and then you see that it takes abcd, takes c tau into c tau plus dc, looks like I left something out here, but that's okay, so now, and so what we have is that this induces a map from sl2z c star cross h, and it will induce an isomorphism with m1 vector 1 analytic, so let's talk about, no I haven't, I could, I'm not going to, I'll explain the compactification when I do the algebraic case, but you can hear, maybe I'll say briefly what's m11 bar, you've got to glue two albifolds together, and you've got to be a bit careful, but it's essentially, I like this picture here, we'll take, so sl2zx and abcd, and then we'll do z, the quotient of the upper half plane by it, up here I'll put this one, and just the quotient of the upper half plane by it, and this maps down into here, and so this is just the punctured disk, the punctured q-disk, and then here I want to map it down into the group which I'll call plus or minus 1z01, and then I take the quotient of the upper half plane by it, and this is the same as taking a cyclic group of order two, and dividing this by it, but that's the trivial c2 action, right, so that's as a, as an analytic albifold m1 1 bar is this, is given by this sort of myviatoris picture here, so to work on it you would work sl2z equivariantly here, you would work equivariantly under this group here, and they have to match up on this here, so let me say something about fundamental groups, so if you have, so if x is simply connected space and gamma is a discrete group, you can define the albifold quotient and pi1 of the albifold quotient, and I'll say what p is in a second, this is naturally isomorphic to gamma, so here what's p, p is the projection from x into gamma mod x, it's just the projection, so in ordinary topology you can use any simply connected space as a base point, right, and so in particular you can use the universal cover, so if this was fixed point free, properly discontinuous action, then the fundamental group of a space with respect to the universal covering is just the group of deck transformations, and this is just an albifold version of that, no you could, for example you can take this to be a point, I mean, I mean you can be fancy and take x cross e gamma over gamma, right, so and then what about, so this tells us that pi1 of m11 with respect to this project analytic is naturally isomorphic to sl2z, and pi1 of m1 vector 1 is equal to 0. I'll put this maps to sl2z, and if you think about it, we didn't take the quotient of a simply connected space, but you get pi1 of c star, so you actually get a copy of z here, or if you want to be really fancy a copy of z of 1, so and this group here you can think of it lots of ways, one way this guy here is the braid group on three strings, on three strings, it's also the inverse image of sl2z inside the universal covering group of sl2r, it's also the fundamental group of the trefoil knot, so it's a nice group, and this is a central extension, alright so let's, maybe I'll start here, so let's now think about them as algebraic, so I'm going to assume six is invertible, so if six is invertible in a ring, so this is a commutative ring, then every elliptic curve over r can be written in the form y squared equals 4x cubed minus ux minus v, so this is in, you can find this in Katz-Meser, there might be some assumption on r, but it's certainly true if r is z, sorry z was six inverted, and so in the identity of course is the identity is 010, if we write it in projective coordinates, it's the point at infinity, and so what you can conclude from this is that, this gives us that m11 over r is isomorphic to a2r minus, and you have to remove d inverse of zero where d is the discriminant up to a factor of four, so d is equal to u cubed minus 27 v squared, it's essentially the discriminant, it's either four times or one fourth the discriminant, I think it's one fourth the discriminant, and so no this is a scheme, and so and maybe I should say what does uv correspond to, uv corresponds to the elliptic curve, maybe I'll call this one e uv, which is y squared equals 4x cubed minus ux minus v plus the differential dx over y, so that's an Abelian differential, and then what is m, what's m11, it's over r, it's equal to the quotient of gm, the quotient of this by and I have to tell you how gm acts, it acts by multiplying this guy here by t, but this looks slightly strange, but you've got, it doesn't look as strange when you realize that this same curve is isomorphic to this with the negative of this differential, because every elliptic curve has the involution that takes a point to its inverse, so what's the gm action, so lambda would take uv and it takes it to lambda to the minus four u, lambda to the minus six times v, and this is exactly the action that multiplies this guy by lambda, and here I mean stack quotient, and what does this mean, it says well if you want to work on here, you can work gm equivalently up here, and again if we want to talk about m11 bar, what's m11 bar, it's equal to a2 over r, it's the quotient by the same gm action of a2r just minus zero, all the points on d equals zero are just the nodal cubic, but with different differentials. So it's a non-mergingous product, it's a weighted product, you have to coordinate to mv and gm action, right, but viewed as a stack and not as a variety, I mean this is, it's the usual m11, it's just, and then also note that minus one acts trivially on everything, because if you multiply the differential by minus one, you get an isomorphic curve, so everything has inertia, and so, and then this is an exercise, and since some of the notes I put up on the web, I mean you can hunt through some of this, but what's the universal elliptic curve, you can think of this as its m12 plus one, this looks bizarre, but this is, you just write down the equation, you know the coordinates are going to be x, y, u, v, and so on, modulo gm. So it's an elliptic curve with two points and a tangent vector and then you, so this will be a scheme and then you just, the gm action multiplies the abelian differential, sorry, yes that's the universal curve, so this will be the universal curve over m11. So this is