 Okay, welcome everybody in Zoom and live here to our talk. Okay, Mathematical Physics Seminar, Physical Mathematics Seminar. We're very happy today to have Hossein Mofassati from IMPA who will tell us about modular and automorphic forms and beyond. Okay, thank you, Johannes. First of all, I am a mathematician. The whole my training was mathematics. It was a great luck to know Murad Johannes to learn mathematics coming from physics. So, well, I'm going to talk about a kind of, that's really a theorem, a point of language. This is kind of a new language. This language is about what in mathematics we have a lot of modular, we have many theories. Modular forms. There are, for example, elliptic ones, a Ziegel, Hilbert, Hilbert modular forms and more. There are a lot of TETA series is and so maybe, and then after the works in string theory, we have a Yukava couplings, and then topological string partition functions and so on. So, there are a lot of, these are explicit holomorphic functions. So, the basic idea, there is a language which you can develop for all these and you can go even beyond these classical cases and get new types of modular forms. And this language has to do with a kind of modular space, T that I will try to describe. And it's a language which must be developed, but if you want in mathematics that I give you immediate applications, I think that the first immediate application is that the differential equations of these objects are ill-studied in mathematics. And so you will, for free, you will get differential equations of all these objects. So, this application will be inside the mathematics itself. And so, the idea is that modular forms in number theory, they have been very useful. For example, Fermat lost theorem has to do with the existence of certain eigenform and so on. So, at some point you feel that you have to do this generalization just for the sake of completeness. Okay, so let's go after this introduction. So, let me talk about the basics of this language. So, first the language is algebra geometric. So, you will start modular space. And in a few minutes I will talk about examples. It will be clear. But at the end you will start with the modular space of 13 projective varieties. So, this will be is most projective variety. Eliptic curves, Calabio, three-fold, Calabio, and four-fold will be examples. But at the end the language you can develop in this generality. And what I am talking modular space, this means that I am interested on the deformation of x, all possible deformation complex structure, deformation x and so on. If I want to do this thing, right, exactly at the end it will be a Hilbert scheme, certain history divided by some algebraic group. But I want to keep this, let's say this physics treatment of modular spaces, not really going too much algebraic geometry. And basically what the intuition that I need from this modular space is that at the end when two projective varieties from this modular space you take, there will be no topological difference. So, this means that x1 as a C infinity manifold will be the same as x2. And this one, but not analytically or algebraically in the frame, this means that this x1, x2 as holomorphic complex manifold, they will be different. And the second one that all possible deformations, actually with the fixed polarization coming from the projective space, but let's write in this ambiguous format, of x is inside m. Okay, for example, modular space of elliptic curve. So any elliptic curve is a tori and that then this modular space will be the P1 minus infinity, so this one dimensional. But in a minute I will, okay, so, so the language that I want to a little bit explain, actually I will work our varieties, x1, x2, x2, x2, over a field, not necessarily complex numbers, but in many instances I will assume that characteristic of K equal to zero. So for this language at the beginning you don't need to work over complex numbers. Okay, so let's start. So, well, in mathematics, let's say 100 years ago, they wanted to understand the manifolds and they invented the cohomologies. In particular, so I want to know all these linear structures that I know from, I can get from this complex manifolds. One of them is, okay, so what is this? So, if you know classical Dironc homology, this is actually classical Dironc homology. This is actually classical Dironc homology. But after the works of Delin, let's say Grotendig and plus Delin, actually the contribution of Delin will come in the harsh filtration, was that you don't need, to define the Dironc homology of a projective variety of a field, you don't need really see infinity form like Diron. You can define it by polynomial objects defined over the field. So this is, let's say, it was important the observation by Grotendig after the works of Atia and Hajj. So anyway, the Dironc homologies are now k vector space. Okay, so let's review what any other structure that we have. So we have the coproduct, so HM2 going to HM. So everything is Diron, M1 alpha omega, alpha cop omega. In the C infinity context, the original one was this veg product of differential form, but when you do algebraically or our, for Czech homologies, usually you use cop, but basically they are the same. So you know this structure on this, this homology. What else, what else you have? You have a harsh