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4/4 Universal mixed elliptic motives
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I have been doing
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this we a and if we we have to do with you you on the other hand OK so this is the last lecture so because this definition of mixed elegiac narrative universal makes it a look the motive as long I will repeat it very quickly this so recall so the 1st thing is a universal mixed elliptic modern and has various pieces and I'll say over and 1 the 1 I once at 1 time only it over and 1 vector 1 and all have some notation there but mn will denied motors of this guy I what does it consist of 1st of all it it's the 1st thing is that of V M T M and by that I mean in TMC so make state motive I and it's way filtration so the whole story is a little confusing because there filtration the way filtration but in M T M is denoted by and .period not the usual W . and that there's another filtration W . which is also weighed filtration and L enters and filtration by some objects in and then the 2nd piece of data is a representation Road but from S L Tuesday which you can naturally identify With Paul I won M 1 1 With base .period Q In 2 the wanna morphisms of VAT the baby you take the Betty realization of this guy and that preserve the filtration W . they don't it doesn't preserve the actual weight filtration of the better and then such that each G R W M all of the baby that is a sound all symmetric powers of age so he just talking better representation of the irreducible representations of SL to the algebraic group adjust the symmetric pals of the fundamental representation and In will be congruent come on to spread you donated this will follow anyway from the other conditions and note that this implies here for example that on TV which is 1 1 0 1 many politically because you have the filtration W and each greater caution is 1 of these and TX unit potently on each graded quotient so we're going to N equal to log to you and I you can add this is a requirement hero later em .period is equal to the relative weight filtration all of In this no pardon in morphism said this is no pardon acting on V uh W . kinds of you filtered vector space in potent will visit you may have a role to wait filtration and here you definitely do part of the requirement and so this also lead you to a filtered local systems W . Over on 1 1 analytic the here to play and right so we want this to be part of a variation of mixed Todd structure and I and I should point out here that the fiber all Q in the sense I mentioned before it is naturally the Betty right you couldn't 2 to define this you'd have to take De Lange canonical extension of this and then put Q structure on the fiber over the cast and it would be exactly this naturally identified with this rational that displays and then I the next part of it may be open to over here it is found 3 you've got a vector bundle this is the Durant story I Over and 1 1 bar of a Q so that's the same out with connections With the flat connection Nabil take the into V tents a
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major 1 and by Walter P. where P is the cast of God logarithmic singularity and I also you want Hodge filtration f Of this guy here and you want that now blood of F P V is contained in F P minus 1 of Griffis treads would sell the and I await filtration so all of these defined over Q by the the survive flats sell bundles and you want the fiber all over the cast he To be to be naturally identified with the did you fix and identification with the Byram realization of you make state motives and then the next thing is you have a nice amorphous if you take Solar City will be isomorphic too the canonical extension all of on the tense of 0 and 1 1 so this this is a flat vector bundle and if you it should be isomorphic to that this guy here always amorphous and should be compatible with the isomorphic the better should respect the isomorphic some of the better chances C is isomorphic to read around consistency and then I can together and then I and together so this this video it's inadmissible variation of mixed solid structure right so we've got this admissible variation I and the last part of the attack the LAT Catalpa What is that said let me destroyed diagram so we have a S L 2 this is isomorphic to apply 1 of M 1 1 analytic over on the hour based .period we have a representation hearing to order the better but I'm know attention with Q I In the lodging acting on sludge a vector space this guy here maps and supply 1 of M 1 1 over Cuba did and this guy this map appears just pro finite completion and I then he'll get bored of the elastic realization and these guys here isomorphic by the comparison saw morphism and so you're getting a representation here Nicolas Roeg had and the last condition is that FreeCell the Opel wrote FreeCell roh had L is GQ "quotation mark variant GQ X on the LME realization here and it also acts on this geometric fundamental group of M 1 1 right and that's the whole definitions so it's a mouthful that's why report repeated and that so the other thing I wanna review very quickly pulsating just 2 words is relative completion this is the 2nd thing to recall and I'll just here stood for SL Tuesday so said is equal to the category of arms finite dimensional representations of S L 2 the Of the queue the representations on a finite dimensional vector spaces on that admits a filtration it can be any filtration and has this property and I such that but the reaction on each graded quotient of this representation it is actually a representation of at the algebraic group L 2 and then S L 2 so this notation is meant to me that the action vessel Tuesday on each graded quotient factors through a rational representation of the algebraic group Texas polynomial representation all this feature using he also so this is this is just an abstract designs for the from users that's right these local systems will be in that there's record that representation will be in here sir Rocco is a here but we can just consider all these representations anyway end of town so in other words each of these years just as some of things mind and so on this is a 219 category so the relative completion of SL Tuesday is that it is that Anakin fundamental group of this category with here this is just the underlying Q vector
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space so I'm and what is this guy look like it is an extension the deal with here we have 1 aim to new rail conclude into S L or cold SLH and we have a cell to the rear of a bit sloppy I should bright Q rational points here is the standard inclusion and be this lived here this lift a risky venture and this guy here prounion followed and can I can't remember exactly how much I told about this last time also a few more words in a minute in a little while the inflation resulting in the the usually exist at all now it is representations that have that admit a filtration which were you off more and more people usually don't you know provisions of for example I could if I had a great cost isomorphic the 2nd refined infiltration the splits of these new different greater caution the identity mapping and its inversion rematch in the category so on this level will compute what this is it's going to boil down to computing what this group is here in a little while and then there the goal is that we want so our goal is to compute pi 1 of With respect useful base .period usually Betty or around I so let me just remind you of some things we said last time