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3/4 Universal mixed elliptic motives
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Title  3/4 Universal mixed elliptic motives 
Title of Series  Les Cours de l'IHES  Spectral Geometric Unification 
Number of Parts  18 
Author 
Hain, Richard

License 
CC Attribution 3.0 Unported: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
DOI  10.5446/16988 
Publisher  Institut des Hautes Études Scientifiques (IHÉS) 
Release Date  2014 
Language  English 
Content Metadata
Subject Area  Mathematics 
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. 
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b beam
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and the and a half the a and we want thank you you and you know the case so last time we were talking about 2 generations of polarized Todd structures and there will talk about 2 generations of mixed structures so all mixed with structures so the 1st thing we need to talk about the relative weight filtration so the difference with the pure cases we now have a way infiltration so the setup here is going to be the is a finite dimensional vector spaces Sandra Q on some field characteristics 0 and W . equals the filtration so for example is to be a mixed taught structure weight filtration and in takes the V W . into itself as an amorphous and so by this I mean that the you'll pardon and enters and open an amorphous moving preserves W so so lectures recall from last time so if you have 0 if you have a vector space burial colored eggs and you have until potent morphism say the figures of milk butter then from this you can construct a filtration which a local W. Fahey .period this is the the the weight all the and it has the property that I see of W this contained inside WK minus and it has the property that the the cat induces a nicer morphism between graded W K a integrated W. blindness cave and you can construct a cells the Jordan form it always exists and then if if I'm a head head it "quotation mark I'll call it yet W. equals W. fee so it's OK because if if free has say wake him so for example the Hodge structure of wait and see what we do we want to do is make center this
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weight filtration at em so shift To get filtration then centered at the weight and so in this case you would have in to the karaoke would induce a nicer morphism that should be a nice morphism here are graded M I am plus carriage of a integrated M and minus care of it again this will always exist so it's always exists so there's no restriction a toll on me and that in the but in this case here what we what we have is we have a we have degraded the said but W and of any risk going to take graded Mariel is a simple limitations I was going to have graded a map on the integrated quotient around right so this is just the induced map and this guy's no pardon so that means it has a weight filtration and we gotta shifted so centered here so we're going to let a M sub . be equal to the shifted so the weight filtration all of them N M shifted well let's just say center there we want a center them because this guy has weighed in right so what's the definition of a relatively infiltration await filtration all of em take me into the relative 2 W . it is a filtration m .period all of the With the property that the injuries filtration satisfying the 1st condition we want is that I the filtration induced on graded W end of the it is Our it is the family In . so in other words it cannot sell the correct way filtration graded quotients and to you also require that end of M K is contained inside and K minds too now so I and the basic simple factor so remarks is said 1 is relatively Phillips said not every
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and takes up the W .period into itself has a role to play .period filtration and 2 if it exists it's unique at relatively filtration this is easy relative weight filtration sigh unique when they exist so let me do a let me do a little example here and if graded W . 2 in there equal to 0 then you can easily check that and on exists if and only if I n of W K is contained in W K minds too and in this case I
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am daughter equals W . and so it's easy to see here maybe I should give an example of where does not exist so I could take care the equal say Page 1 of some code of relative to say X 0 0 and X so see is it will take 6 0 and he is the next 1 and so we have an exact sequence I guess I was and so this is isomorphic to say Q and I suppose I looked at the following an amorphous Simitis take X and I let it travel around Olympia's say gamma so that'll that you'll get a corresponding Alpine in a morphism visits can be trivial here but it's going to be trivial on integrated quotient that the animal will the nontrivial so it can't be you have to give this away filtration so this is going to be say this is going to be equal to W 0 and this guy he W minus 1 then the standard way infiltrations on in the situation this guy does not have sigh so let x move around gamma obvious sloppy so this is an image 1 body equal to 0 image 1 of C and then use in it was log of the automorphism the corresponding automorphism and I and is not equal to 0 gee I W N . is equal to 0 but no no relatively filtration it's a silly thing because for there to be a relative weight filtration and would have to lower weights by 2 different load weights by 2 it would be trivial so and there better example I didn't think of 1 right to its talk about 2 generations of mixed .period structures and so this is going to be very similar to generations of structures I have so the input it is 1 we have a filtered local system of over some base undertake the base to be onedimensional for simplicity you can do all this and higher dimensions but in fact you can test the city something's a variation just by restricting to a code on that increases a filter local systems sisters cumulative and let's assume the dead and the local wanted drumming so Michael couldn't see will be an effort well a possibly a fine coat of this will be say projective so soon the local monitor His unique if this is true announcing this will be true in our situation always true after a finite based change this is the 2nd thing we want is on a hunch filtration you know it'll be it'll be away for the moment just a filter political system India and it will be the weight filtration hard filtration off the and this is defined to be Soto "quotation mark done so that extends to a filtration twice sell bundles all the the canonical Lange canonical extension I think so so what this is giving us as in the case of