 ... and we asked Professor Tomasi Lekeče. So, good morning. Let us recall where we got yesterday. So, yesterday we got to the fact that if we are considering any grocery projectiv variety, but in our case we are working with our favorite modular spaces of cars, In tudi da je tudi možno vse, bo tudi možno vsi in bali, kde je tudi in tudi vse počke. Vse je, na razne, objez, mgn. Zato pa to je vse malo, ali bo, da je vse vse, kot nekaj prideli vse. Na vse deprav. In spoletite, to je srednji, pa je, na kaj je vse. tako začelj, tako tudi miselj izgleda je načinja, da je čutko z materijali izgleda. Zato, da je vse skupaj, v komodiji, v komodiji z kompatijstvom, je, da bi smo izgledali, da imamo načinje vizitive, vzvečovati, v valijo v glosniji skupaj z zelo hožnju. Tudi razmah vzvečenih hožnju. Znamo na notacijo v jasnjih. Vse v mgnju sem zelo v mgnju, ker sem izgleda ozvrčenje. Vse ne jazno ne zelo negativne. Filtacijo v W0. Vse nezelo, da kvosjenje izgleda so vse, je nezelo, da je nezelo, da je nezelo. Zelo vzelo, da je zelo, da se vse vzelo, v orchidovin za glasben iz boaca. V neko nožek mi kisses structure razbitne in begomobna realize. Vsebe moh v nič nič je tudi igrat, danes to nem chacun je poundingizir, the extension of pure-hod structure we only need to take pure-hod structure here. So the generators are the equivalence classes, the isomotheum classes of pure-hod structures. If you want to need to define a relation, so every time we have a long hexa-sequence respecting the mixed-hods structure, the short hexa sequence, Then this gives that the class of B should be equal to the class of A plus the class of C. So in this case this means that, here this will correspond to the sum of the class of W0, that carries a pure hot structure of weight 0 plus the class of the quotient of W1 by W0. le chances of the first graded. It has weight 1. An important thing is that this since every time we have an inclusion, of a closed subvariety, into the whole variety, we get a long exact sequence of informalia in combo support with Haunt weight 5X, which is a closed subvariety of sum Y o komplemenju, kaj je bilo. V open sub set. We have a long exact sequence, mapping the chronology with comback support of u into the chronology of comback support of the big space and then the chronology becomes the support of x. And then this being comback support of the degree I guess should go up. This means because of the fact that every time we have a short exact sequences, we have additivity, this means that if we take the alternating sum of such classes for different values of k, we get an additive invariant. So the idea is that if we take for any variety x, we take its homology with rational coefficients, then we take the class in the growth in the group and we take the alternating sum. I don't care k, maybe navigative and then everything is zero. So this will give some kind of oil characteristic with values in the growth in the group. So in particular using this long exact sequence we find in this kind of situation, I mean even if y is quasi-projective and x is a close variety, so it's not compact in itself, we will find in this kind of situation that this whole oil characteristic see if you take the one of y then it will be equal to the sum of the one of x and u, I see the time for getting all the small c in the notation but I want to have that. And this kind of oil characteristics exist in very many forms in the literature because its existence is a straightforward consequence of the fact that we have mixed structures, I see a question. No, we don't need. You see this long exact sequence is always there, we can reinterpret this piece as a relative homology in this case. So this always works. The problem is that if x is not smooth and you want to have something about homology, you are not allowed to use pioncal ideology to deduce it. So you could write in the spirit of homology but then the third time it will not be exactly this. So perhaps I will call for today hodge oiler characteristics. So how much information do we lose if we pass from the homology of something to this hodge oiler characteristic. Well, if we are able to reconstruct in a unique way from the weight structure, the degree from which it comes from, then of course we have not lost anything. So for instance if we take mg and bar, then we know that the weight will be equal to the degree. So if we simply know this hodge oiler characteristic for mg and bar, then from each class we simply look at the weight of the corresponding hodge structure which is part of the data. From each block we know exactly from which degree it comes. So the information is just the same. At least if you are just interested in structures graded vector space with hodge structures. On the other hand in general you can't do this. For instance for mg and the new invariant will be weaker. Mainly two issues. And we can re-phrase this duely for core module complex support. But let's say we fix a degree k, then we know that the ways that occur are a priori between k and 2k. So anything which we find with the serf and hodge weight in this description can come if we find in something of weight w then it will be somewhere between w divided by 2 but there is no way where it comes from. And this is actually the harmless part of the problem. Yeah, we find some kind of indication of course about the parity from which it is coming from because we have signs but that's everything. But the worst problem is of course that here we are taking differences of classes so there