 It's an honor to speak at this conference for Barry. I'm not a student of Barry, but Barry's influence on math is very large, as you would see. And you'll see that in my talk today. So the emphasis of my talk is a bit different, I think, the way I wrote the abstract. So the starting point of my, some years ago, I made some conjectures that, roughly speaking, they relate derived structures in the Langlands program over number fields to motific cohomology. And at an MSRI program three or four years ago, Michael Harris said to me that we should, originally, I had worked with scientists for singular cohomology. And Michael said we should work it out for coherent cohomology. I actually used that to make it. OK. I hope that. I told Michael we should work it out for coherent cohomology. And so we wrote a paper working it out for weight one forms. And in particular, we gave numerical, there this general conjecture turns it to a numerical prediction. And we tested it for the first two weight one forms, the ones of level 23 and 31. And actually, already Dick Gross had suggested to me that we should really think carefully about the dihedral case. And it took some years. But that's what I, this is something that we were able to do. So Daman, Harris, Victor, Rutger, and I, we proved this conjecture. And I'm going to say a bit about it at the end. This could four weight one forms. So I'm just writing that in case I get rushed at the end. But there's a large chunk of that proof, which is, I think, it's kind of a self-contained piece. And it has a lot to do with Barry's work. So I'm going to try to explain that self-contained piece. And I think really what got this work off the ground is the PhD thesis of a student of Morel, Immanuel Le Couturier. And so a lot of what I say is actually I'm going to explain some of his ideas. And then at the end, I'll briefly come back to talk a bit about how they fit in with this derived Hickey algebra story. OK, so I'm going to start with Barry's Eisenstein ideal paper. So we're going to have n, which is a prime number. And p, which is a divisor of n minus 1. p is also prime. p is bigger than 5. It's not at all important, but just to make our notation a bit more convenient to assume p exactly divides it. OK, this is not necessary. So now I'm just going to talk about one very simple thing of the many things Barry does in that paper. So at level, among weight two forms, at level gamma node of n, there is the weight two Eisenstein series. And its expansion is as follows. We saw it was written earlier. I'll just write it again. Its q expansion is n minus 1 over 24 plus sum. And the coefficients are essentially the sum of divisors functions, but they're kind of modified slightly in a fashion related to prime divisors. n is a prime divisor. So this guy, so this guy lives in this space here. And under our assumptions, this is 0 mod p. And therefore, it's easy to see from that that there exists a cusp form congruent to it. So there exists some cusp form e2 mod p. And now, Barry studies the integral theory of such congruences very deeply in his paper. But the most naive question you can ask is, is this cusp form unique? So in the language of Barry's paper, that is, if you take the Hecke algebra, the cuspial Hecke algebra and localize it, maybe I should say the most relevant thing about the Eisenstein series for us is that it's killed by the Hecke operator tl minus l minus 1 kills it. So if you take this ideal, this generates together the p in the Hecke algebra, Barry studies that the Hecke algebra completed that ring, and you can ask what its rank over zp is. But equivalently, this is just asking if you write out all, I'm sorry, this should have been a cusp form. There exists a cusp eigen form and eigen form for the Hecke operators. So if you write out a list of all the eigen forms and you take their coefficients in qp bar, you can just ask is there just one which is congruent to this or more than one? And Barry gives a criterion for that. So let's say there exists more than one such form. In other words, this rank of this localized Hecke algebra, I'll just write that as ti. This is saying the rank over zp of this thing is 2. So in his paper, Barry describes a criterion in terms of the pairing of a certain pairing, which I'm going to tell you about, alpha beta, is if and only if a certain pairing vanishes. Before I tell you that this pairing, maybe I'll just say a little bit more about this. So this pairing was computed by Merrell. So this equivalence, which is not very hard given various other results. But this was computed by Merrell to be, OK. And here I want to introduce some notation that will stand throughout the talk. So what we're interested in is the p part of this group, z1 and star. So I'm going to fix, once and for all, a map which you should think of as a discrete log from this to z1pz, mod p discrete log. So yeah, so every time you see this, of course, this doesn't matter. I could just more canonically work in this group here. But it's psychologically very convenient to choose this. Whenever you see it, it will either cancel out on both sides of a formula or just asking if it only matters up to scaling. So Merrell computed this, and he found it to be equal to. So this is valid in z1pz. Sorry, I'm going to describe this. Yeah, but Merrell computed it. And he found that it was