1/4 Universal mixed elliptic motives

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1/4 Universal mixed elliptic motives
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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 non-trivial 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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a and a thank you it's a great pleasure to be here and you know what kind of audience I have so be a bit introductory today my goal is to at the end to explain to give an approximate definition of universal next elliptic motive means it'll be more or less complete except there's lots of things to be explained and I'll do that in detail in the next lecture so when the stuff so the 1st thing is history and context and maybe a lot of people know this that I will stop here so very brief comment as some the 1960's look growing discussed the theory of pure motives and so that an effort to make smooth projected varieties into an abelian category and find a universal call knowledge theory so basically come on over this guy he would you feel you can basically the objects of varieties and the morphisms a correspondences and so on I think this theory is still not well understood because of all the problems of all the standard conjectures that basically objects with projected variety and the morphism 4 you more objects because you could correspond to the island poems and they didn't projectors and so on and so you can break up these varieties so the 2nd thing is maybe about 1980 those Beilenson write a paper on regulator and I'm not sure exactly who introduced mixed motives but they the idea appears in this paper and so on and so he defined most of from notice recalled that in the 2nd but he defined it didn't run about the problem of finding code change which was later sold by way what's the but it turns out at was for me was very good conceptual way of thinking about this helps this work all of X's and just 3 Stapleton X to be ballistic expertise but I don't think this is necessary so so you
have the take theory of so algebraic which was defined by quill and and you have Adams operations it's a written so CPA which takes the cake .period X into itself and these can be simultaneously died Wise of detention with Q so that you have the parquet .period X and through with Q is equal to a direct some of the on this right only when I put this guy here and so I think this is engraved the Europe and this is where the CPA the so so you can break up the K theory here and basically the projection of this onto the pieces the nature and character of the Church of the the degree and part of the trend characters just projection onto this effect here and this is compatible with maps told colleges theory and so what he did was redefined I should mention that I know how interested you but I will have made notes in this way and I will put them on the Web somewhere or would you know in the next lecture but anyway so the motive knowledge Of the with coefficients in Q N is equal to is defined to be you look at 2 2 2 end -minus 6 and so this may look odd overtake us I said little What about this is a universal call theory for the following reasons because there maps from K theory say if I took care what he called K M of 6 and I I can map by saying the degree In part of the churn you think of this as being carried out on minors and certain degree minus Samoans will go into into any commentary theory to the degree to and minus of X with coefficients Q in so this is any theory so you can put ordinary colleges you can put DeLeon ,comma algae can put Telcel knowledge and so on and you always have this map here and this will always affected through H M X and but this is this is just equal to if I didn't write this is the motive ecology and agreed to an understanding of 6 Q and so it's showing that they so that this is really the universal colleges theory and so if you want to produce to this is the old-fashioned way and I guess I'm still old-fashioned pretty boy of that new questions about how you produce consistent sets of elements compatible with developments in all these different colleges fears that idea they'll come from the same mold since then the idea was that again I didn't run to the library and checked so I apologize for that but so this may be a little bit revision but I should also point out here it's it's not clear that this college theory vanishes in new degrees of and so vanishing conjectures of sky vanishes in negative degree this is the age age less than 0 motivated all the 6 2 end equals 0 which 1 axis and you can also later blocked showed that these these pieces here rationally the same overview of the various things that 1 can do here that gives you an integral version of history right and so it another comment that effect defined oversee for example then all of these things should be finite dimensional and their dimensions of all these guys are predicting no supposed to correspond to the order of vanishing of certain malfunctions that may not be known to exist and so on the examples the first one is this is really good block it is X the H 2 end I want to to this is a restatement of of result the chair because the chip to good use great portions of the varying degrees Chen another example is to this is more or less Burrell and Beilenson is that the soulless state after being on the field so you'd like to know what the motive ,comma Q