may be cancellations because the fact that taking additive invariant is hiding the fact that we are sort of taking some long exact sequences of which we have not computed the differential. So if the differential have some kind of maximal possible rank then we have more or less the same thing but perhaps there is something non-trivial which we simply can't see because it occurs once in odd degree and once in even degree. If we want to investigate the relationship of the open part and the homology of the compactified modular space we know that when we try to construct or strata we need to identify mark points in some way. If you remember we quotient by the action of the automotive group of the graph. So somehow we need to define this invariant a little bit to keep stock of what the symmetric group is doing. Because if you remember mixed host structures are functorial so this means that if you have an action of the group this will respect the weight filtration so the idea is that the SN action on MgN and M bar gN respects the mixed host structures homology and on homology with compact support so the idea is that we can take any graded part and subdivide it according to the structure as representation so the symmetric group any graded piece in the filtration then we can decompose as direct sums of representations of irreducible representation of the symmetric group. If you remember they are indexed by sure polynomials so we can use them to distinguish spaces so somehow we can take the order characteristic but it's more clever to take the SN equivalent order characteristics in which we are taking values in the growth in the group of host structures with the structure of SN representation and equivalent order characteristic is then the characteristics the order characteristic of the homology with compact support on a space with SN action so in our case it would be either MgN or MgN bar with values in the SN equivalent group of host structures. Structures over here here I am writing SN to remind myself of original host structures endowed with the structure as representation of the symmetric group. I am actually describing how this thing works is more complicated than giving examples so let's think for instance about N04 so if you remember this is just the model of four points on P1 so we can fix the first three to be 0, 1 and infinity so this is just P1 with three points removed so this space is irreducizible so H0 is one dimensional and if you use perpendicularity this means that H with compact support is also one dimensional and if you want to keep pack of host structures then it will carry yes excuse me in additive invariant because somehow it is just the same invariant we had before just restricted to each representation it comes from the fact that if you have a short exact sequence as you had before then taking the quotient by any subgroup of the symmetric group will preserve the short exact sequence and this is one dimensional you know that there are not so many representations of rank one actually just the trivial and the alternating one and this is clearly a case in which we have the trivial representation so this is S4 invariant H1 or H1 with compact support in this case they have not the same thing what I wanted to say is if we compute H1 with compact support then we get something which has dimension 2 because one of the three points kills the class of the point in P1 so we get something of rank 2 left and the idea is that S4 acts according to the only irreducible representation it has of rank 2 this is the one determined by the partition 2 if you prefer you can write it this way if you prefer to have the young diagrams so what is there is for equivalent order characteristic of the of m04 where we need to take the class of everything so we need to take the class of q minus 1 and then usually one denotes the class of the representation by the corresponding true polynomial and multiplies it and then the class of the thevial of such is just 1 multiplied by S22 and now we are making a mistake because this is of course H1 so we don't have to add we should subtract this is the so this is my notation for the notation associated to the partition 2, 2 of 4 and this is my notation for the true polynomial and actually this is the standard notation for the fact for the for the tight-hold structure of word 2 yes, sorry yes, absolutely because I mean I am removing I mean they are morally I mean when I write at least as pure-hold structures in some cases I can also decompose the pure-hold structures of other pure-hold structures and the point is that yes the part of the data of the pure-hold structure where you the weight is reconstructed taking the sum of p and q for any so yes this is intrinsically yes so usually L is used as a notation for the class of the tight-hold structure of word 2 and since one can endow the growth in this group with a monomer structure in the sense that ideas that if one takes the tensile product of two-hold structures it's the same as this gives a product in the growth in this group then if we take the class of any other tight-hold structure it will become a power of L so the idea is that we can define the product of the classes of two-hold structures as the class of the tensile product yes yes I mean it's just one incarnation because you know if you have a motive class L then I mean in general motives can be specialized to different theories and this is the incarnation so it is the same L and it was defined usually as the class of A1 if you look at the