 No, I'm okay. Okay, thank you. I'm fine with that. Okay, so yeah, so this is the workshop on hot theory, mirror symmetry, and physics of Calabria and Montelai, and it's a joint workshop of Mach and Heidebeck and Pipps University of Canada, and it's a second workshop they've organized in this sort of framework. The first one was two years ago on a different topic, and yeah, so I look forward to many interesting talks, and it's some practical announcement, you know, from time to time, but I think for the time being, we should get going with the program. And first, we asked a Gephard Fertner, a colleague and acting director of Mach to say a few words of welcome. Yes, so hello everyone. So Johannes convinced me to say a few words on behalf of Mitch, and so I look, I mean, I see that you all made it well here, and that's great to welcome you all in Neheilberg, and so I want to welcome everyone here to the workshop on hot theory, mirror symmetry, and physics of Calabria and Montelai, and so the workshop is funded jointly by Mach, Pipps, and Structures, and if you all know, organized by Johannes Nevalcher, Simon and Neuer, and Charles Doren, and Noah. Thank you. Very good. I actually didn't know all the organizers. And so before I let you dive deeply into the talks of the workshop, let me say a few words about the Mathematics Center Heidelberg, which I guess is abbreviated by Mach. So this was founded about 15 years ago, and an important aim then was to coordinate research activities, I mean to coordinate the research activities and to have a more outside visibility of the Heidelberg mathematics. And at the time, 16 years ago, mathematics was spread out in three buildings, not near here, but somewhere on the campus. So it was the Mathematical Institute, the Institute for Applied Mathematics and the Interdisciplinary Center for Scientific Computing. And I don't even remember, but I remember that the Mathematics Institute was in the oldest building on campus, and the other one were in some temporary buildings that were temporary for 25 years. So maybe longer, maybe they're still there. And so fortunately, some generous donor made possible that we have this new building in which we can now host this conference. And so since 2016, we all moved in here. So all of mathematics, the Interdisciplinary Center of Scientific Computing, and even the computer science field, we all found a new location. And now that we have this new building, I guess the space is running low again. So yeah, so we have to go and expand again. And also, I should mention that a few weeks ago, so until then, there were two mathematics sub departments, the Mathematical Institute and the Institute for Applied Mathematics. And so we joined forces early April to become the new Institute of Mathematics. Maybe I should also then go to another point. So another aim of match back in 2008 was to support many new hireings in mathematics. I think in 2008 was clear that the next 15 years, many people would retire, and new positions would be revealed. And this was possible due to the role of mathematics in the first two rounds of the German Excellence Initiative. And in the last 15 years, I guess nearly all of the then existing faculty was replaced by new faculty. So there was a complete turnaround. And an important aspect of this process of renewal was to possibly strengthen the link between math and physics. It was one of the goals. And this might bring most of you here at this point. So I think this was a very good initiative. And so as part of this effort, it was possible to hire Johannes here as a faculty. And there was also a new partnership initiative between math and physics in 2015. And I think this became one of the pillars of the many pillars that proved important for the success of the structural cluster at Heidelberg that has funded since around 2019 by the German Excellence Strategy and brings together math and physics in many ways. And I guess also supports this workshop. And since 2019, so I guess we achieved all the goals. We had to look for some new things. So since 2019, maybe the new emphasis of math is on strengthening international collaboration. And it was an excellent possibility to have joined, I mean, to have joined events with the PIMS in Canada and also with other German universities, I think next to Heidelberg, also Münster and Berlin involved in this. And in fact, I think in 2019, there was a conference here between PIMS and Berlin and Münster on analysis, modeling and numeric. So this was one of the areas, I think in 2021. And now there is a workshop here in Mathematical Physics organized by Johannes and I think the same people like them. And there was also another area where we have seen a lot of collaborations, namely the arithmetic geometry and number theory group that just this March had a workshop on Eigen varieties and attic spaces. So I think there are a lot of collaborations now taking place between PIMS and Heidelberg and other German institutions. So I hope this gives you an impression of the