so let me write it down so will need this later on this basically the ingredient for understanding would mix take motives is that on the first one is H J M people destroyed it Q equals 0 in less than 0 2 instead of 1 8 0 it will 0 in on equal to 0 sources of expected 3 the advantages H. f Q and equals 0 J is bigger than 1 of the interesting case and for all put F Q 1 success across tense with key that leaves the remainder witches and on destroyed like this day and age 1 and a half Q so this is going to be in bigger than 1 is equal to 1 plus 2 In 2 here and the degree of the 1 plus 2 0 2 and as usual as the number of real embedding sensors the number of complex conjugate pairs of invading the complex and things like that and Burrell basically Burrell computed the rational like a theory of a ring of images and a number field by computing group knowledge stable College land and Beilenson interpreted Burrell's met with when going think raising their arms Samsung shows you know constructed churning veils and other people constructed chain classes as something akin to an expert on expert on I think will be consumers will use their cell and so I think that's another part hears that surveillance and also I think conjectures the existence of a category that which important for us of a mixed motives In end and such that the motor-vehicle multi as a 6 2 this equal to X in this category also Q and then you take some motive associated to X whatever that is and so the motor-vehicle knowledge would be an extensive some so the voice of skin moving and this category show this guy should be eaten and so basically it means it's an abelian category with cancer products and tools and faithful of forgetful perfunctory finite dimensional vector spaces of so To avoid about some dude DeLeon I guess they constructed a triangulated and the category With these properties such that so this is not what this is incredibly good but it's not it's not a tonight in category so and I should say something about the objects secure and listed in all these theories roughly this is the tape motives these Q 1 is roughly I think it is a it's 1 of the event Q minus 1 is equal to 1 of the multiplicative groups and then you can cure is equal to the a hue of 1 tents and this is included in Europe all queue of minus 1 10th of so that that's some old-fashioned I think in terms of compatible realization so because people take a break after 1 hour of the Samarra for so I will say something about me so it's and it is not
carefully checked the literature history and the history of screwed up but let's take care to be number field and who as contained in string of images and I said the upshot of this work here will be the 1st fields and landing Goncharov for rings of percentages is thanks in category all of the state's most of all Over all but I should these guys maybe also bit more later on but the fact that you've got maps from K theories into extensions of a hard structures will Gallois models tells you I want the realization of this guy should be the Hodge realization of this just the Hodge structure dimension 1 and tight -minus in innocence and the LAT guitar it is just a one-dimensional Gallois representations of GQ was acting on cue held by the instead Of the secret on the character think so so the Q M it's going to be the given by the asparagus acquittal OK so so this is only makes state motives of the successive extensions of these guys here so and then you have the property that page Jr I want online this year the a correct 6 crew of going to the experts to give like a theory and objects have way filtration Minister doing something slightly non-standard here yeah and the way filtration is by some objects of this category and I'm going to denoted by 0 0 in the pieces will be hold em all every mile objectives and the containing an hour plus 1 of the so the Union should be all of the and section should be 0 but later on will become clear I hope when using em rather than the usual W because basically the objects were be dealing with later on an elliptic case have to wait filtration and then I think we normally think of a way filtration being denied by W. and get a 2nd Way filtration which will be be the 1 here this has the property such there so users Julia 2 Of the isomorphic Q months To some power and G R M all Of these sequences to 0 and mix take months review and saying if the grantee quotient obtained monitored and get anything woods but see I know lots of experts here tonight which limit the said let me do this some .period mines will do me a quick review of the the next step in the right place sir but 1 is what secure Hodge structure Q Hodge consists of the 1st thing is a finite dimensional the note :colon cute and the a decreasing filtration after all at this stage of the season with the complex vacations so we'll have to see and it
contains some f and so again the Union of these is all the same in the intersection of 0 and so it has to satisfy
the as the direct sale and mines something of a pain where is equal to the intersect Park here you got complex conjugation because of complex a rational vector space Wright said whoever anymore and so let 1 very quick example is that of the 2 and to this is white lines 2 men and a chairman of a smooth projected world right now I say what most people we just say so it's a mixed on structure and Hugh next structure so again the consists of a finite dimensional here that despair the here a filtration and that's away infiltration and here I'm going to use the letter W just to confuse Paul W M minus 1 V contains W M of it's defined in the queue section 0 Union the whole thing but the satisfying G W M of the services defined W and of the OMB W and it's 1 of with the use of filtration which means you for example restricted by intersection the new project of the quotient it is a hard structures of waiting look I forgot thank you a Honchul the new face of what the all of the city and Hodge filtration to decreasing said this the contains plus 1 again in the 2nd 0 Union the whole thing and I gotta