Thank you very much for the interaction and also thank you very much for the invitation for this opportunity Since Professor Casuala's birthday was earlier this year. I cannot say happy birthday today, but I'd like to Offer you happy birthday. Yeah, and the seven tails. So a quantum curve is Still we don't have a good definition yet. So quantum curve is something like It's something like a family of Reese D modules on a curve See and so that We can construct the semi-classical limit. So the notion of semi-classical limit exists on this and this semi-classical limit produces a family of Specter curves in the cotangent bundle of C. So this is a kind of a definition because we don't really Capture everything in this format. So I'd like to give a talk Two parts one is a motivational part. Maybe it is related to the enumeration of Interesting numbers then that is exactly where the physics motivation comes in So physicist a ganagic digraph claim Marino and Waffa wrote a paper around 15 years ago Who are maybe published around 10 years ago where they introduced this quantum curves to capture some Quantum topological invariance of some manifold then later Gukov-Sikovsky and then many other people Are using this quantum curve technique to understand the not invariance. So that's one part of the motivation in physics Yet those motivation Understanding themselves is hard. So let me just start with a mathematical motivation And then the later on I'd like to relate this motivational consideration to another physics conjecture made by Davide Gaiotto in a geometric setting. So that is what I'd like to talk about today. So Mathematical motivation comes from Hurwitz numbers So these are purely a classical object studied by Hurwitz in 1891 and so on so the definition is that you consider a Meromorphic function of a Riemann surface of genus G So let's assume this is a compact Riemann surface of genus G to a Projective line and then a bunch of conditions you impose so that the enumeration becomes non-trivial. So first F has a labeled pose of odors Say mu 1 up to mu N so the number of poses N They are all labeled and these are a positive Integers and then a df equal to 0 has so this is the critical points set of critical points except for the poles has only simple Zeroes And then one more condition because if you impose only these the space of all Meromorphic functions is the finite dimensional manifold So to make it a zero dimension We want to say that the image of these critical points namely the critical values, which are actually branched point In p1 are fixed So then this quantity would become finite Numbers and then well, of course you have to divide by the automorphism and so on But that's a finite number Russian number. I'd like to denote this by group is number of this notation then just Using this representation theoretic Realization of this quantity. There's a combinatorial Equation Well, actually this equation doesn't play any role in my talk So there's no reason to write it up, but since I love formulas Every talk has to have one single formula at least so let me just write down the formula. So first The number of branched points so each branch point except for infinity are simple. So R is a number of simple Ramification points So this is just a solution of a number of solutions of the DF is equal to zero which is actually Easy to compute by Riemann-Hulbitz. So it's a 2g minus 2 plus the end is a number of poles plus a sum of all mu's which is the degree. This is just a sum of all mu's. So combinatorial equation is that if you multiply R to hgn and then mu as a vector Breaks down in several pieces. So this is equal to first half summation i is not equal to j and then So what I'm doing is that when you realize this as a representation of pi1 of punctured p1 At all points you have transposition your sign and then over the infinity you have a cycle type of mu1 up to mun element of the cyclic group and then you count how many are there and that is this number. So what I'm doing is that one of these Transpositions I bring it to the infinity and then merge them together then two things happen. So this is what this is writing so i plus ij so two cycles in the infinity put together and I remove two cycles so cycle length is less one less and then half summation i runs from 1 to n and then I break now single cycle into two pieces with an merging of the transposition so this would cause the summation like this then alpha times beta and then two pieces again so hg minus 1 and plus 1 alpha beta and then a mu Now mu i is broken into two pieces so everything so if I don't mention anything everything else stays and then Set partition and then number partition so genus g so in this case you are cutting Handle of this image curve I mean source curve and then in this case cutting the neck of the source curve So that it will become disconnected then so in this case hg1 so notation probably is Self-explanatory so I don't go too much into this g2 then j plus 1 and the beta Mu j so this is not a traditional way of writing the equation equation can be called in many different ways It is first a similar expression was discovered by Gordon Jackson so I Just use then a terminology cut and join Equation so this this is the formula now So the number is to be normalized in a certain way like like that the mass divide by the Define the who gets number you have to take into account the automorphism, right? So this is auto automorphism normalized. Yes already. So they are not integers. They are just a rational numbers Right and then with this normalization this formula holds So this is an equation for the rational numbers here So the theorem so maybe let me just erase this part Sorry Yeah, it can be zero so this is all possible partitions the all possible Set