variations of Hodge structure if you have a history coevolving here and his son cost at the it's giving you a vector space you associate with that that cast and now and we're going to have a hard filtration here it is here by Hall Norfolk some bundles that will cut on a hunch filtration on each of these following and the weight filtration is going to behave well and so this and this guy here will have a natural flight connections there characterization of the extended connection isn't it In followed from when I say here so we want on so this year implies that the residue at pH of Nablus is no part will be enhanced and we also require that the connection as regular single points Nablus takes the bus into the bath tend watch the blood cancer and may go 1 city by logging you're allowed to have logarithmic singularities so this this To say that that the economics of the extension of the is just to say that these 2 profit hold "quotation mark so we want this to satisfy Griffis transfer Seldin so far so this means that if you take a section all 4 F P V by new differentiated you get something that's it was 10 in F P minus 1 of the got cancer amigo 1 see for all people OK so the next thing is the obvious condition here we needed fiber by fiber we get makes what structures so each fiber of say it is a mixologist tractors so it gets its rational structure from the key local system and so with it with induced
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Hodgen weight filtration From this adjustment and now the next condition we want is actually a nontrivial condition is that the bridge P In the past let's just let canopy equal to the residue P all the connection to this is no potent we assume that the residues role no blood and on by the way I should say it to the naturally construction implies that all the white bundles extent through as well as the W .period extends as well but this this is not a hypothesis automatic and so the condition we what we want is that that the GNP so it takes the fiber of the canonical extension of review with the weight filtration into itself has a relative weight for fluctuation I tell you what the relative weight filtration to exist end of 5 instead French this In the change in space of key policy but this guy not equal to 0 no it's all colored VVD so explained how to construct out retained in vector accuse structure on the fiber of the canonical extension so this is this is the Q structure on V. P and then with the who the White filtration the Hodge filtration sorry I want take on daughter this is the relative weight filtration and F it is a mixed Hodge structure filtered by W . filtered in the category often extort structures so a more complicated structure and if you go back a good exercise is to look at the definition of the variation Hodge structure In that case there's no W . 4 the W . trivial and then you just have an indoor what's going to be the mondro reweighed filtration so anyway so this is a basically the axioms of inadmissible variations such a thing as cold admissible various the fact that he was very physical and yet because the monitor of the Mott the local wanted Ramiro preserves W . into its unique poet which logarithm world the logarithms just Apollo 9 the monitoring we would have said during his long hair rights look at those that because because the local system was filtered right so each W M of the use of a local system you can take its canonical extension the naked natural just implies that that just gives you a filtration of the canonical extension and so the local modern the pertinent and so it's longer than the snowplow and will preserved the White filtration because the modern drumming because the filtration is by flat some bundles will by some local systems and I suchandsuch a 80 of whatever a V W . in half .period etc. is cold inadmissible variation and this definition may look very strange so the question is is it natural nuances of sound so when you look at this condition it looks very strange 5 when you try to construct these things if you're in the business of constructing it's usually not a problem to construct the and just comes out so and this will become clearer when when I discuss these examples so 1 is so this is esteemed Rankins aka by the way I think the definition of of the relative weight filtration is made by deleting invade too now I'm sure you guessed it from I haven't looked at country guess that from the Atlantic set up the 1st touch theory case where this was considered was by biased reconsider and they showed the local systems the savings with suppose I have an X of the sea and this is our family of smooth but not necessarily projected smooth varieties and I want to be by family I mean topologically locally trivial and so in this case here local systems are flow of and every fiber here has a weight filtration delaying construction of a so here I'm assuming Prairie tenancy in next season's smooth variety to its ecology has a Mix structure you can easily see the white filtration slickly constant and down these guys are admissible variations of McCulloch struck this is also a good source for relative weight filtration Rodale wanted the details and then this the piano no narrative like Puerta I think this is my memory stated the case the more general case I X overseas here but this is just a failed topologically locally trivial family all of them complex varieties so not necessarily smooth not necessarily I when they can be singular open whatever on the of the game here you'll get the same result I K airflow Q this inadmissible variation of export structure the subject love abbreviations a mail another 1 will mean 3 officers myself is that you if you can look at the following situation so we will take a section here for Sigma so this is appointed family also varieties so again family just means topologically lovely trivial here I think so here you can look at the