filtration. So each homology comes with a filtration, fm and then fm plus one equal to zero. So maybe, well, if you put the k equal to c complex numbers, then this, as I said, this is the classical Dironc homology and the Hajj proved this Hajj decomposition, H0M. But he proved the Hajj decomposition by harmonic form, but harmonic form you need to see infinity functions. So something, maybe the observation of Deline, that okay, over arbitrary field, you cannot define Hajj decomposition. This is something that you cannot construct Hajj decomposition by means of polynomials over field. But if you define the Hajj filtration, what is the Hajj filtration? This piece is fm. These two pieces is fm minus one and so on. If you put these pieces together, you will get a filtration and actually the filtration itself will be defined over, you can define it over field. So this means that at the end these will be k-vector spaces. And the main reason for Grotendieg Deline was that they wanted to take everything to this big machinery of algebraic geometry, our arbitrary field, maybe our schemes and so on. So it is desirable to avoid c infinity and analysis, no? For example, for harmonic form, you have to do a lot of analysis. Grotendieg and Deline, they don't like so much analysis. Okay, so what else let's say we have? Therefore, okay, maybe I write, for example, the Hajj filtration satisfies this one. Maybe I will cop, I don't write complete, fghm2 is will be inside f plus g, hm1 plus mt, this h means the cohomology. But the last piece that we need is the polarization. So we start from projective varieties in a fixed pn. This pn is in some sense is fixed. So h the pn, this is one-dimensional. So you just take the generator of this guy and then you restrict it to x and you will get element h to the wrong x. And let's call this one the polarization. So we have attached to a projective variety. We have this structure. And I will write a proposition and actually this proposition will clearly say a difference between mathematician and physicist. Proposition, and actually this proposition went 2.4 in my book. Okay, let's say take two points in your modular space. As I said, these are topological, they are the same, so the analytic structure changes. Or if you want to take, let's say, families of these x varieties. And if you want, okay, even you assume that they are defined over the field k. So this cohomology ring, so this one, what we have constructed, we have constructed the hodge filtration, we have constructed the coproduct, we have constructed the polarization. This structure, even so it is defined in algebraic framework, it doesn't distinguish the two varieties from each other. What it means that at the end this is isomorphic in a minute I will explain to the order structure. So cohomology ring of x0, the hodge filtration, the coproduct here, and maybe the polarization I denoted by theta0. So what it means, this means that there is an isomorphism of k vector spaces from here to here. It respects the grading this m corresponding m goes to the corresponding m. It will respect the hodge filtration, it will respect the coproduct, and it will take the polarization to the polarization. So the moral of the story is that even so we construct all this thing over arbitrary field, it is a kind of topological data, okay? It doesn't distinguish x from x0. Sometimes when we want to say that x is notomorphic to x0, we try to find some invariant of x which is different from that invariant of x0. So this means that this will not do the job for us. But, and actually this is the math, I will need this one to define the modular space. Okay. Maybe I say something about the, not really the proof of the observation. I had written my book which was more than 250 page books, but at the end I realized that there was no rigorous proof for this fact. And actually after many email contacts with Deline, he writes a rigorous proof, but he refers to a lot of SGA, Semine, Geometrial Gebrick, and then you know that there is SGA, FGA, there are a lot of subdivisions and so on. And really myself, I have not looked at proof because it goes through all these machineries of the grotendic and so on. But I started to develop the theory assuming this one because at that time I was like a physicist. I mean, this is trivial. I mean, I don't need to prove it. I have to develop whatever I want to develop. So, but the proof in this generality is not trivial. And of course if you want to work over arbitrary fields and so on, so if you want we can try, but what I am saying, maybe since I started to collaborate with physicists, I didn't care about some basic facts. Sorry though, may I interrupt? I mean, do you need M to be connected for this one? No, exactly. I mean the best one, the M connected. Yeah, exactly. Oh, but you prove this in a differential way? You prove this in... No, actually then in order to avoid the modular space, you have to take families. Because if you prove it, I mean, if you have a connection, the connection gives you a kind of isomorphism. Aha, now in a minute I will take. No, no, okay, just a minute. So, okay. The proof of our arbitrary field goes through K. But let's say, K equal to say, we have a very old theorem, actually it has to do with the Ereshman, but at the end H star XZ is the cop product here also, and also