partitions including the empty set so G has to be greater than or equal to zero and has to be a greater than or equal to one These are the only conditions Okay, so this is what I said equation. I'm not talking about the kind of formula So G and N are peers of also here But of course they are a mu the size of mu is reduced on the right-hand side so from the computational point of view Hand calculation doesn't work Only computer program works because you can reduce the size of mu in this way But that's not my point. The point is the next stage stage so, let me just phrase a theorem and Actually published in three different papers with nijian jang and then also with Beltrand enough and then a broad soft nook and then also Also a Pioto Sukowski says the following number one You then introduce the Laplace transform of this number so define Fgn as a Hormofic function so n Hormofic complex variables to be just a Laplace transform so mu is Just an integer vector of size n and then a hg N of mu then I just multiply irons from one to n of Xi to the power mu i what I said the Laplace transform which means that I'm thinking x variable is simply e to the minus w I Don't use this coordinate. So I just suppressed but this is a Discrete Laplace transform then several things happens then Laplace transform of this equation cut and join equation is a PDE Recursion or induction formula With respect to the quantity 2g minus 2 plus n So the question was that this doesn't allow you to compute a function hg and as a function of integer variable Mu and but here if you do the Laplace transform then you can in actually Change the Laplace transform this equation, which is a PDE but the PDE becomes a genuine recursion With respect to 2g minus 2 plus n. I'm not going to write the formula. This is just a little bit too complicated But nature is exactly the same and then in this formula you do not allow the same gene and then appears So that formula becomes in the PDE So this one is indeed originally conjectured by Marcos Marinho So what is the Laplace transform function of what? What are the space where this this acts ah, so you yes, it's a good question because The question is that does this converge and then what kind of space are you talking about? Yes, you are looking at indeed the simple examples.  We are simply looking at that as a Power series. Yes, there's a radius of convergence, which is non-zero right So at this moment just a XR complex parameters So It's a generating function, of course people don't go into the past also But the reason why I want to say Laplace transform is that if you have a question Generating series. Yeah, but the Laplace transform can transform the equation itself to a creation, right? So that's what I meant. So Marinho and then his group So that so this is a recursion existence of recursion was conjectured by him and then a second thing is That yeah, so this is the after long calculations you realize that let's introduce a new variable T Which is again related to your question. So so T actually is indeed a Parameter of a projective line. So indeed Where it is defined is an end product of P1 that is answered your question So let's introduce a new variable T in this Mechanism so let's Let's T be as above Then so I can just plug all these Variables X of t1 dot x of tn. So let me just abbreviate Later. These are just T's Then is a sorry is a polynomial in T of degree 6g minus 6 plus 3n So so to me that that was a very surprising thing and then Marinho did not conjecture this part either. So this is Somehow what happens to be so it is a polynomial in T and then at this polynomial When you write down the formula, it's obvious in some sense. So this is alpha 1 plus alpha n all these are non-negative integers Less than 3g minus 3 plus n and then this is actually tensor polynomial So each variable has one polynomial here irons from 1 to n and then a d alpha i Apply to c0 of t i so I have to and then I'll have to fill in this Rational coefficient. So here D is a differential operator d dx In terms of t I'm using because I'm applying it to t So this is t t squared t minus 1 d dt is a differential operator Yeah, actually Zagie told me that he knew this kind of thing said 17 years ago and so on C 0 of t is actually d squared applied to f 0 1 t So this is a generating function of the Hulu with numbers of the simplest situation namely Rational curve covering rational curves with only one pole So that number you can easily calculate so you can determine this function easily you apply this and then you realize that this function is actually t minus 1 So this is a degree to alpha i plus 1 polynomial and you Just multiply them together then here the rational number is an intersection number So tau alpha 1 tau alpha n times capital omega g n and then this is the intersection over the motorized space m g n bar so here omega g n in General is what? Maximum the mining code Co-homological field theory so usually you have to choose an Frobenius algebra a and then this n tensor Represent this number n here going to the co-homology ring of mg and bar the stable curves of Rational coefficients, but in our case I don't have to really explain what it is because in our case a is simply one-dimensional Vector space so this is a trivial Frobenius algebra field itself and then how it is defined So in our case this is indeed so since it's one dimension just a value is important. This is actually a Total churn polynomial of what is called a hodge bundle Which actually doesn't depend on n at t is equal to minus 1 Where you do have to introduce this projection? Which well again, let me just write in this way The well that doesn't matter I set it to minus one, but of course this t and that here are different things Yeah, give me some letters, please so