initiatives supported by Mech. Certainly Mech was not the unique player on this. There were definitely many players of this. And in any case, I should also say that I only got involved in this after all the successful stuff was done. So I'm only here to give an introductory speech. Yeah. And but maybe even though we are not maybe at the base of all these important developments, I mean, we're one of the things that maybe having Mech as a center might have led people to develop new visions. And I think only with new visions, new developments and new progress will be possible. And so let me end on this now maybe and let me now wish you a successful workshop with excellent talks and lots of interactions and hopefully many new visions and insights. And I hope you enjoy your time in Heidelberg in many ways. Thank you. And I'll keep it short. On behalf of PIMS, I'd like to welcome friends old and new to Heidelberg. One of my main jobs this week is to talk to you honest about follow up events or events in Canada. And yeah, so if you have ideas, share them with us. We'll work on that next. So okay, okay. Well, Matt's getting geared up. Our first speaker is Matt Curve from Washington University in St. Louis. And he's giving part one of the two part talk with Russell Goldschild on his song. Yeah, from the upper right corner of our screen. Russell, what is talk to you about? Seems the volume is quite high. Or maybe that needs to be done. But it does seem very high. Okay. Can everyone hear me on the thumbs up? Okay. See the title of my talk is differential equations, hyper geometric families, algebraic cycles. And the main theme is that you can use differential equations often to bypass constructions of cycles and evaluation of their Abel Jacobi maps for purposes of computing Abel Jacobi maps, regulator maps, higher normal function construction and checking the balance and conjecture as possible. And this is, as Chuck said, the first in a two talk series, the second of which will be given by Vasily. So I'm going to start by saying a few words about Frobenius deformations. So here, m over p1 with coordinate z minus some finite sense sigma, which I'll call u, will be a variation of hodge structure of weight n. And all the hodge numbers are one. So it's rank n plus one. And I'm going to give myself a section that is nowhere zero on the affine line only has zeros at infinity. So this uniquely specifies this section once I impose additional scaling. So this is the canonical extension of m globally. This is the top hodge filtration. I'm going to assume that I have various things about monodromes. So this set here is going to include zero and infinity, as well as c equals c1 up through cd. So this c will be less than or equal to an absolute value of the other ci's. Those are all conifold points. And I have maximal unit put monodrome at z equals zero and conifold, meaning Picard left shuts monodrome at z equals ci for each i. And for all the monodromes for sigma and sigma, t sigma will be written in the semisimple part times the unit part. Okay. So now I need to tell you what form the differential equation takes. We have a differential equation differential operator rather of degree d. So I can write it as z to the j polynomials in d, where d is z d dz. I can also write it as sum as i goes from zero to n plus one. It has order n plus one q n plus one minus i of z d to the i. And this thing is in c of z d. So q's and p's are all polynomials. And of course, the important thing is to have that l kills mu as a class. And so then l is going to kill all periods. And what are those periods going to look like? Well, I'm going to start with taking the dual local system of m, right? It's a variation. It's a q variation of hot structures. So it has a q local system. And then I take a base point p, not in sigma. And I take the t not invariance. So this is like invariant homology classes about zero. And that will be generated by a single cycle because I have maximum uniform monodrome. And then I have for the conifold monodrome tc minus i applied to mp dual is generated by a single conifold cycle delta. And so then I claim I'm not going to prove that's true. There exists unique basis of mp check consisting of epsilon not up through epsilon n such that n epsilon m equals epsilon m minus one. And all the ones except for epsilon zero are invariant about the conifold. Okay. And so then l kills the periods. These are the beddy periods. Epsilon m is the integral over r epsilon m of mu. So those are my beddy periods that have something to do with geometry. And I want to ask the basic question how to solve the differential equation l of y equals z to the one over r for some integer r in this setup. So here is a proposition that comes out of work by Bloch Blasenko and a follow up paper I wrote just those abbreviations. I can restrict m to a punctured disk about the origin and then the claim is that this extends to a unique variation of mixed hodge structure b over the same thing with hodge numbers that are mixed hodge numbers