stressed that a little slogan is a mixed Todd structures more than the sum of its great quotient there's controversial extensions and that's what we're mainly concerned with and so hard structural and extort structure beneath you got a hunch structure of weight and justifying W M minus 1 equals 0 W M equals the right every hard structures and make such structure a right to something I want to stress here so this important fact all this you can find in in the wings paper input because from ashes the important effect and so intermediaries there are many but here that he had W . using their associated greater Hodge filtration is also expected that and to get all this the definitions of doing so on the police trivial thing here is that doing this with ecology all right complex algebraic varieties natural next Todd structure and compatible with more efficient and so on all right so let me just to
a few simple examples I I understand that it this violated the order of my notes it's enough good things about him say something like that the rights let me when I got here all of the media strategy all objects of make statements of losing 26 something about show the well known around here but when we say just for completeness so I so I think if so well that when I say realizations their various kinds that come under the name of that sort Hodge or maybe I would think you'd around in this case and so let me just say 1st of all discussed and so realization many about abstract motive and then you can realize it always different categories in these different categories compatible and strong ways so equals Q 1 I already mentioned before again apologies for those who know all this stuff logistical to H 1 of the Hugh just equal to queue and all the of a time sequence is my copy of the this might say and then let's stick you I'm going to do the duels so if I looked at age wander around of Aegean of cure it's just a few times a year in the quarter GM said no 1 door OK we're going to take the that it's going to be a few times and now you have this and I so it can what's called this will be very calls this is gonna be varied around the whole thing and then you have the better Kansas City device basic around the room .period around Kansas City and let's see see so it should be that the history of the check should be equal to To should correspond 2 right of about the really think so and then this this is a hard filtration the equals F 1 the door so they've got Hodges you put these 2 together this guy has a hard filtration and this gives you a lot this gives you a hard structure 1 of my so should be equal to minus-1 because when the dual right now in a letter would fail it's just going to be an alternative to this it's supply 1 of the there was and this is just going to be isomorphic Help to what sorry so how 1 about a general the it's a similar story so we can have the better we're going to have a the which is going to have an afterthought and this is a Q is defined as the queue and people and both of these have away infiltration and we're going to have that on the whole are just going to be there attends the Betty W. .period cancel routes as isomorphic too the door W . 2 2 this guy here also has a hard filtration this together gives us the mixed on structure this way graded quotients this is an excellent structure will when greater caution to wait daughter of equal to a direct some of Q event various children's books there something here this is defined of acute case eventually I'm not going to do it in the beginning onto the kid around case because that's world slipped arsenic into Iran gives you splitting we think theory of mixtapes motives the fiber factors that kid around you know it's going to get amend the 1 of the field definitions what should I call just around the around things on sea again so I and so it doing the simple examples so this is the example the next thing you know I guess I could have done their of his son and let's take the to be equal to H 1 oxygen where landed in the "quotation mark so we can draw a
picture of the guy here here's the 0 infinity and his the signal in we used the generated here we don't have all won In land and will take a upon from it you can see here that you've got exact sequence 0 into each 1 interview and then he did say the reduced of Wonderland In this year of which none too hard to imagine that this guy here 0 0 so I can here maybe I should take everything skewed coefficients In this example works perfectly fine of disease well and so here and we can win we still have the form performed using the makes perfect sense will have all in here for will each 1 2 1 land which is the jewel of this we don't have the dual class and hitting the of the tools and here we have seen and here we have the power I have the power GAM which is a lift of the generator was sigh they so what you said to see whether the hard filtration minus 1 is standby by here in is the speech and in the complex colleges and easier to duel said to know what the next hot structure is this extension to know where the hot filtration is inside this with respect to say some interval basis and 2 pi by sigma plus long planned with his religious community from landed in the times so the exact branch of this depends on the family inches but I have minus 1 as determined by the Germans well-defined Vermont the Times so is justified by land and sea knowledge to prioritising the ambiguity here or here whatever this is matched by the financial mess into the the story of what was going and so this really corresponds to the land and sea stars and if you think about this is actually fixed 1 the other end up doing is integrally frequency of these and services so so this is an example of a nontrivial extension