mg n plus 1 goes down to mg n and So this is the projection Which actually forgets the last marked point here Since this is a universal curve. You do have a relative canonical Line bundle, so this is what I write and maybe a plus one is It's easier to understand, but it's okay So this is just a cotangent bundle along the fiber and then you push it down that is Egn, which is just a push forward g n star of this canonical So this is a rank g bundle differential forms Behave very well in the degeneration of curves on this a dream of a compact occasion So this is not really only they are not only the coherent shift It is actually vector bundle rank never changes anywhere. How you degenerate because the generation is restricted here So this is a vector bundle and then you compute its total churn class and then take t equal to minus one then this is a Conceivich Maninco homological field theory which shows up here And then this polynomial is equal to that polynomial after substitution So this is a part of the theorem we proved This is a modular stack of curves, yes And do you have to work only in the range where usually because sometimes there are many Automorphism, so yeah, yeah stable curves. Yes, it's a modular stable curves. So so it's a obi-fold So neither G is larger because for G was even one. Oh Thank you. I completely forgot to mention Yes, if 2g minus 2 plus n is greater than 0. Thank you and then indeed F01 and F02 are not polynomials Of course from this formula, it's obvious After application of this differential operator, it is a polynomial. So it cannot be a polynomial So only after c0 it becomes polynomial and then these are the coordinate Maximiliana using in your formulation tau alpha Okay, so I'm There is a canonical section from here to here So the data here is a curve with n marked points and then a fiber is a universal curve itself So the curve itself is on the fiber which but in particular has a marked point so you can put this data to that point on the fiber that is a Section here then you pull back this Relative canonical bundle by this section you get the line bundle here and Then the first turn class of this line bundle to the power alpha one is tau alpha one And so on so there are n different line bundles by this section and then pull back take a power So this is the Witten's notation in the physics language it is Correlation function so this bracket is of course integration over this mg and bar right of these Comoraji classes Okay, so one this is just a part of that theorem I see How strong should I push? Okay, is it okay? I thought is a pull down, but you can also push up Yeah so Pull back and the push down both can be done Which one is a vector bundle, okay, so Now let's define Sorry, um, do I want to say anything else here? I don't think so. Okay, so just define c of x and h bar Now x is the original x not the t variable Is equal to just the exponential of a gigantic summation or g or n and Then one of our n factorial because I symmetrize everything and then h bar 2g minus 2 plus n Not h so n of f g n So here I do allow Unstable regions g equal to 0 and equal to 1 and 2 and then I put all x to be the same So let's define this as a wave function So if you don't set so this is indeed the principal specialization of a symmetric function If you don't symmetrize then the exact same expression with the infinitely many variables is a kp tau function So that has been known to Russian school 20 years ago, but now I'm using this principal specialization and then find a simple expression then This solves the following ordinary differential equation, which is h bar x d dx minus x times e to the power h h bar Sorry x d dx. I said the differential equation. It is actually diff differential difference equation Psi xh bar is equal to 0 So what I'd like to talk about I mean I say is that this is an example of a quantum curve It is a vial quantization of some sort of algebraic or analytic curve here So this psi is a solution So this is indeed a rather simple to prove in this particular case And then so what I mentioned is that not theory people are doing is that here you bring colored Jones polynomial You have a difference operator that kills and then this difference operator is a quantization of the a polynomial Things like that have been speculated It's never been proved in those cases. So here semi-classical limit here. What is h bar? I'm going to give you the definition of h bar later in geometry Okay, so here h bar is a parameter Is it real Complex and then here yeah important point is that this expression never converges with the specter h bar And then also it is really strange because if g equal to 0 n equal to 1 h power is minus 1 And then the rest is positive and then I'm still taking exponential So how do you define this kind of mixture of exponential of negative powers and positive powers? So it's all legitimate question at this moment is a formal expression and I'll make sense immediately out of it Okay, so at this moment. It's a h bar is an Parameter where it is not expected for any convergence anyway later. It's a WKB expansion So here semi-classical limit makes sense. So how you do that the question was This one starts with a negative power of h bar and it goes to zero positive and so on so how you make sense Well, I separate out negative power from here. It's exponential. So it's a multiplication and then Sandwich with a negative power there.  