in the following digger. So we start with the hodge numbers of m and now we're going to extend them like this to get a variation of mixed hodge structure with negative minus one minus one minus two minus two. All of those mixed hodge numbers are one. Differential equation d infinity l. So if m as a d module is d mod dl this is d mod d infinity l. The idea is you apply l to a class here and that kills the stuff up here but then you still have to keep applying d to kill everything. Local system v closed under tc. I'm not saying it extends to a variation of mixed hodge structure on p one minus sigma but we at least have this and then that's the uniqueness statement and then there are further statements. VP check has a basis epsilon m greater than or equal to zero with tc epsilon m equals epsilon m and m not epsilon m equals epsilon m minus one for all m bigger than zero and we can encode this. You can ignore the table. We encode this set of mixed betty periods if you will by a e of sz. This is going to be a generating series. What do we satisfy? We have tc minus i applied to e is psi of z which is the integral of mu over a non-fold cycle. So that's the first statement. t not of e is e to the two pi i s e and l e the current Fuchs operator in z gives s to the m plus one times some power series alpha of s times c to the s where alpha of s takes the form one plus o alpha of s one plus o of s. Okay so it's some complicated generating series and I can also rewrite it formally as a k of s z to the k plus s. Now this looks pretty mysterious but I'm going to tell you what it is explicitly in one case. So in the hyper geometric setting where d is equal to one that's the order sorry the degree of the differential operator and number of conifold points so sigma can be taken to be zero one infinity l to be d to the n plus one minus z times the product j goes from one to n plus one d minus some aj's so those are some coefficients typically between typically between zero and one and symmetric under reflection about a half but one doesn't have to insist that but you can see that then ak of s is the product over j of gamma of k plus s plus aj over gamma of aj times gamma of k plus s plus one and the leading period epsilon not of z which by the way mu is specified completely by assuming that always epsilon not begins with one so this is the sum k greater or equal to zero ak zero z to the k is for these ak of zeros they're simply given in terms of gamma functions or you can rewrite them in terms of pockheimer symbols k plus one z to the k and the moral of this is that the classical Frobenius method actually encodes betty periods so what do i mean by that i mean you start with this hyper geometric series then you formally write you formally replace k everywhere by k plus s that gives you the generating series of these and then the statement is that recovers all these mixed betty periods by taking derivatives with respect to s and setting s equal to zero and so this gives you the betty periods not some Frobenius periods which i'll discuss in a moment but it tells you geometry without having geometry at hand so that's the magic of this now i believe what do i need to get at this thing where the right hand side ah not so okay so so these epsilon m's from epsilon zero up to epsilon n are killed by l but for higher that's no longer true and you can see that already from le gives s to the n plus one times some series times z and v s so you see as soon as you go above m equals n when you go to m equals n plus one you start getting experience and so then there are solutions to dl d squared l d cubed l so on and so forth uh not yet i have to tell you how to solve how to how to do that question yeah the question is what can i erase maybe here okay so now um we want to take epsilon of sz divided by alpha s and call this phi of sz and if i formally write this as phi m of z s to the m then these are called Frobenius periods and they are uh sort of you can uniquely specify them more or less by saying or you can characterize them by saying this okay so there are a lot of interesting constant terms in the analytic parts of the epsilon m's these don't have and accordingly this alpha of s though i don't have time to say anything about this is telling you something about the limiting mixed odd structure not just at m but of the whole variation of mixed odd structure v and the origin so it's power series coefficients tell you the entire limiting period matrix effect so phi can be in fact uniquely specified by the properties alpha is s to the n plus one z to the s and t naught by is e to the two by is and then this leads to proposition two immediately just by looking at that and that is that if i put w r of z to be r for the n plus one by one over r z this is the unique solution of l of law equals z to the one over r with w r of zero equals zero and here's again the hyper geometric example w r of z is some k greater or equal to zero product number j of polkhammer symbols alpha j plus one over r k over one over r k plus one