Modular Symbols, K-theory, and Eisenstein Cohomology
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Stevens, Glenn
In this talk we will give an adelic construction of an object that we call the Kato-Beilinson modular symbol for GL(2), extending constructions of Goncharov and Brunault. We obtain a modular symbol Ψbelonging to the compactly supported cohomology of arithmetic subgroups of GL(2) and taking values in a group of distributions valued inK2of the tower of modular curves. We interpret Ψas a “universal” L-value for modular forms and explain how it specializes to Kato's Euler systems, as well as its role in Fukaya and Kato's proof of Sharifi's conjecture. Our hope is that these ideas will also help us understand a conjecture of Darmon and Dasgupta about “elliptic units” associated to real quadratic fields.
 So let me start with my distance. So I'm going to keep my notes so I don't stray. So what is a modular cell? Well, so you start with a subgroup of GL2Q. Sometimes it will be GL2Q, but usually it will be a congruent subgroup of SL2Z. M is a gamma model. So I'm going to say, let me say, they usually have your ring sitting around, and so R will be summary. And then delta zero, that's my notation for the set of divisors of degrees zero on P1 of Q. So P1 of Q, I imagine, as being, start with the upper half playing, and P1 of Q is the set of rational numbers on a real line together with the point that I fit. And so then let me just tell you now, so that's what I'm thinking about P1 of Q. I don't know if people can see that board, so well, I'll move over here. And then, so what are the modular symbols? So modular symbols over gamma with values in M, sometimes I'll write it that way, that is the set of homomorphisms equilibrating over gamma from delta zero to M. And so if I have a modular sample, I'll be thinking about things like this. So I'll tend to write things like this, phi of R to X. And when I write that, what I'm, you might, in this interpretation, that would be phi of the divisor of degrees zero S minus R. And I'll be thinking about R and S and a path from the upper half playing that goes like that. And so when I write phi R to S, I'm imagining it as an integral from R to S. And so that's the reason for this notation. So there's a remark, in fact, this captures an important part, a remark from the modern as, for myself, a long time ago, that there's a naturalisomorphism from the compact you supported called modular gamma with values in M to the modular symbols with values in M. And so that may be the justification for this picture. Now the way of gamma sort of visualizing these things, it's fun to visualize these things too. Let me try. I'll do it here. So what you might think about is zero, you have i infinity up there somewhere. i infinity. And you think about the path down, 1, 2, minus 1, minus 2. You can draw these geodesics connecting those guys. Take a half, make one and a half. You make what's called the fairy tessellation. My picture's not great. But you can think about what's called the fairy tessellation of the path. And the idea is you draw the fairy sequence down on the real axis and then you connect up. So let me just say it very simply. If I take the path from infinity to zero, which is this path, and then try to apply the elements of SL2Z to it, there's a 1 to 1 correspondence by PSL2Z, I should say. There's a 1 to 1 correspondence between the elements of PSL2Z and the edges of this thing. Where if I have a matrix gamma, this is A over C, B over D, where gamma is the matrix, A, B, C, D. And so those are two adjacent terms in some fairy sequences. The idea. So I'm going to leave this picture on the board. This, by the way, is something that I learned a long time ago when I was in high school about fairy tessellations and fairy sequences. And one of the first things that I actually fell in love with was this idea of continued fractions. And I think that is the initial reason why I got so interested in this project. Gary showed me how to use continued fractions to compute periods of modular forms, compute modular symbols. And so of course that caught my eye. Of course now there's very little continued fractions in the theory anymore, but still I like drawing that picture. So that picture, I'm going to leave on the board so I can point to it now and then. OK? All right. So the idea is that a matrix symbol, and gamma is a confidence of room, a matrix symbol with value can have is the way of labeling the fairy tessellation. Where each edge of the fairy tessellation, I say like labeling, I mean just labeling the edges. And giving each edge associated with some element of the module now, but doing it in an equidriot way. Gamma equidriot way. That's a picture of the module itself. You'll see that if you do that, there are certain relations because we need to have, when we go down and over and then back up again, we better get zero if this thing represents divisors of degrees zero. And so in order to make such a labeling, such a matrix symbol, like there are certain relations that have to be satisfied. And they're the ones you see in the picture and they're called the binary relations. OK? And so you will use them without a lot of saying here, but they're important. OK, so now one of the things that Barry taught us is that when we're studying values of L functions, at least for modular forms, really what we're studying in some sense are these modular symbols, or if you like, the homology of the modular curve, and you're making certain homology classes in the modular curve and evaluating modular symbols on them, that means maybe you're elements of a model that correspond to values of L functions. I'm not very good with these names. OK, so in particular, there are things like they're the universal L values. So the universal L value is the path from infinity to zero. OK, so if you like, it's the difference between zero and infinity and delta zero. But that's the path that you integrate a modular form over in order to get the melanchor transform of the modular form. And so that's how we compute L functions. Or another thing that you might do, if you have a character to R, the units in R, a character, say a Dirichlet character, then you might write lambda of chi. This you see is an element of delta zero, or it's an element of the relative homology of the modular curve. The lambda of chi would be some sum, a equals zero to m minus one, say a, and z mod m cross of chi bar of a times a over m. And if you integrate a modular form over sections, then you'll get the value of the L function, actually. For, twisted by the Dirichlet character, chi. OK. And so we think of lambda of phi, that means phi of that thing. Lambda of phi chi is just a notation for phi of that thing, let's say. OK. And so it somehow changes the context for thinking about the value of L functions, thinking of them as homology classes, which is very natural. OK. So let me just give some examples. So this is a basic example. If f is, say, a cost form of weight k for some group gamma, where k is greater than or equal to two or even, we let omega f be the difference in people. Two pi i times tau x plus y to the k minus two f of tau d tau. That's a differential form on the upper half thing. Taking values in some space that I will call w k minus two, where w k minus two of any ring is just r x, y homogenous polynomial in two variables, x and y, to reach k minus two. OK. And so this is the key thing. If the depth is a cost form, this defines a gamma invariant differential form. This is something that's invariant under the action of the group gamma. And so I can integrate it over paths and make them all resemble this way. And so I want to do that. Then we define pi of f of any path from r to s to be the integral from r to s of omega f. OK. So that's the model of the symbol. And notice that it takes values in the homogenous polynomial in the context for this is the degree k minus two. OK. Now there's a theorem now of mononim. I just want to mention. And then I'll pick up the page a little bit. OK. So mononim theorem. So for f and i, I get more. And then there exists a minus and two model symbols actually apply a plus or minus model symbols for the group gamma with values in wk minus two of the ring over f. This is the ring of HECA eigenvalues of the form f such that n and two periods. So I call omega f plus or minus such that. Well, if you put lambda of pi f, so the universal for the l value of pi sub f, is equal to omega plus times something of i equals to over i, i even between zero and k minus two of some algebraic part of the cuss form with values in i plus one times x to the iy to the k minus one. Where these guys are plus, sorry, plus also some minus part. The point here is that the phi sub f, the phi sub f can be written as, I mean first say this, omega f plus times pi sub f plus omega f minus pi sub f minus. So we can sort of split this guy over here into this plus part and its minus part such that, so these are like the algebraic parts, the omega or the psi f plus and psi f minus are the algebraic parts of the modular symbol associated to f. And when you take l values, you get something similar. The l values of the cuss form into