the polarization, the polarization also it is defined as Z integers. This object is the same as X0, cop and theta zero of the X0. This is almost the same thing that the topological between X and X0 are C infinity isomorphic, but the point is that, okay, this cop and theta, when you complexify it, it will be the same as this cop and theta here, but the Hatch filtration is missing here. And if you take the isomorphism here, it will not give the isomorphism in the level of Hatch filtration. So, this piece of Hatch filtration in this topology, I think you were trying to say this... And this has to do with the monodrome. If you have a flat connection, you have to take the flat sections and then you get this isomorphism, but let me say in this way, and actually how you get this one, then you have your modular... Well, let's, instead of modular space, maybe I just take a parameter space and then forget these difficulties of constructing modular space. But anyway, if you have M, you have one point X, you have another point X0, you just connect by a pass, and actually this H, depending on this pass will be unique, and this is the isomorphism you want. But from here, actually my wrong argument is starting from here and a little bit try to massage it and to get the isomorphism here was not really working. So then at the end of the day, you have to be careful that you are working over... This is the isomorphism K vector spaces. And then anyway, this topological picture will not precisely will go through the statement here. Johannes, it's okay. Yeah, but I thought you had the connection also under the Rammcommology, right? Yeah, I have the connection of the Rammcommology also. That's the same as the one when you look at... When you're complexifying, you look at the integers with the integers of the flat section. But these are the morphisms using the connection how you want to get. You want to get as flat section, taking the flat section, no? For example, yeah. But flat sections are not compatible with the Hodge-Felcher. Exactly. I mean, you have the Griffith transversality, so some other... I'm really surprised by this isomorphism because the Griffith transversality says that... Okay. What I can say, these things at the end, in the case of modular form, will produce for me SL2Z, whereas these things will produce an algebraic group of this form. Okay, my name is... an upper triangular. If you want, in the case of elliptic curve, I can say. So the type of the isomorphism is different. Since this, in the case of elliptic curve, this respect to Hodge-Felcher, this will give two-by-two upper triangular matrices, but this guy will give me the whole SL2Z. Anyway, if you want later, if you want, we can discuss and to see if you can give me a proof. What I am saying, that the only rigorous proof that I have apparently it goes through this SGA and so on, and I have no time really to invest my... Well, I can do, but anyway. Like a physicist, I set the goal, and sometimes I make a lot of ansatz that they must be trivial. And then after reaching the goal, and then maybe I will be back and try to make everything rigorous. Okay, so maybe let's... Actually, maybe to have... When I started? 4.15? Okay. Later, no? Okay. Anyway, so the modular space that I want to say that it is... contains... Using this, you can produce a lot of modular form theory in mathematics. This is the modular... Modular of X. So X itself will be very in this M, but you enhance it with alpha. With a fixed... So fix... X0, so you fix one point in your modular space, maybe the best... I mean, if you are a physicist, sometimes you take a variety, you call it Landa Ginsburg, or whatever, apparently it is the most... you fix that one. And then you... So this is the modular of X together with isomorphism, alpha. Alpha is from the cosmology in your variety to a fixed comolar ring of X0. And... Anyway, you kill... Well, maybe I have to say that you can also try to do fixed-touch structures. You can start adding more structure. For example, if you want to get modular form for congruence group, you have to start adding torsion structure. But anyway, this enhancement is the basic one that I want to speak, but of course you can put more data in order to get other classical objects. Okay. I'm a little bit scared to put this as conjecture, but even in the case of, let's say, Calabiao 3-fold that I worked out with you, Murad, still it is a conjecture because we treated modular spaces like a physicist, not like a mathematician. But... T... Maybe I put the most... D, A... Quasi... Affine... Variety. And what did... Well, at least... Or if you don't want to start from this general, at least for Calabiao 3-fold, for which we have a lot of computation which supports this one. And let me say why... So, when it is something quasi-affine, this means that you have some affine variety. Affine variety is given by what? It is given by... So maybe start... Let's... Okay. So it will be some polynomial ring, T1, T2, Tk. There must be some polynomial ring divided by some ideal. I mean, the zeros of this ideal are your affine variety in... in a T1, Tk space. And then this means that T must be some... open. Basically, basically what this... from this... geometric term that is zero, locus of some ideal in T1, T2, Tk, in Ck... Let's write down the... So if you want