and you can see I was getting interested contained inside so I will actually be can grasp the extent extensions after a couple .period excellence much less trivial example so although the details actually handling case that details of various things the definitions and so on and this is due to leave entree let's look at that and so if we can look 1 lines 0 1 infinity of we have to make sense of this but we can talk about conflict Paul equal to the G on the whole pie unit price equal to hi 1 of P 1 minus 0 1 and 4 the exit from the union power in complete notified materials that few would know what this means this will take but let's just take it takes to be an element of few minor you cry some of the site stated to be in any complex .period of so let's 1st of all think of this as being part of the of pipeline in the complex case so give you a brief explanation so so far I haven't group so what you mean by completion of the later but it is the quickest definition so I have a group again as a a discrete and I will get a Category C let's call it an honest assisted equal to the category of the good representation of doing so this is not the intent of Bloomberg this was a field of character so this is this is so you've got a map from seeing them into a forgetful font into vector spaces of the care and you can just sort of high 1 once again the unit posted completion of gamma over it is just equal to the fundamentals of Yemen with respect to the that's the quickest and most abstract way the sale saying concrete ways here Bloomberg In fact victims of the son of the sewage just the automorphism to this fiber found to the compatible with that of products that or I can do it in his most but no way I know how to do it I can look at all I just look at all representations of gamma into a unit of group so this is you Over it and I look at the role and now you can look at the the closure of the image that's also a this is a risky dance representation and now I just look at the inverse limit overall or that it is risky dance the year this is going to be equal to a gamut of completion and in this case here you take I'll do this in more detail later on so this 1st starters infinite dimensional here the Free Willy algebra in this case .period 1 union power in is isomorphic to a freely helped run 2 generations X 0 Annex 1 and you have to complete its treated we palace series this thing sits inside
so how serious is known to shooting it on In the got power series and so so DeLeon Goncharov said that only algebra of pot Union in a crowd objects all of state model of a Q but in fact it will generally be followed 1 last week at the base .period what I wanted to know what the effect of covered not a mix tape motive of disease so because I want to say that while the case if we look at he 1 of this and we take some X here we now it's down to like and now you've got some point here acts if you look at the picture here suspects you've got to 0 1 infinity every such finite base .period will intersect you know the 0 1 the Infiniti itself someplace so if you think about if you try to pick up I won here to say this crime here at the base .period is become 1 which runoff off into the varieties of of pie 1 with the based .period make sense of all the force of the year award winner was forced to do is introduce tangential base .period let me just be very brief here so if I take if I take the romance of inducing high-dimensional procedures because of the fighting romance and say I've removed some points here and I suppose I wanted so I can use a tangent vector said 1 of the missing .period online access is going to be some expand minus some finite said this I'm going to take a decision in enhanced and Singhvi inside the detention space 6 5 and equal to 0 it is my view right here what do I mean by hi 1 with this tension vector I I take positive leave you with this philosophy vector travel around between 0 1 1 the path never believes X and then I can return here with negative that the Davos guys model home a topic that definition way I got infected is conceptually nice is which produces let's take X had to be equal to the real into blocked all right the real oriented below on sex at all so that we use the .period here I I replaced that with all the different directions I can leave this point it's when I blow this becomes a circles so I never got a border agreement here so I the boundary effects that is equal to 1 which is equal to the change in space this export mines 0 no plot and now so when I do that the picture up here I would have replaced the bicycle everything else saying the government missing .period than others tension vector determines a point here which I'll call the brackets and you can define the the fundamental group with this change vectors based on just being part of 1 part 1 X was based .period really Yukon defined to be part 1 of this blow up disoriented blow at the mine and mine come on that's close to saying the usual topological fundamental groups and while I'm here let me mention the sometimes we don't have a variation Hodge structure of such a kind and we want to build told the fiber over this days .period it's going to be a link on its talk structures sir I'm here to what you can do is that we have he won only 1 refiner based .period that is non-zero might pay for everything we look at the 1 0 1 and infinity and that they want hear that is 0 . 0 1 1 definitive infinity and let's look at what means that we would take the easy and we're going to take this guy here elements of the tangent space hero of the 1 and that's not here visibly on 0 at every prime and if we take the theory of the winning Gonchar office that hi 1 unit part of P 1 minors 0 1 this is also commonly written 0 1 DEC is a pro object the so so there's some remarks in order and on view at the moment so remarks the Hodge .period 1 that the the mixed on structure so on the Hodge realization so this will be this is an easy kind of limit makes Todd structure here so on the hunt realization are multiuser about so Zeta of anyone in this is something like this 1 of K 1 to the anyone can to the end I think any day now bigger than 1 0 was 1 so the standard multiuser have written