the even and odd parts multiplied by the same. Okay, so these are the algebraic parts of the l function. Now the message here, which is one that very popular, very well back in the old days, is that modern g l functions were something that we all know have something to do with arithmetic. So we think of some arithmetic logic like an elliptic curve or something like that. We make an l function and that then is a way of encoding local information into some global analytic thing and then we try to retrieve global analytic information out, or global arithmetic information out. But the message here when we do this, that not only are the values of the l function complex common, you can tell us something about arithmetic, but you can actually do arithmetic with the l values. And of course by now that's a thing that we know very, very well, but I just want to point it out because it's what I'm most interested in. Okay, so, okay, now, so let me just, the next topic I want to talk about in this context is Eisenstein-Compagnen, which is something that very, also taught me a long time ago, can we go over here? Oh, my very tussillation to stay there for now. Okay, so, a section of the two columns. Eisenstein-Compagnen. And so the idea is to think about cases where cuss forms are congruent, modulose on prime, license on the surface. And then can you say something about the associated ellipads? This is another very old story and this is a typical standard example. The first example is one that we've seen already in this conference, is to start with f and s2 gamma zero of 11. So f is q times some product. Oh, I've got a picture. q, 1 minus q to the n, 1 minus q to the 11. Yeah, square. Yeah, these are going to square. Okay, and so, and e, the Eisenstein series, this is a model for gamma zero of 11. There's an Eisenstein series of weight two also for gamma zero of 11. The cuss forms are one-dimensional and the Eisenstein series are also one-dimensional and this is this. Plus some sum, n equal to 1 to infinity, some modified version of the sigma function times q to the n. And the thing that one notices here is that the f, it's not obvious, but it's true that the f is congruent of the q expansion to the Eisenstein series, modulo five. And so in particular you'll notice that the constant term of the Eisenstein series is the visible white white. Okay? And so that's a congruence between two analytic objects and the question is, does that tell us something about the, these L values? Are they congruent? And the answer is yes, they are. And so, in fact, and I'll write this down rather politely, but I'll say more about it in a second. So what one ends up with, they're all right. I'll just state it in this case. Five, size of f I should say, size of f minus is congruent to size of e minus mod five. The next question would be to ask whether or not there's a conference between the two relative symbols that reflects the congruence between the cuss form and the Eisenstein series. And there are two periods. The first period, the one in terms of the average, there are two. But for Eisenstein series there's only one and it's this. Well maybe e minus is in fact the tie up, that's the theory. There's no even part of the micro symbol. And so we can't even, it doesn't make sense to frame this. Okay, now, so this is something that's interesting because in fact people like Rob Pollack and Preston Wake have been thinking about congruences even in the even case. And they do write down congruences between micro symbols associated to f and micro symbols associated to e. But when they talk about the micro symbol associated to e, they're thinking about a boundary symbol. Boundary symbols are things that only depend, they're actually extended to homomorphisms not just from delta zero to m, but also to gap from delta to m where you don't take the divisors at degree zero. That's much easier to make such things. And there is an, there aren't bad elements of that thing that have the same idling value as the Eisenstein series does. And so that's what people are, when they talk about those Eisenstein series, that's what they mean. But they're not talking about periods of Eisenstein series. They're very similar things. And so the question then is what happens, what do you do? What happens when you're working with an Eisenstein series and they have, you're thinking about the even part when you're working with an Eisenstein series. Okay. And so, so that's the question. What does that mean? And how do we make sense of that? Okay. So, well, let me just think about an example. I won't write down the actual example that's all described as best I can. I want to start with a cost form. By the way, this, let me say right at the front, that this is the, this is the question, the original question. I don't know what, where is Romgart Scherrithy actually starting with this whole stuff. But Romgart, certainly the simplest case of his theorem, or his conjecture, which is now theorem, given for Caillat-Paco mostly, the conjecture, he gave a conjectural formula for besides the best plus, but P in the case where P is an Eisenstein problem. In the case where there's a concoct like this. And so, that's really what I want to talk about. Okay. So, so I'm going to start with a simple case. He says much more than what I'm going to say right now, but I'll try to say something that's his full conjecture problem. So, start with a cost form on SL2Z of weight K, K greater than or equal to 2, and P prime, it will always be greater than 2 for me, prime, such that pair Pk is a new regular pair. Now, this position that it should be an irregular pair basically says, by the way, I want the P, P would be bigger not just than true, I want to weigh one small prime. Okay. And so, and so this just means that there's a congruence between this cost form. I want to say this is an I form, it's a Hecker I form, and then I'll have an Eisenstein series. There's only one, the one space of Eisenstein series of weight K on SL2Z is one dimensional. And so I'll take a generator of that thing. And suppose, so even he then, think of it as minus Bk over 2k plus again some sum, n equals 1 to infinity, some sigma K minus 1 star of n, q to the nth, that's q expansion. And I want the cost form to be congruent to E modulo sum prime P, or P sum prime say in the Hecker algebra, in the region generated by the Hecker algebra, by the Hecker value of F. And I want F then to be congruent to E as a q expansion. And that's going to happen, then the only way they can happen is to take it, it must be the case that the constant term is infinity. So P then is some prime above P. That's a little like a P for terms. Okay. And so the condition that Pk is near-regular pair, I'm thinking of it really as a state, what I really want is the cost form in an Isis line series, and the Isis line series. So then, okay. And so to run our game with a nice conjecture for what the value of the algebraic parts of the L value should look like in this situation. So I want to just describe it, quick. So Charini's conjecture. So let R, let R be the bracket Fp through the very end invertly. Okay. And you, I don't know if you are right, the units in R, module of these powers. Okay. And so that's just some Fp, finite dimensional Fp, that just takes a bit more flexibly. That is. And so, okay. So now we take the U and write it in decomposing. I'm going to use the same notation I used before. I equal 0 to P minus 2 U 1 minus, oh, okay, sorry for the weird way of arranging this. But, and so this means, by this I mean the part of you, we can decompose you in the eigenspaces for the action of the Galawang group of Q zeta P over Q. So that's just some type of a character. So we're taking, this is the competition of the components on which the Galawang group acts by the U 1 minus I power of the type of a character. Okay. And then we let a i be equal to the projection of 1 minus zeta P, this is the element of R cross, to U 1 minus I. Okay. So that's just some, some element in the 1 minus I piece of that thing. And, okay. And so the conjecture then is something like this. Let me get it exactly right. So what we think about is we want to take U tensor U. Of course this maps down to, by, uh, tumor theory. That's where they, one, uh, F, U, P. Uh, and that then goes by cut product to H2, H2, H2, H2, H2, F, uh, my P. Okay. And then at a time of Tate he says that there's there to A, A-gencerly P. And so our A is the class group of the field S. F is the field of fractions of R, of course, module P. Okay? Okay. Now our column point says that Pk is an irregular pair, and so Pk, irregular, implies by Eberon-Ribbitt that A1-k is not zero. The one minus k component of this A reaction, the same, the same group of definitions, is not zero. And so what we do then is we let lambda R for each R, for each R, we let lambda R be the, I don't know my column, do I get myself, what do I call it? A-R. A-R, A-R, cancels for A-R, what I want to say. And that's right. Okay. And so this, this then is an element of u tends to, u, I'm thinking of an element of u tends to, or u. And then I take, I call it, not this method, this one up here, rho. I take rho of lambda R, that is in this A tensor mu