the 1st is that we have pie 1 what I call geometric the geometric part of pie 1 of and I'm happy to supply 1 no map Paul I 1 of mix tape motives so if I take a representation of that at some extent motive when it pulled back here to geometrically constant mixed elliptic motive this section here coming from the base .period because we know the fiber of the based pointer mix Tate Soviet taken mixed elliptic motive to forget everything that the initial you've got an object of this category that's giving you a section also I'll be sloppy here just to save time if I put QL here just to give you an idea of what might be going on here I could put I 1 of In 1 1 overstate Cuba plight 1 of and 1 1 of which 2 and then GQ here so but the standard exact sequence that's going to be a map here this is going to correspond to looking the moment basically this representation here Of those this a natural map here and get induced appears so this the exact sequence really does look like the usual exact sequence of the geometric fundamental group the arithmetical fundamental group and the Galois group said right so that's the end of the world and the yeah so and that's the end of the review now the little tests in this is just let me do the density argument because I was trying explained that in a clumsy way last time and watching complete and also it I have to do this stuff anyway so a start with the fear and it's a time to special cases all of a general result which I don't wanna state and and the general results actually with some him in trying to write up the EU's with various people undergoing right with great post on it was going to be with unmatched model and terrorists but I discovered that when you try to write a paper with 4 offices nothing much gets written so I accept on but anyway I and so this version is written up I put a paper on the way the other day colder hunched around theory of modular groups and proved in their work sketched in there so at the firm says this for each base .period and I should point out here at the base .period we most wanted uses DDE DQ I also M 1 1 analytic but the coordinate ring of Jere L. so implicit in this is that I'm identifying S L Tuesday with the fundamental group of M 1 1 analytic with that particular base point in the case the the way have set things up SL naturally isomorphic to Part 1 of this With this space .period right and this is the 1 I'm going to be using so you might as well just think of this basic .period has 12 it is they hope algebra the category of unions mixed unstructured so that by this I mean it has it's a direct limit of mixed Todd structures in the product in the product and the postal preserve Hodgen way infiltrations to structure and so on it's so for all finite base .period approve this in a much older paper In the 2nd part is so let's let M H S 1 1 a this is going to be equal to the category admissible variations of mixed solid structure over M 1 1 who's white graded quotients I as some solved sounds as variations of as polarize variations of Hodge structure some solved symmetric pals of H. 10 said with a with this is a constant Hodge structure so not on allowing more
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than just take things here if you restricted to just Q and you get this result is not true this it doesn't kill Eisenstein variations a case so and then the fiber at the base .period induces an equivalent of categories it's going to take the MHS In 2 when H. Rap g rail and what do I mean by this so this this is Hodge representations so what do I mean by this Hodge representation would be a V equals a mixed Todd structure plus an action movie into well if we write it down algebraic group right so this is if the is that a mixed Hodge structure then intermixed on structure the statement and if I V is a representation of this group you have this map here and now this guy we know as a mixed on structure that's what part 1 says and so we require that this be a morphism so this is a more efficient you can also say that if the groups connected as it is here if you took the lead Aldsworth the group that's means for this to be a large representation so implicit in this is somehow that you know it when you take the fiber at the base .period you get a mixed touch structure that the monitor Rami representations amorphous and and conversely if you have a mixed taut structure enhance representation you can construct the variation the variation or be unique by the the theory of the fixed and so that this version of the Cyprus Steve Sacker almost 30 years ago .period from improved to 4 years ago published in 87 the unique case but this is the general case so this is useful to us on up and so let me restate this in another way said the Coral area instead quiet 1 of M H S and 1 1 h is isomorphic to high 1 of mixed taught structures similar direct product Jere so why does this make sense so this is it to knocking category this is the fundamental group of Justin acting category this group has a mixed Todd structure which is the same saying this group Exxon so I can form semi direct product if you think about it a hunch representation of this group is the same as a representation of this group said that yeah representations of this representations of this whole group equal Hodge representations all it's a similar situation To this mean what saying a represent you know what gonna splitting here coming from the DQ what's what's what's a representation of this group you can think of it as saying GQ echo variant representation of this group on a module just the exactly the analogous statement arrange so Of so His an observation is that we can take em E M and we can take it to M and we just take we just take a mixedup Witaker mixed elliptic motive which is going to go to the underlying variation of fixed on structure and now if we use the fact that the Hodge realization mimic statement is fully faithful you can see this guy heroes fully faithful and so and so another choral area of this form of written down Zafira Mennonites he insisted on to Sephira is that I GE railed the natural representation a constructive last time from GE rail into pie 1 geometric of mixed electric motors this objective and this is this is good because we're going to be able to compute what this guy is and that's going down the size of this share remember that somewhere on those boards here we have Paul I wanted them again an extension of pipe I wanna make state motives which we know but by 1 of India so and so let's do the theorem proof this statement here the statement right here says that style is equivalent to saying that I hope I will 1 M H S 7 redirect chronic rail subjects on to apply 1 of Indiana this subject and now and then you just look at the but diagram he'd have Jere apply 1 and page essay now the idea is pretty straightforward