z to the k plus one okay so i'm going to leave that world for a second and tell you what a higher normal function is so i'm going to start with this slightly different notation so let h over u still the same u a different h different variation be a variation of pod structure of weight w less than zero then higher normal function on u with values in h is equivalently either of the two things a variation of mixed pod structure of the form zero goes to h goes to e goes to q of zero goes to zero short exact sequence of mixed hot structures or a holomorphic horizontal section of j of h which means h with complex coefficients over f not h plus the q local system so sort of bundle of generalized intermediate Jacobi edits the way you go between these things is you take a lift of one from here to f not e c and to the local system over q you take their difference that lands in here and so you get something in the numerator of there and you quotient by f not h and h to excise the ambiguity is in that lift okay so it is admissible i'm just going to keep things simple here with respect to p one if and only if all the limiting mixed hot structures of the variation of mixed hot structure e exist okay if those all exist and i will write new for my higher normal function in a and f u of h and i'll also write e you can write e new for the extension you can write new for the section of j of h etc etc and this new has invariance they're called singularity invariance sing sigma of new at each point in the discriminant locus that lies in palm q of zero this is on mixed hot structures psi sigma of h which is my shorthand for the limit mixed hot structure okay and here the idea is you take a local lift of new about sigma and then you apply n sigma the monodromy longer and i'll need a slightly more sophisticated notation so for you not containing you contained in p one i'm going to set a and f u not of h to be those admissible normal functions such that sing sigma of new is zero for all sigma in u not intersect sigma then there are a bunch of facts that are useful for working with these things these look very mysterious but you can compute readily so w less than negative one that's the true higher normal function case admissible normal functions with no singularities all are just zero because i'm working in modular torsion if w equals one minus one that's the classical normal function case and then all admissible normal functions have no singularities sort of the opposite it presents difficulties in going back and forth between those two cases and then an f u not of h this is how you actually compute these things there's a class of an admissible normal function which lands in palm q of zero into i h one parabolic cohomology of h on u not and if you're used to thinking about parabolic cohomology on all of p one that's always pure this is not because that this is not an all of p one there exist exact sequences zero i h one p one h to i h one u not h to the direct sum overall sigma in the complement of u not that's sigma not um size sigma of h limit mixed hot structure t sigma co invariance twisted by minus one and then finally there's the Euler Poincare formula which says that the dimension of i h one of u not h is the sum over sigma in u not intersect sigma delta sigma minus rank h times the Euler characteristic of u not where delta sigma is the rank of t sigma minus the identity okay now i want to take an admissible normal function over gm so no singularities apart from zero and infinity with values in m twisted by p so now i'm back to my original weight and variation twisted by p so this is isomorphic to hum q of minus p into i h one gm m for let's say p is between n plus one over two and n plus one those the values it can possibly take and i'll take a multi-valued lift new tilde so you think of this you think of new as being a section of some generalized Jacobian bundle and i could just take a multi-valued lift to the variation v uh h in this case m and um so this is a lift to m on the universal cover and then i put v for the resulting multi-valued function in z which is i take this lift evaluated at z and pair against mu z that was the section of the punchline bundle um and finally i'm going to write f and e me um that's a line bundle on p1 so it has some degree h this will almost this will basically always be one press punchline proposition three this comes from a paper with vasily and my student tokyo sasaki this is postdoc in miami now um the inhomogeneous term lv so i take this higher normal function v and i hit it with l that gives not just a rational function this is a polynomial of degree less than or equal to b minus h and if the singularity and variant of nu at zero is zero then t divides g sorry z divides g um and you can say other things like if the entire setup if the variation comes from a variety a family of varieties to find over q bar then g is a polynomial with coefficients in q bar um and if the higher normal function is not identically zero then g is not identically zero so so here