actually m12 and that's equal to what I like to call e prime, e prime is equal to e minus the zero section, thank you. But you can also write down the integral of the universal elliptic curve this way, and it's in somewhere in the notes. Alright, so to compare the algebraic and analytic, what we need are Eisenstein related functions, so Eisenstein series, and I'll just, one reason for writing this down is I want to give them the normalization I use. So for me g2k tau is equal to 1 over 2, and I'll take k to be greater than or equal to 2, but if we work on m1 vector 1 then we need to k equals 1 as well, we need g2, and this is the normalized guy, so it's minus b4k divided by 4k plus the sum n equals 1 to infinity sigma 2k minus 1 n q to the n, q as always is e to the 2 pi i tau, and this is the divisor function here, so this guy's normalized. Yeah, I think that's what I wrote in, so, and it's a modular form of weight 2k, and so then the viastrasp function, so p tau z, so it's the usual formula, but let me write it like this, it's 2 pi i squared, I like to isolate out all of these because of all the tape twists and so on, and then 1 over 2 pi i z, and then z squared, so I guess what I'm saying here, a better variable here would be 2 pi i z instead of z plus the sum m equals 1 to infinity 2 over 2 m factorial g2 m plus 2 of tau times 2 pi i z to the 2 m, this is just the usual formula, it's just the, it's 1 over z squared plus the sum prime 1 over z minus 2 m plus the sum prime 1 over z minus 2 m, and then this lambda squared, it's just the usual formula, but, and then, again, I'm writing these down just because I want to have the normalizations, g2 tau is equal to 20 g4 tau, and g3 tau is equal to 7g6 over 3, and so why am I doing all this, so what's the significance, the significance is you can embed e tau into p2, and the way I'm going to do it is slightly non-standard, it's, I want to put 2 pi i to the minus 2 times p tau z, 2 pi i to the minus 3 times p prime tau of z, 1, this is what you need to do to make the q theory come out correct, the q to run theory, and so the image is the elliptic curve y squared equals 4 x cubed minus g2 tau x minus g3 tau, and the discriminant up to a factor of 4 is, it's just the cusp form equals, you know, q times the product, so it's the cusp form of 812, and the important point here is that the differential here is dx over y, the abelian differential is equal to 2 pi i times dz, and maybe I, that's why I put all these factors of 2 pi in here, and why did I do that, it's because when we degenerate this class to the nodal cubic, the nodal cubic is basically p1 with zero and infinity glued, you know, and the identity at one, what does this differential become on the nodal cubic, if we choose a coordinate here, which is, you know, zero and infinity at these two points and one here, this degenerates to dw over w, which is a q to run form, if I didn't put the 2 pi i in there, it would be 1 over 2 pi i, and you can see this just by seeing what happens to the period over a, because dz has period 1 over the a curve, the a curve here goes to the loop around zero, and, right, and so, right, so we want to, we want to map from the analytic picture to the algebraic picture, and so this is a map into c2 minus d inverse of zero, so this is equal to, and you just take c tau goes to c to the minus 2 g2 tau, c to the minus 4 g3 tau, and this induces an isomorphism from, this gives you a map from m1, 1 analytic, maybe I should have, better here, I should have written this, this induces a map from, so this is how you compare the complex picture here with the analytic picture, all right, so now, that's probably more than enough on moduli, let's talk about the local system, so remember what we have here, we have the universal elliptic curve over m1, 1, and h is in r1, f flow star of q, with suitably interpreted in whichever theory, so, I'm hoping it's m1, 1, analytic curve with a vector, sorry, this side, yes, correct, yeah, that should be a vector there, yeah, because the question of this by SL2z is this, this has weight minus 2, this has, sorry, this has weight minus 4, this has weight 4, so this is a map, this map is SL2z, where there's a trivial action on this side, all right, so Betty, so let's just go through all the realizations, so we'll discuss all the realizations separately, so Betty is the most trivial, it's just topological local system, and we've more or less already discussed it, so except I'm going to play a little game here with ponk radiality, so the fiber over e is h1 of eq, but that's equal to harm h1 e into q, maybe I'll stress that some h1, and now we know we have, so again, if we work on the upper half plane, so we're going to lift h to the upper half plane, and so then we can talk about the fiber over a particular elliptic curve, so let's look at, we've got eab, this guy here is a point in the upper half plane, because this was framed elliptic curves, and so let's take, let's let a dual, b dual is a dual basis of h1 of e, it's a basis of h1 dual to these guys, but this guy here is isomorphic to h low 1 of e by Poincare duality, so under Poincare duality, a dual is equal to minus b, and this is minus b, and b dual is equal to a, so in Betty there's no difference, in the other theories there's going to be a tate twist, but I'll still use this basis, so I'm, so what's the SL2Z action, so we have h here, so what I want to think here, and we have h over it, it's a local system, it's a trivial local system, and h is really just equal to say q times a check plus q times b check, but that's the same as q times a by Poincare duality plus q times b, so I'm going to think of it this way, but I'm thinking of it as a cosmology, and so what's the SL2Z action, so SL2Z is going to act on everything inside, so the local system downstairs is just a local system upstairs with an SL2Z action, and so a, b, c, d takes, this is a little exercise, a minus b into a minus b, a, b, c, d, so it's acting on the right of frame, so it's acting on the left of the bundle, so it's, and so like I said here, the quotient is the local system h over m, one, one, analytic, so that's the