common you find only this so we know that we can produce mg and bar by multiplying together mg prime and prime and then dividing with automorphism group of a graph and this means that mg and bar is somehow dominated by the structure of the graphs once you know what happens for mg and so this is related to the fact to the structure of modular operas which was studied by Gessler and Kaplanov in the 90s paper appeared much later in 1998 so the idea is that mg and bar is constructed from mg prime and prime taking a set of values by gluing them together according to graphs and this can be rephrased formally by saying that mg and bar in some sense is the free modular operas generated by mg prime and prime Well I have not defined for you what a modular operas is but somehow you only need this example but anyway operas is a topological concept I think originally coming from topological applications and from applications studying the deformation theory and Gessler and Kaplanov used this concept to give a very compact description of the relationship between the Hodge Euler characteristic of m bar g n and the equivalent Hodge Euler characteristics of the open modular spaces occurring in the boundary so the idea is that we would like to study the Hodge Euler characteristic of m bar g n by keeping track of the structural SN representations as rational Hodge structures so if we want to encode everything in a generating function we need some kind of dummy variable that keeps track of the genus we don't need to keep the track of the number of points because the SN action already tells us that but since we are going to glue together products of components somehow it could be more natural to work with disconnected curves so curves which are the whose connected components are stable curves so this is just all possible and so the idea is that we want to obtain this from the same thing but for the open state so the first thing is that if we want to glue together things so here we are just starting with the data about all smooth curves but then we want first to produce all possible normalizations of stable curves and they are going to be disconnected curves so we need to produce from this formula the corresponding formula for the generating series for the whole order characteristics for spaces of disconnected curves there is actually an operator called EXP that transforms this generating series in the corresponding one for curves which are smooth and of the correct arithmetic genus but possibly disconnected so up to here we are controlling the normalization the possible normalizations of elements here where we fix the genus we just have a finite number of possibilities so I don't know it does not matter so much what the meaning of the construction is so somehow up to here we can control the possible normalizations of the curves so they will work in this way I do think that one can make sense of this but it's not the yeah but if they algebraic anyway the total number of components is going to be finite are you saying that in the mark points will still be there so yes but they are not smooth so I don't I mean with mark points so I don't oh yes so the idea is up to here we are we are dealing with the possible normalizations of curves and then I mean this EXP is some generalization of the usual exponential and it's kind of operation that is often defined when it is working with regenerated series the problem is that I have not defined for you the I mean it's so if you remember how EXP is defined in general you need to take something like a power of a variable and then divide by the by the factorial but instead of taking the power of the variable you need somehow to take the class of the symmetric decade symmetric product of the classes you are taking so this is what morally is happening instead of taking variable and then it takes its kate power to define the exponential you need to take the symmetric product and this has something to do with the fact that if you have disconnected curves of the two of the same genus then you get some if there are no further data you can exchange them so that's why you have to take the symmetric product for this kind of operation but then we need something that keeps the act of the fact that we want to be able to identify pairs of points to create the nodes and the idea is that there is some kind of operator that works for power series with coefficients in the growth and the group multiply by symmetric functions so if we apply this delta is some kind of Laplacian operator and then the exponential here says that we can apply this Laplacian operator in central number of times so delta is some kind of Laplacian so if we apply it once we are yes I don't think I can explain I mean that I don't think that the Laplacian operator has a special meaning in itself but what is interesting is what the exponential of this guy is doing because this operator corresponds to the operation of forgetting two labors in a symmetric way so we are identifying if we had something like N mark points in some disconnected curve this operator tells us well it's encoding all possible choices of two of them in a symmetric way and then because of this if you do the operation once you get this but since this is an infinite sum you are allowed to identify in a symmetric way pairs of mark points in any number of pairs of mark points if you want if you just have one pair of points the idea is that it has something to do with the fact that inside s inside the symmetric group sN there is a subgroup