are some examples if d is one so we're back in the hyper geometric case essentially then ih1 of gm m is one just because the drop at each conifold point is one so you count the number of conifold points that gives you the dimension of this and um psi not m for invariance uh at the origin is just a q of minus n minus one because i have maximal unit put monodromes so i just look at the top of the n string and that has weight two n and then i twist by initial minus one uh so that gives you that ih1 gm m is ih1 p1 minus zero m is q of minus n minus one and so that means that p is n plus one and the singularity has a singularity at zero and lv is some constant so if we go now to d equals two then one of two things can happen either we have an exact sequence ih1 p1 m to ih1 gm m to this q of minus n minus one which persists to this case that comes from similarity at the origin um and this says that you're going to have a non-trivial classical normal function the other possibility is that ih1 gm m is the covariance at zero plus the covariance at infinity and these both have right one and this will be again a q of minus n minus one because of the mum and this will be some q of minus p and so we want to understand um this situation so coming from the normal function coming from this class let's call it v0 is going to satisfy lv0 equal c0 and by this proposition the normal function coming from this class because it has no singularity at zero will look like let's call it the infinity c infinity times z where both of these in our constructions will be uh in q bar okay so we sort of get out of thin air motivic solutions to these inhomogeneous equations so where do these come from geometrically they come from regulator maps on algebraic cycles but not usual algebraic cycles rather higher cycles unless you're in this case so this is the geometric realization part we have x over p1 lvl infold family and it's hn let's say xu over u contains our m as an irreducible subvariation now an admissible normal function over some u0 with coefficients in m of a as class let's write it like this class of nu is in we're going to shorten palm q of zero into something to hodge hclasses in ih1 u0 m of a in hodge classes in ih1 well let's say actually this lives in h n plus one x u0 q of a well then the balance in hodge conjecture tells us that such a class such a hodge class should have a geometric realization of the form z in a typical homology in degree n plus one x u0 q of a maybe you if you're more comfortable with higher chow notation here is that this tells you the sort of degree of k theory that we were in um twisted by q with cycle class of z equals the class of a normal function and so then that tells us that restricting to fibers apple jacobi on xz of all right just zz for the restriction just to fiber xz which is represented by classes of regulator currents associated to these cycles um recovers new ball new new z as it comes from that in the Jacobian of m of a okay so taking fiber wise apple jacobi of these higher cycles gives us our normal function back and now what does the balance and conjecture say the first balance and conjecture says that it affects over q is an n-fold b equals 2a minus n minus one is at least two and script x is an integral model then if i look at cycles extending to the integral model if you prefer chow a of xb then the size of apple jacobi maps to j of h and n of x q of a which is h n of xc over f a of that plus that with q of a coefficients and i can project that to the real deline co homology h n of x r of a minus one over f a plus f a moment and now inside that there are the so-called duraman variants which is what you get by composing complex conjugation on the points of x together with conjugation of coefficients and then if i look at extension of this to r this is supposed to be an isomorphism so let's call that rb and the conjecture furthermore says exactly what the ranks of the two sides are so the rank of kab of x should be the order of vanishing at n plus one minus a of the l function and for n odd this can be computed by looking at the arc median parts of the gamma the arc median parts of the l function n plus one minus a less than or equal to p less than or equal to n minus one over two hp n minus b it's pretty straightforward okay and then the leading Taylor coefficient for which you have to differentiate this many times of the l function at n plus one minus a is equivalent up to q cross to the determinant of rb and this determinant is taken with respect to the q structure defined by the determinant of the duraman variants and here by the determinant of the algebraic duraman cosmology fa hm duraman x over q yeah um so yeah i am i do want to assume that's defined over q bar actually over q so that when i special specialize to uh integer points in there i have an integral model of so-called or to rash i mean okay so let me just do a quick example so if x is globia three hodge numbers are one one one and l of s means l of h3 of x s then we're supposed to have that