to let's... even let's say the complex language, because really I will not go to Arithmetic application, so in C to the K, given by... Okay. The zero locus of this ideal, maybe minus some... some sub... Okay, maybe let's... Actually, this open is a rescue open. Anyway, classical modular space in general in algebraic geometry, they are not affine. But... But there is this... even this intuition from algebraic geometry, when you make enhancement of your object, always the things... the modular spaces becomes better and better. And in that... in this case, when you add alpha, when your modular space suddenly will be affine, this means that you can find this polynomial ring, this ideal, and so on. And... Okay. So maybe from now on, in order not to make the confusion, let's put K equal to C, and then let's proceed with... with K equal to C. Okay. And actually this polynomial ring that will contains this algebra of modular forms, algebra of Ziegler's modular form, Hilbert's modular form, this... this algebra of topological string partition functions attached to a Kalabiya variety, and so on. So... this is... So this will be interpreted at the end as modular forms. Okay. Maybe I make the summary of... of, let's say, articles and results supporting this fact that this is affine, or in many cases, quasi-affine, and this algebra is really the algebra of modular or quasi-modular form. So... let's say a case of elliptic curve. In this case, your variety is just a complex story. And... when you work... maybe for this one, actually in 2005, we started to do this one after works of griffices. But basically, there are some ideas in Nick Katz's article. Some ideas in the Cytos article. It's called the Cytos perimeter form. But at the end, the modular space is not there. But then... maybe I... in 2012, I have lecture notes describing how one gets quasi-modular form theory from this one. Then this T is actually the spec Ct1, Ct2, Ct3, this guy. T... T3 to the 2 minus T2 to the 3. So... So, I mean, this is the full ring generated by 3 elements. So you see that this is of dimension 3. So classical modular space of elliptic curve is of dimension 1. So this enhancement will give two more freedom for your object. And, well, this game I have written lecture notes of almost 100 pages explaining that this will give the theory of Kaneco-Zagir theory of quasi-modular forms. And again, this T1 will be interpreted as the IZN series E2, T2 as IZN series E4 and so on. So I don't want to go really into detail, but there is some machinery to develop this geometric language into quasi-modular form theory of Kaneco-Zagir. And maybe if returning back to this comments of Johannes, if you, in the case of elliptic curve, you look this isomorphism, this one will give you, all this isomorphism will give you SL2Z, whereas this isomorphism will group your triangular matrices. So, at least in the case of elliptic curve, it shows that they are too distinct isomorphism. Okay, sorry, this isomorphism says that some of the hot numbers are the same. But the point is that hot numbers usually are transcendentally, but by some miracle, they are topological invariant. It tells you basically the dimensions of these quotients. Yeah, but the dimensions are the same. It is not enough to make this argument. Just the numbers. That's what you have to prove. Okay, so I mean, it's a general theory going on, maybe just I write a summary of the results. Maybe let's say 13 lattice polarized K3 surfaces under open Calabio varieties. This is Alim and many of his few of his students. They compute this modular space. They get some algebra that then they interpret. They find the this is algebraic. They transcendental interpretation of all these TIs as transcendental objects like modular forms and so on. And as I said at the beginning, I will just mention the summary of which person where has done some works. And Calabio three-fold again Alim Murat me Yao Emanuel Shahid. So maybe at least okay, so for Calabio three-fold again I don't want to go too much into detail but but let me say this there is this let me just explain the case of this topological extreme partition function. So we can find object let's let's call topology algebraic topological extreme partition function. So and it is possible to make the precise translation of this one into the same object computed in physics so what I am saying a lot of nice functions which comes from mathematical physics, they can be interpreted as functions. So this let me say this will be interpreted as interpreted as a global function in this T-modulized space. So you have to fix the topological type of Calabio variety and then you consider your modular space classical modular space you construct this bigger modular space and then you can interpret these things maybe in the case of in order to so the general theory of Calabio three-fold it is written in this article but let's say in the case of there is a very special famous Calabio three-fold called mirror quintic Calabio three-fold so for example this modular space of this mirror quintic Calabio three-fold is of dimension one but once you start computing this this modular space for this mirror quintic this will be suddenly seven-dimensional then you start computing coordinates T1 T6 and as usual there will be some discriminant that doesn't matter what