the sum from i, let me write it two ways, i squared log i, or equivalently this log of product i to the i squared. OK, so this kind of fantastic number occurs in Merrell's work, and it occurs other places as well. So when these equivalent things happen, so by the way, the first interesting example where this happens is when the level is 31 and the prime p is 5. And in this case, Frank and Matt showed that the p part of the class group, the class group of this guy, is not cyclic. OK, so for example, n is 31 and p is 5. This class group is z mod 5 times z mod 5. Actually, the whole class group has 5 power 1. OK, so let me tell you what these invariant did. Now, there are several different ways to describe this. However you say it, one of them has something to do with the existence of x1 of n, which is a covering of the gamma 0 level structure. And the other one has something to do with that Eisenstein series. OK, so there are different ways of describing this. And I'm going to give you, so you can do this pairing in singular homology. I'm going to say it in a way that takes place on the mod p fiber, which is I think it's less nice, but it fits into what I want to say later. So OK, so what is alpha? Alpha is going to be a class in H0. It's just a differential on the special fiber of this guy. And it is the one that comes from E2. OK, so it's kind of E2 of z, d z. Now, the point is that because the constant term is the little by p, this is cuspital when you consider the mod p fiber, so it gives you. It's actually regular at the cusps. OK, and now what can I pair this with? I can pair it with someone by seriality. I'm allowed to pair it with something that lives here, fpo. And when I pair them, I will get an element of, so this thing lives in z mod pv. Sorry, I've crowded this board, but this thing lives in z mod pv. I can tell you how to find beta. So beta comes from, as I said, the existence of the covering x1 of n over x0 of n. So now in this whole talk, I'm going to be sort of sloppy about stack issues, but they don't really affect us because p is not 2 or 3 anyway. So this is a covering of degree n minus 1. And it's certainly covering in the interior, but it's sort of a nice fact that it extends to be covering over the cusps as well. So it defines a class, and the Galois group in this covering is z mod m star. So it defines a class in the etalc homology of this guy, x0 of n. And we can do this over the special fiber, fpo of p, with coefficients in this group, which I can push forward to z mod pz because by my log homomorphism. I can just sit in this etalc homology. I can push it into zersky homology. OK, so this is a strange construction, but it sounds more natural in various formulation, which is a v pairing. But OK, this is, I think all this is important. This has to do with the Eisenstein series, and this has to do with x1n. So I push this. I sit this inside the structure sheaf, and then the etalon zersky homology of the structure sheaf are the same. OK, so this is, now, are there any questions? So the reason this had no direct relation to what Michael and I were doing, except for, as I said, we numerically computed some things. And in this numerical computation, we needed to use Morel's computation, OK? Like, without it, we could have only somehow done things up to this unknown constant. So what was missing in our, to try to go any further, was some understanding of what this thing actually meant. OK, so I want to explain that next, and then I'll come to telling you about the work of Lycatoria. So now, I'm going to say a little bit about, I'm sorry, I'm not using these boards very well, am I? OK, I'm going to say a bit about this sort of section too, about the paper of Mazer and Kate that was mentioned this morning by Glenn Stevens. OK, so in this talk, what is going to happen is we're going to sort of understand what this thing means and where it shows up, sort of why it should show up here. And then we'll explain how it shows up in a different context related to super singularly curves. And then I'll sort of put those things together at the end. So there's a paper of Mazer and Kate. And what they do in this paper, there's another paper of Mazer and Kate in title bound where they're considering the Pianic L function of an elliptic curve. So you have E in elliptic curve, and you consider the L value. So chi is a Dirichlet character. And this is just motivational for me, so I'm not going to be very precise. You divide it by some period. And as you vary chi, Pianically is the Pianic analytic function. And then you try to say what the Birch-Sputter and our conjecture is in this context. But there's this other paper, as I said, where they do something very beautiful. They say, instead, let's, OK, so now I need maybe I'll be helpful to call this group G. Use the modern notation. G is the modern start. So instead of considering characters with round flight of P, I can consider just chi being a Dirichlet character with conductor n. And it's convenient for me to think of these now. Everything is being valued inside QP, so we can talk about congruences, not P. And Mason and Kate have this idea that this is just a finite set, so we're going to think of this as a function on this finite set. But nonetheless, we can make sense of the derivative of this function. So the point is, what is a function on this finite