P, 2 minus k. Okay? It lives in there. Now under special conditions, and this, this group might be, this rig, this is a, an Ft vector space if you like. It will be one dimensional in examples, it's always been seen to be one dimensional, it could be higher dimensional, in which case what I'm saying is it's too weak. But the point is that, that I can say take a linear functional of this thing and identify a piece of it as a, as sort of a line. So I'm going to think of the rho of lambda R as actually being an element of that thing. Up in choice of the basis. Okay? Up in choice of some linear functional if you like. And so lambda R's conjecture says that, let me actually get it exactly right, lambda R's conjecture says that if you take slice of F plus, appropriately normalize, and then evaluate on the universal special value, you will get modulo P, the sum over even R's of k minus 2 choose R, lambda R, x to the R, y to the k minus 2 minus R. Okay? So it's a bit of a strange statement. So I have to have this ability to think of elements of the class group as elements of that key. Okay? But other than that, I could normalize things. With the statement, if I take that out, that's what I'm going to get. So it's kind of an interesting statement. Maybe you can identify coefficients, sort of the coefficients of this modulo sample with certain elements that are built out of k2. Okay? It's a nice statement. So this was just, to be honest, I don't know exactly what motivated it. I find it an amazing statement. And so I had a student at that time, her name was Cecilia Bousouyock. And I said, well, this is a fascinating conjecture. Why don't you use the computations? Let's see if we can see it. And so she did some calculations. She's actually first to have to figure out where it all ends. So she took a couple of months. I didn't hear from her for a while. A few months later, I got an email from her. I bought nine o'clock one evening. And she said, I've done the calculations for these two examples. I don't remember which ones. And it's checked out. Okay? So that was great. And so I went to bed and I woke up the next morning and I had another email from her at 3 a.m. that said, and I can prove it. And so I was kind of surprised. Her statement about autism might have been a little bit strong. But the point was that she saw the idea of what had to be due to prove at least something about this that was interesting. I should say up front that I communicated this with Ron Kamshari if he, and he said, he had just discovered the same thing. So it seems like a simultaneous, close to simultaneous. I don't know exactly. A discovery. But the argument turns out to be fairly elementary. And so let me just write it down the other way. Okay. And so I already got my fairy tessellation. There it is. Okay. And so what she does is she defines a micro-simple. It's a micro-simple. I'll write it down in terms of the theory of tessellation instead of this other one. And so here it is. So psi of, if I have a matrix gamma in SL2z, then if I take gamma times, I take this path from infinity to zero, and I apply gamma to it, that will give me some edge. Okay, every edge, the horizon in that shape for exactly one gamma. It's about to do something. And so this is, well, if gamma looks like this, if it's something, something x, y, as congruent to that, my p, then the volume can read off what it is from the bottom row. It's just, you take 1 minus zeta px, 1 minus zeta py. This is the Steinberg symbol happening in k2 of the ring r. And so this, what she said is that if I put that, label the edges of the fairy tessellation, that, then it's equivoc. That's actually what she put. And it's not so hard to prove. And you end it at the bottom of the symbol that satisfies the monium. The fact that it's equivoc, the second range, not so hard, but part that's interesting is that it satisfies the monium relations, and not just a matter of writing down some particular relation in k2 of that ring, which was not very hard to do. She just came in and showed it to me. It worked beautifully. It was amazing. And so, this is the monium sample. Okay, so, satisfies the monium relation. Okay, now, Shalisi's conjecture, of course, much more than this. So let me try to state a little bit more of what he did, what he actually conjectured. Oh, I should say something. Part of Ron Yar's conjecture is that this should reflect the Meisner symbol that we're writing down should be corresponding to the Eisenstein series. So in particular, this Meisner symbol, I should be, part of the statement is that it's a heck of an item set. And that the eigenvalues are the eigenvalues of the Eisenstein series. And that's the part that turned out to be really hard. Okay? The evidence was there, and you can see that the conjecture says it worked. So it was right in the example, but Shalisi sat down and she tried to prove it down what it meant for t2 to be the right eigenvalue, and she found some relations in the Steinberg, Steinberg's samples that implied that. And so she proved that this is actually an eigenvehicle for the Hacker operators t2 and t3. And that's it. And they were hard. They were pretty impressive. Ron Yar did the same. I'm not sure. I'm not sure. I'm just saying to me, I remember some of these saying they had a proof like this for t5, but I'm not sure. Okay, but the point is that proving that this worked in general, that it was an Eisenstein example, was really, really hard to prove some reason. Okay? All right. But that is just a double. This can make work rejected anyway. And so, so let me just say that, that similarly, we can define, I haven't written it down even in the one case, sort of, let me just say, since I've been in close. So let Hn plus be equal to the homology of the modular curve x1 of mp to the n, with values in zp. And I'll take the plus part of the action of context conjugation, where n is some level, pain level. And so these guys are relatively prime. And suppose, again, they're written in case where you have this irregular, or pk, an irregular pair. And then the idea then is to understand what happens as you go up the modular tower and as you go up the cycotomic tower. And so the omega n is defined to be, we can define some map, Hn plus to an minus, what's this one? So it's the same an, but now it's the class group, the p power part of the class, actually it's the p to the, it's the classroom on p to the field fn, it should go up the tower, the cycotomic, yeah? Okay, this is a classroom. Okay. And so it's a map, it's defined not so differently from this. And what he calls this, well then he says, this factors through the Eisenstein quotient. So Hn plus modulo and Eisenstein ideal are i, where the i is what you think of it. Yeah, it's a little complicated, but I'm going to write it down. Eisenstein, I think. Okay, he calls this thing Pn. And then the omega n goes, think of the omega n, and really it's something from Pn to An minus. And that's the first part. The first statement is the factors through the Eisenstein thing. And so I'm just making a remark that even in the simplest case proving that the factor through the Eisenstein quotient was not obvious. And so, but the conjecture is that it always grew, happens. And moreover, moreover, on the omega n, or quantum omega n, I'm just feeling anybody else would call it pi n, to An minus of 1 is an isomorphism. So that's the second part. And he then, I'd wait until he actually said that, hold on a second. I don't actually have that statement written down here, I want to be careful. Moreover, the right thing to say is that if I take a limit, as n goes to infinity of the Pn plus, that goes to some projective limit with respect to the norm mass of An minus of 1. And this is the map that he describes as an isomorphism. That's the contention. Okay, so, so I did a little bit of a mess to read it all. This is part A and that's part B. And so, he said more. He built some sort of a piatic L function associated to all of this, which is I'm sure important. But I don't want to talk about it now. And part of this, what happens here is that Sharifi, I'm not Sharifi, but Foucailla and Cato, throughout this time, came up with a proof of this conjecture, pretty much as I'd written it down, under the hypothesis that the Komodo-Level piatic zeta function associated to the level n, to the tamed level n, has no multiple zeros. I think that's the only hypothesis that they needed. They wrote down in the inverse to this map, omega, and then computed it both directions and said they explained the identity of the either direction. Okay. So, okay, now, what happened here is that Cillian and I were very interested in trying to prove the statement that this guy was in Eisenstein's class, even in the simplest case. And so, we did actually some stuff, and I think I might have enough time to jot some notes about what we tried to do. I want to write down some minor symbols, and this is the first one. This is the first one. It's not so obvious, but it's not so hard to get. And so, so what we did, we were trying, we said, okay, how do we define that object? Well, we're used to thinking of them as double coset operators. And so, we'll write and use some