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here and this is Paul I 1 of and in both cases you have a splitting given by everything and check everything fits together this map was just taking it is induced by the Hodge realization of MTN this is subjective this Osijek given you can this guy another Correla in this this may be obvious from tonight considerations that we get it for free here for his on this well I know explained this guy cannot be an isomorphism so I'll just give you a little preview the unity pardon radical of this is generated by Eisenstein's years and cost forms the radical this is generated only by Eisenstein's series and so the Cardinals generated by cost forms and they're going to impose relations down here In here this is huge and and that is you might ask what's the smallest the category of mixed Todd structures you can put here and Francis has something that is pushing it is our recalls mixed module motives that basically the hard structures you got out of module forms and so that's so it's going to be bigger the region injured just by 1 of MGM I think it's going to be big enough to to contain the hard structures that correspond to cover walls so yet so Larry on just put it over here this instead the colonel all Paul I 1 and P M there's a map to GE elevate should remember this is the fundamental group of the category of send simple mixed elliptic motives so I know the statements this is really about is prounion vote but I think that should just follow because the Cardinals what controls extensions of these and can extend by stuff of lowwage failing and certainly implies this result all right so we're going to denote the cardinal so the colonel all denied by you and the the new enemy might think so we have got an extension 1 into you the interpol I won in medium into G Help and again you can Split this guy apart into its geometric part but you'll have you got you and in the end will map to what I call this is the likely algebra no 1 we algebras here so they'll be the geometric so I can break it into its the party coming from a mix tape motives said this guy here is essentially the freely algebra on the 3 received 5 the 7 and so on and this guy here's the geometric but we can see that this guy up to the action of S L 2 is generated by Eisenstein's series the atomic school right so let's understand I so start here you've got 1 in 2 new rail and maps into Jere L. maps into S L H into 1 and if it and the skies prounion and now lebanese theorem says we can choose a section here 11 says we can do this so this says Jere L. it is isomorphic too S L H 7 direct product you rail not only that but I should say this is unique up to conjugation 1 element here isn't unique how to conjugation applied you you don't have to love no not that I know of you can write and you can prove their natural choices but not canonical choice as far as I know OK so it was once you put this in here this group Exxon that's so it says that that you Rail that's the LEA Aldsworth McConnell this acted on by SLH so it's a pro Neil potently algebra in the category they're cell to representations right so this is prior to no buttons finally algebra in red of SLH I should say so if completed representations Jason now we want to write down the presentation of this so and I should also point out this guy's exactly new rail so knowing this is exactly the same as well knowing that you lose no information by going to the Union but radical so so the next thing to do it is you've got a new rail and it maps into H 1 a few this is subjective and also as SL to wage they could very so you can choose and NSL too equal variance section so this is SL too worse still H. equal variance section just choose 1 so this this will give you a suggestion from the LEA algebra generated by each 1 of you rail little matter you this is the section induces this and have to complete time and this will be subjective so this is the 1st step to watch running down a presentation in fact will be
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minimal because you can't make the generating set in a small here because they're too nice a morphism on each 1 and now what about so let's let the Connell we are sure this is some ideal so this is contained inside L bracket because of the maps and isomorphism on H 1 and story the yells freely Algeria on saving short so I'm so hot this the standard argument for group the love of my age to a group as a formula due a halt it's easy wrote that applies equally to Lee Aldsworth refuse this condition here we see that page 2 of you rail is isomorphic to our law and our are L the and so if you think about this is all relations Margalo consequences of all relations right so it's essentially a minimal federal agents I want to stress that this sort of us I can gives you by the way should take this hop doses just tell you that they the college's very freely algebras trivial degree greater than 1 so what you can do here but it is you can you've got a map of ah onto this choose an L 2 SLH section and now I'm a little exercise as you can see the image the image of this section will be a minimal set of generators of this ideals of what you'll see is the new rail this isomorphic to the freely algebra on each 1 of you completed Margalo I let me like call the section city Greek and look at the image of city closure and and the image of this guy is isomorphic to H to you right because the image here exactly that has to inject Media it's the ideal generated by the image of city said the relation between H 1 being generators and age to being relations as much stricter for private it's perfectly straight for parental potently algebras when so the reason I wrote that up is because the game is we're trying to find his presentation of on the unified radical of supply 1 of the millions of media during the CDs unlawful solution of the idea was not by the way this discussion works equally well I just didn't hear for new rail but it works well for any primal partly algebra in the category of representations of a reductive group all right so what's the so is Sephira and it's by this person called easy is that 1 is that H 1 review rail Chen with S N H and if you take the as L H part of this invariant part this is isomorphic to H 1 vessel to and I should point out here that this vanishes when In is all that's standard the whole point is that minus the identity is central in here this at you can see with marginal forms but from group colleges and argument called center kills and minus 1 x nontrivial LeeOnTheSolent has to kill was when it was odd and so this is good because we want to find a presentation here well we just have to understand these college groups vessel Tuesday which is going to be like which Mora and now we need to at least managed to do so 2 Is that a H 2 you right yet so this is telling about the SNI so typical pop out of the sky here to representation still too still this and this guy is contained in age 2 SL Tuesday S & H but this group here is trivial because this is because S L 2 Tuesday it is virtually free 7 the group has virtually a property that has a finite index
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subgroup with his property sell to Zinni has a fine in subgroup which is free just take any modular code of the corresponds to a torsion 3 groups to their flank of such fundamental group is free perfectly standard and then a little spectral sequence vitamin shows that as long as the scrupulous divisible on the Coalmont adventures so what's this telling us telling us that age 2 0 so there are no relations would still use the UNimposed radicals free following you rail it is 3 so you rail it's isomorphic but not naturally isomorphic too fuel tank completed and you have taken place you have to take a look all the finite dimensional no pardon quotients and take the inverse limit and I should point out here this this year so the Beyond that .period have something here which I am maybe I don't have to do it but I will because I think it makes everything clearer observations and notation so remember H is the fiber of delays canonically extension of the vector DDC and so age as the 1st ones that age has a natural wastage so age we know is isomorphic to Q plus a few of minus 1 that's calculation we didn't know only lecture and I can ride out the door and a writer but the