So I just kept in that way. So f 0 1 so f 0 1 is a function in x and I make this conjugation and then compute So if you do that So after that you can take limit h bar goes to zero. So the honest computation So this is exactly what we learned in the quantum mechanics first year course Then the result is that so this change into y and then this change into e to the y is equal to zero Where y is definitely here is a d applied to f 0 1 So d is x d dx So this is what happens and then this is the curve now this one can be solved x is equal to y e to the minus y So this is equal to that and then that has a name. It's a it's called the Lambert function And then this appears in the tree counting problem anyway So semi classical limit is indeed the Lambert function, which is Essentially the generating function of zero one invariance. That's what it is happening So these are the theorems we found then So this is an example. So so this curve We call it the spectral curve and then The deformation we'd like to consider here So I said at the family of some sort of d modules So we'd like to consider the family of this kind of object. So here the Deformation we want to do is a following. So we just to make a limit lambda goes to zero of Lambda to the power 6g minus 6 plus 3n and then fgn and then inside you just to scale everything with this parameter Lambda and then so this one, let me call it fgn Tilda which is just a function in original variables. Now, this is the homogeneous polynomial Because I picked up only the top degree terms and everything else is killed So it's a homogeneous polynomial of a degree 68 minus 6 plus n which is indeed having a simple expression The sum of all alphas are equal to now 3g minus 3 plus n and then intersection numbers are exactly what Maxim Considered many years ago and then so it's a tensor product Irons from 1 to n D tilde alpha i c0 t i tilde and then here What I'm doing here is a t very large asymptotics Which means the operator d Change to t tilde as t is large. So this looks like simply t cube d dt and then c tilde zero So t minus one doesn't count so it is equal to t so you plug it in here, then you do get The generating function of intersection numbers exactly what Maxim considered many years ago, and then What I mentioned here is that Laplace transform of cut and join equation Can be restricted to t large and Then this is indeed a virus or a constant condition So this gives a simple simplest proof of the Famous theorem of Maxime's but of course done many years after the discovery of the theorem itself So all that this virus all comes from is this combinatorial formula where you merge transposition into the general type Cycle or Permutation so that's all that was behind the scene is was our understanding of this thing So this gives the motivation one thing I'd like to say is that what did I do here? So in this limit process what happens is that this original omega GN actually change into little omega GN where It is indeed if there were Froben's algebra, then I'm making this to H zero part of mg and bar q So I'm picking up not the co-homology ring, but just a zero dimensional zero-degree co-homology So this gadget is called the two-dimensional Tqft so from the co-homological field theory degenerate into the degree zero part gives you this Computation allows you to make this computation So that is the process I'd like to consider in What is T and what is T i means I didn't understand this fire But T i and T are the same T i and Sorry, yeah, I can say Well, yeah T i is it Better What is the operator then Right, so I applied the operator to the function in T where T is replaced by T i Okay, so each operator you can put T sub i Right, it's a tensor product So you do the same operation and then name these variables in a different way Right, okay, so let me just hurry up And then so this is the motivational part and then the relation between this curve or the Lambert function over there and the differential operator That is the quantization we'd like to study so geometry So this actually comes from One of the one conjecture that's the guy auto made so since most of the things are defined already by Philip earlier this morning So I just rely on that and then now change the gear completely so it doesn't reflect any who's situation now It is a guy auto situation So see is an um ribbon compact women's office It's not really algebraic curve structure. I need I need really analytic structure so compact women's surface and then genius is For the moment greater than two. Well, we would generalize it later Then what are already so let's me use notation G for the complex simple Simply connected again complex regroup but for the Presentation with respect to time. Let me just restrict to the case SL and see then Already it was introduced a double modular space. So this is a modular space of Higgs bundles and then it's important that they are stable ones So I am not considering the modular stuck here the stable ones because I want to change into analysis So he's bundles where he is a vector bundle and the Phi is just a So K is a canonical again. So this is the all linear all module Homomorphism here is a Higgs bundle the stability simply means that if you have any Hormorific vector bundle Which goes the vector bundle F goes to F tensor K then that vector bundle has to have degree purely negative That's the stability condition and then he also mentioned drum visualize space and drum so this is the modular space of Flat connections. So here I forgot to say since I'm talking about SLN So determinant of V is a fixed as a structure shift. So here. This is the Hormorific connection Since it's everywhere Hormorific the degree of the vector bundle has to be zero and then I also Impose the condition stability is a reducible reducible means that the monodromy representation goes to the Zalski open part of the structure group So then I'd like to say the following so non-nabidian hodge correspondence that Phil was talking about Philip was talking about this morning was an a different morphism from Dorbo To a drum and then how it goes is the following so you consider the data