representation theory in this business. And so, what we did is we introduced some ring, I'll call it R1, some ring of trigonometric functions. So, either the two pi i z is an example of one. The sine of z is an example of one. It really looks pretty much the only one that we use. But the other one, that you know, there, there, there, there. Okay? The point is that these are our power series in Q, where Q is either two pi i z, you can specialize Q1. And if you do that, you might have to be fixing it in a roots of unity and things like that. So, we tried to compute K2 of this ring in a way that we can remember, in a way that had an action of a roots on it. And so, the gl1 q action on that thing. There's an actual action. You can imagine what it is. And so, we first made a distribution on Q, taking values in, what K1 of R, K1 of R is just R1 cross. Okay? And so, I won't say what a distribution is, but it's, it's, it's kind of what you think it is. You, you make 10 functions on Q and you make homomorphisms from the, from the group of test functions to this thing. And it's, that's the way of expressing sort of distribution of a ring. And then we, there we go. That is, a unique mind and a sample. I'll call it 5 from delta zero to distributions on Q2, taking values in K2 of some ring, which I also won't write down, some ring of, of tracelet response in 2 here. Okay? Okay, now this is not exactly right. We had to divide by some ideal. I can tell you what it is, but for now, let me just try to get something on the board. This is an ideal in the, it's just, it's actually the part, take K, the moment of this ring, K ring, the moment of K ring of the ring R2. That's a graded ring of head logic, graded pieces. I'm just looking at the graded two parts. And there's a part of it, and then taking the ideal generated by what I call Q1 and Q2, I call it to a Q which is Q1 and Q2 or the two variables. Okay? That's a pretty trivial submodule. But if you modify that, this thing becomes a much more simple, where, where two things are true. One, five is GL2Q of a, GL2Q of a grade. And B, if I leave you to your imagination, the Q2 acts on this, and it also acts on the R2, and it's a mixing of that. Okay? And the other thing is that the, the five, I just have to tell you what is L value. And this L value is mu1. Okay? In other words, I should, to be very precise, I should be saying like in pi1, you pull back pi1. I think of Q2 has two projections to Q. And I'm taking, I'm taking a distribution on this, I'm pulling back one of the projections, and then taking some convolution product of that, we're going to pull that according to the other projection. Okay? Okay. So there are, and, and now the idea was, and I should, I should, I should say it's a, whatever I've written down is true so far, but this isn't what we wanted to prove. We wanted to use this to prove that the thing she's written down was an eigen-assembled for HECA. And we never got back to that. We wrote this down, we, I'm not even able to tell you how, how the idea was that we would be able to specialize this at Q1 even with certain points and be able to say that they, that they would make a class out of this that's a eigen-assembled way upstairs and then specialize it to say that it's an eigen-assembled downstairs. We had never completed this project because it was about this time that Foucaillet and Caillot announced the general proof and we were getting lots of details and stuff. So it seemed like a good place to study quite a few down at Foucaillet and Caillot were doing. Okay, so the idea, let me just, since I'm out of time, let me just say that what Foucaillet and Caillot, what Foucaillet and Caillot did was they made a quantum sample out of valence and elements built out of Ziegler units. And so, so specifically, this was actually, in my world this was due to consular off. You take the Ziegler units, you take the Ziegler that's what the P0U back with G0V and when I write this let me be careful, it's u over n, p to the n. This is a V over n and then P to the n, takes the Steindor element of K2, but I'm going to think of it not living at level n, so it's fine to think of it. Let me write it at level n. It gives an element of K2 of that much of a curve. And what they do is they then make a minus, they take a minus symbol from H1, X1, and P to the n to the n. That's specialized, and then they specialize at infinity. In other words, they make a map. You can specialize these guys to make elements of K2 of these rings, of these rings. And then they specialize at infinity. You take this guy, and so it's an element of K2 of this thing, but you specialize it at the infinity cost, all right, infinity curve. You specialize in that element of K2, and then you specialize it at the infinity cost, all right, infinity curve. You specialize in that element of K2, and when you do that, the idea is that specialization at a point like infinity sort of is an eigenspine map. So it respects the Hecker-Arborism, the eigenvalues of those, of the eigenspine situation, in fact, of the boundary symbol, VAT, which is a thought that just occurred to me recently. But because you're only using one cost, that's the thing that's interesting. Okay, so, but the boundary symbol is made as follows. I was looking again at the various facilitation. I take the U and the V, it's given by, I take the U and the V, and I associate to the same thing. I take a gamma UV, this is an element of SL2, this is actually the other element of SL2Z, where this thing is congruent, my little n, t to the n, does something like this, and you put that, so for that guy, you might just might resemble to be at this, this path, this new line is a path, and you find it to be that. Okay, so I'm going to stop except that I want, there's one more thing that I just want to at least say. The thing that I personally am interested in right now, which I'm not able to, it's hard to talk about, is a related modular symbol, it's built in the same spirit as many people have seen this thing that Kudall had in ice paper, and we should wrote down a modular symbol taking values in a certain, well he didn't describe it as distribution theoretically, and so our formulation of this is to make a modular symbol of this type, it's a distribution, instead of Q2, use matrices of, two by two matrices with rational entries, that's a four dimensional Q vector space, so you talk about distributions on that four dimensional Q vector space, taking values in K2 of the modular tower, all the way up the tower, and this is due to Von Schropp, by the way, didn't state it, but it's equivalent to this, and it's also Bruno gave a different proof of Von Schropp's theorem with some extra restrictions, but the statement is that there's some, in the form that I like it, is that you're getting a distribution on M to the Q taking values in K2 of the modular tower, it's built not so differently from this, except that you don't require the zero for the first coordinate, this is just so that you can specialize that infinity, you'd use A and B, so the first column of the matrix over there, A, B, and the second column of the matrix that I'm talking about over there is C, B, and you get some big micro-sample, and I believe that this micro-sample deserves in some sense to specialize to the micro-sample that's still here on our own down, but more, I think to be a lot more than that, and let me just say a little bit more about it, when you specialize at the cusp, one of these valence examples say, you have got to worry about whether or not the modular units are holomorphic or they're non-zero and holomorphic at the cusp that you're specializing in. And so when you take, even though this thing I call the concovalence in the micro-sample following the zero suggestion, it doesn't give you micro-samples, it's honest, but when you try to specialize at the cusp, make something Isenstein from the boundary, you have problems because you don't know that you can specialize at all points, and so you only get something called a partial, we might call it a partial micro-sample, and I'm sort of tending to calling them pseudo micro-samples because there's something about them that feels like they're going to smooth them, in the same way that you smooth the periodic measures to make them bounded, things like that. So I'm interested now in this relationship between these two things, and I should say, I'm trying to, my main interest is right now is depending on this Darmo Daskutik injection. And so the Darmo Daskutik injection, the Darmo Daskutik made a periodic measure and they gave a formula for a conjectural unit above some, in some great class field of a real quadratic field, and that formula involved piano integration with respect to that measure. Now that measure that they write down can be described in terms of this thing that's a Yenakian, it's an easy description of it, but it's just another description of the thing, that distribution somehow needs to be enriched, that's the field, if they're enriched somehow in order to make these models, you never make a unit, you have to get some arithmetic out of it. And so that's basically my interest right now, is just to examine this kind of variance and by the simple to see if it has enough information in it in order to recover in some concrete form of the conjectural unit. OK, so I'll stop there, I'll stop here.