Duron version and the Betty version so the Betty version this is queue of plus Q of W and in innovative devotion it's queue of class Q all of the and the relation between W and B is W it is equal to 2 pie all right the inverse times and the and so USA right end we but we actually computed that if we used to tangent vector if instead I use land at times DBQ well land was not plus or minus 1 I couldn't say this because this would be a nontrivial extension of this by that and you wouldn't have a natural basis but because of Split may have an natural base and I so there should be a minus minus W increases you look at the reason on extension to direct summit saying so the point I want to make is that the S L 2 as a natural torso and we have a natural birds in all of this so and I mean the representation theory is trivial but that we want to nail down the Taurus and so on so I and also if you take into the law Our change is equal to a a D and that the With the Duran version is equal to minus it's a mindset minds a GED that Israel's followed from calculations we did earlier and it lowers waits by 2 lows and diehard Wang's by 2 it's getting confusing because we have SL 2 weights and weights and W. weights they're all related diseases and so are we going to do this way Gunther said he 0 equal to this guy here minus a indeed and so other at the end of the season usually out of the lack leverage in the unit because I wanna know how all to 2 elements of the methods so I'm so what does this tell us says there is a natural choice of Of
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course so I will have had a will have weighed 1 the will have waived minus 1 it's natural to give a it's got away L on the other 1 of the other way around with wasn't sorry as weight minus 1 India's quite plus 1 write to this has lower M Hodge weight than that want an end and the 0 as weight minus 2 has L to ways this is a S L 2 In another notation I want that France's doesn't like is that so this is defined to be h module we've highest weight Victor suu again need lots of different copies of this and I want to be able to label them and write down their elements so this is just going to be the span Of P 0 2 the J times in and but the zeroed to the end plus 1 thailand's eh equals 0 so let's look at a quick review of cyclists to more of Mandarin filled with Jack is name in there as well because so a lastsecond belongs in here and I think I think it that he does because I use the way he does the Hodge 34 is still too not to put the haunted the hunch theory has to be put in the framework of the elderly machine I in plus manager filled so so what is the age 1 In 1 1 in the lake s through an H which is the same as H 1 as fell Tuesday as 2 on page a natural mixed on structure think and lights when in a 0 0 leaving few H 1 and H 2 both Wright said 2 I'll take into the positive and then you have an exact sequence the casket local which is the same action is the intersection of all 1 1 by the coefficients Tunis to an age this Napster H 1 and then maps to Q it's actually useful to think of this as I S 2 2 and H knowledge the 0 times as to an ancient naturally isomorphic to this twisted by minus 1 the poem variance and so I have colored chalk it reasons so this spot here this is a hard structure of choice 2 end plus 1 0 0 and 0 2 10 plus 1 and this despite here and it is if you put them all this part here is of type 2 n plus 1 2 n plus 1 so it's really a copy of cubed minus 2 and minus 1 so you have a question is it was the 1st person to say this should be the Hodge numbers of this write it down and explain and that's what I guess why gives hooker credit I right and said this but here these are the cast forms all of of cast form wait to end plus 2 we have and weights w weights weights of Council forms in a cell to ways I and these are the AT whole Moffat they're all the same day complex conjugate and despite here is corresponds to the Eisenstein's G 2 and plus 2 and also a little bit more about that in 1 2nd on 3 is that this is Mandarin felt I this this extension of Split by havoc you I mean I'm viewing is askew makes taught structures and unseen extolled structures so such direct sound the middle guys naturally a direct some splintered using Hector correspondences you will have people here said that the and that's correct so I guess I big man and prudent for in wait to or something and then I guess the drink felt probably realize the same argument applies in general so I and let me just say a word about how cost forms determined
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or module forms determined Mountain class so as as the module form the whole Norfolk model form of weight phase to and plus 2 you can define a major effort to be equal to I like to think of it this way it's F of tale times W to the 2 twoinch I defined it's got to be a section of the bundle arrange In remember W at is equal to but to Part II times Jeezy which you think it was an element of h 0 a a major 1 of Seymour and landed tale Of the lattice associated with at times DQ of acute end of and if you recall if this looks different from the usual picture this is just minus to pie by I B plus a long queue times what if you like to buy I tell jokes the usual formula and this guy is an element of the 2 1 form on the upper half playing with coefficients soonest willingness to NIH and it's SL to variant but that's the same thing In this all before all world as a 1 form on M 1 1 analytic with cold Tunis to win directions :colon Catherine became so I and this gives you a class In f 2 n plus 1 H 1 and I wonder what analytic that explains these things here and the other classes of the complex conjugate civilians no yeah wiser it in this level of the Hodge filtration this is in F 1 so it's 2 and a half hours in half to win a new picking up the sky guys giving you an F 1 here OK so I I'm gonna let it be to and plus 2 this equals the normalized take care of the writing test forms you never know had punctuated of weight to and plus 2 so we get a basis yes and so I'm so what's the basis of of page 1 you did the Iomega effort Iomega F bonds f in To in plots to blow up union and the Iomega corresponding to G 2 and close to and if I need to all notice spice side to him and to the Ugandan natural basis of pulmonology out of our normalize customs but you need a cost conjugate the joining need 1 Eisenstein's series and this guy spans the copy of this guy here a right so I there may be all all write this down very quickly and then I'll stop for a break but let's try to put this together so what it is I'm so let's let the effort prime the effort double prime F P and B 2 and plus 2 a basis I duel to the above the basis of H 1 dual :colon I duel to that To this 1 up here rent so when you remember that each 1 of you recall that age Lowell 1 of you Ralph it's going to be isomorphic to the direct product of H 1 and 1 1 analytic S to end H duel Cancer S to so visages vector spaces and this is where sold to Ryan and you rail is unnaturally freely generated by this guy here so know that we have lots of copies of S 2 when we have won for different buying so basically the trick is to find a basis of this so yeah also we have a new rail is isomorphic to the freely algebra on H 1 of you completed so this is telling us that new rail is isomorphic as an S representation but not naturally which isomorphic to the freely algebra generated by the direct sun and greater than or equal to 1 S 2 the 2 and plus 2 that's saying you take a copy of this to win for the Eisenstein's series this is the guy duel to the Eisenstein's years plus plus the sun Of the Eigen forms Oregon Coast forms just try to give you some idea of how big this nasty this thing is I was not nasty it's actually very beautiful but for every Kassebaum you get 2 copies of the same because you've got because the accounts funded you twodimensional Hodge structure and it's as to when the F doubled prime