Hormorific data from the Dorbo modular space, so it's a Hormorific object then this the stability condition is The system of partial differential equations that has been studied in the 80s by Donson yaw Urenberg those people So if you do the same thing here the stability condition becomes the following so you first Look at the topological part of the vector bundle since determinant is trivial. This is actually topologically trivial bundle So you introduce Hermitian Fibrametric H and then with respect to Hermitian Fibrametric H the complex structure E is encoded into the Chan connection D and then you have this Hormorific phi so you just Yeah, I'm sorry. I'm mixing two things together. So here Chan connection D and then a phi is itself here which satisfies the following non-linear equation the curvature plus Phi and then Phi Hermitian conjugate with respect to H is equal to zero and then the Phi is Indeed the Hormorific. So this is the system of equations called Hitching equations and then at this one is Indeed the the flatness of the following. So D is zeta. This is called the twister line So this is a Chan connection which is a unitary connection plus one over zeta phi plus zeta phi dagger is flat for all zeta inside the C cross so that is the way of encoding the Arzbeck definition of stability into the differential geometric Condition of PDE. So that is what is known. So this and then from here What you do is a solution to so well, maybe this solution HD Phi with a flat D of zeta So that is the equivalent information and then from here What you want to do is that first you look at the zeta equal to one So zeta equal to zero picks up the information of Phi. Phi is a Hormorific one So zeta equal to zero goes back to the original because Chan connection and cause the complex structure of E and then now zeta equal to one what I'd like to do is that the zero one part On the topologically trivial bundle that gives you a new complex structure and then since I define the Hormoric structure by zero one part and the One zero part is automatically Hormorific so that that is the point on the Torbo Mojirai, I'm sorry Duran Mojirai and then this is indeed and the isomorphism is approved by the small-run cases Dornson and Hitchin and then Simpson and then Coret did all the general situations of a high dimensional base so this is what Non-Abelian Hodge correspondences then what Davide Gaiotto said is that let's look at the following family You introduce Real Parameter well, it doesn't have to be positive, but let's me just take positive real parameter And then define this as the same Chan connection plus R over zeta phi plus R times zeta phi dollar So R appears not in the same way as zeta Just R is a risk rating of Phi and then a Hermitian conjugation doesn't Change R at all. So so this is a definition and then now that I ought to say that the limit the scaling limit zeta goes to zero R goes to zero and then this Ratio is fixed So the question what was H bar here you fix the ratio to be a complex number and then this look at this quantity exists So he said that this exists and is an Opah so to be precise this is a H bar Parameterized the family of opas. What is an opa? So opa is a particular point on the drum modular space. So E nabla, which is a connection is an opa Means that there is so three conditions. There is a filtration zero is equal to f n goes to f and minus one Sorry, it's not the exact sequence F1 f0 is equal to e and then two Griffith transversality So the connection restricted to each piece if I goes to f I minus one tensor kc And then the third condition is a real dimension cutting condition so that It generates the grading Gradation so if my which is which plus one goes to f I minus one over f I tensor kc is a OC linear isomorphism. So when you have filtered vector bundle with the connection satisfying this it is called an opa so that's what the Gaiotto said and then this one should be an opa and then so this so what I really would like to do is Try to relate to this with this with a Condition but of course here His conjecture so the so this is just the definition of the procedure Actually conjecture is that if you choose E nabla Sorry, not nabla Let me just write more precisely e0 phi Q is on the hitching section then this limit the double scaling limit Scaling limit is an opa So it is never true For for arbitrary e and phi. Yes. Yeah So if you suppose fix the R and then take the residue of the d zeta R then then You will it is not evident that age bar is fixed and suppose for a moment to fix the R Then take the residue at zeta equal to 0 of this This d d come in a d of zeta R. Then you will find that R is the residue That's so I don't understand why age bar fixed then it would be Well, this is a particular limit. I'm talking about I want both zeta and R go to zero simultaneously Hitching equations it's for R one equal to one. It's not a flat connection. So I have to change the hitching equation here The where did I write? Here R square comes in here Right, right So this equation is modified and this one isn't yes, so to fundamentally we are solving that equation and then therefore Argos to zero would allow would actually force the metric to explode That's right and then so what happens is that maybe this is what I want to say in the end But I may not have time. So you look at this those drum Mosulite hitching section. I'm going to explain in a minute Would that go to by no I'm feeling hodge the real slice of the drum Mosulite SL and R? Connections but then as soon as you change the complex structure from the original a minute later It will be all the same as the drum complex action So it is hypercalous So there's a p1 parameter family of complex structures about everything else is the same as here except for this