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local system so you can look at the local system may be all electrical at P so over tea here you would look at Part 1 XT and you would look at it with the base .period segment of the new take tuneup by completion and maybe take the queen wouldn't bring about overly algebra of still need this local system here this so that this is actually in general infinite dimensional but it'll be a proper thing but this is inadmissible variation of will be interested in this way and this is the universal it because of our minds and 0 section was really just a section of tension vectors along 0 section start with the board you can pretty well this is by P I mean the local system fiber over tea is the unit of the fundamental group and I will eventually give a quick definition of this I'm going to say something about tonight in business and a little bit His 1st in the interests of efficiency and basically this is the best year poem approximation to the fundamental group of the could do whatever the rights and now I I can define universal Nextel acknowledged finally and to do that I needed the notion of a on variation of fixed on structure list and relatively filtration and so on and I thought that would be a said that was a bad place to start 1st lecture hall with so dollar remind you this is joined with Lakota medicine OK so I'll be very briefly recall the there were other rough definition the words of probably make a little more sense so what is a universal so it is a compatible said all filtered local systems and I'll be more precise about this later but for example you would have daddy queued around this together gives you Hodge and you would have elected so here you did shave I and there are 2 basic conditions are with G I W . yeah maybe all call it and basically gonna a call at V W . 1 and with Jiang W . of V isomorphic to a direct sound all up S & H R so wages Giselle basic local system and it wants something so when I say Hodge implicit in this I mean that I didn't admissible variation of fixedodds structure and then I also we have a filtered I make statements others the and so let's call it the and it's got a way filtration this is this is the weight filtration in MGM and it's going to be filtered by by another filtration W . in and the right so it's going to be a filtered make state motive and the mixed take narrative weight filtration called that's because it's going to be a relatively filtration such that the fiber all of the these are over the integral based .period DBQ is I is V and daughter WE door song explain what this all means but just this is the other view it so let's look at the official definition so a Universal mixed elliptic motive 8 it consists of so it's best to start with the Mix state motive a filtered object viii W. John but also make state relatives of disease I and denied the weight filtration all of the while I enjoy so I did that to drive France's knots Everybody notes 2 a representation Road which takes SL Tuesday In 2 the automorphism all the baby realization of the and it preserves this guy here and this guy I should point out as I hope I explained clearly before this is naturally isomorphic to apply 1 of M 1 1 analytic With through with the base .period Didier Cuche you can lift the base .period the tangent vector DBQ to the upper half plane by as the imaginary axis so so what you know ,comma near while you only need the germ of up near plus infinity in Surry Hills New which the last 1 by a statement that the leaders but what if it's part of the day it's part of the definition of the consolidated his money from the Kate let let me make well now you could have all sorts of other representations of cell itself to z is virtually free to take a finite index subgroup it's freakin map at just about anywhere that'll give you a local system on a finite covenant and push it down the history of the United News of India I can construct ones that come from geometry sorry and I want to stress
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note that we're not allowing things like SNH tinted with some simple Hodge structure that's not a queue of our right this this is a different category it's going to have a different To lacking fundamental group and I'll try to explain that later I mean this is a bit of a surprise to me I so what I want but if you assume that those there are you can hear the parliament like the hostages at the end of the meeting I'm sorry it implies as to be mix tape but it doesn't imply that it's in object of them you know that was the year of now you can get just taking a mix tape motive that's not in the best periods that animal so you just take mixed hard state structures periods on Maltese a mail to the constant variation of rent 1 1 without its fiber that won't be in this category because the limit will be that constant hard structure but it's not the Hodge said the Hodge realization something that's in all so I'm wary of mostly black which all adjusted this summer up to erase this go ahead right now so the losing most and hide in here I said somewhere I think said it it's an object all see not just someone structure that of tight is greater caution to replace the outermost region conditions so you can then say well other an interesting examples get to that in a minute I am so I wear each G I R W M all of I just lost track of where his on well it is right so so this this will give us this will give us here a local system B W . 4 over and 1 1 analytic right and we want each it is save some this is just as a local system all of this I N H the hearing various end where In this congruent to I am not to blame so just as old topological local system we see if I can get into the tune I and now the next thing 3 a filter vector bundle I'm W . over on 1 1 5 over Q so this is going to be the kid around With connection now below which takes into the chance a major 1 M 1 1 of the 2 Long take and this this guy here's the cost the cast Q equals and the filtration unit Hodge filtration such that the connection satisfies Griffis transfer salad I think so you only lose 1 degree and and the residue the head of the of the connection is will actually I I needn't have added that that's going to be automatic and so now we can combine this idea With a flat connections if you go to say we won a flight connections and so on and we want the obvious ISO officers of acute around set up so what we wanted to freedom fee W died I Nabila and we with see this and we restricted to M 1 1 analytic understand 1 1 see this should be isomorphic too and it's natural flight connections right to this just giving a cute around version of the vector bundle underlying this guy here you have any together they they form an admissible variation of the Yukon structure in 1 1 and so In the last part the Eleni part In fact can probably leave out some of these pieces still when I think about I think some of the existence of summer consequences of the other parts 4 5 so the representation Rodale which is going to take part I 1 of M 1 1 of a Cuban In DQ into order field so L is going to be the ln it realization of the mixedrace motive induced by roh so this guy here is naturally isomorphic to the pro finite completion of a sultan's the this a cell to the head so on so the representation from row on the betting version of this land use of representation here