 So let me start with my distance.  So I'm going to keep my notes so I don't stray.  So what is a modular cell?  Well, so you start with a subgroup of GL2Q.  Sometimes it will be GL2Q, but usually it will be a congruent subgroup of SL2Z.  M is a gamma model.  So I'm going to say, let me say, they usually have your ring sitting around, and so R will be summary.  And then delta zero, that's my notation for the set of divisors of degrees zero on P1 of Q.  So P1 of Q, I imagine, as being, start with the upper half playing, and P1 of Q is the set of rational numbers on a real line together with the point that I fit.  And so then let me just tell you now, so that's what I'm thinking about P1 of Q.  I don't know if people can see that board, so well, I'll move over here.  And then, so what are the modular symbols?  So modular symbols over gamma with values in M, sometimes I'll write it that way, that is the set of homomorphisms equilibrating over gamma from delta zero to M.  And so if I have a modular sample, I'll be thinking about things like this.  So I'll tend to write things like this, phi of R to X.  And when I write that, what I'm, you might, in this interpretation, that would be phi of the divisor of degrees zero S minus R.  And I'll be thinking about R and S and a path from the upper half playing that goes like that.  And so when I write phi R to S, I'm imagining it as an integral from R to S.  And so that's the reason for this notation.  So there's a remark, in fact, this captures an important part, a remark from the modern as, for myself, a long time ago, that there's a naturalisomorphism from the compact you supported called modular gamma with values in M to the modular symbols with values in M.  And so that may be the justification for this picture.  Now the way of gamma sort of visualizing these things, it's fun to visualize these things too.  Let me try. I'll do it here.  So what you might think about is zero, you have i infinity up there somewhere.  i infinity. And you think about the path down, 1, 2, minus 1, minus 2.  You can draw these geodesics connecting those guys.  Take a half, make one and a half.  You make what's called the fairy tessellation.  My picture's not great. But you can think about what's called the fairy tessellation of the path.  And the idea is you draw the fairy sequence down on the real axis and then you connect up.  So let me just say it very simply.  If I take the path from infinity to zero, which is this path, and then try to apply the elements of SL2Z to it, there's a 1 to 1 correspondence by PSL2Z, I should say.  There's a 1 to 1 correspondence between the elements of PSL2Z and the edges of this thing.  Where if I have a matrix gamma, this is A over C, B over D, where gamma is the matrix, A, B, C, D.  And so those are two adjacent terms in some fairy sequences. The idea.  So I'm going to leave this picture on the board.  This, by the way, is something that I learned a long time ago when I was in high school about fairy tessellations and fairy sequences.  And one of the first things that I actually fell in love with was this idea of continued fractions.  And I think that is the initial reason why I got so interested in this project.  Gary showed me how to use continued fractions to compute periods of modular forms, compute modular symbols.  And so of course that caught my eye.  Of course now there's very little continued fractions in the theory anymore, but still I like drawing that picture.  So that picture, I'm going to leave on the board so I can point to it now and then.  OK? All right.  So the idea is that a matrix symbol, and gamma is a confidence of room,  a matrix symbol with value can have is the way of labeling the fairy tessellation.  Where each edge of the fairy tessellation, I say like labeling, I mean just labeling the edges.  And giving each edge associated with some element of the module now,  but doing it in an equidriot way. Gamma equidriot way.  That's a picture of the module itself.  You'll see that if you do that, there are certain relations because we need to have,  when we go down and over and then back up again, we better get zero if this thing represents divisors of degrees zero.  And so in order to make such a labeling, such a matrix symbol,  like there are certain relations that have to be satisfied.  And they're the ones you see in the picture and they're called the binary relations.  OK? And so you will use them without a lot of saying here, but they're important.  OK, so now one of the things that Barry taught us is that when we're studying values of L functions,  at least for modular forms, really what we're studying in some sense are these modular symbols,  or if you like, the homology of the modular curve,  and you're making certain homology classes in the modular curve and evaluating modular symbols on them,  that means maybe you're elements of a model that correspond to values of L functions.  I'm not very good with these names.  OK, so in particular, there are things like they're the universal L values.  So the universal L value is the path from infinity to zero.  OK, so if you like, it's the difference between zero and infinity and delta zero.  But that's the path that you integrate a modular form over in order to get the melanchor transform of the modular form.  And so that's how we compute L functions.  Or another thing that you might do, if you have a character to R, the units in R,  a character, say a Dirichlet character, then you might write lambda of chi.  This you see is an element of delta zero, or it's an element of the relative homology of the modular curve.  The lambda of chi would be some sum, a equals zero to m minus one, say a,  and z mod m cross of chi bar of a times a over m.  And if you integrate a modular form over sections, then you'll get the value of the L function, actually.  For, twisted by the Dirichlet character, chi.  OK.  And so we think of lambda of phi, that means phi of that thing.  Lambda of phi chi is just a notation for phi of that thing, let's say.  OK. And so it somehow changes the context for thinking about the value of L functions,  thinking of them as homology classes, which is very natural.  OK. So let me just give some examples.  So this is a basic example. If f is, say, a cost form of weight k for some group gamma,  where k is greater than or equal to two or even, we let omega f be the difference in people.  Two pi i times tau x plus y to the k minus two f of tau d tau.  That's a differential form on the upper half thing.  Taking values in some space that I will call w k minus two, where w k minus two of any ring  is just r x, y homogenous polynomial in two variables, x and y, to reach k minus two.  OK. And so this is the key thing. If the depth is a cost form, this defines a gamma invariant differential form.  This is something that's invariant under the action of the group gamma.  And so I can integrate it over paths and make them all resemble this way.  And so I want to do that.  Then we define pi of f of any path from r to s to be the integral from r to s of omega f.  OK. So that's the model of the symbol. And notice that it takes values in the homogenous polynomial  in the context for this is the degree k minus two. OK. Now there's a theorem now of mononim.  I just want to mention. And then I'll pick up the page a little bit.  OK. So mononim theorem.  So for f and i, I get more.  And then there exists a minus and two model symbols actually apply a plus or minus model  symbols for the group gamma with values in wk minus two of the ring over f.  This is the ring of HECA eigenvalues of the form f such that n and two periods.  So I call omega f plus or minus such that.  Well, if you put lambda of pi f, so the universal for the l value of pi sub f, is equal to omega  plus times something of i equals to over i, i even between zero and k minus two of some  algebraic part of the cuss form with values in i plus one times x to the iy to the k minus  one. Where these guys are plus, sorry, plus also some minus part.  The point here is that the phi sub f, the phi sub f can be written as, I mean first say this,  omega f plus times pi sub f plus omega f minus pi sub f minus.  So we can sort of split this guy over here into this plus part and its minus part such  that, so these are like the algebraic parts, the omega or the psi f plus and psi f minus  are the algebraic parts of the modular symbol associated to f.  And when you take l values, you get something similar.  The l values of the cuss form into the even and odd parts multiplied by the same.  Okay, so these are the algebraic parts of the l function.  Now the message here, which is one that very popular, very well back in the old days, is  that modern g l functions were something that we all know have something to do with arithmetic.  So we think of some arithmetic logic like an elliptic curve or something like that.  