set? A function on this finite set is meant to correspond to, so this thing is going to be equal to, a function on this finite set is the same as an element of the group algebra. So it turns out that there is some element, let's call it L, in the integral group algebra, such that you can get this guy by pairing L with the character chi. And now this guy, if you think of it in characteristic P, so in characteristic 0, the spectrum is a bunch of points, but in characteristic P, it has some thickness, and you can differentiate in that thick direction. So you could agree this has the following meaning. So for example, let's say, for example, L paired with 1 is 0, let's imagine this function is 0 when chi is the trivial character. That means L lies in the augmentation ideal, which I'll call I. Augmentation ideal is things here whose coefficients sum to 0. And just as an algebraic geometry, I'll declare the derivative to be the image of this in I mod I squared. So sorry, I'm not really using this board badly, so you can't see tell me. Image of L in I mod I squared. And this I mod I squared, it's isomorphic to the group G itself, or rather it's P part. It's the first homology of G with ZP coefficients. So this is their definition. And then they studied if this thing was, and it was, related to the rank of the elliptic curve. And they made many beautiful conjectures, even about the leading term of that. But what's going to be relevant to us is something actually even simpler, which is I'm just going to do this without the E, just for the zeta function. So I'm going to differentiate the zeta function in the direction of a tamed character like this. OK, so let's do the same thing for L, the directly L. OK, so just to be clear, I'm going to now consider the function, which is sends a character to L minus 1. This is still a directly character, a character of Z1 and N stock. There's the value of the zeta function of minus 1. But just to be clear, this is when chi is not trivial. And when chi is trivial, you put the reasonable limit of this is not the zeta function, but the zeta function that's for getting the Euler factor N, which is N minus 1 over 12 explicitly. So that's the function. And again, we can think of this as coming from an element. It's a function on, so this is a function, kind of on the dual of this group G. But we think of it as coming from an element in the group algebra. OK, so this comes from an element of L of. And that function is explicitly, it's given by you. So this is given by pairing chi. So it's very easy to evaluate these things in terms of Bernoulli polynomials. And so maybe I should do that first. So explicitly, this is the sum from i equals 1 to N minus 1, chi of i times some polynomial B of i. And explicitly, it doesn't really matter, except for this quadratic. It's negative 1 over 2N i squared minus i plus 6. So this is given by pairing chi with the following element of the group algebra, this sum 1 N minus 1 B of i times the class of i. So we're thinking this now as being an element L inside VP of G. And now we can, again, differentiate it. Oh, OK, so before that. So it's value at the identity. It's value at the trivial character is this, which is, as I said, N minus 1 over 12. So it's 0 mod p. It's not 0, but it's 0 mod p. So what I can do is I can just, let's say, I reduce this group algebra to Fp. So I have to change it about the Fp coefficients. And then I'll take its image in the augmentation. So the image, so I'm going to sort of declare d by d chi. This is a definition. OK, so I have no mathematical content, just a definition. It'll be the image of L, but in i mod i squared. And here, i and i squared, I'm going to write this, but you first reduce 1p. Now this thing is isomorphic to, it's isomorphic to z mod pv. And this is the easy computation. I'll just write down the explicit isomorphism. So you send sum of a, g, g in the group algebra. Every time you see a group element, you replace it by its log. OK, and that's just unwinding the usual way of identifying this with the abelianization of a group. We'll get to that. This object, I will, what? I'll take some of i log i from our elbow, because it could be the abelianization of a group. Let me say something about that in a moment. It's right, I stated, but I'll say something. Was there another question? So i is related to chi, right? So this i here is the ideal in fp of g considering elements who sum the coefficient to 0. And then that would be equal to here. Yes, yes, yes. OK, so yeah, sorry, let us call this thing the derivative of the zeta function in the sense of measuring p at minus 1. So you should think, yeah, it's the derivative of l1 chi in the chi direction evaluated as a trivial character. OK, so what is this thing? If you use this formula, zeta prime mp of minus 1 is using my formula. It is the sum of this polynomial, this b of i times log i, i equals 1 to n minus 1, which is the sum from 1. If you just buy some fairly simple manipulations, the linear terms don't matter. And this is a number mod p, by the way. This is in z mod pz, and it's congruent to 1 mod p. So you easily get this is minus 1 1⁄2 sum of i squared log i. As Michael said, this coincidence was realized by Frank Caligari and I sitting in a coffee shop maybe two years ago. It was also realized by Emanuel, like Arturia at the same time, and he took