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we want this to be GQ variant so the group backs on this because this is based .period here's but Q rational and also acts here because the Eleni realization of these uh Gallois module yeah right so that's the end of the definition right so a couple of quick reminds me sorry for 5 years in the Senate you need people to know that we started with the mix tape motives and make state motive has analytic realization this is the elastic realization that comes equipped with a Galois action will what is the source ,comma well because because this is a variation of weighed 1 and I mean is this conditions forced this saying this guy a variation of the same age is a variation of wage and and then we take twisted the parody of the White doesn't change anything easily proves that because this is simple local system can occur as a variation .period structure the way because of variations unique kept to take twist she can't couldn't even power in an odd way so so the remark here on display can obvious from like you can do something similar there are a few technical difficulty with their own so 1 is similarly can define what makes still the motives of say on 1 hand plus you have to choose the Abass pointing at the show to independent of the base and basically 1 choice and you can also do it you can also do mixed look motives of things like and 1 In plus our and you can take a level and on land structure and maybe you have to walk over some bring of integers and so on this guy here I the notice categorically by the can't but in plus are the many samples again the very simple the category I'm calling and medium is really be equal to a and B and 1 that's the 1 thing talking about the dignity of the tension vector the asleep in the samples so on so the first one is these the this is the simple In any of these categories this of the simple dollars young simple dollars it is you can just take as in page are the maker of already showing you that these guys to generate to direct some of statements of this and we've already verified this In the 2nd example local geometrically constant so they suggest you just take prior just take a V In MTM and you just look at that end up pullbacks so I think of the mixed money was being a local system use pulled back here and Libya as a variation of Hodge structure will just be a constant variations make solid structure and as a leash she have trivial monitor the on the geometric part of the fundamental group will just be a Galois representation In case of these guys 1st example In the actual toll rose slightly more general situation on the the of Polly logarithms all Beilenson 11 so be it their 1 thing I think they called elliptic Polly logarithm but I like to splintered into pieces so this is some variation this this is some variation safety he on the Universal Akiko we could also defined mixed elliptic motives here I think you can pull back along 0 section To get something over here so there restrictions 2 0 section give did you get the simple extensions which for form as to NIH will in a bit but these correspond Eisenstein's series a nontrivial extension that corresponds to and the normal G and close to the Eisenstein's series and these again to be the most basic these old simple extensions so I like to think of these as the analogs you know if you think what we're trying to construct an extension virility because if this is the generalizations of the values of the Riemann zeta function now that you get in the theory of mixed take notice Tunisia values so 4 and this this example is important for us said let's look at 1 1 China look at anyone vectorone here I have to look at that so of a typical element of this guy here would be an elliptic curve plus a nonzero tangent vector and gonna look at the local system again I'm going to call it and the fiber of this guy over here is when I'm going to pull Pete being the so this is not a primal pointedly algebra and so he the is defined to be the LEA algebra of Paul I want you to be pardoned Of the elliptic curve mind its identity so this is a minus 0 the so this is a freely algebra so this is this guy he just give you some idea is isomorphic too the freely algebra on page 1 of the completed this is not natural In the wake filtration albeit slower
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central series so and so this is a proud and here this is 1 reason I need some decorations because I need base points so you want a coffee break currently going many of the 1 I throw something up a little more than we can take 3 of sorry yes well these things again occur everywhere and so yeah they certainly occur inside this guy here the from so I the next thing is so 290 fundamental group also in the you can also do this some decorations here they're very they don't change when you change decorations this guy the fundamental group of this guy changes and some predictable way so I yet so all this stuff with a remark Is that it the categories M P In plus ah ah of this means that the carrier Air of representations of sound the following the group scheme or the same it's also a pro algebraic group and and this leads us to the following problems computed Paul I won of so let me stop here I'll start up after the break will do a very very quick where may something's about to knocking categoric and infect an old rigorously constructed completion in a way that Francis sinks is eliminating that I think otherwise ruled morning noon and and then I will go on to discuss this problem and in fact we know a lot we don't know it we can't completely computed you run up against some standard conjectures in number theory at some point but we have very good evidence that the status of presentation there we can write explicitly so was older than after the break so this will be very brief so but at the close of field of characteristic 0 and I so unusual category of a F it is a rigid abelian tends category city but let me also say that about this where did this a trivial object which will deny by 1 is isomorphic to so attention category as attentive product and here it's going to be commuted of associative and so on all sorts of actions have to be satisfied these are exactly the axioms the dissatisfied by the category representations of a group you have an associative tenser a product and so on on and it has to admit an exact and faithful From and they usually denoted by major from into vector spaces into finite dimensional this is going to be finite dimensional vector spaces over over half not that preserves tainted products so so the 1st example 1 G is a groups and you can look at the F representations of gene so this is just representations of Jeong a finite dimensional vector space category called finite dimensional Jean modules find dimensional however G. modules and to make state motives and mixed elliptic motives In plus and I should say here the trivial objects here is just equal to Cuba the trivial representation should the trivial representation here the trivial object 1 just 2 0 0 0 another good example is next time structures and here the trivial example it is our cue from 0 again and for you can look at Mix hard structures X so it's equal to the category admissible variations and so Xia assessment variety and so on the basic fact we need is that I should say such a functor school the fiber functor and they can be many of them the basic factors that therefore a major takes city In 2 vector is a fiber functor His any fiber functor so that's 1 of these exact and faithful functions then so let's see is equivalent to Red where she Is the order morphisms the 10