it is let me write so it will be of dimension seven and and then for example let me try to write for example FG FG then this FG algebraic will be of this format it will be QG T0 so this is T let's call this one T T4 minus T0 to the 5 let me just write it down a structure theorem just and then T5, 3G minus 3 and you will see that this topological string function becomes a kind of rational function and actually you know QG homogeneous of degree what it is 69G minus 1 and then you have to give some weight for your TI's degree of TI is 3I plus 1 and this is I equal to 0, 1 until 4 and maybe just a degree T5 the weight of T5 is 11 the weight of T6 is 8 anyway what it will be clear you will see that FG is a homogeneous polynomial of 13 degree divided by this one so in order to discover to compute FG you need really find that number of coefficients in QG to compute FG experts in modular form why modular forms are useful but most of the time you need just in order to compute a modular form you need to compute a finite number of coefficients because this finite generatedness of modular forms and so on if you have not seen this one, two, three of modular forms of Zagir is there are a lot of examples of this kind and at the end of the theory of Calabiomodular Form attached to Meere-Quintic but unfortunately I don't have this miraculous applications of modular forms in this context just these things which comes from physics and okay maybe let's say there are other works also maybe I don't mention it but I'm not writing explicit explicit data but for this, just for this one after my work with Muradio and Shai-Dige I wrote there is a book for this one called Gauss Manning Connection in Distance Calabiomodular Forms anyway and there are many other ones and okay, this is a theory that is going on so the idea is that you can combine you classical object of mathematics with objects coming from physics and actually in the case of objects coming from physics you can say that what is the corresponding modular form theory what is the corresponding algebra of functions that your objects leave there and in some sense this structure in the physics literature it is hidden in Yamaguchi Yao article which at the end without this language they observe that this FG has polynomial structure of course they make this other variables such that the things becomes a polynomial but anyway I have written more more examples the point is that well when you want to sell your product in mathematics you have to prove theorems and you cannot just say that these are nice theories going on and I want to just a little bit change the topic but before changing let me say so at least in this case algebraic object something very important also most of the time for some object for example like FG you have three different incarnations for example now I am talking about this algebraic one there is a holomorphic holomorphic so there are precise dictionaries translating algebraic to holomorphic object and also holomorphic plus anti holomorphic even in the case of modular form you have some many some C infinity modular forms and there is some correspondence between the Cnity and holomorphic one in the topological string partition starting from you will see that this FG has two different incarnations holomorphic and something which is mixed holomorphic and then holomorphic and there is a dictionary between this one and this one and what I am saying that this third object algebraic also there is some precise dictionary to pass from one to another gromo-fitting invariants are hidden here I am not really familiar with these things in the physics because apparently in the Bershachi, Ogrivafa they first compute this one and then they get this one and then they get the gromo-fitting invariant but anyway what I am saying that these arrows are precise dictionaries of to pass from to another okay okay this is some theory going on and and okay I think more 20 minutes at least 20 minutes two blackboards also the good things that I have been in Germany before that I know this system otherwise when I was in Max Planck the foreigners comes and they make a mess okay so since really this is a kind of general language to be developed until it reach some some nice applications maybe either in physics or in mathematics I started to a little bit related these things into something else and what I want to say next that in this modular space the concept of Hodge-Losai okay there is this concept of Hodge-Losai in algebraic geometry and the best place to see this object is this bigger space not classical not classical modular space and I will try to explain this one this Hodge-Losai one in just one example but this is just for any variety and okay anyway the modular form part I will not talk more so the rest of it I want to say that these modular spaces are good to study something called Hodge-Losai and at the end let's say maybe the Hodge conjecture itself so in the case of product of elliptic curve let's say Calabio one fold times Calabio one fold and okay so let's say my X will be one elliptic curve product with so the theory is in general but I took this example just to as a kind of preliminary to understand the whole general picture and in this in this case this modular space so there will be modular space of coming from E1 E2 then this will be P1 