it much further, as I'm about to say. But the one thing I want to say, which was not obvious about it, is Morel's formula. This is not actually how Morel writes his formula. Morel writes a formula involving i log i and going up to n minus 1 over 2. Those two things are equivalent. And then this is related to the fact that in classical, if you look at classical versions of the class number formula for imaginary quadratic fields, you will find two versions. One involves adding up quadratic residues and quadratic non-residues and going all the way to your discriminant. And the other one, you only go halfway, but you count them. And those two versions are equivalent. So that's related to the fact there are two versions of this formula. And Morel wrote the other version, which made it quite a bit harder to identify what it was. OK, but anyway, so we at this point, we see this thing Morel computed to be this guy. And this makes sense, right? You can imagine now that whatever the rank of Hecke algebra might be connected to some Galois deformation theory and then via the main trajectory to something like this. But OK, so now what I want to do is tell you the first thing from the work of Immanuel. So many things in this thesis. And I'm only telling you two small pieces of it. But in particular, what I'm going to say will imply, it will give you a very direct proof that if this mesotage derivative vanishes, then the, did I give that number a name? Then the rank of the Hecke algebra is bigger than one. OK, so I'm going to write down a formula. But in terms of what I've been saying, it in particular gives you this direction. But as you'll see, it somehow gives you much more. So the idea of what we're about to do is we are going to lift the computation of 2 by what is 2? The section I just did, to modular forms. And by lift, I mean the things we just did will now be all interpreted as the constant term of modular forms for gamma 1n. OK, so I already mentioned the modular form e2 for gamma 0n. And because I'm going to write down some, so let me not write its Q expansion again, but I'll just write the formal Dirichlet series for it. So if you take its coefficients and you form them into a Dirichlet series, you will get zeta of s times zeta of s minus 1, where you omit the factor n here. Now, there are two ways, kind of two families of Eisenstein series on gamma 1n. There are kind of two ways to deform this, the gamma 1n. OK, the first I'll call e chi 1. And the associated Dirichlet series for it, I'd rather write the coefficients, I'll just tell you the Dirichlet series, is L of s chi. You can put a character in either of those zeta values. So I can do it here. Or the other one, I put it in the first coordinate. Yeah, it's a reasonable word, I think, but it doesn't matter for our purposes. Oh wait, well maybe I'm going to say what it is. When either one of these are specialized to the trivial character, you recover this one, at least mod p. OK, there's some slight subtlety about this missing factors at n, but mod p, both of these give you back this. OK, so now for a moment, let's look at these. I'm going to take the difference of these guys. Now, this is some kind of formal, just think of it as a q expansion. Each of the coefficients of the q expansion are functions of chi, which as I just said, they vanish when you specialize back to the trivial character because they both give you this. So what that means is I can differentiate them with respect to chi in exactly the sense that I just said. OK, so this is going to be some formal expansion with z mod p coefficients. OK, so it's a z mod pz q expansion. Let's call it e prime. It's easy to check, actually, the following things. This thing is actually a modular form. This is a modular form. So e prime is a weight 2 modular form for a level gamma naught n with z mod pz coefficients, basically because you're just by both fairly formal properties of this derivative construction. The second thing about it is that it's like when you differentiate a family of eigenvectors where the eigenvalue is changing, you get some kind of generalized eigenvector, even in linear algebra. So this has the same feature. Namely, it's not an eigenvector, but it's tl minus l minus 1 applied to e prime gives you your original e back. Or I guess I call it e2 over there. e2. In particular, this guy is killed by the square of the Isomstein ideal. All right? And the last thing is, well, oh, by the way, oh, no. I said this totally wrong. Sorry, sorry, sorry. I'm missing something important. It's not this. It's l minus 1 log l. Yeah, it's killed by the square of the Isomstein series, but it has this factor. And this factor is already in Barry's Isomstein paper. But the last fact is the most interesting, which is, what is its constant term? So you can tell the constant term of these things, because the constant term is the negative of this series evaluated at 0. So if you have a sort of a modular Q series, the kind of the formal value of it when Q is 1 is always equal to 0. So the constant term of this is the negative of e1 chi. Is this negative of this evaluated at 0? Negative l0 chi zeta s minus 1. Yeah, I probably should have said that, you know what? I should have always made chi even. The group z mod n star should have been, it's quotient by plus or minus 1. So this thing vanishes. Sorry, it's a