one so You just change it and then I bring it back Zeta goes to zero as close as original Rescale it then it becomes tangent to the Open Mosulite so that's what is happening H bar is anything H bar gives you actually the opa as a deline connection H by is equal to lambda of the dreams connection So it's a family of Lambda connections Yeah, yeah, so so I have to I want to say a lot of things here So let me just say first I have to define hitching section. Okay, and then yeah come back to your point So hitching section. Yes, hitching section here. I do have to use the constants principle TDS So for the quantization so we call this a procedure quantization completely analogous to the Who is and so on has to depend on the hidden existence of SL 2 So SL 2 is indeed essential for the quantization here. So constants are principle TDS I don't have the time to define but the constant was intrigued by the Poincare polynomial of Simple simply connected the group and then he wanted to Explain the exponent of that factor and in terms of representation theoretic way and then discover So this is the for any such regroup or The algebra of that group they exist up to conjugation a unique three-dimensional TDS Satisfying such a such condition for SL take in case. It's obvious because there's a unique and dimensional representation of SL 2 Any question over there? Oh What is TDS three dimensional simple sub algebra SL 2 principle SL 2 So every Simple I mean a simple V algebra has a I see Well, sorry, I'm just a coping from constants paper So using that so in this case a constant previous It's generated by X plus X minus and an H and then here SLN case is easy to write down X minus is so let me just use the representation This appears in constants paper. So I just a copy down from there So n minus one.  So I have to choose a spin structure or say the characteristic whatever so this line bundle and Powered with diagonal matrix, which means that just a Kc R1 over 2 plus and so on all these diagonal entries Kc minus Rn minus 1 over 2 so this is the vector bundle and then Phi let me just write the phi of Q to be just a X minus Plus let me just cheat here because of the sake of the time So 2 to n and then Q J X plus to the power J minus 1 The reason is because I do have to introduce positive roots positive simple roots to make this Expression going through arbitrarily Regroup G, but I don't do that. So here SLN case this works and then the Q is Q2 and so on Qn, which is just an element of J runs from 2 to n H0 of C Kc Tensor J. So this is denoted by so I want to denote by B Felt both wrote it H, which is a hitching base. So indeed from hitching base I am giving you a unique hitching Higgs bundle E0 and then phi 2. Now E0 is Very far from stable vector bundle, but this pair is stable. So it corresponds to the It is a point on the modular space stable Modular space and then for this one and you compute this procedure Defining D of zeta R and then are changing the equation hitching equation with R squared here So and then solve it so then the dot of two parameter families flat again, and then you do have Limit so a theorem is that a lot of people here. So theorem Olivia Dometrescu and Laura Fredrickson Gelbius Kido Narciss and then Laugh Matzio and then myself and the night ski and with myself So we show that the conjecture is true So what is happening? So the question is how do you? Find the limit so quickly how you proceed of this kind of limit First you just choose q equal to 0 and then I ease 0 and then just x minus This is exactly what the hitching considered in his 87 paper So this one is also stable Higgs pair and then you plug into the hitching equation So this actually gives you Harmonicity condition For the fiber metric H, but please look easy. Oh is a direct sum of Canonical to the powers so canonical has a canonical metric if you fix the reman's Surface structure so remain a metric G on the surface namely conformal class I just write it as a lambda squared dz dz bar and they're using this lambda So lambda is a metric of Kc minus half So lambda to the power appears and the fiber metric becomes a just diagonal and then you plug it into this hitching equation And then there are no cues are involved there. So you realize that the solution is Constant-covered geometric So this is a conformal class. I chose has to be fixed as a Constant-covered geometric therefore curve C is realized as a quotient of the upper half plane by the pi 1 of C and then that also allows us to introduce unique Projective structure Projective structure on C coming comes from the solution of the hitching equation then what happens is that around that Metric and then that was obtained by assuming q equal to zero now q is not equal to zero you just that use the Banach analysis to solve this equation, but then a limit is very easy So the formula another formula I want to say is that the zeta goes to zero r goes to zero and then r over zeta is equal to h bar fixed of D Zeta r is actually is a two pieces E h bar is a deformation of easy as a vector bundle and then a nabla q of sorry h bar where H bar is a unique Filtered extension Parameterized by h bar now the definition of h bar here So h1 of C KC so h bar is an element of this so n equal to two case filtered extension is easy I'm talking about sorry. I mean it so KC So I'm talking about the unique extension h bar I'm talking about this one so any element here would give so this is a isomorphic to extension group So this gives you the unique extension and I'm using that so for higher-rank case or general Regroup case you do have to work a little bit, but that's filtered extension idea gives you such a Unique extension and then the nabla h bar q Is as simple as it could be so in this vector bundle Which is no longer the