minus infinity times P1 minus infinity and that then this infinity has to do with the generation of elliptic curve maybe for just not to make my notation long just let's compactify it let's call it P1 over P1 so what is Hodge-Losai in this specific context this is very simple let me write EG1 so so this means that if I take G1 so let's say so this point here there is a J invariant of elliptic curve so let's say G1 J1, J2 here so this will be J1 J2 in my modular space I say P1 and P1 such that AJ1 is isogenous to EAJ2 this means that there will be isogenous of degree degree degree N this means that there is a I mean this is the best possible this is that there is an N to one map holomorphic algebraic respecting the group structure and all the beautiful things that you might imagine from EG1 to EJ2 this is holomorph this is some algebraic structure between EG1 and EG2 if you take G1 and G2 generic there will be no such map but at the end maybe I write other versions other versions of this the other version that EG1 times has a non trivial curve so when you take the product of two elliptic curve there are two trivial curves you take the coordinate constant so EG1 times points will be a curve point times EG2 will be a curve so so the condition is that there will be some curve inside which projects to each coordinates non trivially, non-constantly and if you want I want to write a version which has to do with the Hajj Loci and actually it will be this one so it will be okay so the non-trivial and if I want to write it so this is a calabi-1 fold it has a 1-0 form you take the product you take this this 1-0 form here with 1-0 form here and the integration of this for over some cycle over some cycle in H2 E1 EJ1 EJ2 Z so the statement is that okay you choose this omega 0 EJ1 and omega 1-0 in EJ2 make the tensor product so some integration of this guy over some cycle in the product must be 0 and actually the translation from here to here is that you take the topological class of this curve inside your product and well if yeah maybe maybe just in this way so and actually this is the concept of let's say this Hajj Loci is that certain piece of the Hajj filtration of this guy over some cycle must be 0 but maybe I write some well known fact this is the union so you see that at least in the first version there is an N going on this union will be over N in the other versions you don't see this N explicitly union of merable set of highly singular curves inside inside p1 times p1 this this union rebel you will see it if you for each N actually there will be one reducible curve and actually let me write the things are not so much mysterious in a variable let's say call this y0N or maybe here in a variable set of highly singular curves curves let's say a curve I call it and and actually these curves are algebraic curves they are called well singular models of modular curves so y0N this is by rational so after resolution of singularities this will be gamma N co-sciented by H H will be the upper half plane imaginary tau bigger than 0 and gamma N will be all a, b's this modular group L, a, b, c, d a, b a, b this is in sl2z the subgroup of sl2z such that this c is congruent to 0 mod N anyway this in this very simple example the concept of the heart loci is the union of singular models of modular curves in p1 times p1 and if you look the literature there are very in number theory they are very well studied for example Fermat-Lost theorem has been the consequence of this fact that these guys are defined over q and every elliptic curve modular this means that every elliptic curve must appear in the Jacobian of this guy and so on so there are a lot of history going on here but let me try to say maybe just more 10 minutes or okay so maybe so if you start doing this bigger modular space in this case okay you have two elliptic curves I said that the case of elliptic curve is modularity of dimension 3 if you take 2 the dimension v6 but for some geometric reason that I am not interested in to do modular form theory I am interested to study these curves you will have some results like this and then I will finish the talk in this case so this is different from a kind of modification of the modular space that I wrote but anyway you define modularity of so e1 two elliptic curves pairs of elliptic curves and then alpha so in your modular space now e1 and e2 can move also and alpha what it is alpha is as before alpha is the isomorphism between and in the case of elliptic curve it can be simplified into h1 but lets me the structure and then h star e2 so in the previous version I fixed one variety and I took a modular space of one variety together with alpha in this version I take two elliptic curve and the modularity of two elliptic curves and I make the enhancement together with alpha between these and then I make the homologering so maybe I just write down the kind of theorem well it is published just this year and so let's just try to explain so this this modular space is fine maybe right over again x2 x3 it is of dimension 4 one can write down explicitly coordinates over this modular space so I have to take this this what was it discriminant 27x 3 to the 2 minus x2 to the 3 and the same for 27y y3 to the 2 minus y so it is this affine variety c4 basically c4 minus delta1 equal to 0 union with delta2 equal to 