We make an l function and that then is a way of encoding local information into some global  analytic thing and then we try to retrieve global analytic information out, or global  arithmetic information out. But the message here when we do this, that not only are the  values of the l function complex common, you can tell us something about arithmetic, but  you can actually do arithmetic with the l values.  And of course by now that's a thing that we know very, very well, but I just want to point  it out because it's what I'm most interested in.  Okay, so, okay, now, so let me just, the next topic I want to talk about in this context  is Eisenstein-Compagnen, which is something that very, also taught me a long time ago,  can we go over here? Oh, my very tussillation to stay there for now.  Okay, so, a section of the two columns. Eisenstein-Compagnen. And so the idea is to think about cases where  cuss forms are congruent, modulose on prime, license on the surface. And then can you say  something about the associated ellipads? This is another very old story and this is a typical  standard example. The first example is one that we've seen already in this conference,  is to start with f and s2 gamma zero of 11. So f is q times some product.  Oh, I've got a picture. q, 1 minus q to the n, 1 minus q to the 11.  Yeah, square. Yeah, these are going to square.  Okay, and so, and e, the Eisenstein series, this is a model for gamma zero of 11.  There's an Eisenstein series of weight two also for gamma zero of 11.  The cuss forms are one-dimensional and the Eisenstein series are also one-dimensional  and this is this. Plus some sum, n equal to 1 to infinity, some modified version of the  sigma function times q to the n. And the thing that one notices here is that the f,  it's not obvious, but it's true that the f is congruent of the q expansion to the Eisenstein  series, modulo five. And so in particular you'll notice that the constant term of the Eisenstein  series is the visible white white. Okay? And so that's a congruence between two  analytic objects and the question is, does that tell us something about the, these L values?  Are they congruent? And the answer is yes, they are. And so, in fact, and I'll write this  down rather politely, but I'll say more about it in a second.  So what one ends up with, they're all right. I'll just state it in this case. Five,  size of f I should say, size of f minus is congruent to size of e minus mod five.  The next question would be to ask whether or not there's a conference between the  two relative symbols that reflects the congruence between the cuss form and the  Eisenstein series. And there are two periods. The first period, the one in terms of  the average, there are two. But for Eisenstein series there's only one and it's this.  Well maybe e minus is in fact the tie up, that's the theory. There's no even  part of the micro symbol. And so we can't even, it doesn't make sense to frame this.  Okay, now, so this is something that's interesting because in fact people like  Rob Pollack and Preston Wake have been thinking about congruences even in the  even case. And they do write down congruences between micro symbols associated  to f and micro symbols associated to e. But when they talk about the micro symbol  associated to e, they're thinking about a boundary symbol. Boundary symbols are  things that only depend, they're actually extended to homomorphisms not just from  delta zero to m, but also to gap from delta to m where you don't take the  divisors at degree zero. That's much easier to make such things. And there is  an, there aren't bad elements of that thing that have the same idling value as  the Eisenstein series does. And so that's what people are, when they talk about  those Eisenstein series, that's what they mean. But they're not talking about  periods of Eisenstein series. They're very similar things. And so the question  then is what happens, what do you do? What happens when you're working with an  Eisenstein series and they have, you're thinking about the even part when  you're working with an Eisenstein series. Okay. And so, so that's the question.  What does that mean? And how do we make sense of that? Okay. So, well, let me just  think about an example. I won't write down the actual example that's all  described as best I can. I want to start with a cost form. By the way, this, let  me say right at the front, that this is the, this is the question, the original  question. I don't know what, where is Romgart Scherrithy actually starting with  this whole stuff. But Romgart, certainly the simplest case of his theorem, or  his conjecture, which is now theorem, given for Caillat-Paco mostly, the  conjecture, he gave a conjectural formula for besides the best plus, but P in the  case where P is an Eisenstein problem. In the case where there's a  concoct like this. And so, that's really what I want to talk about. Okay. So, so  I'm going to start with a simple case. He says much more than what I'm going to  say right now, but I'll try to say something that's his full conjecture  problem. So, start with a cost form on SL2Z of weight K, K greater than or equal to  2, and P prime, it will always be greater than 2 for me, prime, such that  pair Pk is a new regular pair. Now, this position that it should be an irregular  pair basically says, by the way, I want the P, P would be bigger not just than  true, I want to weigh one small prime. Okay. And so, and so this just means that  there's a congruence between this cost form. I want to say this is an I form,  it's a Hecker I form, and then I'll have an Eisenstein series. There's only one,  the one space of Eisenstein series of weight K on SL2Z is one dimensional.  And so I'll take a generator of that thing. And suppose, so even he then,  think of it as minus Bk over 2k plus again some sum, n equals 1 to infinity,  some sigma K minus 1 star of n, q to the nth, that's q expansion. And I want the  cost form to be congruent to E modulo sum prime P, or P sum prime say in the  Hecker algebra, in the region generated by the Hecker algebra, by the Hecker  value of F. And I want F then to be congruent to E as a q expansion. And that's  going to happen, then the only way they can happen is to take it, it must be the  case that the constant term is infinity. So P then is some prime above P.  That's a little like a P for terms. Okay. And so the condition that Pk is  near-regular pair, I'm thinking of it really as a state, what I really want is  the cost form in an Isis line series, and the Isis line series. So then, okay.  And so to run our game with a nice conjecture for what the value of the  algebraic parts of the L value should look like in this situation. So I want to  just describe it, quick. So Charini's conjecture. So let R, let R be the  bracket Fp through the very end invertly. Okay. And you, I don't know if you  are right, the units in R, module of these powers. Okay. And so that's just  some Fp, finite dimensional Fp, that just takes a bit more flexibly. That is.  And so, okay. So now we take the U and write it in decomposing. I'm going to  use the same notation I used before. I equal 0 to P minus 2 U 1 minus, oh,  okay, sorry for the weird way of arranging this. But, and so this means,  by this I mean the part of you, we can decompose you in the eigenspaces for the  action of the Galawang group of Q zeta P over Q. So that's just some type  of a character. So we're taking, this is the competition of the components on  which the Galawang group acts by the U 1 minus I power of the type of a character.  Okay. And then we let a i be equal to the projection of 1 minus zeta P, this is  the element of R cross, to U 1 minus I. Okay. So that's just some, some  element in the 1 minus I piece of that thing. And, okay. And so the conjecture  then is something like this. Let me get it exactly right. So what we think about  is we want to take U tensor U. Of course this maps down to, by, uh, tumor theory.  That's where they, one, uh, F, U, P. Uh, and that then goes by cut product to H2,  H2, H2, H2, H2, F, uh, my P. Okay. And then at a time of Tate he says that there's  there to A, A-gencerly P. And so our A is the class group of the field S. F is the field  of fractions of R, of course, module P. Okay? Okay. Now our column point says that Pk is  an irregular pair, and so Pk, irregular, implies by Eberon-Ribbitt that A1-k is not zero. The  one minus k component of this A reaction, the same, the same group of definitions, is not  zero. And so what we do then is we let lambda R for each R, for each R, we let lambda R be  the, I don't know my column, do I get myself, what do I call it? A-R. A-R, A-R, cancels for  A-R, what I want to say. And that's right. Okay. And so this, this then is an element of  u tends to, u, I'm thinking of an element of u tends to, or u. And then I take, I call  it, not this method, this one up here, rho. I take rho of lambda R, that is in this A  tensor mu P, 2 minus k. Okay? It lives in there. Now under special conditions, and this, this  group might be, this rig, this is a, an Ft vector space if you like. It will be one  dimensional in examples, it's always been seen to be one dimensional, it could be higher  dimensional, in which case what I'm saying is it's too weak. But the point is that, that  I can say take a linear functional of this thing and identify a piece of it as a, as  sort of a line. So I'm going to think of the rho of lambda R as actually being an element  of that thing. Up in choice of the basis. Okay? Up in choice of some linear functional  if you like. And so lambda R's conjecture says that, let me actually get it exactly right,  lambda R's conjecture says that if you take slice of F plus, appropriately normalize,  and then evaluate on the universal special value, you will get modulo P, the sum over  even R's of k minus 2 choose R, lambda R, x to the R, y to the k minus 2 minus R. Okay?  