backwards and I'm going to finish this remark then the 2nd market when instead the only way to stop you but so making remarks as a few important remarks I want make so the 2nd 1 is what we know that AT and T and understanding this the statement of the disease this is to the system knocking that means this is the category of representations of some Proell to break group and I'll give a brief explanation of what this looks like an hour so we will have a high 1 so the category of representations all of a sudden group this is a pro algebraic group but part of his right as part of what eventually will be vague about the fight budget I'm the common 1 is the drama before so what you want to think about and there's also you could look at the core of the category on the list mixed motives of so what with the screen these one-to-one by splintered direct sun of say a Q M 2 some power in and you would you would someone over and there's no extension of justice and a simple thing so what was the fundamental group of this category B it's just the categories high 1 of them category and is just equal to GM I just need to know way of a mix tape might to Kewanna have GM act on cue just expert at the end of the standard character all the defining character so I'm so G said accused's inside Q crosses you think the and that and that representation tells you that you an answer this is a subcategory so if you think about representations things going the opposite way so we can get enacted G so if we had a representation of this affected through this completely Split and the Connell again you can think about is what I like to call it can ensure friends of different nations that have so this this guy here is pro-union part and so on and it tells you it's telling us about the government's extensions so it's just a little calculation but let us try to complete what it is and I could just write it down but it's a good exercise because we needed an elliptic case so we we know that next and next to the mode of execution also here here by and it was computed so this is something think done long ago by the likes of doing and held them for about what they should be so but this should be some text in the category of representations were prone to break group was given by a group called logic in defiance of a profile during groups and we're going to take part in 1 of and so with coefficients in Q now if you look at spectral sequence this group extension will actually see that this is a good 0 of GM a few of care and secure and why is that because GM's reductive right to every representation splits is no higher a genuinely denied 0 this is because Japan through and now look at this year and strictly speaking should be dealing with so this is this is going to be a nephew of appropriate pardon group it's come on the same sentiments Lee algebra instead no again tenses with Q an elegant invariants and made means so right and so what we know as we know what we know of we know this is equal to 0 if Chen is bigger than 1 so this implies this implies that H. of care equals 0 J. bigger than 1 because I and that implies giving references for this this is all very easy stuff this implies that carries free I should say strictly speaking it's the completion of a freely algebra but I and you can watch hearings because of the exactness of G the exact cause of the White Crane cautioned you can walk you with everything and then you actually worked with regrettably algebra here so the the 1st thing was we noticed a free hand to nowadays we need to calculate the generators and stuff on so so this is where to find to be leader of care it is topologically generators although the of it's going to be but How do we know this is here well sorry I want GM and do all the things you basically want to pick up the generators because they they're telling us about the 1 expansion and we know because we know that next 1 of few by IQ but in mix tape motives of his who remember this is given by the K theory this is equal to the Q 4 all and bigger than 1 and it's equal to 0 for an even think about this for a 2nd what this tells you this implies that the can is isomorphic not naturally you would have a statement here but know that I'm
aware of his OK case so what happens within equals 1 it's 0 and pulled 1 because I guess I did for field before maybe I should have done for the images because K 1 these is equal to the into plus minus 1 tents with so this is isomorphic but not naturally to it Fred LI algebra the completion of the freely out with 1 1 generator for every order integer greater than or equal to 3 and so "quotation mark famously algebra through the work of various people including France's and should also say this extension up here .period various reasons basically living theorem implies that this is hear but also this comes from the on W . org is the various ways to construct living here and so that means this this Lee algebra as a GM action the source with wasteful during which it was agreed the completion of grated with algebra and yeah and held GM actually get the same direct product allergy and so GM acts on 2 and 1 by that too In health of the defining character so so that tells us GMX here so tells us what this fundamental provisions of the Senate direct product of GM in this group