same as before but I'm using this Projective structure you just write down the exterior differentiation minus 1 over h bar and then Phi of q phi of q is that expression that expression itself becomes a connection form In this a new vector bundle. So this is the limit open then a final thing I want to say is that So this one is a joint work with Olivia do me to rescue This correspondence from phi to this final connection does not really depend on that Assumption that the q is a holomorphic section here. You can put any poles over there Each bike or to zero goes back to the original right and then if h bar is not zero immediately All the complex structures are the same so h bike or to one or zero is only the case, but of course you do have a Family and then as you can see this is the doings lambda connection if you multiply h bar everywhere This is the dream's connection. So Since it's time. I just one word Because of this a filtration and then there exists a filtered filtration I mean h bar parameterized filtration this D module here is Principally generated and then you have a single Differential operator globally defined again with respect to this Projective structure and then you apply the original idea of semi classical limit as we did for the Hurwitz case It gives you a global spectral curve, which is indeed characteristic polynomial of this phi So in that sense WKB works semi classical limit to works and then you do have a quantization here. So that is the story. I wanted to say Thank you very much question remarks Just a clarification when you say some limit exists and is an opera in several places. Does it mean that? the Filtration that in the definition of an opera is Is exist I mean is it uniquely defined or is it a part of you have the cost part of the definition Part of the data. No, but when you say the limit exists, how you construct Filtration yes satisfying all these conditions, but you don't claim that it is unique. Yeah with respect to h bar Yes, there's a unique Construction right But he doesn't mean that another filtration cannot satisfy the same property Property not structure, but if you have a if you have a connection, the fact it's an opera is the property of the structure There's a theorem that if such a thing exists is unique Yeah, but the theorem proof of the theorem does contain the actual construction of this filtration satisfying all this right?  Yes, always that's a question. That's Place we want to go but at this moment we don't have it You see here everything is algebraic and there it was kind of essentially homomorphic system. It was essentially normal So you do want to go to the analytic situation They're all not situations and the grammar of written situations are all over there in the analytic case not in this case There's a question or remarks Okay, so let's sing the speaker again


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.  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.  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.


Hello everyone, it's Gabrielle and I'm here at Oxmer Toyota. You guys, guess what? We have the all new, highly anticipated 2022 Toyota Tundra. We just got this in and we're super excited.  It's definitely different from the older style Tundra. Yes, it's a brand new truck. Right. The only real thing is in name. It's a brand new frame, brand new body, brand new grille. Everything about this is brand new, brand new engine. This is a twin turbo V6. It's a 3.5 liter, gives you more power. This has more towing capacity than the previous generation. Wow, great. So we're 11,000 pounds on this one. This thing's awesome. I get excited about it. It looks fantastic, more power, different build, different setup. You got the halogen lights on here. This has Toyota Safety Sense P, so all the pre-collision safety stuff, the lane departure alert, all that cool stuff that you love about Toyota. But inside the technology is really what I want to show you all. Because this thing is awesome. It is stacked. Now this is the crew max. This is the limited. It's the four-wheel drive crew max. So you've got the two bigger doors, lots of space in here. It still has that back window that rolls all the way down that people love on the Tundra. This is the five and a half foot bed. It's got the composite bed liner in here with the halogen lights back here. And then if you want to hop in the backseat, I want to show you all this screen. It's got a 14-inch display screen. I hopped in, it recognized my phone. It said, do you want to connect a new device? I said, yes, added my phone and it right into Apple CarPlay. So I'm going to shut this here. So it's wireless Apple CarPlay, which we only have in the Super. So this is the first all 100% Toyota that has that wireless Apple CarPlay. So I didn't have to plug in. All I did was pair my phone, Bluetooth, and it popped in here. So my navigation, all my apps, everything for Apple CarPlay. It also has wireless Android Auto, which is cool. And then up here, you've got that, we've seen this in the RAV4. This is that digital rear view. So if you have people in the backseat, if you're loaded up, you've got a camera so you can actually see beyond the inside of the car. You don't have this. But the thing that I got most excited about is look at all the options back here. I'm going to put the car in reverse. And then the camera angles, right? So we've got the bird's eye view over here, the 360 degree view, the normal, with the projected path back up. But then look, you've got a front camera. You've got the front two corner cameras, the straight back up camera. You've got the side camera. So if you're going to tow something, if you need to look and see what's back off those back pillars there, you've got that. This is really cool. You can look into the bed if you want to see. That's nice. So it's got this great, just different views.  