the second derivative of l at zero is up to a rational constant the determinant of a two by two matrix given by you take the balance and regulator of two cycles um divided by two by i cubed and you integrate them so here's z1 and z2 are supposed to be a basis in k44 of the integral model and gamma one and gamma two are a basis for the f infinity covariance that's the complex conjugation okay and you can say something similar about l prime at one and that has to do with k2 of x but it's a bit more complicated to state and lastly we'll say something about that this is all i need for now how much longer do i have you on this okay that's for 45 yes okay all right 10 to 15 yeah i want to say one more thing about geometric realization it's important because there's a case where you always have a k n plus one cycle for free and this goes back to case of k2 of elliptic curves work of Fernando Rodriguez de Agas and then for k n general work i did with Jack Doran so we have the notion of a tempered Laurent polynomial so we assume that our Laurent polynomial is defined over q bar in n plus one variables and so that's out an n fold but what i want to do is take its Newton polytope assume that's reflexive and then take one minus t phi of x equals zero and take its closure inside some blow up of the associated torque variety cross p1 and then i have a section of the hodge line bundle given by one over two by i to the n res xt of the wedge of all the delogs of the coordinates over one minus t phi that gives holomorphic forms and that extends across the canonical extension as well now i take the symbol x1 through xn plus one that lives in k n plus one n plus one or if you prefer this what typical homology group and this we assume this lifts assume all these are assumptions assume this lifts to a class c in h n plus one motivic of x minus the fiber over zero q n plus one then we call phi tempered and finally we assume that our if you look at h n x u over u that contains our m as a one one sub variation so there exists some sub variation of that type there and we set v not equal to a lift of the associated higher normal function paired against mu and this automatically satisfies the following things class of c equals mu c is in hodge of i h1 p1 minus zero m n plus one and l v not equals c not is some nonzero constant and also nabla d of the regulator classes of xi gives mu on the nose and so what you can say here since mu satisfies l you can sort of say that r satisfies uh l d whereas v not satisfies d l so you sort of you have the perbenius duels the perbenius dual operators and that corresponds to dual extensions of the original variation pod structure okay so finally i'm ready for this table what does the table mean those are the 14 hyper geometric cases summarized where the 14 hyper geometric cases come from originally uh diran morgan classification so we have 14 hyper geometrics with mom at zero and a single conical point at one so basili will tell you a little bit about examples with no mom orphans but i won't um so the assumptions are m has type 1111 d equals one equals h h was the degree of the hodge line and they all have tempered models one of them singular jump with a four class c in chow four of x minus x not comma four just like above okay and now we have an order to this is where the magic happens there's an order to pull back that we're going to do where i take the coordinate on the upstairs p1 to be z hat that goes to z hat squared equals z and then i take upper star of m and call that m hat and the point is that you get cycles with values in m hat that you don't get with values in m this has degree d equals two but still h equals one because there's this mysterious fact that the deline canonical extension does not pull back under finite monodromes unless all the monodromes under finite base change unless the monodromes are all unibode in which they're not here so there's a second there's a second higher normal function in addition to the one here of the form lv infinity equals c infinity z hat which is c infinity square root of z so now i'm going back to the beginning of the talk with r equals two and looking at things that satisfy this inhomogeneous equation l of law equals z to the half okay now assuming the balance in hodge conjecture we should expect cycles as in the table okay so let's go through the table first you have the case of just finite monodromean infinity which is called 1a there are seven cases like that so the limit mixed hot structure type is pure the covariance at infinity are zero there aren't any so that means that you get an extra classical function and these have been found in many cases they should be k-naught cycles they've been found in many cases by yoannis valser together with some people morris and cruffle type 2b you have after you do some order base change something that looks like kind of full monodromean infinity so the co invariance there's always a half a half in the middle so if you do just the order two base change you get a covariance twisted by