minus 1. And the other one gives you, what does it give you? Minus ll negative 1 chi times zeta of 0. So this first one vanishes, and the second one, when you differentiate it, it gives you the major k, this thing that I just said. So yes, thank you. Yeah. So when I differentiate this, the first one is 0, and the second one gives you the major k derivative. So you get zeta of 0, negative zeta of 0, times the major k derivative. So what just happened? I wrote down an explicit generalized eigenvector, or I, this is an manual thesis, under the square of the Isenstein ideal. In particular, if this thing is 0, you've produced a second cusp form, which lies in the Isenstein part. So the rank is more than 1. So that's the implication here. So you can say what this thing is. Let me say what this is slightly more intrinsically. So if, in fact, actually, let's put it up here. Is it congruent by p to the e to, isn't that what you want? No, no, no, it's not. It's not. You don't want it. It's something independent from each. It's independent. Yeah, so sorry, let me write. In fact, if you take the space of this level of mod p, and you take those that are killed by the square of the Isenstein ideal, I'll call iis, means the Isenstein ideal, this thing, you can deduce from Barry's results that it's rank 2. And modulo e2, it's spanned exactly by, it is spanned by, it's the span of this e prime. OK, so this answer is, I think, a fairly natural question, which makes sense. Namely, this thing is always two-dimensional. So what is the second thing in it? And it's that one. What happens in characteristics of 2 and 3? I do not know. So now, OK, are there any questions about that? No, no, that was just, in fact, it might not even be necessary the way I set things up. It is definitely not necessary for anything. Super singular points. So now, there's another version of this space of modulo forms that exists, which is we can take, so let's call ss, modulo forms on the definite paternian algebra, ramified infinity at n and n. OK, and by the work of Eichler and Jacke and Leiland's, that that gives you another, functions on that set gives you another incarnation of this space as a HECA module. But for me, I want to think of this slightly differently. So in terms of functions on super singular points, which is, it's well-known, this is kind of more algebraic way of thinking about that. So ss will be the set of super singular elliptic curves over some algebraic closure. Let's call x the modulo, not right. So I think, maybe at this point, we're already just gonna stick to mod p coefficients, okay, rather than zp coefficients. So x will be the functions from x to z mod pz. Oh, x is, that's not a good definition. Let's try again. So now, as I said, there's a fairly old, I mean, from ideas of Eichler, these things are the same as HECA modules in characteristic zero. The mod p under my assumption is a beautiful paper of math which shows that these things are isomorphic as HECA modules, okay? So m2, gamma naught n, z mod pz. I might just call this m later on in the talk. Is isomorphic as a HECA module to this x? Okay, so this is, and in fact, in math's paper, he writes down this canonical isomorphism, right, well, I think that the formula is not due to math, he showed it's an isomorphism, which I'll come to later. But for now, given that they're isomorphic, you can ask for the analog of the elements that I just constructed, but on this side. So what are, in other words, they must be, so what are the analogs of E and E prime on x or in x? So just, maybe I should say this more clearly, right? The properties of E and E prime are E, or E is E2. E is killed by this and E prime is a generalized eigenvector and that sort of upper triangular entry is this L minus one log L. Oh, by the way, I should, one thing, I'm not, there may be some constants, like twos and threes in particular. So I'm not totally sure I have these constants, right? Do we have some? So there must be, because they're isomorphic, there must be some functions, let's call these analogs f and f prime, there must be some functions f and f prime on the set of super singular points which satisfy the same relations. And f is really easy to find because f is the constant function. Okay, f, this only characterizes it up to a constant, but okay, so f, you can take it to be the constant function one. So f. I think it's okay, but if I think of them as functions. Yeah, fine. Now that was also analyzed in Emanuel's paper and he gave a formula that said, roughly speaking, f prime is the derivative of the Hasse invariant. Okay, so this was, I think this was really the inspiring thing in Emanuel's thesis for me. The idea that you could even hope to find a formula for this and it might be an algebraic nature. Okay, because it just simply would not occur to me that you could hope to have a formula for this thing. Okay, so we gave a kind of different version of this which I'll tell you about. But as I said, it was inspired by, so in Emanuel's version, you have to add auxiliary level two structure. So in fact, you can take delta and I'm going to explain, the first way I write this, it requires some making sense of, you can take f prime to be the logarithm of the modular function delta. Okay, so I'm going to explain what this means in a second because this is not, it's of course not a function, it's