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and so I think about what elements of this army an element of this would be something like EDF prime time's E 0 2 the J so the usefulness of the zeros generates all these pieces and so Everything in here so this is somehow it's generated by all be very sloppy it's generated by the zeros to the J times the 2 and plans to the 0 2 the Jays times the of primes and eh 0 2 the Jays times India double prime you have to take all lands and all efforts and so on we want to win note take twist while I'm going to put these guys in the natural place so let me on the other so I've got my culture here again this part here is isomorphic to test to and age to plus 1 and so this is this is gonna have W Wait the "quotation mark this said W. weight but minus 2 in minus 2 that's why I put this to imports plus to their urged it gives you negative the way and these pieces here these have W. wage equals minus 1 yeah that and this is actually on the 1st of expanded tended to be very good and in the notes I want to put it on the board and a picture of the hunched so I and all the stuff I'm doing here is in this paper I recently put on the archive around theories and in detail I'm so let's take a break a coffee break and then there the next goal is to see what this says about so after the break the goal is to understand basically you and me and we we're not there yet I mean and we've been saying for example Francis and myself I quit trying to understand what this guy is and we can we know how big the relations are unless there's something very desire that violates the order of the universe so we know what do we know so we have an exact sequence so this is this is the unique potent part of pie 1 of MTN yeah I'm concentrating on the podium parts of the proindependence parts because the reductive parts role well understood the reductive but here's Jeanne reductive years G L H and reductive part here's cell each 1 and so these are the unique potent parts of pie 1 enemy and the geometric by 1 of Indiana and this is the geometric part of the state would and so we have a corresponding exact sequence of prior appallingly algebras you I think I might just call Jeong next to you and on and it will map to care and remember this is the free Guy Elzy 3 the fight and so on and right and so facility little proposition is that you M E M and you g project all and and this is silly because I watched the proof we have Paul I 1 of mix tapes so mixed yeah of mix tape motives it maps into the order morphisms of pie 1 of mixed elliptic motives by the base .period DQ and it preserves the homomorphism to Yale age so it is that saying that pie 1 event acts on this and also acts on that so therefore they're both mixed trade motives so so a 1 right yeah so this is this is a choral areas let's put it this way a Okoro areas that H 1 of you and me and and they each 1 of you Jeong are you have crew of 2 weeks Todd structures of taste all greater caution serve type pp Brian because their Hodge realizations of mixed notice but there also quotient they How well but it is a quotient all I need your help sorry about all these abbreviations and we have many kinds of weights and we have many kinds of adjectives we apply the LEA algebras said so Aprile potent Lee algebras in some sense generated by its H 1 so we need to understand the map on H 1 so I would guide you rail maps onto you remember this
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is to make it absolutely clear this is that this is the more timid guy here this is the relative completion of sale to and now I meant to put H 1 in here so the 1st thing we 101 Astana's how can this guy being but we know this guy here is the direct sun 2 from what I did every year up there is that this is the direct sun and greater than or equal to 1 s to end to end plus 1 that's the Eisenstein's Parliament plus it's got the Castel plant I'm
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a writer like his V F
1:13:16
dual tenser S 2 and h and I'm here the sequel to the spin of I I'm going to be slightly sloppy these guys are really defined over a number field but totally real number field blowing pretend that defined over the reels Will rational so maybe the class of a major effort plus the class of a major effort about right this is this is the 3rd 1 form that represents the logic cost .period so this is a twodimensional on structure once the work sorry when you all know it's of tied 2 and plus 1 0 0 and 0 2 and plus 1 so and this guy here's take when I say it's it's it's table I mean it at the this is where it is confusing we gotta think of it at the base .period DBQ at DBQ the symmetric Powerade she becomes mixed so but this guy can never be mixed take this generate to on a direct some of copies of Quran you for some Jr this guy will never go away you can never make these Hodgman was equal rights so this part here has to be the Connell despite here and so this is a key thing to understand this part he goes to 0 this is the Conlon H 1 we're going to be careful because the H 1 is not the canonical generators but we have to use the fact that in the land of Hodge serial The Land of motors the functors GI W Ng odd m both exact so so what does this implies this implies that I H 1 of you know Jiang the to guy here is a quotient all of the direct sound and greater than or equal to 1 has to be an age To end plus 1 so we know the generators there so studies to sewage generated by Eisenstein's series I'll put you go correctly interpret the statements but there is that correct suitably impressed so and so cop so choral area of this is that well all right down to proposition is in fact the page 1 you you and we yeah is equal to the direct sun and greater than or equal to 1 Q of 2 n plus 1 plus the direct in greater the equal to 1 S 2 N. H 2 end plus 1 so this is the Eisenstein's so this and this is also the geometric pot because this is is H 1 of you John and this part here's the pipe coming from cadences H 1 of cagily algebra generated by the odds a values so this is a so this is the mix tape motive story To the Eisenstein's series of geometric pattern the a the values of like the Galois part sorry is the days will give generous they want live canonically 2 generators so so the next thing I talk about discusses the Eisenstein quotient I'm sorry you know I found him not so obvious sorry I forgot to explain so on this so approved you we only need to produce and makes elliptic motive where all these guys at nonzero and so on proof 1 is well the best 1 I know is that I in the beginning I wrote down a mixed elliptic motive so Pete which is the mixed elliptic
1:19:03