So it's a bit of a strange statement. So I have to have this ability to think of elements  of the class group as elements of that key. Okay? But other than that, I could normalize  things. With the statement, if I take that out, that's what I'm going to get. So it's  kind of an interesting statement. Maybe you can identify coefficients, sort of the coefficients  of this modulo sample with certain elements that are built out of k2. Okay? It's a nice  statement. So this was just, to be honest, I don't know exactly what motivated it. I  find it an amazing statement. And so I had a student at that time, her name was Cecilia  Bousouyock. And I said, well, this is a fascinating conjecture. Why don't you use the  computations? Let's see if we can see it. And so she did some calculations. She's actually  first to have to figure out where it all ends. So she took a couple of months. I didn't  hear from her for a while. A few months later, I got an email from her. I bought nine o'clock  one evening. And she said, I've done the calculations for these two examples. I don't  remember which ones. And it's checked out. Okay? So that was great. And so I went to bed  and I woke up the next morning and I had another email from her at 3 a.m. that said,  and I can prove it. And so I was kind of surprised. Her statement about autism might have been  a little bit strong. But the point was that she saw the idea of what had to be due to  prove at least something about this that was interesting. I should say up front that I  communicated this with Ron Kamshari if he, and he said, he had just discovered the same  thing. So it seems like a simultaneous, close to simultaneous. I don't know exactly.  A discovery. But the argument turns out to be fairly elementary. And so let me just  write it down the other way. Okay. And so I already got my fairy tessellation. There it  is. Okay. And so what she does is she defines a micro-simple. It's a micro-simple. I'll  write it down in terms of the theory of tessellation instead of this other one. And so here it is.  So psi of, if I have a matrix gamma in SL2z, then if I take gamma times, I take this path  from infinity to zero, and I apply gamma to it, that will give me some edge. Okay, every  edge, the horizon in that shape for exactly one gamma. It's about to do something. And  so this is, well, if gamma looks like this, if it's something, something x, y, as congruent  to that, my p, then the volume can read off what it is from the bottom row. It's just,  you take 1 minus zeta px, 1 minus zeta py. This is the Steinberg symbol happening in  k2 of the ring r. And so this, what she said is that if I put that, label the edges of  the fairy tessellation, that, then it's equivoc. That's actually what she put. And it's not  so hard to prove. And you end it at the bottom of the symbol that satisfies the monium. The  fact that it's equivoc, the second range, not so hard, but part that's interesting is  that it satisfies the monium relations, and not just a matter of writing down some particular  relation in k2 of that ring, which was not very hard to do. She just came in and showed  it to me. It worked beautifully. It was amazing. And so, this is the monium sample.  Okay, so, satisfies the monium relation. Okay, now, Shalisi's conjecture, of course,  much more than this. So let me try to state a little bit more of what he did, what he  actually conjectured. Oh, I should say something. Part of Ron Yar's conjecture is that this  should reflect the Meisner symbol that we're writing down should be corresponding to the  Eisenstein series. So in particular, this Meisner symbol, I should be, part of the statement  is that it's a heck of an item set. And that the eigenvalues are the eigenvalues of the  Eisenstein series. And that's the part that turned out to be really hard. Okay? The evidence  was there, and you can see that the conjecture says it worked. So it was right in the example,  but Shalisi sat down and she tried to prove it down what it meant for t2 to be the right  eigenvalue, and she found some relations in the Steinberg, Steinberg's samples that implied  that. And so she proved that this is actually an eigenvehicle for the Hacker operators t2  and t3. And that's it. And they were hard. They were pretty impressive. Ron Yar did the  same. I'm not sure. I'm not sure. I'm just saying to me, I remember some of these saying  they had a proof like this for t5, but I'm not sure. Okay, but the point is that proving  that this worked in general, that it was an Eisenstein example, was really, really hard  to prove some reason. Okay? All right. But that is just a double. This can make work  rejected anyway. And so, so let me just say that, that similarly, we can define, I haven't  written it down even in the one case, sort of, let me just say, since I've been in  close. So let Hn plus be equal to the homology of the modular curve x1 of mp to the n, with  values in zp. And I'll take the plus part of the action of context conjugation, where  n is some level, pain level. And so these guys are relatively prime. And suppose, again,  they're written in case where you have this irregular, or pk, an irregular pair. And then  the idea then is to understand what happens as you go up the modular tower and as you  go up the cycotomic tower. And so the omega n is defined to be, we can define some map,  Hn plus to an minus, what's this one? So it's the same an, but now it's the class group,  the p power part of the class, actually it's the p to the, it's the classroom on p to the  field fn, it should go up the tower, the cycotomic, yeah? Okay, this is a classroom. Okay. And  so it's a map, it's defined not so differently from this. And what he calls this, well then  he says, this factors through the Eisenstein quotient. So Hn plus modulo and Eisenstein  ideal are i, where the i is what you think of it. Yeah, it's a little complicated, but  I'm going to write it down. Eisenstein, I think. Okay, he calls this thing Pn. And then  the omega n goes, think of the omega n, and really it's something from Pn to An minus.  And that's the first part. The first statement is the factors through the Eisenstein thing.  And so I'm just making a remark that even in the simplest case proving that the factor  through the Eisenstein quotient was not obvious. And so, but the conjecture is that it always  grew, happens. And moreover, moreover, on the omega n, or quantum omega n, I'm just  feeling anybody else would call it pi n, to An minus of 1 is an isomorphism. So that's  the second part. And he then, I'd wait until he actually said that, hold on a second.  I don't actually have that statement written down here, I want to be careful. Moreover,  the right thing to say is that if I take a limit, as n goes to infinity of the Pn plus,  that goes to some projective limit with respect to the norm mass of An minus of 1. And this  is the map that he describes as an isomorphism. That's the contention. Okay, so, so I did a  little bit of a mess to read it all. This is part A and that's part B. And so, he said  more. He built some sort of a piatic L function associated to all of this, which is  I'm sure important. But I don't want to talk about it now. And part of this, what happens  here is that Sharifi, I'm not Sharifi, but Foucailla and Cato, throughout this time,  came up with a proof of this conjecture, pretty much as I'd written it down, under the  hypothesis that the Komodo-Level piatic zeta function associated to the level n, to the  tamed level n, has no multiple zeros. I think that's the only hypothesis that they needed.  