which is isomorphic to s2 and s minus 2 and so somehow we can restrict any representation here to this pair and then take the three variable representation here to represent the points so it's somehow taking the contribution of sN coming from things that become three when they stick to s2 in some way so once we do this so once we identify point pairwise we have something which has a finite number a curve which has a finite number of connected components and all connected components are stable so somehow if we take what we get is the exponential of this thing so to get exactly what we want for mgN bar we need to take the logarithm which is the power series which is the inverse of the exponential of this very, very, very big formula actually there is an explicit formula for the logarithm but in the only implementation I know the formula for the logarithm is so slow that actually it's quicker to teach the computer to guess if the explanation is approximating what we want to take but anyway that's the idea so the idea is that if we allow to take disconnected curves then there is this operator that tells us exactly how to transform things so first we need to transform smooth connected curves to smooth possibly disconnected curves so we need to apply the logarithm to go back to what we need other questions about this scary formula you see it's an infinite series so if you look at it like this it gets a little better if you know the definitions but it does not get much better definitions I mean even to see that if you fix g and n and you want to trankate at that point you only need the g prime n prime that occur in the boundary you know you need to look at the formulas a little bit to keep to check that it's true but on the other hand this is part of the proof of the formula because the explanation I give you is the reason why this formula holds without taking the dummy variable but then there is some kind of book keeping to check that one is keeping track of the genus of everything in the correct way yes yes and it's called h but so something like so let me think a little bit so there is some there must be a reason for this but I don't I took the formula from the paper because for my point of view the thing about the power is just some kind of book keeping thing and it's not exactly what I'm most concerned about so I guess that the only important thing is that the behavior becomes different when you are working in genus zero so it has something to do with where your unit will lie and having a unit is very important for instance because if you don't have something with the unit you can take the logarithm so there may be some technical reason for this otherwise one could say yeah people in mathematical physics are always a little bit strange why do they have to call the dummy variable h bar for instance so anyway it looks complicated but you can think of it as some kind of black box if you take a truncation of this formula according to the to the symmetric group acting here and to the degree of the dummy variable then you are able to get the the homology of m bar g n from the hodger Euler in the hodger Euler characteristic of all the strata over the yeah exactly yes yes yes yes yes yes so let me write it here because I said it but I didn't write it, this is the inverse of log yes it would be strange otherwise so x is more natural but log is also well defined at least if you have something with a unit yeah exactly because you get the disconnected ones just by identifying in any possible way pairs of edges the only thing is that one needs to keep track in the correct way of the geners because identifying edges will change the geners of water I define marked points creating more edges in the graph will change the geners but anyway if you have all of them together you get all of them together delete this so what is the meaning of this from this formula we have an efficient way of computing the homology of m g n bar with all additional structures coming from hodgers structures and as a representation of a symmetric group starting from with a priority in this formula one needs all of them but what one needs, what one truly needs is the s n equivalent hodger characteristics of the m g prime n prime with g prime at most g n prime at most n plus 2 g minus g prime so these things happen in the boundary so somehow somehow this is the only information that needs to feed in here to get the tonkation you want of the formula and using this Gessel was able to give explicit formulas for instance for the homology of m 0 n bar and for the homology of m 1 n bar but he had to stop there because these are the cases in which he had a recipe for computing the homology of the open starter from genus 2 there was no general such a general formula which would be valid for all values of g and n for all values of n if you take genus 2 so these are the ingredients but then comes the second question well what is the standard approach to computing the homology of the open parts and everything we care about we don't even need the final invariant we just need this kind of oil characteristics but how do we compute this well the point is that there is a standard way to compute this using a spectral sequence because by definition m g n maps to m g so the problem of computing the homology m 0 n will solve long before in the 60s because it follows from work of Arnold about the homology configurations of points on the complex plane if you take g equal to 1 you can make this work by mapping to m 1 1 but I'm not doing this