motive which was defined to be fully algebra all of hi 1 of the prime look at minus the identity all of a DBQ so essentially the firstorder takeover I'm with a suitable tangent vector as an object I mixed elliptic matters will probably take weeks to look at what is now should stay out of the thing with the vector and then I there various ways to see that all these classes a nontrivial here but I have this paper called Notes on elliptic KCB equation so it can sell this work of 11 resonated you see all these extensions and on 0 2 this implies it's even get a representation save you Cheonan into their P and in a little while all right down where the 2 the 2 and plus 2 goes to something not equal to 0 you can write down explicit derivation can I you also get a L of I haven't checked everything although or I the elliptical at logarithms I'm sure everything is in the paper Beilenson 11 everything I need to check this right but I mean I know they're looking Polly logarithms emotive In this sense so and they realize so the handful of people a lot rhythm restricted to the 0 section realizes they can't 1 of these all right so so what we have here is we have Jere El Bulli algebra of the relative completion and this guy is definitely not tape Hodge structure which got a Hodge structure it's not too late that not take at say DBQ because you have these it's pure quotient you get model of you get the hard structures of cast forms appearing and now it's mapping into G all the well sales maybe the geometric pie I wondered maps onto this 1 here and this guy here is the to make stated to take hard structure so you might ask yourself what's the maximal take a guy you would put here and I wanna call it the Eisenstein quotient right so what's what's the biggest quotient of this world the graded portions are of the form as age 2 sums of things like this where I do not allow things like this tainted with media I these guys are out these guys are in and see when I go to the base .period DBQ h becomes state services describing mix tape at the base .period so that's what the next take the Eisenstein quotient so so GIC is equal to the maximal quotient and Jere L. In the category all groups With mixed Todd structure whose white graded portions I should say the way greater quotient of Lee algebra that'll be sloppy and ah sounds all of this to SAN page hours and it's also and you can check that it's Paul I 1 of the categories admissible variations of mixed Todd structure Over and 1 1 with G R W daughter of the US at a sum all In Page AZ you can have SNH tented with a constant Hodge structure like Nunavut so this this is all well it's interesting that this guy is not going to be the same as Jarrell because of the cost form so so we have
1:24:49
factorization the good cheer L matched to the Eisenstein caution is called this because it involves Tarom theory on involves Eisenstein's series we're gonna have the geometric pie 1 although medium we have this factorization this is subjective and now you might ask if I was going to say about 1st what can you say about GIC and what can you say about this matter and all jumped the gun a little bit and say this quite a bit of evidence to suggest this is a nice amorphous and so on I let's see if we can understand this you gonna jump ahead a little little Britain jump back well maybe not alright so we want 1 so let's get a some sort of lower bound on what it's like and so the little except project in understand In the case of fact is that if I looked at home 1 of family M I'm sorry news from scifi like at GE rail and I'll call it 1 vector to this equals the relative completion all of Part III 1 of em 1 vectorone the 2 wild appropriate the base .period fear yeah I don't want to write all right this year I hold 1 1 crossed with what cold G A of 1 just to copy of G8 with I that the corresponding Hodge structure is of type minus 1 minus 1 and I also want Part 1 of a medium 1 vectorone precisely to find 1 of em you and the usual 1 crossed 1 so this is prudent and manuscript I've put on the archive that you get this product here and you can also proves this it's not hard the reason I introduced the base .period is I want to consider the action on In a pardon fundamental group of the takeover that I wrote down over there so we have we have an action saying of the G 1 vectorone into or water P the appropriate base .period and so this means ,comma 1 union and because of this statement up here the map down to G 1 1 actually displayed so we have a representation and here
1:28:57
we can look at Lee algebras but I should point out this guy here remember is it's a it's a mixed elliptic motive so everything here is tainted the problems at the start of the year of the yet so I'm taking if you like the takeover code on on the desk and I II the think In Hodge theory I think of it as being the fiber over the tangent vector DBQ Q here on Galois theory you think images overstate Chubais hew to the 1 over in give me elliptic curve of this guy here so I need this is a morphism of McCulloch structures again this fall follows from an old result while so proven these Olympiad KCB nodes and so you get out of the map here from new rail the election rail near you get a map from new rail In 2 and the derivations on on on this page collected scoreless P and now I can take the way graded view rail entered into the way grated here and I end up in what's called day 0 on graded wakes P is naturally isomorphic to freely algebra on age so the white filtration for this Lee Albers of slow central series it's H 1 canonically age which gives us this thing here so I'm looking at the freely algebra on this is canonical and now I because GI W is an exact functor Ulick newsman lose no information by going from here to the associated graded this is standard punch stand much to be type argument we have the means to end plus 2 this here they go to certain derivations here and I want they're easy to write down I won't do it but let me so I tried to understand if the city's satisfied any relations and I should say before I do this this thisfactors through G R. W .period you ice into here and now the to end plus 2 I was going to go to a derivation and just to get the normalization right now yeah what would you some going to define Epsilon to end to be equal to the image as 2 times To and minus to factorial divided by 2 I believe the correct normalization so just a rescaling of these guys here what do you fuel but thank you so write to this mapping class group G 1 1 you can figure it's the mapping class group of a genus 1 Cove with 1 boundary component and so on those mapping classes really on the boundaries were actually using a tangent vector same thing they don't know where to turn the attention vector so that's what they it's hard to write down that conditions he would on the associated graded you looking at the derivations Delta such the delta of the book Bracken and that the 2 generators is equal to 0 so that's would dare 0 it's 0 means you Kilby commentator yes and so H remember has naturally as this basis and in the end right so when I started doing this which believe it or not was right in 2007 some undergraduate named Aaron Pollack walked into my office and 1 of the problems and for reasons I'll explain in a little part our series now postrock at Stanford got a Ph.D. at Princeton but I bet you just this young undergraduate and there are reasons to believe which I'll explain in the manner that these epsilon should satisfy relations 1 for each cast form of each degree so for example the cost for Delta will give you a quadratic QB Kordic and so when relations between these derivations because we expected them to maximize to hold in the mud to be case we couldn't prove it but they should hold here and lo and behold he found the relations Salemme right down the quadratic ones so he says the summer J. plus cash equal to end positive siege Epsilon to J. plus 2 Epsilon to came close to equals 0 In there L H if and only if there's a cost for please check Inc it's not quite ready except salute the money going quite radical and there's a cast form but as of late To
1:35:11