a form. Okay, but the idea is where the function you would evaluate it at a super singular point, you have something in fn star or some extension and you take its discrete log. And maybe I should say this discrete log, you can extend it, you can extend it uniquely to this larger thing which I need to do. So in fact, you can take this and as I said I'll explain what it means and if you sum this guy over super singular points and here I do have to wait by automorphisms, this thing gives you, and I'm not totally confident in the constant, again the masertate derivative at minus one. So what this formula says, modulo the interpretation I'm about to give is that the productive values of delta at the super singular points detects this derivative of the zeta function. Okay, so now let me explain how to, just how to make sense of this, or actually I can explain it over here. Delta, over these super singular points of course you have this line model omega. Delta is a section of omega to the 12. Okay, so delta is a section of. But as Sarah pointed out, firstly, I'm just, this is a finite set. Okay, this is like a a priori, it's some Fn bar vector space over each point of this finite set. Yeah, I'm gonna ignore stack issues but they're not a problem for what I'm about to say. Now you can trivialize the n squared minus the first power of this. This was pointed out by Sarah because you can descend it, there's a way to descend everything to being over the finite field n squared elements. In fact, for our purposes you can do a little bit better so you can find there exists a section p of omega to the n plus one over the set of super singular points which behaves in a very simple way under isogenics. Okay, such that p evaluated at the elliptic curve. Okay, so if you have an isogeny, whenever you have an isogeny phi from e prime to e and you have a differential form here, let's call it omega e which pulls back to omega e prime under this map. Then p of e prime omega e prime is the degree of the isogeny times t of e omega. So there's a, sorry, I shouldn't say this, everywhere non vanishing section of this which has this behavior and this is unique of the constants. Now there's a very explicit way you can do this. You just take the weight n plus one, Eisenstein series. Okay, that gives an explicit such thing but it's sort of important, we don't really care what it is. Okay, so the key displays an auxiliary role, it just trivializes this enough that we can make sense of the value of this. It's unimportant for us that there's actually a formula. But for what it's worth you can take t to be this weight n plus one, Eisenstein series. Once you have this, you can just fill around, you can sort of use this to trivialize enough of this to make sense of it. Okay, so you do something like, you take, so now you can form delta to some power, I don't know, n plus first power and divide it by the right power of this t, what power would that be, 12? This is now a function which you can evaluate at every super singular point. It gives you an element here and I take its log and I divide this by n plus one. So this is what I mean by log delta. Okay, so it is log delta except I use this trivialization to make sense of it. You can ease this, the fact that this thing exists, you can see easily, maybe it's not obvious that the Eisenstein series has that property that the fact that there exists such a trivialization is not hard. Okay, so that's the, so you should see this. So as I said, these are analogs of E and E prime and this formula is an analog of the constant term of E prime involving this major k term. So this, now, all right, so I'm going to very briefly sketch. So we give a proof of this and I'm gonna briefly sketch the proof of it because it's quite interesting even if I don't really understand it. Are there any questions? Is there any more in the situation where there are, are you assuming we're in the street where this derivative manages that there is? No, this guy always, so I should have clarified this. This thing always exists. It has nothing to do with the vanishing. Vanishing tells you whether or not this is cuspital. So correspondingly, these F and F prime always exist and vanishing tells you whether it's cuspital, meaning is it orthogonal to the constant function. That's exactly what that says. But what's very important for us is not just the fact that these two things happen at the same time but that the actual numbers are the same. Like you don't just care about these numbers up to non-zero scaling. Okay, so actually I might not have time to get to the derived package stuff but let me try to finish this for any other questions. So I'm gonna give you a very brief proof, a sketch, and it really is a sketch of the proof of this formula. And Le Couturier approves a very similar formula in the context of derivative of the Hasse invariant. But I think the proof I'm about to give is somehow more conceptual. A sketch of proof. So the main point is, right, on the one hand, this thing in the modular world came from the constant term. It came from something to do with the behavior of Eisenstein series of cusps. This thing has to do with evaluating something at some super singular points. Okay, this has to do with the sort of order of vanishing. This has to do with an evaluation. But you can relate those two things. So the idea is