at the moment so this kind of theory also works for g equal to 1 just technically so m 1 does not exist in our category but m 1 1 is perfectly fine and the fiber of the c is just the configuration space of n distinct points on the curve which I denoted yesterday by f and c so it sits inside the nth product of the curve which this reflects the fact that m g n lies inside the nth fiber product of the universal curve of the energy so this is a vibration and we can look at the respective sequence associated to that and exactly in this example with cross-videration alters the i-chuta would be determined by the homology of m g with coefficients in some local system and then once we have this this converges to the homology of m g n in degree p plus q so somehow the first question is what is that local system it's going to be a local system because this is a vibration so there will be no strange things happening so in general local systems in some space will be determined by a q representation since we are working with the components of the of the fundamental group of our space q1 of the base space which is in this case m g so this would be all possible representations of the mapping class group so this is going to become rather I mean the mapping class group was a discrete group as if the representations are not known so well but actually in this case we can see with the homology of the curve the only local systems that will occur are those that are related with the representations of the symplectic group in this case this guy is going to be an extension this thing is going to be a symplectic local system well if you take it as an object well that's because you can realize m g as the portion of tachmular space by the mapping class group tachmular space is contactable yeah that's it so yes it's not easy to see this in algebraic geometry but if you take an analytic construction then this is fine so practically in this case if you look at what the local system is of some curve c then the fiber will be the cute cohomology and this has a structure representation sp2g and the structure is coming from the fact that we already had an action of sp2g on the cohomology of the curve so what happens if you want to know sort of the monodromy representation you need to take in this case you say well I start with the curve I consider any possible loop starting there but anyway whatever the monodromy did it has to respect this structure the symplectic representation so the only thing it can do is to act on it as the symplectic group and so in this way we find a class of local systems that are controlled by the representations of the symplectic group we want to take representation of this as an algebraic group because it's some kind of motivic construction should be compatible with the extension to the complex numbers so what have they reduced representations of sp2g well if we look over the complex numbers it's clear in general that's the theory for Lie groups these are indexed by partitions lambda1 up to lambdaG so the length is equal to the size in some sense these are the data in which one reconstructs the weight of the action anyone has such a partition when defines the weight of the partition should be the sum of all of all parts so for instance v0 is the trivial representation I'm writing v0 g times if I don't write it I'm only adding zeros at the end so v1 which is as I said my notation for v1, 0 up to 0 is the standard representation what we will find in the h1 of the curve so this will play a role when we are considering mg1 and in general if we have a v lambda then we can write it in a natural way to a sure a symplektive sure function into the tensor product of a number of copies of the standard representation equal to the weight so there is a natural way to cut all the v lambdas inside the space so if we know the comology of all local of all the symplektic local systems of weight up to n then we can compute the comology of mgn conversely if we know the comology of mgn not just for n but also for all up to n points then we can reconstruct all local systems of a certain weight so the idea is that there are explicit formulas for the comology of the space of configurations so that is not the issue it's more finding an effective way to package that information so there it is that we need to first decompose the comology of a general fiber into symplektic representations once we have this we can compute the e to tan using the comology with values in the local systems and then we need to study the differential in the spectral sequence and if instead of computing mgn bar we simply need to want to know as an equivalent Euler characteristic of it we need this only for we only need the equivalent Euler characteristic of all the fibers because anyway and then we can skip the last step because we are working with an additive invariant so we don't care about the differential we will get the same outcome we simply take the alternating sum of the e to pq just the sum of the of the Euler characteristics of the e to pq in a natural way so the idea is that if we just want to know the whole Euler characteristic knowing it for these spaces equivalent to knowing this Euler characteristic for the local systems and if we work here at most local systems will appear because of the expect that we are taking all the things that are contained at most in the end product of the standard representation so you may wonder how this is used so this is used both ways if one happens to know the chronology of the local