simplest case, right, so what does it mean to have a local, see this guy here is a space, we know it's the j line, it's simply connected, so local systems there would be boring, so what is it, one way to think of it is it's a local system on, it's a, it's a orbital universal covering together with an action of SL2Z. Well, what do you say quotient is, of course, ambiguous? Well, orbital quotient, so, you know, when I say quotient I'm always going to mean stack quotient or orbital quotient, but when we're on m one vector one it's going to be okay because that's a genuine variety or a scheme, so what about hodge, well that's just the same but with more information, so this is going to be a variation of hodge structure, and this is, the hodge realization of h is the most canonical example of a variation of hodge structure on the planet, it's so canonical it's atypical, in the sense that it's, in Wilfrid Schmidt's work on variations of hodge structure, this thing called an SL2 orbit, and this is the most fundamental SL2 orbit. So, so again we, we want to talk about the vector bundle associated to h, so this is going to be h tensed with o, and I'll always be vague about o, if we're on the upper half plane it'll be out of the upper half plane and so on, so this is, this is the associated flat vector, holomorphic vector bundle, actually we can take a little break here if you want, pause cafe, all right so we're about to do the hodge story, so this, this bundle has a connection, and so a canonical flat connection, so sections of the local system are, have derivative zero, so this says that nabla a and nabla b are equal to zero, and so we want to, to write down the hodge story we need to write down the hodge filtration, and from this we'll extract some features, so we need, so first of all I should say if we look at h1 of e, this is equal to f0, this contains f1, which is equal to c times, say the Abelian differential, and this contains f2 equals zero, so the only thing to describe is f1, so we, we need to describe f0 of h, and this is a dumb little, this is a dumb little calculation, so on e tau, that's quotient of c by the lattice associated with tau, dz, so you just draw the usual picture here, here is a, here is b, here's 1, here's tau, so you just look at this and dz is equal to a dual plus tau b dual, but we're using Poincare duality, so a dual was minus b plus tau a, and so, and so I'm going to let w, bold w be the section of h, so we'll just take the value of it, w at tau is going to be this, it's going to be 2 pi i times this guy here, minus b plus tau a, so this is, again the reason for this 2 pi i is so that it's a q rational section of this, this bundle will be defined over q, and maybe I can even write that as, I can write it as minus 2 pi i b plus log q times a, because e to the 2 pi i tau is q, so just take logarithms, and so let's look at the derivative of w, it's equal to a dq over q, so this is a dumb calculation, but this is somehow fundamental, this is, this derivation arises here, and so what's a formula for the connection, so if we use the framing, so write h, we can, instead of using the framing a and b, we're going to use the framing a and w, so this is over h, and there's several reasons for doing this, we'll become apparent in a minute, but one is that this is adapted to the hodge filtration, so this is going to be our, this is f1 of h, and you can see it varies holomorphically, because this w is a holomorphic linear combination of a and b, so this is a holomorphic sub-bundle, so this is a, in fancy hodge theory this is always true, if you have a family of varieties and you look at the corresponding local system of cosmology, the hodge bundles vary holomorphically, and here it's we've just written it down, it's trivial to do by hand, and so we're also interested in the local system snh, and so here snh tends to o is isomorphic to sn of h, sorry I forgot to write down the connection here, so with the respect to this trivialization, the connection is d plus addw times dq over q, so this is a formula for the connection on this bundle, and here it's just linear algebra to get from this situation to here, but it's worth writing down, here the hodge filtration's longer, we're going to have snh will be equal to f0, will contain f1, will contain all the way down to f, maybe I'll write fp down to fn, and this will be spanned by a to the n minus p, w to the p, and so on, this will be spanned by w to the n, when I say that I mean you take all monomials of degree greater than or equal to p and w, and so these all vary holomorphically, and the connection is given by the same formula, S means symmetric power, yeah S means symmetric power, so these are going to be our basic, these and their T twist will be our basic simple elliptic motives, and we're going to be interested in extensions of these in various categories like the category of variations of mixed hodge structure, and so there's some important observations here, the first is that the connection takes, if you have a section of fp of snh, you get a section of fp minus 1 snh tensored with omega 1, and I'm going to write m11 bar log of the cusp, so these are forms with the logarithmic pole of the cusp, and you see the connection has a singularity, and this is called, this is, if I put, so this is one of the standard axioms of a variation of hodge structure, and it's called Griffith's transversality, and again it holds in great generality, and we're going to be imposing this condition on the variations we have, this is a condition you get in variations of hodge structure and mixed hodge structure, another is that the intersection form is flat form, so the intersection form is, you know, a a equals b b equals 0, and a b equals 1 equals minus b a, so this is a flat inner product, and it satisfies the Riemann bilinear relations, is flat with respect to the connection, and satisfies, I won't write them out, the Riemann bilinear relations, and so this is called a polarization, so I'm not going to dwell on this, except I'm going to use the word polarized variation of hodge structure, and in reality people often drop, sometimes drop the