0 and these are this is delta1 and this one delta2 is this one and so the vector field so this is let's say the first part so I didn't mention I didn't really define the concept of hajlosai here but you will see in the second part that the only the only algebraic maybe I write algebraic solutions out so there is explicit vector field and this explicit vector field maybe I write it this is because this is 2x3 okay maybe I might write that this is published in Moscow mathematical journal maybe just I write a few it is explicit degree 3 6x2 actually no 2 3 minus 1 6x2 minus y2 x2 okay I will not write but anyway there is explicit vector field in this modular space so there will be tdx3 explicit again y2 y3 and only algebraic the only algebraic solutions of v r s0n so this guy has innumerable set of algebraic solutions and actually these guys are bi-holomorphic to to this modular curve that I wrote somewhere there I don't have really time to define these things as hajlosai in this bigger modular space but anyway so the model of usually you when you how many of you have done a course in ordinary differential equations I don't know so when you do a course in ordinary differential equations usually for example in ordinary differential equations there are vector fields and then you start saying that okay there is a vector field but there are solutions but the solutions are vector fields are usually transcendental curves not algebraic curves for example when you look y2x0 equal to x0 this is an algebraic curve this is a cost the picture is like this or if you make the formation the picture will be like this these are algebraic curves but if you look for the solutions of vector fields still you have local holomorphic objects curves but they are not necessarily algebraic and the theorem says that the only it has innumerable set of algebraic curves which are solutions of this vector fields and actually these are all modular curves that I have in the previous version in the previous version that in p1 times p1 this is just the jungle of curves that you don't see any structure but what I am saying is that in this bigger modular space which is of dimension 4 they are characterized just by one vector field this vector field defines them quickly at some point if I have time I will try to give some more arithmetic applications to this one but this is something that it is not well known for to number theorists and maybe just two minutes and then I will finish maybe just by hand and actually if you want to to know what are why the Eisenstein series for example E2 E4 E6 the classical Eisenstein series they are holomorphic function from upper half plane to C they have explicit formulas and actually you can characterize these curves also using Eisenstein series maybe I say S0 N will be the image of this kind of map and then I finish so my mode life space C4 minus discriminant and then you construct your map basically by Eisenstein series up to some constant again up to some constant this is tau going to something like this E2 oh sorry E4 tau it is E2 again there are some constant that I am not caring but really I have to care otherwise this minus N E2 to the N tau again I will just write one coordinate but anyway what I want to say here is to the to you can construct the transcendentalities curves using Eisenstein series and using this combination of this E2 Eisenstein series and so on but the the point is that even so this will be the kind of holomorphic description of these curves but these curves are defined over Q this means that you can make mode B and this relation with Fermat-Lass theorem actually starts from this fact that this curve this modular curve is defined over Q it has some nice model over Z you can make and so on anyway I wrote this one because as I said for this big big theory going on I cannot claim that I can surprise somebody this is just some theory going on but at least when you look for this kind of geometric applications something a single vector field polynomial vector field of degree 2 becomes responsible for all modular curves embedded in dimension 4 and at least at some point in the future I hope that I find more applications of this particular example in the case of let's say modularity theorem and by the way I can do this for any Hodge-Losai and maybe again in the case of general Hodge-Losai there are consequences of Hodge-Conjecture saying that Hodge-Losai must be algebraic defined over a certain field and so on these consequences of the Hodge-Conjecture are as difficult as the Hodge-Conjecture itself and at the end I hope that this study might help also in this direction but I think this example clearly shows me why I want to study Hodge-Losai in this bigger modular space in a more modular space I think I finish Thank you very much this is an inspiring talk so we have time for questions in Zoom or here I have a question I mean the last one so the N is an integration constant for this or how does N appear you have this vector field and the vector field is independent of N what I am saying maybe just imagine with this vector field let's say there are two ODE versions of the vector fields when you take a course in ODE you will say that there are solutions and so on okay but sometimes algebraic objects algebraic curves also might be the solutions of some ODE what I