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steroidal morphisms of America In case so 1 what Watson automorphism here these unnatural transformations show should say natural From major to a major and that preserve that respect and support and so I will do no so basically it saying that sees the category representations of this group this group here is going to be it's are fine it's also pro algebraic probably near algebraic actually and so our goal is to write this group when the category just a media mix elliptic motives and so we gotta do notation Is that G is equal to Paul I won the With based .period this fiber functor so let's do an example of a more interesting example I really pardon completion said How do you define the potent completion of a discrete groups or 1 way is news to knocking categories I'm sorry discrete and so the equals of field of characters to 0 In this case wouldn't take To be the category all finite dimensional you need guidance represented representations of and these old defined over there so there 2 ways to say we're unified representation is it's a homomorphism of gamma 1 way to say it is so this will be contained in G L and F it's a representation that can be conjugated into are many potent matrices you could also say just a representation that at that admits of filtration people's feet 0 contains the 1 contains the 2 and so on where when buy Damaso models where each VGA monitor VJ plus 1 is a trivial gamma module that's a more toward a free way to say it it's just that that you can filter that the modules by some modules and all the associated great it's just trivial modules and now this category is tonight so on major see into effect f it just takes a representation goes to its underlying vector space "quotation mark clearly faithful clearly the exact and so on the definition of beauty Foden completion is definition is fed up the gamma you need over is defined to be Paul I 1 of this category with respect to this here with respect to this fiber affronted if you're as good at enacting categorizes Francis's you can easily walk with this definition there are others maybe I should say something just the all remarked so this isn't all the way of thinking about it but 1 way it is you can if gamma as finitely generated so that definition there does not assume the group's findings generated but you can look at their game which equals the group algebra and you can take out this 2 ways you can go here 1 is you can say what is of gamma you need posed Nova so what's the quartering of this and it's actually it's equal to the I'm home continuous so what does Hong continues here so you've got an augmentation from if gamma into air the standard augmentation and you've got an ideal which is the become of Exelon which is the augmentation and it defines a topology on despite the perils of this ideal so this is just so we can give the CIA topology and so this is just also the direct limit both the therefore gamma log or lied to the ever and I this is a hobbled ridiculously check so and this moralize comes out Quillen Long time ago Quillen wrote a paper on rational homotopy theory and buried in the appendix is the section on what he called melts of completion but now basis Golding potent completion yet another another way to say it too is the unique if you take the universal enveloping algebra affiliate algebra of Part I of the gamma you need pardon over after this is just equal to if gamma completed and this is just the inverse limit I should say the completed enveloping algebra of of F Gammon 1 hour into the and so I'll problems it is to compute applied 1 all and he E and his sometimes all be precise about the fiber functor because of matters other times it doesn't matter because I'm you always get the same group you could just get different uniforms of the group if you different firefighters you get the fundamental group try some offered to an isomorphism unique up to conjugation I think so but let me start with generalities so I am so relieved most of this is very soft and wood with 1 exception I'm the 1st the 1st taken as well we have a fighter from makes Tate Modern's into next public acknowledges this just takes a mix tape motive to the constant the geometrically constant things this is we've got em 1 1 and we mapped down to suspects and you can
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take a mix tape motive over here and pull back here to the to the geometrically constant guys and so this gives us come geometrically constant so this gives us a map from Part II 1 of em Inc L 2 Paul 1 of and T M and this guy is you can argue directly the subjective but it's easier to see you also have a font of from mixed elliptic motives In 2 Mix Tape monitors you this is just the fiber Oliver Didier Cuche you think the definition of a mixed elliptic motive consisted of a mix tape might have passed all by other stuff so I 2 years take That forget everything else mistakenly detained monitor and this is going to give us a splitting of this guy here so this is going to give us this will give us I 1 of mixed motives In 2 part 1 of em and and this is if you think about is clearly a splitting of this because if you take a constant geometrically constant mixed elliptic motive it's on the base .period sends its fiber everywhere so so I while we get what we can do is defined Paul 1 geometric all mixed Libyan motives to be equal to the Connell Paul I won next Olympic motives mapping to Paul I 1 of mixed take I should remind you of what this guy is here so we what we do is we have an exact sequence pie 1 next Olympic motives it matched applied 1 of mix tape motives and gotta be subjective the because analysts Paul 1 geometric also make still looking motives and we have a section here this section is given by the base .period didn't you In fact we can compare this with we have so over example the tell fundamental group of M 1 1 100 it's compatible with this