term polarized polarization, but it plays a key technical role without it, a lot of the results don't hold, and so this also polarizes the SNH, so they're all polarized, they've all got an invariant bilinear form, and they all automatically satisfy the Riemann bilinear relations, so what this says is that SNH, SNH with its connection, and with its hodge filtration, and also this guy here is a polarized variation of hodge structure, well PVHS if you're a professional, and basically the axioms of what a polarized variation of hodge structure are just generalize these, and we won't need any other polarized variations of hodge structure other than these guys here, and it's actually of weight in because every fiber of this local system with the induced hodge filtration is a hodge structure of weight in, so it's a whole family of hodge structures that move holomorphically and they don't move too fast, that's what Griffith transversality says, you can't, I guess this is slow food movement in the US, right, this is, there's slow variations here, the derivative is not too big, so let me quickly say something about the Aladdin case, so yeah I'm going to take a quick and dirty route here, but so E is elliptic over Q, say, and this gives you a point in M11Q, and so we get an exact sequence of fundamental groups, we'll get pi1 of M11 over Q bar, that's the geometric fundamental group here, I'm going to call this point E, this will map to pi1 of M11 over Q, this is the arithmetic fundamental group and this will map to GQ, the Galois group of Q bar over Q, the elliptic curve gives you a section because this is pi1 of spec Q and you get a map back in here in the way I've set it up, it's base point preserving, so this is induced by the elliptic curve E, and so this tells you that this group here is just isomorphic to GQ, semi-direct product pi1 of M11 over Q bar E, and this group here is also isomorphic to the usual fundamental group profinitely completed, and now what's an elliptic local system here, it will be a representation of this group and what representation will it be, so the middle group here, pi1 of M11 over Q E, it acts on h1 of E over Q bar, so we can think of this as just Tom of pi1 of E over Q bar, so we described this last time into QL of minus 1, and so you can think of it as an action of the semi-direct product, how does this part act, how's the geometric part act, well it just acts on this guy here, that's just the fundamental representation of SL2Z, and then you have to have, but also the Galois group acts on this guy as well, so it acts on this, and together they give you an action of the semi-direct product, we can also do it by looking at a tile covers here coming from the L to the end torsion points of the universal elliptic curve, but anyway it's just to give an idea, so anyway so there's this elliptic least sheaf, and then let me say something briefly about the Q to Ram theory, at first I was going to leave it out, but it turns out to be very useful for the same reason in the mixed tape case is that it gives you a very good fiber functor, it's the best fiber functor for mixed elliptic motives, and so I'll briefly sketch this here, so I'm going to do it over M1 vector 1, and I'm going to write down a GM invariant connection, and then therefore it will give me a Q to Ram story on M1 1, right, so we're going to look at M1 vector 1, so we're going to trivialize H over this guy here over Q, and how we're going to do that, so remember this is going to be A to Q minus D inverse of zero, and we're going to have two sections, I'll call them S and T, and so what are their values, so their values over UV, so this is a point in here, so UV which corresponds to the elliptic curve Y squared equals 4X cubed minus UX minus V, and so the value of T at UV is just going to be DX over Y, this is clearly a Q rational section, and S of UV is going to be the differential of the second kind, X DX over Y, and it also if you want to write, so this is actually going to be the way I set it up, this is 2 pi I DZ, and this guy here is going to be 1 over 2 pi I P of tau Z DZ, so this is a differential of the second kind, and it's Q rational, and and these guys you can calculate there, cup product is 2 pi I times the fundamental class, so they're linearly independent on every fiber, they're both defined over Q, and so this tells us that H is equal to O of M1 vector 1 times S plus O of M1 1 T, and now the fact is that it's not hard to prove it, an exercise is that with respect to this trivialization, the connection is D plus minus one-twelfth D D over D, tensor T plus 3 alpha over 2 D, I mean the exact thing is not, which is visible, I have to tell you what alpha is, so where alpha is equal to 2 UDV minus 3 VDU, and D is always is equal to U cubed minus 27 V squared, right, so this is clearly defined over Q, and if you, not only that, it's also, it's GM invariant, I write it up, I won't write it down, I write up the weights of everything under the GM action, but just to get, I set it up so the GM action multiplies, lambda multiplies this by lambda, and it'll multiply this by 1 over lambda, and you just look at everything else, I think U had weight minus 4 and V had weight minus 6, and so on, and you just work it all out, so it's GM invariant, so it descends to a connection, a Q rational connection on M1 1 over Q, all right, so that, I mean this is sort of the end of the setup, and you can find a proof of this, it's sort of, it's not hard, just an exercise, but I have these notes on the elliptic KZB equation, in the last section there I worked this out, together with what I really wanted, which is another Q rational connection, anyway, so, oh yes, sorry, there's a tensor S missing right here, right there, anyway, it's in the notes which I'll put on the web later today, there, so, let me mention a little history, so, in the abstract, Wilfred Schmidt sometime about 76 proved that if you have any polarized variation of hodge structure, it has a limit mixed hodge structure, he didn't even assume it came from geometry, and then Steen, sorry, Wilfred Schmidt, and so then, and so beautiful piece of work, and then