They wrote down in the inverse to this map, omega, and then computed it both directions  and said they explained the identity of the either direction. Okay. So, okay, now, what  happened here is that Cillian and I were very interested in trying to prove the statement  that this guy was in Eisenstein's class, even in the simplest case. And so, we did actually  some stuff, and I think I might have enough time to jot some notes about what we tried  to do. I want to write down some minor symbols, and this is the first one. This is the first  one. It's not so obvious, but it's not so hard to get. And so, so what we did, we were  trying, we said, okay, how do we define that object? Well, we're used to thinking of them  as double coset operators. And so, we'll write and use some representation theory in this  business. And so, what we did is we introduced some ring, I'll call it R1, some ring of  trigonometric functions. So, either the two pi i z is an example of one. The sine of z  is an example of one. It really looks pretty much the only one that we use. But the other  one, that you know, there, there, there, there. Okay? The point is that these are our power  series in Q, where Q is either two pi i z, you can specialize Q1. And if you do that,  you might have to be fixing it in a roots of unity and things like that. So, we tried  to compute K2 of this ring in a way that we can remember, in a way that had an action of  a roots on it. And so, the gl1 q action on that thing. There's an actual action. You can  imagine what it is. And so, we first made a distribution on Q, taking values in, what  K1 of R, K1 of R is just R1 cross. Okay? And so, I won't say what a distribution is, but  it's, it's, it's kind of what you think it is. You, you make 10 functions on Q and you  make homomorphisms from the, from the group of test functions to this thing. And it's,  that's the way of expressing sort of distribution of a ring. And then we, there we go. That  is, a unique mind and a sample. I'll call it 5 from delta zero to distributions on Q2,  taking values in K2 of some ring, which I also won't write down, some ring of, of  tracelet response in 2 here. Okay? Okay, now this is not exactly right. We had to divide  by some ideal. I can tell you what it is, but for now, let me just try to get something  on the board. This is an ideal in the, it's just, it's actually the part, take K, the  moment of this ring, K ring, the moment of K ring of the ring R2. That's a graded ring  of head logic, graded pieces. I'm just looking at the graded two parts. And there's a part  of it, and then taking the ideal generated by what I call Q1 and Q2, I call it to a Q  which is Q1 and Q2 or the two variables. Okay? That's a pretty trivial submodule. But  if you modify that, this thing becomes a much more simple, where, where two things are  true. One, five is GL2Q of a, GL2Q of a grade. And B, if I leave you to your imagination,  the Q2 acts on this, and it also acts on the R2, and it's a mixing of that. Okay? And  the other thing is that the, the five, I just have to tell you what is L value. And  this L value is mu1. Okay? In other words, I should, to be very precise, I should be  saying like in pi1, you pull back pi1. I think of Q2 has two projections to Q. And I'm taking,  I'm taking a distribution on this, I'm pulling back one of the projections, and then taking  some convolution product of that, we're going to pull that according to the other projection.  Okay? Okay. So there are, and, and now the idea was, and I should, I should, I should  say it's a, whatever I've written down is true so far, but this isn't what we wanted  to prove. We wanted to use this to prove that the thing she's written down was an  eigen-assembled for HECA. And we never got back to that. We wrote this down, we, I'm not  even able to tell you how, how the idea was that we would be able to specialize this at  Q1 even with certain points and be able to say that they, that they would make a class  out of this that's a eigen-assembled way upstairs and then specialize it to say that it's an  eigen-assembled downstairs. We had never completed this project because it was about this time  that Foucaillet and Caillot announced the general proof and we were getting lots of details  and stuff. So it seemed like a good place to study quite a few down at Foucaillet and  Caillot were doing. Okay, so the idea, let me just, since I'm out of time, let me just  say that what Foucaillet and Caillot, what Foucaillet and Caillot did was they made a  quantum sample out of valence and elements built out of Ziegler units. And so, so specifically,  this was actually, in my world this was due to consular off. You take the Ziegler units,  you take the Ziegler that's what the P0U back with G0V and when I write this let me be careful,  it's u over n, p to the n. This is a V over n and then P to the n, takes the Steindor  element of K2, but I'm going to think of it not living at level  n, so it's fine to think of it. Let me write it at level n.  It gives an element of K2 of that much of a curve. And what they do is  they then make a minus, they take a minus symbol from  H1, X1, and P to the n to  the n. That's specialized, and then they specialize  at infinity. In other words, they make a map.  You can specialize these guys to make elements of K2 of these  rings, of these rings. And then they specialize at infinity.  You take this guy, and so it's an element of K2 of this thing,  but you specialize it at the infinity cost, all right, infinity  curve. You specialize in that element of K2, and then you specialize  it at the infinity cost, all right, infinity curve.  You specialize in that element of K2, and when you do that,  the idea is that specialization at a point like infinity  sort of is an eigenspine map. So it respects the Hecker-Arborism,  the eigenvalues of those, of the eigenspine situation, in fact,  of the boundary symbol, VAT, which is a thought that just occurred to me  recently. But because you're only using one cost, that's the thing that's interesting.  Okay, so, but the boundary symbol is made as follows.  I was looking again at the various facilitation. I take the U and the V,  it's given by, I take the U and the V, and I associate to the same thing.  I take a gamma UV, this is an element of SL2, this is actually the  other element of SL2Z, where this thing is congruent,  my little n, t to the n, does something like this, and you put that,  so for that guy, you might just might resemble to be at this,  this path, this new line is a path, and you find it to be that.  Okay, so I'm going to stop except that I want, there's one more thing  that I just want to at least say. The thing that I personally am  interested in right now, which I'm not able to, it's hard to talk about,  is a related modular symbol, it's built in the same spirit as many people  have seen this thing that Kudall had in ice paper, and we should wrote down  a modular symbol taking values in a certain, well he didn't describe  it as distribution theoretically, and so our formulation of this is to make  a modular symbol of this type, it's a distribution, instead of Q2,  use matrices of, two by two matrices with rational entries, that's a four  dimensional Q vector space, so you talk about distributions on that four  dimensional Q vector space, taking values in K2 of the modular tower,  all the way up the tower, and this is due to Von Schropp, by the way,  didn't state it, but it's equivalent to this, and it's also Bruno gave  a different proof of Von Schropp's theorem with some extra restrictions,  but the statement is that there's some, in the form that I like it,  is that you're getting a distribution on M to the Q taking values in K2  of the modular tower, it's built not so differently from this,  except that you don't require the zero for the first coordinate, this is just  so that you can specialize that infinity, you'd use A and B, so the first  column of the matrix over there, A, B, and the second column of the matrix  that I'm talking about over there is C, B, and you get some big micro-sample,  and I believe that this micro-sample deserves in some sense to specialize  to the micro-sample that's still here on our own down, but more, I think  to be a lot more than that, and let me just say a little bit more about it,  when you specialize at the cusp, one of these valence examples say,  you have got to worry about whether or not the modular units are holomorphic  or they're non-zero and holomorphic at the cusp that you're specializing in.  And so when you take, even though this thing I call the concovalence in  the micro-sample following the zero suggestion, it doesn't give you micro-samples,  it's honest, but when you try to specialize at the cusp, make something  Isenstein from the boundary, you have problems because you don't know that  you can specialize at all points, and so you only get something called a partial,  we might call it a partial micro-sample, and I'm sort of tending to calling them  pseudo micro-samples because there's something about them that feels like  they're going to smooth them, in the same way that you smooth  the periodic measures to make them bounded, things like that.  So I'm interested now in this relationship between these two things,  and I should say, I'm trying to, my main interest is right now  is depending on this Darmo Daskutik injection.  And so the Darmo Daskutik injection, the Darmo Daskutik made a periodic measure  and they gave a formula for a conjectural unit above some,  in some great class field of a real quadratic field,  and that formula involved piano integration with respect to that measure.  Now that measure that they write down can be described in terms of this thing  that's a Yenakian, it's an easy description of it,  but it's just another description of the thing,  that distribution somehow needs to be enriched,  that's the field, if they're enriched somehow in order to make these models,  you never make a unit, you have to get some arithmetic out of it.  And so that's basically my interest right now,  is just to examine this kind of variance and by the simple to see  if it has enough information in it in order to recover  in some concrete form of the conjectural unit.  OK, so I'll stop there, I'll stop here.
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