it'll be maps discuss that later right just on just recall here hi 1 Sofia look at Part II 1 of mix tape motives it maps to GM so this tells it this is basically the fundamental group of the category of Split hardens these direct sums of Q events and the colonel I like to call it Kennedy and is 7 this guy here's prounion potent in every prounion potent group is isomorphic to its algebra the exponential so I can a little K will be William of K and this guy is isomorphic to freely algebra on 3 the 5 disease 7 on all things here that cuisine online complaint completed been staying power series Lee power series so on proposition it is the 1st statement is that and there is a natural homomorphism from S L 2 disease into the geometric fundamental group of next elliptic motives and here all the more precise I should be the firefighters the baby fiber functor and I'm going to take the cure rational point of this group and so the baby fiber from 2 takes the fiber so a major Betty takes V the mixed elected motive and it takes it to on the fiber of this already DQ then takes the Betty realization of and to it is too risky debts so this is the on this guy here's the army this is not bad but it it believes that everything else something as solve this requires some water so what's Fleming sketch the proof to proof of 1 is so we have moved we have we have flunked his from Mix Tape motives we can take it to the next elliptic motives the stakes mix tape motive to the constant geometrically constantly extorted motive and now here we can map into representations of S L 2 0 so secure representations sorry quality of course yes well I mean if you remember the debt the way I wrote down the definition there was a V in Eldorado and a whole bunch of other things I'm just taking it to the road and now a few look at what happens here so the fundamental group of this won't be a cell to Kewell's and be much bigger but there is a natural map this this picture will give you a natural map you've got if you took a major beta here would take effect Q and then just take the forgetful funk here this community so you get a map in major said and you get connected to him and why does it land in the geometric putt well if you continue on Paul I 1 of mix tape .period of thank you but while we know is this map here this is trivial it takes everything every 1 of these guys goes to the trivial representation and that's telling you that this year's trivial which is telling you that you landed in the geometric fundamental group so this but is completely solve the 1st solution to so what sort of got an old version new version of this page years to get deleted and so this follows from is news and the theorem of the 5th spot what this theorem
1:24:26
says in this case is about if you have a variation of fixed structure which is trivial woman from me it has to be constant which implies that there a fixed budget says that if you have any variation of the sun exits is 8 0 affects the match to say the fiber over X This is a morphism of mixed on structure and you can use that to prove the following statement implies that and variation of mixed Hodge structure With trivialize drumming it is constant and so this implies so if you think about saying If you map here it's telling you that this is a risky closure of this has got generate inside the normal Clojure if you if you look at the normal closure of that the SL Tuesday in here and take it's a risky closure of the whole thing so this Subpart also relatively soft and this implies that but they Mazursky closure all of the normal closure Of the image of a sultan's it is but we have to see that the major vessel Tuesday the closure of its normal and the way we do that is you have to use the fact that so the assistance so this is the nontrivial part the existence of the limit makes taut structure on the relative completion which I'm about to define implies 50 years TV curative basis .period implies that this year the the jurors the is normal In right and so this implies the written put these 2 things together you see that the so this will see then jurors the closure of the West inside part MEN this part 1 Jerome you don't like that I've got a version of this review very this is and this is important because general ousted ban on the size of the colonel OK so we got to call a mixed elliptic motives last if it is a direct sound all and page there's and simple these are the simple objects of mixed elliptic motives and so display it was the single if the situation were lose a guy of sauce this this is the same as simple makes 1 said the same is simple said so simple next elliptic motives His it's not entirely .period it's a subcategory of the whole thing which is clear and then apply 1 of semi simple mixed elliptic motives if to think about it because just by some to jail Ford's actually equal to jail agents isomorphic to jail too this is on you may wonder about the tape twists but if you look at it if you look at for example jail H that's the division section the inverse of the determine the tone the look at this representation in Jeanne this representation corresponds to queue of what 1 that's because we put a chain weight plus 1 but with age and weight minus 1 it would correspond to what we know so far right so now that we have so we have we have Sammy simple mixed elliptic ignited this includes intermixed elliptic motives so this is going to give us a up a map on fundamental groups the other way we also demand that because we can't take G I W . right if you have a mixed elliptic motive it's associated great weight graded isn't simple and so I want this gives us some this gives us we're going to have I 1 of em leave em and we can use any fiber functor mapping into I G L H and study the subjective because we have a splitting and splitting is she W . and so on they'll be a colonel and mobile see Sooners that that Connell's prounion poet and will say so remote so said we'll see that the cardinal I am against stands to reason because the call should be telling us about extensions and very emotive Djukanovic extend stuff of lowered from Ingalls even a little more direct way the Connolly part 1 of mixed Philippine motives In 2 G L 8 prounion OK so let me know and I'll go up the issue relative completion Celeste selected the healthful relatively poor In this idea is unlikely in the 1st person to write about the idea came to me in a letter from doing so may be sent
1:33:49