end plus 2 such that I our if plus of a AB 7 I'm thinking of this as the module symbol I always think of these things having meaning an all right down the definition this is the even part of the module symbol of therefore is the son Jay plus K equals and his Jane K. greater than or equal to 0 so this can have this doesn't have the highest water apart and C. J a to the tune J. B To this to end monastery so what's the modulus symbol so while there are they have of a baby is equal to the integral from 0 to apply at infinity Soviet Union enroll up imaginary axis of a major air and I'll remind you if I write that out that's equal to the integral from 0 to i infinity of after tale I'm actually leaving out effective to which as a matter of Bminus tale a 2 2 in detail by end I'll put to plug on to go the way I normalized something they'll be a power of 2 . it doesn't matter because the multiple of relations relations and plus so this is a polynomial and 92 element and are half plus a B is equal to the good part at the Times where on the degree the degrees of a and and B or even so let me give you a simple example the of sorry you know it's not obvious that have no because and if you have it doesn't matter it just every relation corresponds to a cost for doesn't have to be right so if I wrote down every relations quadratic relations going to be a sum of relations and you can find if you whatever the coefficients 1 to correspond I informed his right to express your reference that basis and it's true story others know you don't need to specifying basis I'm so for example if you look at the way the 12 Cust for what you get is easy for the 10 minus 3 times epsilon 6 epsilon was equal to 0 and I could drive the White 16 months in the but said this is just the 1st 1 and now so the question is on are these relations "quotation mark what I mean by that I mean do they hold that the corresponding relations hold between are the guys to and plus 2 hold in you G on 1 of the I should also mention he found out in some truncation of the derivation algebra he found relations of the but every possible degree for every cuss form all right so let me try to do this and that may be the next 20 minutes so I want the answer is going to be yes and I want to explain why so could it gives you some of the procedures needed yesterday right I couldn't write it down reunited write down perspective on but that is not yet they appear at a in so if you look at the paper of 11 Rossoneri they appear there they appear in all the work hero locking Nakamura and there In fact I'll tell you have to find not belter I want like and formula not good with formal if you look at dinner 0 village and you look at the power of the you look at grade weight minus 2 and minus 2 of this thing here it contains a unique copies of this to it I think I put it in the right place yet this unique opportunity take the highest weight factor so I got a written this up somewhere but but sitting in a drawer so this is that they just very natural may come up if you had to our writedowns divisions alright so I have to write talk about college even extensions so 9 we have forgetful functions you've got em E M and you've got to say I'll call them Hodge a medium you also got a landing and they're both faithful so for example here you would look at the underlying variations Todd structure he would look at the underlying the machines so what's so this story starts with maximum those exact and exactly so so what we have is on these induced ejected office actually on the Atlantic side I have to explain why I get subjectivity but I've got high 1 of Indiana and then I got Paul I 1 of Hodge some highly motivating 1 doing here and what subjects onto this is going to be Taiwan of makes Todd structures direct product with GIC and on the other side this
1:43:42
part I won the LAT In the end the correct thing to put over here it is L weighted completion all of the high 1 the M 1 1 over 1 of the did you so this is where the story started actually and I 1 day set down In the spring of 2007 computer this guy got a stranger that we don't understand and that was the basis couple weeks later I talked and Pollack and he knew nothing of model forms of freely algebras wrestle to but a week later he had the 1st form of relations it's my Allen said I so on both sides of the yeah so I'm going to talk generally in what I say will apply equally well on this side of the sort get the same answer that is consistent with various conjectures in number theory about but on both sides have and I assume office on H 1 of the geometric part which is the public trying to understand OK so I'm relations correspond to H 2 so the H 1 is 6 1 so on the hard side what you did is that and I don't have time to explain this analogous singer Miguel West Side that might be clear to and I'm gonna take half Infiniti this means in invariant on real for because everything is defined over diseases defined over our action and I think this is going to be isomorphic to direct sun direct sun F in the 2 plus 2 the next 1 again a lot of this is explained in this preprint I've put up an the the the Via Plaza OK he because they're they're actually real hard structures so I can write them down as these twodimensional real hard structures and I have to take at infinity and so if you pull a surprise and I should say this guy here is of this is our this is this is the haulage but this 1 here so it's got this is going to be roughly in in what I think I called M H S M 1 1 h except it doesn't matter whether I restrict the grated cautions here on so anyway What is this this turns out to give you all the things that correspond to the arms on get on with it this is a different strain ordinary even in here it's going to be our class and the sound of F N B To and plus 2 V F the class so the city this is that this is a onedimensional thing its corresponding to the past part of the module symbol and this is for our even and it's just equal to the direct some of the F indeed To end plus 2 the F minus this is for our so real for effects on this guide twodimensional has a plusone Eigen space in a minus 100 space and they correspond to the plus and minus parts of the module symbol already now appear in technical I am an only Delaware so it's a similar story limit is right the gal guy here at the end of this would be Page 2 therefore while page 2 of M 1 1 of policy which we have to think about what this means S 2 N H so it means on Ramaphosa if Israel added coefficients unratified outside Ellen I'm told was to be crystal and then hear this would be H 1 after all will all the begs the all of them Q L minus to in minus 1 and this 1 here would be 1 she GQ I in the if right to the Selma groups and you see this field at the spectral sequence maybe if I'd been a bit more honest I would put 0 1 of rail or something here if you look at the spectral sequence from this from geometric knowledge into this you'll see this is the age 2 so this is the Hodge analog and found that both work the same way and so make a little remarked here that I am not that these things shouldn't be 2 surprising because if you look at the realization that from X 1 mixed motives whatever they are before France's is favorite category triple and multiple sorry makes modular motives cancer IAF look at the these guys you went into the room the real X 2 1 makes told structures are the F. f infinity this guy is supposed to be a conjectural unites office and and you got the Atlantic 1 you've got text 1 M M Hugh L. the F sorry hear these guys is supposed to be hopes to anything in this basic engine isomorphism so the what what should happen here is that we should be getting relations this is this is telling us that this age 2 they're all these different classes and age 2 coming from and they should give you relations and this can be 2 kinds of relations and I'm only going to talk about 1 this is these guys here should correspond to relations of the form you can't z to