systems then one computes the chronology of mgn but if with some geometric construction one can compute the chronology of mgn independently still it's more efficient to market the information into the local systems because they could be used also for other vibrations so they are more widely applicable so this kind of sequence of operations sometimes is also applied in the converse order to start with the chronology of mgn and obtain the chronology of local systems because this is the way, as I said in which people so these are the ones that people would prefer to understand but if you have them you can apply them to understand the chronology of the geometry of endpoint curves but not necessarily so this case in which everything is understood is again in the stable range exactly as the chronology of mg is completely understood in the stable range long before, so ten years before the characterization of the stable chronology of mg is available by work of maths and vice Lorena already explained how one can deduce from this what happens if one wants to study curves with markpoints and the idea is that the chronology of mgn stabilizes in the same range as the chronology of mg the limit object should be generated of course because we have the forgetful mark to mg but we will need additional generators which are simply the couple classes so this also means that in this kind of situation the tautological ring is exactly the extension of the stable of the stable chronology so it's what is generated by the stable chronology outside the stable range and this is known actually he also put the corresponding theorem for v lambdas where it's intended that if you increase the genus then you simply increase the lambda by adding zeroes, three values as I was doing before stabilizes but you need to reduce the range according to the weight of the local system so if we had three half k plus one now we have three half of k plus lambda plus one so g should be larger and then there are explicit combinatorial formulas so if you wish also the chronology of mg with coefficients in the local system has some kind of tautological chronology which would be obtained by taking this combinatorial part and taking the natural extension of this there are papers about this or preprints because by Peter Sennan and Tavakor for instance then there is a single good case in which the the single local system has been known for a long time this is the case of genus one the same construction carries on and the source of this information is Isher Schimura theory the idea is that there is an explicit description of the chronology in this case in which the basic part was done by the linear but then an important detail appeared later in a paper by Elkik so the idea is that in these cases it's slightly easier to work with chronology with complex supports so that's the formulas for the way some of uniform so if the here the genus is equal to one so we get just one integer and it's easier to prove that if the weight is odd then the chronology is trivial if the chronology if we are taking v zero we know what happens so here we are taking k at least one and then the chronology of the local system v2k which is just the kth symmetric product has two paths a whole structure so a pure whole structure of weight 2k plus 1 which is usually denoted by s2k plus 2 and then a one dimensional path of weight zero and this follows from the from work of the linear we should have some kind of motivic interpretation in the fact that some is a direct sum of what a lkik and there is also a natural way to find so this is a two dimensional and there is also a natural way to find a generator here so this is a pure whole structure of odd weight and in this case the two pieces of odd composition are in the extreme position and the first one can be actually identified in the space of casp forms for sp2 but this is the same as sl2z of weight 2k plus 2 so you may wonder what a what a what a casp form is doing here and if I have some time I do have an explanation for this but I already know that I will not have so much time so perhaps I should permute things and try to explain to you now so the idea is that if we want to have a casp form here and something which is non-trivial the first non-trivial case of the casp form is the casp form of weight 12 so what is a casp form this is a holomorphic function half half plane this behaves well in when we take this value and we consider the action of sl2z gamma in sl2z then it will act on tau in the usual way by taking a tau plus plus b divided by c tau plus d and then one has that f gamma tau should be equal to c so the denominator here to which has certain power and this is going to be the weight of the casp form and to be a modular form in this case it needs to extend holomorphically to the only to infinity so to the boundary of each one and vanish that so once we have such a thing what can we do with it so the idea is that here the first non-trivial example is a casp form of weight 12 the discriminant form so it's a functional variable z but actually it factors to its periodic so one writes in this way as a function of q given by the exponential of z and then the classical way to identify it to write it as an infinite product by taking 1 minus q to the power n to the power 24 so I'm not an amateur so I'm not going to explain to you why this kind of thing is well defined but the point is that it transforms nicely when we apply the s and 2z but for genus 1 this is exactly the group so if in genus 1 if you