from this very place so this is a generalization of Unity pardon completion and a half years equal to field of characters tire is equal to a reductive and you don't have to take it to be reductive but this now a real difference from under the plan example here are equal to the trivial group this is going to give us a unique but completion so this is just going to generalize what I did earlier here are we're going to have road takes gamma 2 the effort points of our is a risky direct precipitation now I'm going to take see it's going to be the category all the finite dimensional representations the of gamma what selected Gammon here's a discrete group I hope that Denmark was described In all of this is also a pro finite and all of this to his use you for a long time this is 1 of them this means we have to worry you have to get elected but I can't take it I can take what's the fiber founder I can take it to be graded him here yeah I have to rewrite have take graded I would say that we will in of user or you just got no I think I think France's rising 500 but I can't take the weight graded but the front but I can have any any firefighter the go this way to get 1 back this way the analog of the picture topology have got on base .period hailed got a base .period here we may choose a section this a different place .period so there's a F representations of that admit the filtration to this filtration the way I'm setting this up this doesn't have to be you To some filtration on 0 equals the zeros contained in the 1 contained in V. it will fail I like themselves modules on such that she the Associated graded as they are on his module safeguard it got an action of the algebraic group the reductive group associated graded In gamma acts while on the G I don't the virus you'll have Indiana will map into and this Exxon the G & W and so the action on the group all the graded portions really come from representations of our and the way damn Exxon's vise this rational representations of so this is tonight and so on there's an obvious fiber found just take the underlying vector space so the relative completion it is all colors G right is defined to be applied 1 of this category and here this is the obvious fiber from through so as important information about this group is Proposition 8 is out 1 it is John Burrell it is an extension but 1 into what I'll call you rail in Taji rail Inter are into 1 where you realm please prounion pilots machine case I haven't said prounion pardon this means an inverse limit of Union pardon groups he will hold you to work rail lines right rail here for Realtors completion call me again I doubt isolation of the situation Huan Huan a move it back here with got 1 group that we care about matches still choosy so sometimes you go with the independence of the group in their there is a natural homomorphism From Alabama into and to be accurate I should put air freshener .period it is a risky debts now if you have a we are here all the states and things just leave that blank there and Sosa relax it's that I surprised if you is prounion pilots well maybe I'll start at if if if you it is unique buttons and you as equal to the quarterly algebra view then you can easily check that you continue to fulfill the exponential maps polynomial it's a polynomial by ejection and the got a welldefined logarithm that is because all made every year he pardoned group can be moved inside this guy here and then it becomes clear so if not you have a unified group this guy's going to be mail buttons if you it is prounion patterns still true every every prounion pardon group is isomorphic to its Lee algebra by the exponential logarithm maps that means to know this you deserve to know that and
1:43:20
3 and I can explain this in more detail later on and if it is primal pardons you has presentation has a minimal presentation you is isomorphic too the freely algebra on its H 1 of you you have to complete this there will be a map there will be an injective map front page 2 of you In 2 now let's call it safely see intellect ,comma of algebra and you take a look at the image of the city taken close to ideal that generates isomorphic right so this is again I I write versions of his injuries papers but this is just some generalization a result of Stallings but it's not canonical but basically you know united in a regular Group H to conserve the bigger than the relations somehow now this a tight this a tight relationship between minerals that generators for the relation ideal an age to write so up here working them the 3 so we care about what H 1 and H 2 hour of this guy here and so for all I'm modules the by that I mean all rational representations of our 8 1 of you tested with varying take our invariants is isomorphic too H 1 GAM IV and H 2 of you intensively Our invariants injects into aged 2 the government With this here so now arm so this is telling us something about the presentation and some may be a good place to stop today it will be called to the example of a cell Tuesday so will have a lot more to say about it but let me just start with the basis so we know that the relative completion of SL turns me is that what you call me in the laughter of 1 year this is this is colleges 7 what's in singular homology in H 1 ages you know the colleges the dual ophthalmologist In this theory you have to take continuous tools so the H 1 assumption of thing and the H. Lowell 1 will be a pro thing and so the big thing is that homology so let's just look at work on the peaceful use the use of your U.S. you know so so example will take that f equals Q Gamma is equal to the S L 2 Art is equal to S L 2 over Q and road takes gamma wrote this just the inclusion cell Tuesday In 2 S L 2 Q you might think the completion is just as Celtic Q it's not because this is a group of rank 1 Real rank 1 so it's not rigid and so what we get here is that 1 of you all destroyed the EU on this result says that this is isomorphic the direct song you any greater than or equal to 1 H 1 of S L 2 yes 2 and a half tension with and I like to write this way asked to and age tool I know that this to NH duels systems eschewing but I wanna ride it this way and and the next thing I know is that s fell Tuesday is virtually free that's easy to say because I'm mean actually standard because you can choose some torsion free subgroup of SL Tuesday on the upper half plane divided by that isn't on compact romances therefore has 3 fundamental so and this implies that age 2 a vessel to Xavi with any coefficients if this is divisible you know this is a key module the 0 and they when you assemble that with this statement the statement here that tells you that age to you 0 sigh so this is telling us that age 2 of you is equal to 0 but this discussion I had been here of age to a 0 that same year was so therefore and so and so therefore you it is free OK so we know unnaturally is said so this is saying that you this use isomorphic to the freely algebra the direct sun H 1 S L Tuesday s to end age so