Steen Brink proved the thing if you, in the case of geometric origin, although there was a gap, it's correct, but there was a gap and various people fixed the gap, but ultimately I think it's by Sito, but so in the geometric case though you can explicitly write it down, and in this case we can just do it because things are so simple, so I'm going to do it, but that's why I didn't put limits of hodge structures in the definition of a polarized variation, it's automatic by the work of Schmidt, in the case of mixed hodge structures you have to talk about the limits, so I'm going to talk about limits of hodge structures then limits of mixed hodge structures, and then you'll be probably ready for lunch, all right, so, and I'm just going to do this in the context of h and its symmetric powers, all right, so let's, so the question is what's, so what's, what's the issue here, we've got d star is sitting inside m11, and you can think of this, you know, we're going to think about this analytically, right, so this is really the upper half plane mod 1z01, including into sl2z, and so this is the punctured q-disc, and we want to know what happens when we get in, so the first thing is we want to understand the behavior of this local system on d star, so I said a little while ago that I was going to explain why we switched the framing from a and b to a and w, so let's check the monodromy here, so what's the monodromy in the q-disc, yeah, local variable is q, so just to be very clear, q is equal to e to the 2 pi i tau, and so what happens here, we can just see, as you go, if you're in the q-disc, q-u-i, you go around from some point, you go around like this, so this is d star, but what's happening in the upper half plane here is that some tau here is going to tau plus 1, and so here's zero here, here's my say 1, so this will be, oh, we do have colors, okay, so we've got here as a, and here as b, and now what happens when you go around the disc is that b goes to this one, which is a plus b, right, I mean it's not a surprise, it's, we already wrote down the formula for the action, is, is, you know, b a goes to 1, 1, 0, 1 times b a, which equals a plus b b, right, so we've got this locally unipotent monodromy, and so the first thing to observe is both a and w are invariant, so a goes to a, so note a and w are invariant, so this is one reason we switched to w, b is clearly not invariant, and you can just check it out by writing out the, sorry, yeah, yeah, that should be an a, thank you, so a goes to a, and b goes to a plus b, and you just see it from this picture here, yeah, yeah, and so, and you, I'll leave it as an x, I'll just write down w, or it's also just clear, it's just, there's only one abelian differential, it's determined by its period on a, so it, it, it's invariant, because w takes the, always takes 2 by i on a, so what's this saying, it's saying that so h on d star is just equal to o of d star a plus o of d star w, so if you have a bundle, it's trivial on a puncture disc, and you have a framing of it, gives you an extension to the entire disc, so we're going to extend to h bar over d, we're going to extend it, so this is just going to be equal to o of d a plus o of d w, so no, there are many ways to extend, because I could instead, I could have used the framing, some power of q times a, and some power of q times w, but there's one, there's a canonical choice, and Deline was the one, I guess, who realized this, so when you extend the connection, so when you extend the bundle, you're going to get a meromorphic connection, so let's calculate what it is, and I should say a, similarly for s and h, it extends just as the nth symmetric power of this guy, okay, so what, and so we already know the formula for the connection, in both cases, nabla is equal to d plus adw dq over q, so what do we notice about this, this is, what's the residue at zero of nabla, it's equal to this endomorphism, yeah, it's equal to this endomorphism which is adw, and you should think of this as an element of the endomorphisms of the fiber over zero, so I'm just going to call that h, this is the fiber over zero, or I could put it of h bar, so different people think of limit mixed hard structures in different ways, I think of the limit mixed hard structure as living on the central fiber of this extension, so this is related to D-PAM's question, okay, so, and it's a nil-poten, and this equation has a regular singular point, because the pole is just a simple pole, so these two conditions alone uniquely determine this extension abstractly, it's just the fact that the connection extends to a meromorphic connection with a simple pole and nil-poten residue, and that's called Deline's canonical extension, okay, so the next point is that, so what we had, we started out with basically h over D star, this gave us h over D star with a connection, then we extended that to h bar with a meromorphic connection over D, all of the different, now this guy here has a hodge filtration by holomorphic sub bundles, maybe to make it more interesting I can put an sn here, now we, well the next thing is that we observe that the hodge bundles extend through sub bundles, this is why it's important we take the correct extension, if we take the wrong one they won't be sub bundles, so observe that the fp of h over say D star extend to holomorphic sub bundles of snh bar, just the same way, you just give, basically you give w type 1 0 and a type, well all you have to do is, so fp, I'll just write it down, of h bar is equal to the sum of O of D times a to the n minus j w to the j, where j is greater than or equal to p, that's visibly a sub bundle, yeah this, yeah sorry, yeah thank you, it's fp of sn, right, so this is a sub bundle, and so this was in the abstract case, this is what Schmetz-Nilpoten orbit theorem says, it says the hodge bundles extend to holomorphic sub bundles of the canonical extension, so what's the limit mixed hodge structure, so what do we got so far, so what do we got so far, so so far we've got snh, so it equals the fiber of snh bar over 0, so somehow that's to me that's where the limit mixed hodge structure should live, right there, and