to lift what we already said in, which section was it? Three, the Ziegle units. Well, maybe that's not actually the idea, but it's more something you can do. So you can find, so yeah, as I said, at this point I'm starting to, I'm gonna skip several details. But you can find some Ziegle unit, x1 of n, which recovers the previous Eisenstein series under the D-log map. Okay, so more precisely, D-log of u chi gives you e chi one. Now, what is this, u chi? This thing has an action of z mod n star. Okay, so as long as you put this unit in some slightly bigger space, you can sort of pick out its chi isotopic component, as you usually do. So it just needs to product over group elements of u of g tensed with chi of g. Okay, so now what you do is you restrict this to the special fiber. Okay, so you restrict u, well, let me put this in quotes, in the special fiber at n. And the special, but inside the special fiber is the igusa curve. Okay, in particular to the igusa curve, n. So this igusa curve, something in characteristic n, and it is a ramified degree n minus, if I'm thinking about core spaces, it's a ramified degree n minus one over two cover of p1. I say restrict because when you can do this naively, but it acquires poles at the super singular points, and you multiply by some multiple of the hassa invariant, which I won't go into. Anyway, when you do this, the resulting function has to prop, the property that the norm from i1 of n to x of one, so restrict in with this is one. Okay, so now what happens is something which is kind of inspired by Thane's method. So in Thane's method, there's a unit in a cyclotomic field whose norm is one, and you construct a differentiated class. You can do that here as well. You can differentiate u chi, and what you get is a class downstairs. It's in the function field of x of one, mod p. But the thing that actually happens in Thane's method is that the ramification of this class is related to the position of what you had up here in the residue field. Okay, that's somehow a critical computation, very simple, it's a very simple computation. But so this is, in this metaphor, this is playing the role of a cyclotomic field, and this u is playing the role of a cyclotomic unit. So it turns out that the order of vanishing of this differentiated class at a super singular and elliptic curve, just as in Thane's method, is the logarithm of the value of u of p, u of e. E is a super, we're looking at this at ramification points. And now it turns out, firstly, that this function is the same as that function that you get by evaluating delta. Though this u is some kind of deformation of delta. On the other hand, when you add this up over all the super singular points, you can also, that gives you the total of vanishing pole or total order of zeros and poles in the interior. That's the same as what happens at the cusp. And at the cusp you can read it off from the constant term of the Eisenstein series, which we already computed. Okay, so the key thing which happens here is the sum over super singular s of this order is the negative of the order of vanishing at infinity. But this guy, as I said, it was designed to lift the Eisenstein series under d log. And so this order of vanishing you read off from the constant term of the differentiated Eisenstein series. And so this is exactly the e prime, which I talked about earlier. So this thing gives you, up to various constants, this makes it take derivative. Yes? This is sort of the sum, but is it S, U, B, C, or is it d log U, P? Yeah, so I have to admit that I don't exactly know directly how to show that the value of this and the value of deltas are the same, but sort of, I mean, you check it using their hecke properties. But wait, say what you said again. I think I just didn't understand. Yes? Yeah, okay, you go aside to the client. Gamma naught to gamma one thing is important. Like it's, because at some point you're switching evaluation of super singular points for order of vanishing at a cusp, and that's where. Okay, so that's the story. And now, are there any other questions? I will very briefly say how this helps. And so unfortunately I can't give you much background on this derived hecke story. But so in this paper of Michael Harrison myself, we showed that these general conjectures about derived hecke algebras on these cobalt conjectures imply that the following statement. For any weight one form, so you take a weight one, so that G is gonna be a weight one eigen form. N is gonna be a prime number, just like we had before. N and P are just like what we already had. And I form capital G, which is little g at z, times little g at nz. So whatever this is, this is a weight two form. It's at some higher level, but I project it to level gamma naught n, project to gamma naught n. So this is a weight two form at the same level we've been working. And what these derived hecke conjectures imply is the following. Again, you can take this g and you can pair it with what I called beta. Okay, so again, you can interpret this g as being a differential on the special fiber, fp omega one, it's a differential. And beta I defined earlier, it was h1 x naught of n fp O. So this expression, it's exactly the same expression we saw earlier, but where the Eisenstein series was is not a product of two weight one forms. Oh, these are both cusp forms, sorry. These are