just told you how the group back from algebra the yield on its key to cure its Q 1 and if if you asked me to put a GM to give you the way I would do what I do component .period minus 2 ever since the 2 input power Of the standard representation Virginia but some the next From this year so we now understand CPI 1 of them to the end is isomorphic to GE and Sammy direct product we can think of .period could the 3 the FAA and by this I mean the pro-union group with this this algebra I think it's the 1st remark is there so we know that high 1 -minus 0 1 infinity so if we look at high 1 unit of this guy here with the base .period alright DDC is a pro project and so more accurately I should say then the quarter ring of and in Japan the statement of and so therefore it is a representation of the group so Paul I 1 of empty and excellent no you could say hello Richard as a mix statement of how representatives it could be really boring could be completely splits no interesting extensions or there's the fear France's Brown 1 of the 1st thing that this action was faithful the importance of this result is telling you that you see everything you want to see can construct all of these weak statement is busy inside this uniform potent fundamentals and we'd like to have something like that and it would be case we don't want you that's what what that's the direction I wanna go and so I should say cholera so this was an open problem says the periods all 3 objects In the end are multiuser right because this is basically saying that the sum portions of this guy here generators category so the period periods of 2 or more "quotation mark intend to buy another point I want to make here so I think of this stuff in terms of what life faces could let me fix some notation I so let's let's let's just let him and I'll say more about this new look taste an excellent the Westland mg In plus b the marginalized all said so here we are going to require that the G 1 2 plus 2 plus in so far and in In his praised the requests 0 and I put an arrow this this is what a stack of others drop picture I think this smooth projected to Jean G. all have In points so as looking like a 6 1 down to 6 on and then I have a point said here all have vector non-zero attention that the 1 up to the side so it's it's plot the point plot I change vectors the inconsistent about how I do and every the church this not equal to 0 and they lot points where the included distinct and distinct from all these guys so again and again in the face with a few small numbers but I thought it through the Senate spent a chunk of my life thinking about these guys to these guys are defined the and so the way I like to think of that but the geniuses said M T M this is the sort of its closely related to so the MC In plus story and to to bring home a little bit here 1 minus 0 1 and infinity this is equal to M this year 4 so the
next step basically that it's it's genus 0 we think that somehow lodging in Europe so said the next step is the elliptic and I claim to understand all these hygiene is once I may say something about this at the very end your need to know genius 0 1 1 goes back to an old resultant John Henry if it'll ever people so let's let me start time at the Olympics while cold classical I'm not choose worked on this Goncharov said they worked on it so you'd like to look at the motives that come from that because up in the key thing is he's done it with elliptic curve and it you would associate emotive say H 1 of the and maybe the center predicts that would represent the 1st call Mauldin so let me talk about what this will be so let's just say let's suppose that we've got the aim is to find the a cure and and has equation we can always right in the form y squared equals excuse minus Q X minus the so here we are Q and we have a duty which shoulda known as is defined to be set up a new Q minus 27 the square which is up to a factor of 4 of the discriminative this polynomial we want this to be not equal to 0 so so let's let's talk about what all call 1 of the wonders of and people talk about just the various realization then of course I just going to be a H-1 the analytics you let's talk about all still call acute around the around and said it's all call it let's just call a local Oracle and HP it's too you can write the of the abelian differential here's the external Wiesner dealing differential and you can also expand the call mild you with a differential of the 2nd kind of 266 of why this is the 2nd kind it still has a politician residues and 0 so it gives you a well-defined commentary and I don't even know everything his hold office this is the this is 1 for this is that this is the abelian differential and now so if you future a tents with the it's isomorphic too Duran and tends with the and this gives you a hard structure is the cues from the underlying rational vector space and this gives you the filtration we got this is this and this this is a hard structure it can ride so it's 0 1 has not necessarily been by that time right so I look at the we have said Miguel I propose said Cuba over Q Exxon can look at the yield to the end toward you and this guy here so this guy will be isomorphic or essentially equal to the L elderly and torsion Of the complex .period because of and that is going to be a H 1 the complex analytical c mon help to the incorporation some let's say Paul I won the time Cuba 0 so it is going to be the 1st of these guys here so people fleeing the of these guys here so it's equal to the inverse slimmer these guys here and each 1 so this is the follows from France machinery but here it's pretty basic and said this is just isomorphic to age 1 still I answer right so the college if we want 1st pulmonology H 1 of so tell him with Kewell coefficients it's going to be isomorphic to home this guy here In its Hume L all money what I'm sorry this will be the Proell maybe I should just put the probe well completion is appealing so we can splitters right we have 1 of the oldest gives Segawa action here June this guy here so what should we buy Nextel mode they are