1:52:14
m plus 1 the 2 and plus 2 the should be another geometric quantity and these relations here should be relations between the should be geometric relations meaning their relations between between the cities so I so I'm not good at computing periods but there's this guy France's Brown I don't an N and part of this special case which says that I the cup products so I maybe I need a whole Blackmon to write a book the guns thanks 1 makes solid structure in 1 1 this Tuesday 2 J. plus 1 for this corresponds to a generator the geometric generator of the Union Poland radical and now another guy annexed next 1 I R S 2 K and then this guy's going to go into Annex 2 next to no variations of mixed torch structure object alders right the X 2 by what I have over there it's really next 1 restored structures are body the F of pain these generators of Romania's invariant so they will land in the 4 venues is invariant part yeah sorry find it's screwed up here right next structured standard being asked to what I have to fix these notes here so it's X 2 X to air it sorry is going to go through the /slash yesterday it does because of the guy here all right I'm but I'm gonna fix up where God here S to J cans as 2 K 2 and plus 2 J. plus because and now I you can project this to an expert that's the appropriate category here next 2 1st of all of our by I 2 and minus 2 are to end minus plus 2 write you this visit these the various projections because the tents a product of these guys it is I am not irreducible it decomposes some of these guys here and that this guy here is now you project onto some 1 makes Todd structure all of the are body you know V F and this is I'm subjective you got up with the correct twist and not be a fire the F. whatever subjective every time this group is nonzero right and so on what this tells you start just try to summarize this quickly stopping 1 2nd and I'm going to try to give a very very brief is very concrete explanation of what's going on in this is old so that strike ,comma logical argument I but the car Larry what's the whole point the car Larry public relations all 4
1:57:19
Paul agrees not just the quadratic wants and that the higher order once with a few exceptions welldefined and some quotient we didn't even know they lifted to relations inland .period Chanel lift to I relations in new Jeong a medium strictly speaking I should say in the way graded quotient of this right so all of these relations and so on so they are material and I remarked instead I give standard conjectures hold some of these are the ones that say these regulator match their eyes morphisms then G 5 the match isomorphic on onto apply 1 John a medium there's also the question of understanding the relations coming from Eisenstein's series it's being worked on so and now I could stop here but I would like to give you a very simple picture of wine politics 1st squad Reddick relation holds a meeting all this stuff that's abstract but it's a very concrete reason why it holds an stop right this was the only well yeah the 1 coming from the Council form of weighed 12 so so what's going on you know why going getting she has forms went from on this remind you cast performs work generators Of the relative completion this with the free guy but they become relations In New why why did this happen so long as drop picture here we have we have S to so inside you rail we have this is freely generated by a whole bunch of stuff that includes to for 6 so it is 46 press 6 E a S uh E 10 and we also have as I hope we have the Delta duel pretense 10 right so this is really equal to answer s 10 E F prime last in we have doubled from was a really concrete thing we gotta freely algebra and among the generating stuff we've got all this tough year this has wait watch the weight the W played office's minus4 minus 6 minus 8 minus 10 and this guy has weight minus 1 if that's confusing says Wade tenderness as white mindful of and so and in some sense we want work in the past 10 ice by so typical part of the LEA algebra so let's look here and how can we get things in here we can bracket this guy and that guy we're going to get something in the tense a product of these we can project on the highest wage we're going to get an answer 10 with highest weight vector before the tent plus we're going to get this with highest weight vector the 6 year and well looking in this as 10 fiveyear period is up if you stare at this you'll see you getting an extension this is a subset of makes torture actually getting some extension here he had matched V F the delta duel tenser as 10 and McConnell it is this stuff here now it turns out everything his eyes to typical as 10 Isa typical you can get rid of the S and 1 way to do it is to go the highest weight lowest weight vectors so you do that and you get an extension of what weight this this is in weight minus 14 so you gonna get something here which is on basically we've got times he 4 the 10 Q times it's 6 the you're going to get some different extension inorganic again Delta duel and when you put the correct way to and so on note the difference in the way here was 13 on this gives Unix an element of X 1 of Q the Delta 12 I 2 copies of it won't do you get 1 extension from this and you get 1 extension from that if you believe that everything here is not to make the motivated but typically this this group was only onedimensional so those 2 extensions have to be proportional and if you believe the extension is nonzero if you take a homomorphism into another algebra that since this to 0 if this extension on some element here is nontrivial and has to drag this guy 0 with that so basically getting relations because of nontrivial at extensions like this occurring between accounts for which goes to 0 and pulled out the junk with that because it's sort of inextricably linked to a that's stopped there so apologies for running time mm
00:00
Quelle <Physik>
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Titel  4/4 Universal mixed elliptic motives 
Serientitel  Les Cours de l'IHES  Spectral Geometric Unification 
Anzahl der Teile  18 
Autor 
Hain, Richard

Lizenz 
CCNamensnennung 3.0 Unported: Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen. 
DOI  10.5446/17060 
Herausgeber  Institut des Hautes Études Scientifiques (IHÉS) 
Erscheinungsjahr  2014 
Sprache  Englisch 
Inhaltliche Metadaten
Fachgebiet  Mathematik 
Abstract  Universal mixed elliptic motives are certain local systems over a modular curve that are endowed with additional structure, such as that of a variation of mixed Hodge structure. They form a tannakian category. The coordinate ring of its fundamental group is a Hopf algebra in a category of mixed Tate motives. This course will be an introduction to universal mixed elliptic motives, which were defined with Makoto Matsumoto, and a report on more recent developments. One focus will be on the structure of the tannakian fundamental group of the category of mixed elliptic motives over M1,1. In particular, we will explain that it is an extension of GL2 by a prounipotent group whose Lie algebra is generated by Eisenstein series and has nontrivial relations coming from cusp forms. We will also discuss the relation of mixed elliptic motives to mixed Tate motives via specialization to the Tate curve and the nodal cubic. 