take the portion of h1 by this group we obtain exactly m11 because this is a modulated space yes yes it does not make sense if my students would write that then they would not pass the exam so yeah thank you so once we have a cast form there are ways to use it to construct a differential form of m11 how? well in this case for instance so of m1n but in this case we have weight 12 so we can work on m11 now we want to use the upper half plane so ziga space to describe m11 you can say well if we are working on m11 the first point is known so we need to take the universal curve over it and ten copies of it to keep track of the other ten points so what we are doing to keep track of the curve then we need to have h1 and divide it and this will give the isomorphic class of the curve this is the standard thing for ellipticals and then the idea is that so e will be done just given by the quotient of c by the quotient by the lattice z times 2 and this is of course of two lattice so the idea is that if to give ten points ten points on this plane and then we will have an action of the ten product of some z2 where there is a semi direct product comes because of course we also need in the process to identify z2 with this lattice so the idea is that what we need to define is a differential from here that descends to the quotient and because of the symmetric property we can use the delta as a coefficient Are you saying that I'm taking this the other way round or let me think about this No, I do think that when I write like this the normal set group should be the one to the left because the same thing is one and this is the one that needs data from here for the identification so if z is the variable here and we note by theta1 option theta10 we simply take everything and one can check that this is invariant with respect to the group action so we get a differential form on m11 and in particular we get a class in some decoromology of the shift of of allomorphic differential forms over x and by the way in which pure host structures are defined this is exactly the eleven zero part of the decoromology this is not x this is m11 because that's the space in which we are doing things and by doing the bookkeeping I was saying before this is actually exactly the contribution of the local system v10 in this case so this is giving us the let's say the holomorphic part and then h011 is simply the complex conjugate of this which is generated by the complex conjugate of this form and this is not some kind of lucky coincidence the idea is that this is the way in which it works for genus1 for genus2 and genus3 this kind of construction where you need to replace h1 with ziger space sl2 with a symplectic group and then what you get is the modulate space of principally polarized bilimvaritis but h1 gets them up from ng into the modulate space of principally polarized bilimvaritis h1 extends to a larger space in which elements are symmetric matrices of the size g rather than just complex numbers and then we are divided by the symplectic group and for g equal to 2 and 3 this map is dominant indeed for g equal to 2 we get an isomorphism because this map extends to clouds of compact type and also the automorphist groups of the corresponding Jacobians are the same so this is the identification and so we can do exactly the same kind of considerations and we can use it to take modular forms in lucky cases and construct this kind of interesting host structures which are the most non-tortological in this case for g equal to 3 this map is still dominant so we can control things in this way but this is generically 2 to 1 because here all abilimvaritis have an automorphism so the automorphist group of the Jacobian of the curve is the extension of the automorphist group of the curve by an evolution so the idea is that for genus 1 and this way guess that gave an equally nice formula for the chronology of m bar 1n in the form of whole-jojlet characteristics in there is if you increase the genus and go to genus 2 you need to there was it was more difficult to know how to replace the description of the local systems and in this case most of the information is actually coming from point from counting the number of points m to n divided over finite fields and in this way producing at least an estimate of what the whole-jojlet characteristics are there and this gave some kind of conjectural description of the relations to spaces of Ziegler modular forms which are the Ziegler cast forms which are there extensions of what we discussed now and this has sort of been proved now by Peterson so I guess in this case one could now work on the characteristics more in general for genus 3 the description is completely is completely open yet is the whole-jojlet characteristic is only known up to 7 point and here part of the issue is that if you take many mark points you may because of the fact that some information gets lost in this map you may get contribution of some things that can be interpreted as modular forms that are directly related with the Taishmuller space so they are called Taishmuller modular forms rather than Ziegler forms and it's something I think Bergsturm, Faber, van der Kierer working on so I hope I could give you, I mean I wanted to give some kind of introduction to the fact that there are there are natural ways in which also things coming from number theory in non-trivial and non-totological things in this way and I hope in this way I haven't made your picture not complete so my aim is reached, thank you for your patience