we also have a hodge filtration, we have the, it's just fp of h is just defined to be fp of s, sorry fp of snh, fp of snh, it's just the fiber at 0, right, you just, the extended, the fact that the hodge bundles extend means it cuts out a hodge filtration on the central fiber, so you might get optimistic and think that well this would define a hodge structure, so we're missing two things, one a q structure, this is where the tangent vector is going to come in, and two we're missing a weight filtration, so let's, so I'll deal with these now, this one's tight, all right, so, so let's just set, let's just set n equal to adw, and this guy is an endomorphism of snh, it's an endomorphism of h and therefore an endomorphism of this, so it's a nilpotent endomorphism, and so a little proposition here is that if n is a nilpotent endomorphism of some vector space v in characteristic zero, then there is a unique filtration, and I'll call it w n dot of v such that one n of w n, say k, is contained in w n k minus two, so this filtration is going to be called the weight filtration of n, and the first property is that n lows it by two, and the second property is that if you look at say n to the k, it's going to take graded the, it's going to take it, you'll certainly get a map here for every k and you ask that this be an isomorphism, and so this thing is centered at zero, and let me give you a very quick, I'm not going to prove it, but I'm going to just, once you see the idea it's trivial, so the proof is the first thing is without loss of generality, I don't know what the French equivalent is, without loss of generality, it's the same, okay, n is a Jordan block, so it's going to look like zeros and then just ones, and two, just write it down, so you've got a basis here, and for a single Jordan block you'll only get non-trivial graded quotients either in odd or even parity depending on the degree of nil-potence, but you just, you just write it down, so, and once you've got its existence, the uniqueness is easy, and you can write it down in terms of kernels of powers and so on, all right, so now, this is just undergraduate linear algebra, yes, I'm seeing everything is pollinate, yeah, because B does not extend, or not in any obvious way, so, yeah, that's right, and now, so what we're going to do here is, so, we have AADDW, and now we want to get it, so this is going to, which takes SNH into itself, so this gives us a weight filtration, which we'll call W, you know, ADDW, but we want to do one other thing, this guy's centered at zero, we want to shift it, so it's centered at the weight of the variation, and I want to, once John Conway gave a lecture in Duke, in the middle of the lecture, he just, you know, a big guy, he just dived onto a table in front of the room, but he just did it to get people's attention, so the best I can do is to hop, okay, but we're going to shift, so this, this guy is centered at weight zero, right, that's what it is, it's symmetric about zero, and we're going to shift it, shift to get, and I, you know, you put some square bracket with something, so that it's centered at N, which is equal to the hodge weight of SNH, it also happens to be the SL2 weight, but it's the hodge weight, it's the weight of the variation, so now what you have is you're going to see that ADDW to the k is going to go from GRM N plus k of SNH, and it's going to go isomorphically into GRM N minus k SNH, so it's now symmetric about the hodge weight, and you may say why do that, well we'll see in a second, this is the new, so I'm going to use M and I'm using it, this is so last week I called one weight filtration M, and this is it, it's, and I use M because it's the monadromi, I guess I called the other one, yeah it's the monadromi weight filtration, so we have locally unipotent monadromi, if you take its logarithm it's in fact this, this operator here, and now it gives us a weight filtration, and we get a weight filtration, and so the proposition, sort of doing things differently from the notes here, so what we've done now is we have, we've got a weight filtration to put on the middle fiber, so so far we've got, we've got a vector space, namely the central fiber of the canonical extension, we have a limit hodge filtration, which is what we get from the extended hodge bundles, we've now got a weight filtration, I've got to give it q structure, and so so far I haven't used any tangent vector, this is the hodge, this is the hodge filtration, so we have a hodge filtration, a weight filtration, now what about the q structure, and I'll just say we, we get one just a preview for each v in the tangent space at zero of the disk, and there's something I've deviated from my notes here, all right, so how are we going to do this, it's going to look kind of strange, but so what we're going to do it will do for v equal to d dq, and maybe I will explain the general thing here, otherwise this is going to look so strange, because this variation is so special, but I, I need the same construction for mixed hard structures where it ain't so cannot, you know, it's not going to look so nice. So yeah, but okay so, so let me just talk about the general situation, so we're going to have say a v over the disk with unipotent monodromy, like we do here, we have unipotent monodromy, given by the nilpot, you know, the unipotent matrix, you know, called, and then we're going to get v delta over d, that's the corresponding flat bundle, then we're going to get its canonical extension, and so what are the key features here is that the residue at zero of nabla is nilpotent, and right, and nabla has, I should have put it in the reverse order, has a regular singular point, has simple pole at say q equals zero, so in other words it has a regular singular point, right, so now there are various ways you can think of this, and the way I like to do this is to use just standard old-fashioned ODE, this is a, this is a good book by Vasov or Wassow, I'm not sure how to say his name, but you can always choose a frame, so you can always trivialize this, so let me name the fiber over zero, so, so v is going to be the fiber over zero, so when I trivialize