am claiming here that the only algebraic solutions of this ODE you can look any vector field as an ordinary differential equation are just this algebraic curve which is isomorphic to the modular curve for all N together you will get again a zoo of curves zoo of algebraic curves and these algebraic curves apart from these algebraic curves any other solutions of this vector field is a transcendental curve not algebraic it does not appear I mean is N somehow so you have a family of solutions and then it is polynomial when N is integer okay maybe again I explain the picture here maybe N must be integration constant for this differential equation exactly when you have ODE you get initial condition and sometimes for example you take initial condition and then you get the solution and then initial condition here solution okay these Ns are discreet set of initial conditions for your differential equation so generic initial condition will give you a transcendental curve just some innumerable set of initial conditions will give you algebraic curves and what happened to the X1 in this you had a T1 up there what did X1 go? oh okay so as I said if you start from the same modular space that you I told at the beginning in the case of elliptic curve this is classical elliptic curve this modular space of dimension 3 so you take two modular you take two elliptic curves so it will be two copies of enhanced elliptic curve so you will get some six dimensional okay but the concept of Hajlosa in that content that I get they are two dimensional sub varieties and six dimensional objects and after that I realized that there is some algebraic action of algebraic group going on I took the quotients of the six dimensional space with some two dimensional algebraic variety I got four dimensional object and everything from six dimension descended to this four dimension and but this Izaneshtan series E2 well if you this algebraic curve X1 and Y1 they have disappeared but if you write down the parameterization of this algebraic curve you will see that at the end Izaneshtan series E2 will appear which correspond to X1 and the Y1 will appear in the denominator in this format and but but originally I wrote everything first in this six dimensional space for example at some point I would be one of the things that I tried a lot and I failed these things of Izaneshtan and so on has to do with the HECU operators acting on the algebra of modular firms at some point I tried to do the same thing in the case of mirror quintic there are for example take a mirror quintic times mirror quintic or Calabria 3-fold times Calabria 3-fold is there any interesting algebraic cycle in this case in the case of elliptic curves I know sometimes there is and it's called the graph of the isogenic but in the case of Calabria 3-fold I mean this question related to algebraic cycles is complicated I cannot immediately reproduce the same things in this case because I don't have a concept of isogenic and well you have all these equivalences some kind of correspondences and so on and in this case it will be also conjectural because they have to do with most of the time I work with hutch cycle and then at the end if there is something algebraic there must be I have to assume that the hutch conjecture is true and so on and I would love to do something similar in the case of mirror quintic but the things are not as easy as expected questions would you draw the line between modular forms and automorphic forms can you say in this perspective what is the line between the two I think the difference well when you try to generalize modular forms well what is the classical generalization you take Hermitian symmetric domains and then then automorphic factor and then you forget the name modular and then the names become an automorphic form and actually I wrote an article with do you think those for the kalabia this is like modular forms or are these already automorphic forms no this is the point there is none of them there is no Hermitian symmetric domain in this case this is the point why I am saying that these theories are parallel to classical mathematical theories for example for a long time Zagir tried to relate this yukova coupling this topological string partition function to classical modular forms and as far as I know he failed there is no relation so what I am claiming that there are some parallel theories going on not necessarily you can write yukova coupling of mere quintics in terms of classical modular form so you have to develop the corresponding modular form theory and I give it geometric language maybe I mean classical people in modular forms will not like it because it is too much hutch theory language and worse than that is that I don't give you Hermitian symmetric domain and the people in Langland's program will like to have Hermitian symmetric or some Lee group with some maximal compact group to start automorphic form theory from this point of view but this generalization does not start from there it starts from let's say kind of this modular space T and yeah any other questions from zoom could you hear us from the zoom maybe we get some feedback how was the experience this is the first time you are transmitting not quite but apparently nobody is hearing us even are they even there well let's just think more same again okay thank you