You don't want to scratch up your brand new tundra. Everybody's going to want to look at it. They're going to be distracting you. All kinds of different camera angle views. That's on the back up. Let me show you that. It's just super cool. The different capabilities they have there, very much a capable off-road vehicle. It's got four-wheel drive, low and high, of course. Different drive modes, different tow haul modes. So many cool things. This has blind spot monitoring on the side here with rear cross traffic alert. But this also has, it can learn, if say you haul a trailer, right? It can learn the length of your trailer and then also give you warnings about trailer merge if somebody's in not only your blind spot, but also where that trailer is. It's very smart. Very, very smart and that's available on this one. Very, very cool. I'm going to put this back in the park. The other thing that I like about this, sometimes on bigger screens and vehicles, what you have is the same amount of area that just condenses it. So they split the screen so they have something over here and here. This is formatted to fit this large 14-inch screen. It's like having a TV in your car. Your navigation, you're not going to get lost using this. It's just fantastic. And this thing has so many different features. You can start the car with your phone. It has remote connect. It's got safety connect up here. This is how you can open and close that back window. Okay. We also have the panoramic sunroof too. Panoramic roof, you can close that. And it's a truck. It's a full-size truck, but listen how quiet it is. It is very quiet. It is extremely comfortable. Yeah, extremely comfortable. Heated, I think this one. I think we have cooled seats too. Yeah, heated and ventilated seats. Dual air control. And we were driving this the other day, just testing it out when we got in yesterday. It's so quiet on the road. Like you can hold a conversation like this as you're driving a big truck. And it's one of the most comfortable things I've ever been in. So I want to show you this. Yes, this. I love this one. This one's probably my favorite. It has your jewelry safe. So all your jewels, your necklaces, earrings, diamonds. Your heist or something. Yeah, it keeps it all just right here in this box. I love that. Right up. Yeah, that's great.  It's just flying. Yeah, I put my rings on there. My diamonds are everywhere. Oh my goodness. So that's a fantastic feature. You got the Qi wireless charger that stands up there. So if you want more screens to look at, you got it there too. Right, that's actually convenient. I've never seen a wireless charger that stands up. So I like that. Oh yeah, very, very cool. You can have. And so everybody can, it's just fantastic. The way they thought through this, the heated steering wheel, everything about this has been like thoroughly planned, thought through, very convenient. It's just an awesome truck, and we're going to sell the heck out of them. We've got three people that have bought this one so far, I believe, so you're going to hurry up and get in and get a deposit on it if you want it. Right. But yeah. OK, and one more thing that I noticed just walking around the vehicle was this button. And I tend to always push things and touch things. But this is the actual, it's called dampened tailgate. So all you have to do is just lightly press on that. And it drops the tailgate down in it, doesn't throw it down. It's very like an easy release. So that's great. Very convenient to have that. Oh, very cool. Yes. All right, you all, I hope you enjoy the little walk around on the all new 2022 Toyota Tundra. If you have any questions, if you like to place an order or stop by, while we have this one to take a look at it, I'm sure we won't have this one very long. But you all, if you're watching this, we're running out here, and hopefully it's still here. But all of our contact info is listed right here. We hope you all have a wonderful day, and we'll see you soon. Bye.


The National Mall Tidal Basin in Washington, D.C. is America's front yard. People come from around the world to experience its beauty. The views, the monuments, and of course, the famous cherry trees. Built in the 1800s, the Tidal Basin was originally a public space for recreation. It has evolved to include memorials, monuments, and the flowering trees that now attract 1.5 million visitors a year for the annual Cherry Blossom Festival. Here's how the basin is meant to work. Twice a day at high tide, inlet gates open and fill the pond with 250 million gallons of water from the Potomac River. Then, at low tide, outlet gates open and the water exits into the Washington Channel, sweeping away silt and sediment as it goes. But over time, sinking land and rising sea levels have led to daily flooding that threatens the Tidal Basin's stability. The seawalls are collapsing, its sidewalks are cracking, the cherry trees' roots are compromised. The area needs as much as $500 million worth of upgrades and conditions will only worsen as the water gets higher. This place is unique, beloved, and irreplaceable. So what can we do to save it? The National Trust for Historic Preservation, together with the Trust for the National Mall and the National Park Service, is sponsoring an Ideas Lab. Presented by American Express, this forum is inviting five world-class teams of design professionals to envision bold and holistic solutions for the Tidal Basin. Their mission? Don't reinvent or erase this iconic place. Rather, make it the best it can be for a modern world. You too can help keep this iconic place around for another 100 years. Sign our pledge today and help protect the National Mall Tidal Basin.