it, I'm going to trivialize it using the fiber over zero. Oh, these are different fonts, this is blackboard bohulf, this is calligraphy, and this is roman, this is, this is just an ordinary v, this is script, right, I use script for the, that's, but this is a blackboard, there's two of those, and so now, so, so ODE, and I think this is more or less in Deline's book on differential equations, but it's also, I'd say CF, you know, you only have to read about three pages in this book, there exists a trivialization of V bar, so V bar is going to be isomorphic to D cross V over D, such that the connection, when you've got a trivialization, the connection you can write out more or less as a one form, is equal to N times DQ over Q, and this is constant, and this is in fact the residue, and it's equal to the residue at zero, right, you can always do that, that's the result of ODE, but what does this do for you? So let me draw a picture, here's the disk, this is V cross the disk, here we've got our central fiber, and we have some sections like invariant sections will extend through, and those with monodromy sort of behave logarithmically, they blow up, you know, they behave badly, but what does it do, you know, and let's suppose Q1 is in our disk, so what it'll let, sorry? You can see there's your horizontal section of your connection, they consider, where do you say section, or is it a section? Yeah, sorry, yeah, horizontal sections, yeah, flat sections, so what this, but what does this trivialization allow you to do, it allows you to compare the fiber over one to the fiber over zero, because this is still, this is still V of one, which equals a fiber of V, by the way, I mean there are the technical people think of it, I mean everybody can, we can communicate, they think of different ways, but I really like to think of it downstairs, they like to think of things on the universal cover, you know, people in the, like Wilfred Schmid or all these young guys like Paul Stein and so on. Yeah, but anyway, so what you're going to do is we're going to use this trivialization to put the Q, we're going to take the Q structure here on the point of this disk corresponding to the tangent vector, right? DDQ corresponds to one, you might say what happens if one's not in here, you can always do it for sufficiently small tangent vectors and then you get a formula, and I'll show you what the formula is in a second and extend out, but let's not, in our case we'll be able to do this, and this gives you a Q structure, so we use the trivialization to transfer the Q structure or, well, to get an isomorphism, a linear isomorphism from v1 into v0 equals v, and if this has a Q structure, this is going to give us a Q structure over here, all right? So this is a key thing, and so what we'll do in our case, it's going to be absolutely trivial, and this is, by the way, how I think of regularizing periods, and it's equivalent, I think, to what everyone else does for regularizing periods, all right? So let's do it for, so let's do it, let's take v is equal to h, well, our trivialization already has this property, we've got h cross d over d, so h is just going to be cA plus cW, right? And the connection, nabla is equal to d plus A ddW times dQ over Q, so this is constant, we've already got a deline, you know, one of these nice trivializations where this is constant, so now let's write out what's the flat structure, so we know that W was equal to something like minus 2 pi I B plus log Q times W, right? And now we just have to turn it around, sorry, log Q times A, and so what we get is that 2 pi I B is equal to minus W plus log Q times A, did I do that right? I think so, and now what is, and we're going to use this trivialization, so it's just going to amount to setting log Q equal to zero, when you set Q equal to one, you get rid of the log Q, right? So v, sorry, h of one is going to be, it's QA plus QB, that's, by this I mean the fiber over one, right? And the trivialization says that's going to be our structure, right? Because, how do I say this? It's, right, and so this is actually equal to QA plus Q times W, right? Because Q equals one, yeah, where is it? 2 pi I B, yeah, I have to put a one over 2 pi I, and one, is that right? Because it's going, B is, if I said, if this is zero, W over 2 pi I is equal to B, up to a sign. And this is coming around because we've got, it's really a Q of minus one, right? So this is our Q structure on the middle. And so what have we got? We've got, so what's this saying? So it's saying, so what's the mixed hodd structure on, let me just do h, first of all, associated to d, dQ. Oh, by the way, I should say here, this, the Q structure here, this is an important remark. So it's, and this is a little surprising when you look at the construction, because you have lots of choices here. The, the Q structure on V equals V of zero depends only on d, dQ. In other words, the parameter to first order. If you change the parameter, if you think about it, you use little exercise, you actually change this trivialization. The only way you preserve this trivialization is if you did a silly change of coordinate, like, multiplied your coordinate by a constant. Right? But if you take any non-trivial change of coordinates, you've got to compute a new trivialization. And when you check everything out, it sounds hard, it's actually easy. You find out that everything depends only on the parameter to first order. Right? So here, the mixed hard structure on H associated to d, dQ is, well, we've, we've got hQ now is going to be equal to Q a plus Q w divided by 2 pi i. And now we've got, and we've got a hodge filtration, hC is going to be CA plus C w. And this is going to be our F1. And if you stare at this, you'll see we've got a split hard structure. This is saying that h d, dQ is equal to Q plus Q of, of minus one. And that, right? So this is spanned by the Dharam part of, this is spanned by w. And I actually made that a Q Dharam class, which is equal to, and the, the betty part spanned by b, which is equal to minus w over 2 pi i. And this part spanned by a. And it's split. It's visibly split. So this is a very basic