cusp forms. So g is a cusp, g is a cusp form. Okay, so this expression is something that comes from studying the derived hecke algebra. This is a torsion thing. Okay, this pairing is in Zeemot PZ. I think Mark is saying I should write higher notes. So this thing is in Zeemot PZ, but what the conjecture says is these various sort of torsion things are kind of reductions of some single characteristic zero story, which in this case has to do with some stark unit. Okay, so the conjecture said that, sorry, I should have put this before. There exists some unit u, which it's a unit in the field associated to g. Okay, so weight one form, it has a finite image gallery representation, it cuts out a field. So it's a unit in this field. And here I have to be imprecise. It's the log of u mod n. The conjecture said there's a single unit. There exists a unit u such that for all n and p as we've been discussing, this holds true. When I say u mod n, okay, so that's, I don't have time, they might be more than one prime above n, but there's some preferred way of choosing one, reducing u mod n so you can apply log to it. So this is a kind of different. So what you see here is you see a single unit and you see its reductions modulo many different. Okay, so now at the last two minutes, I will say how this follows from the story that we've been describing. So what we do, so what we do is we prove this for g dihedral, meaning that the gallery representation is induced from an imaginary or real quadratic field and under the condition that the major k derivative is null and zero. Okay, so under this assumption, so we prove that this statement holds for all data n and p for which the major k derivative is null and zero. Okay, and I'll just say very quickly the idea of the proof and really the key point in the proof is to compare the computations that happen on the modular curve and the super singular locus. So there's a map, okay, so what do I call it, x? So as I mentioned in Matt's paper, comparing these two modules of modular forms, what he actually should write down, some explicit map from functions on super singular, tensor functions on super singular, modular forms. And I think he mentions a deep growth suggested that this might be integrally interesting to study. Okay, so it's a very simple map, it goes something like this. The Q series is the sum over a, t, m, b, q to the m plus some constant term, which I won't write down. And he shows that under the assumptions we'll be talking about, this is actually an isomorphism. So what we're gonna do is we have something that occurs on this side, okay, so we have to compute something in the space of modular forms, the t, x to m. Now, well, let's call this map theta because it's basically a classical theta series. Over here, we wanna compute this pairing between something which is a productive weight two forms with two weight one forms and this class beta. Now it turns out this G, it's a preimage of, you can give an explicit preimage of it under this map. Okay, productive two weight one forms you could write as, so this G, so this G turns out to be theta of something, let's call it a tensor b, where a and b are supported on cm points. This, yes, I hear as you use the, in fact there's a different argument for the real dihedral case, so I'm abusing. So because it's a productive two dihedral things you can explicitly, and cm points means they're cm points which you then reduce to mod n and they land in super singular locus. So you can pull this back over here because a tensor b and then the pullback of beta. So this means that you think of beta as a functional on this, you can pull it back to be a functional on this. And you can almost figure out what this beta is by means of thinking about how the Hecke operator act on it. It turns out to be f tensor f prime up to a constant. It's f tensor f prime plus f prime tensor f. So now, just since I'm speaking with a negative amount of time. The key point here is you have something related to evaluating delta at some cm points. So you get, in other words, the value of some modular units of some cm points. Okay, so that's where you get your unit. But somehow actually the hardest point here is this constant because you know what this is only by some abstract argument. So you need to normalize it somehow. Okay, and that normalization in the end, the fact that this constant really doesn't depend on n and p and so on. It comes from the fact that the two computations I did earlier both have the same answer, the major k derivative. Okay, if they weren't the same, then I would get some sort of irregular thing here. But you normalize by testing exactly. Exactly, yeah. And that's why I need this assumption that it's norm zero. Okay, so I'm out of time. So maybe just the last thing I said is, the last thing I should say is there's, you know, Barry has written this paper, very nice paper about the theme of peatic variation. But there's a kind of parallel theme which is this theme of tame variation. And the whole story we sent here took place in this team, this team of tamely ramified variation. And I think like the Taylor-Wiles method and the theory of Euler systems belong to that tamely ramified story and this derived hecke algebra lives there as well. And I think there's some relation between those three things that still has to be discovered. Thanks.