I left out for I look at the audit powers because the college you don't on pair vanishes by because the center wrestled Tuesday X nontrivial was a standard I got Mariel explained in more detail next on 2 1 1 argument that I wouldn't take duel of that intensity with this to NIH so we're taking and it's something I'll explain a little bit more detail next times SL to acts on this and the claim is that we have to complete it and then to a complete by taking the inverse animal finite dimensional portions to this that have Nestle to action and then I G His unnaturally isomorphic too S L 2 send me direct product you and then the final comment is that all pursue next time is like Alicia Mora tells us that these colleges groups here basically just module forms so you're getting a copy all of this to an age for every for every normalized form and you're also getting another copy for ever the complex conjugate of each normalized last fall but I'll discuss all start next time we like Lucian Moran and I'll explain in more detail about this and then there's the fear and that says the quartering of this for any basis .period including I didn't care a based pointer then 1 1 has a natural mixed solid structure and the Hodge representations of their admissible variations it's all stock
00:00
Finitismus
Theory of relativity
Generating set of a group
Multiplication sign
Characteristic polynomial
Gradient
Algebraic structure
Weight
Bilinear form
Category of being
Morphismus
Vector space
Large eddy simulation
Oval
HodgeStruktur
Figurate number
Field (mathematics)
Subtraction
03:52
Morphismus
Category of being
Theory of relativity
Divisor
Uniqueness quantification
Quotient
Right angle
Weight
Limit (category theory)
Condition number
09:41
Thermal fluctuations
Sheaf (mathematics)
Weight
Mereology
Cumulant
Glatte Funktion
Mathematics
Casting (performing arts)
HodgeStruktur
Fiber (mathematics)
Physical system
Logarithm
Theory of relativity
Spacetime
Gamma function
Generating set of a group
Structural load
Moment (mathematics)
Automorphism
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Vector space
Hausdorff dimension
Mathematical singularity
Right angle
Bounded variation
Resultant
Directed graph
Point (geometry)
Variety (linguistics)
Heat transfer
Canonical ensemble
Regular graph
Theory
Morphismus
Goodness of fit
Natural number
Subtraction
Condition number
Algebraic variety
Standard deviation
Algebraic structure
Axiom
Residual (numerical analysis)
Field extension
Exact sequence
Quotient
Fiber bundle
Game theory
Family
Local ring
28:18
Logical constant
Fundamentalgruppe
State of matter
Sheaf (mathematics)
1 (number)
Complete metric space
Mereology
Weight
Group representation
Casting (performing arts)
Plane (geometry)
Manysorted logic
Stress (mechanics)
HodgeStruktur
Fiber (mathematics)
Physical system
Theory of relativity
Infinity
Geordneter Körper
Price index
Automorphism
Connected space
Category of being
Arithmetic mean
Vector space
Lattice (order)
Right angle
Bounded variation
Imaginary number
Trail
Finitismus
Existence
Heat transfer
Tangent space
Congruence subgroup
Frequency
Natural number
Knot
Queue (abstract data type)
Subtraction
Units of measurement
Condition number
Algebraic structure
8 (number)
Limit (category theory)
Approximation
Subgroup
Ellipse
Residual (numerical analysis)
Vektorraumbündel
Universe (mathematics)
Object (grammar)
Local ring
Nearring
46:55
Axiom of choice
Group action
Fundamentalgruppe
Existential quantification
State of matter
Multiplication sign
Sheaf (mathematics)
Complete metric space
Mereology
Group representation
Mathematics
Manysorted logic
Analogy
HodgeStruktur
Fiber (mathematics)
Physical system
Logarithm
Rational number
Product (category theory)
Closed set
Sampling (statistics)
Bilinear form
Functional (mathematics)
Category of being
Vector space
Funktor
Order (biology)
Module (mathematics)
Right angle
Identical particles
Bounded variation
Directed graph
Point (geometry)
Finitismus
Divisor
Presentation of a group
Variety (linguistics)
Theory
Power (physics)
Tangent space
Energy level
Integer
Field (mathematics)
Condition number
Series (mathematics)
Standard deviation
Element (mathematics)
Algebraic structure
Axiom
Local Group
Ellipse
Elliptic curve
Field extension
Number theory
Charge carrier
Object (grammar)
Local ring
1:05:32
Discrete group
Logical constant
Fundamentalgruppe
Group action
Beta function
Multiplication sign
Direction (geometry)
Sheaf (mathematics)
Propositional formula
Water vapor
Complete metric space
Mereology
Group representation
Heegaard splitting
Matrix (mathematics)
Conjugacy class
Positional notation
Manysorted logic
Homotopie
Fiber (mathematics)
Rational number
Gamma function
Power series
Basis (linear algebra)
Automorphism
Category of being
Proof theory
Funktor
Vector space
Module (mathematics)
Uniform space
Point (geometry)
Geometry
Transformation (genetics)
Event horizon
Morphismus
Goodness of fit
Pi
Natural number
Betti number
Ideal (ethics)
Theorem
Field (mathematics)
Uniqueness quantification
Exponentiation
Morley's categoricity theorem
Mathematical model
Limit (category theory)
Local Group
Ellipse
Exact sequence
Summation
Network topology
Homomorphismus
Fiber bundle
Game theory
1:24:19
Discrete group
Fundamentalgruppe
Group action
State of matter
Multiplication sign
Sheaf (mathematics)
1 (number)
Propositional formula
Inverse element
Complete metric space
Mereology
Weight
Group representation
Heegaard splitting
Analogy
HodgeStruktur
Fiber (mathematics)
Logarithm
Theory of relativity
Gamma function
Basis (linear algebra)
Category of being
Funktor
Graph coloring
Vector space
Chain
Module (mathematics)
Right angle
Bounded variation
Point (geometry)
Polynomial
Existence
Morphismus
Frequency
Natural number
Algebraic group
Associative property
Field (mathematics)
Exponentiation
Algebraic structure
Independence (probability theory)
Division (mathematics)
Limit (category theory)
Local Group
Ellipse
Field extension
Algebraic closure
Network topology
Homomorphismus
Object (grammar)
Exponentialabbildung
Matching (graph theory)
1:43:17
Free group
Homologie
Group action
Finitismus
Presentation of a group
Multiplication sign
Direction (geometry)
Parameter (computer programming)
Complete metric space
Inverse element
Regular graph
Theory
Power (physics)
Group representation
Inclusion map
Plane (geometry)
Conjugacy class
Ideal (ethics)
Ranking
Physical system
Injektivität
Compact space
Theory of relativity
Gamma function
Generating set of a group
Basis (linear algebra)
Algebraic structure
Bilinear form
Local Group
Torsion (mechanics)
Isomorphism
Mathematical singularity
Module (mathematics)
Right angle
Iwasawa decomposition
Coefficient
Direktes Produkt
Invariant (mathematics)
Bounded variation
Resultant