calculation. If you wanted to do a different tangent vector, I, I won't do it. You would replace Q by, if you wanted to, you'd do lambda d, d, dQ, just replace the parameter Q by so lambda, lambda over Q. And do the calculation, you'll get a non-split extension here. Okay, so there's a question of what to do. Last time I stopped on time and then Francis said, oh, you can go longer. I figure it's hard to listen for two hours. Let me see what I, I'll let you vote. Yeah, I'm not going to do, I was going to do limits of mixed hard structures. I've got another 20 pages here, but let me just mop up this and I'll stop limits of hard structures. And next time I'll do limits of mixed hard structures, which is, oops, I'm right, shouldn't have erased that. Okay, so, yeah, and I should say, okay, so I just said that the, what I call h, when I write h, d, dQ, I mean the limit mixed hard structure associated to that tangent vector is Q plus Q of minus one. And this is essentially expand by W and that by A. And if you look at SN, SNH, d, dQ, it's just equal to the n-symmetric power of h, d, dQ. Everything here behaves well under symmetric powers. And this is just going to be Q of 0 plus, plus Q of minus n. And this will be spanned by W to the n, this by A to the n, and things in the middle by things like A to the j, W to the n minus j. So this is also split. Right. And maybe I'll state this here. Before I do that, I should say, why is this reasonable? I mean, what's going on here? Is this alchemy? Is this out? A exponent n minus j and W exponent j. A to the j and W to the n minus j. Yeah, or? No, you have to exchange d to n minus j. Right. If you want to get something which corresponds to Q of minus j. Yeah, you'd put A to the n minus j, but I cleverly didn't write anything there. So, all right. So is this reasonable? So instead of writing the whole thing, so let's look at the universal elliptic curve over the disk. All right. So here's how I draw it. So I think it's sort of like this. It's our family here and here's 0. Right. So this is essentially the Tate curve. And now this total space here, maybe I'll write it as E of D. So basic fact in the topology of varieties is if you include E 0, that's the fiber over 0, into E D, this is a homotopy equivalence. It's an elementary. You can just see you can just squeeze it in, but I can give you a rigorous argument. I can give a rigorous argument lunch. Sorry? Yeah, in this case, it's elementary. You can write it down. But in general, that's a little trick. The one I know uses the triangulation of a variety. It's a very general fact about degenerations. But we also have E 0 star, maybe I'll write NS, meaning the non-singular part, is this is isomorphic to C star. That's the smooth locus of the nodal cubic. And so what we have is that E 0 non-singular includes into E, I want to say this includes into E D D Q and I'll explain how I think of this guy. And then this guy maps into E D, which is homotopy equivalent to E 0. So just suspend belief for a second and pretend there is this fiber that's easier actually to write this down in the etal situation. So this is a C star and this is equal to P 1 mod 0 identified with infinity. Excuse me. And here, let's look at just H lower 1 or H upper 1. We can do it either way. Let's look at H upper 1 of E 0. It's actually even equal to a Z of 0. Let me do all of this integrally. You can imagine, I mean it's a general construction of mixed hodstructors, but basically this class is coming through, passing through a point which has type 00. And then we're going to get a map here into H 1 of E D D Q. And then there's going to be a restriction map into H 1 of C star, which is equal to Z of minus 1. So in the limit you think this guy should be an extension of this guy by Z of 0. That's just a heuristic reason. But this is actually a property of the limit mixed hodstructure that is compatible with all of these things. You know, it's built essentially on the singular fiber. You're sort of backing off infinitesimally. And maybe the last thing I'll say before lunch is I'll tell you how I think of the fiber over D D Q. So how do I think of this? So and this is probably related to log schemes. I'm too old to know about log schemes. So if we looked at, so I want to say how can we describe the fiber over D D Q? So this is, so what I'm going to do is I'm going to take the real oriented blow up at zero and infinity of P 1. And what is that? That's just really equal to the real oriented blow up, so which I discussed last time, is like this. Here's infinity and zero and I've replaced them both by the circle of directions. Yeah, and the real oriented thing. And now I'm thinking here instead of doing this, suppose I have a tangent vector here. So the Hessian gives you a map from the tensor product of the tangent space down here to the tangent space here. So what you can do is over the, so I'm going to take also the disc blowing up at the origin. That's this here. And now I'm going to, so here's the disc, the real oriented blow up. So every point on the circle corresponds to the origin, but they tell me all the different directions. And now how am I going to construct the fiber over D D Q? So D D Q actually has a length, but actually arithmetically this picture is okay because we can't double or triple the length of this, otherwise we'd have bad reduction at wherever. So we don't have much choice. But anyway, and so what we're going to do is we're just going to identify the two ends of this and over say e to the i theta times this, I'm just going to put a rotation by theta in. And you can see that C star includes into this. It's everything except the circle. And you can see it also maps to E0. How do I get E0? I just collapse the circle. And I actually wrote down the equations for this and prove that you get a nice family here. I mean it works for stable families of curves, but I wrote it out at the end of these elliptic KZB. It's some manuscript where I park all sorts of stuff that I don't want to put in papers, but I think it should be out there. So I'll stop there. So I think of the limit mixed odd structures being on this guy. So I'll stop.