minus one of type q of minus three so you predict predict a equals three and a k2 cycle and so as far as i'm aware these have yet to be constructed but we sort of know what to do i'm going to concentrate on the last case because i'm running out of time uh type 2 again you don't have finite monodromean infinity but there are no co invariance because there's nothing there's no a half there and so the upshot is you again expect a classical normal function and yoannis has done these cases if i understand correctly um and then type 4 is the last one and in some ways it's the simplest but in some ways it's the most mysterious so you have this after you pull back a mum at infinity so mum at zero and a mum at infinity and a q of minus four at infinity as well as it's zero and so there should be two k4 cycles one emanating from zero with a singularity there one emanating from infinity with a singularity there and by the uniqueness result in all cases we have a formula for v infinity it's given by w2 right without even knowing if there's a cycle we can say this v infinity of z is w2 of z is c infinity squared of z some k greater than or equal to zero product j equals one before these pochheimer symbols half k plus four fourth um z to the k that seems of interest for interpreting results in open mirror symmetry yoannis and i have discussed this um but what about for the balance in conjectures for the balance in conjectures you can use the fact that dr of the cycle emanating from zero is equal to mu to compute for example this period of the regulator class this is going to be 2 pi i to the n log t uh plus some k greater than zero of constants and powers of the rock polynomial which will again be things like this just some slight variation there on and then you can find um epsilon 2 or epsilon 1 epsilon j is by Frobenius deformations pending an s to k and differentiating and setting s to zero but we really only know um we have a second like suppose we wanted to check the balance in conjecture one for k4 we have exactly one class where we have a case where we have two k4 classes and so there's one case where we can try to check balance in conjecture one right away and that is this case four um there are two tempered models let me just tell you one of them so upstairs we have one minus t hat phi hat equals zero where phi hat is product as i goes from one to four of xi plus xi inverse um that's well okay you can do minus it doesn't change anything and then there exists a birational involution i from x hat to x hat over i'm switching back and forth between z and t uh one is just an integer multiple of the other i don't want to go into details so here x one through x four goes to one plus x two over one minus x two minus uh one minus x one over one plus x one and so on we just do the same thing with the third and fourth coordinates um and you take curly z to be the pullback by this birational evolution of xi so the cycle imitating from infinity is just the pull the base the pullback by i of the cycle imitating from zero and then we can stay um using formulas like this um and uh from the fact that this is a base change of that you get also novela d rz is i of her star mu to some rational multiple equals z hat mu and so this tells you right away that two pi i minus three epsilon not rz is square root of t sum k greater than or equal to zero two k choose k to the fourth over k plus a half t to the k whereas two pi i to the minus three integral epsilon not r c is um given by this thing but with uh two k choose k to the fourth one in here okay so i'm out of time but let me just say the basic claim at four points namely z hat equals one over uh one a quarter a sixteenth and a sixty fourth um both cycles extend across um an integral model of xz and you need one of those four points um basically because you you end up in the integral model with the mum fiber appearing at some point and then you have a non-trivial residue for the cycle and at those four points z hat let's make sure I have the right values equals four for the plus or minus k yeah one two three and four not at one then um z and c in chow for x4 and balance and conjecture numerically checks out and that is the point for which vasily will take it thank you for one question all right go ahead yeah okay so um but that uh case chow for pi comma four uh the balance and conjecture should hold a variety of complex numbers over complex numbers that's right yeah but this is this is balance and conjecture one this is the right okay this is the special value of the l function yeah so yeah all right well I mean so we've constructed the two cycles yeah and now I'm claiming that more is true and actually you didn't need the cycles to check the balance and conjecture one that's the whole weird thing about this you could do everything with differential equations and iterated integrals and so on all right all right thank you okay so um let's take math again and there's um copy and retraction to the common rule and let's say we we could mean let's say i can't 45