extensions all Metropoulos so we can take uptake twists of symmetric powers and by this I mean the instrument repairer With with you In so the category of Nextel maybe we should depend heavily on on depending on what the code for example has complex multiplications get a lot more algebraic cycles for example I don't know much about this but what I want to do is look at the Universal case so so what I like to do is just now give you any of you I it's reasonably complete if I tried to dot every i and cross every t it'll take a long time and won't be clear what's going on said so I should stay here most of this joint with many Mercado Madison going where some I joined with some the people and some joint itself so so what's the eye so I won't be they said these are at moment I'm defined his bank statement for all you would think so what should that mean so what we're going to do here is 2 more precisely and their their motors associated to To the Universal Optical sir comment here so OK the Zionist indicate there's an absolute theory can't at you know schemes of something Robertson snaps would theories and so on and so I think the theory of motives maybe there's some version of relative motion but the most it's just absolute the 3 years when I say that this is a relative thing we want look at this guy over this and so while we're going to do so let's give this a name so a guy what's the analog of the H over there Wiggins said so I use this flexible bowlful local systems it's equal to 1 flows you again to be absolutely precise have to explain what this means in all these different categories so have to be suitably you can see and hear the most people live factions Lee is the nominal says he go back to the case of mix tapes wanted something the ones we understand that the ones where wages working write to him at the end of the thinkable level might be to work with rings of images in some other number field the 1 you can define it for having level and can define it as a bigger range but it's going to be harder and this is like the 1st case at least I think but should be attack both the main characters in Europe reaction it certainly would work with levels but I will when the general set up OK so what is this this is the local system the same 1 1 fiber Over the module I point of analytical is simply it's going to be a H 1 of the but again interpreter service on the boiler approaching take here is that this is all I'm going to look at compatible realizations so then although at the time now we actually get actual mothers that through some strange full at all OK I go backward and forward on this if I wanted a small 1 but I all just take each 1 so that it is somehow my brain functions better if I put aging white 1 so this is this guy here so it's a local system of white wine in all senses of the word so for example if you think was the variation parts fractured variation of which 1 rights but let's talk about realization of the wind again I'll do we need to do more detail here which we need to do computations yet in some sense someone he said it would have been what M 1 1 and you can view it many ways that the way that's most convenient here is that the caution of the upper half .period by the AP dismisses is the upper plane it's really all right at all full watching so it's all the fault was what that means in practical terms you just working on the upper half plane but is still Tunisia covariant so what everybody does in the theory portal for for a base .period I can choose anything but in the upper playing here we have no we can take a strip high enough and we can have some fundamental domain here looks like this reaction vessel Tuesday we can say take the stuff from mansion atop the efficiently lodge and then when we take the caution by a cell Tuesday it's just taking the caution of the spot the subgroups people think they're inside I'm going to sell the stock is basically the quotient of this trip up here this is 1 of the 0 1 the quotient this strip bigger than about 1 ideas and this is and so altered Q this is all very standard to the
2 fire fighters and this disk is the stuff it maps into M 1 1 and all but I want to take the sexual discrimination can you can take every point in here has stabilized minus the identity so it could do something else by what I want genuine punctured am so what can I do for for a basis point I can use the base .period Didier Cuche the base .period so my base .period it's going to be equal to and it naturally I could if I look at the imaginary axis here I can think of it corresponding to the stock here the imaginary axis and so I'm going to include this if you fly into the upper half planes the imaginary axis maybe on being a little too fussy here but in on foot the fussy about faith .period but anyway so you can also use the upper half planes based on what I'm saying is a matter between the space .period with this convention what I'm trying to to say here is that if I look at high 1 of them 1 by 1 analysts the the key to this is isomorphic to hide 1 of em want 1 and we have the projection here from age in M 1 1 analyst video call was going to use may be bad but all hold the and this is just a cell Tuesday and much to the annoyance of algebraic plays the topological convention multiply piles and called of for I write so that apply 1 with these choices you can work at all levels naturally isomorphic Hilton's so local systems 1 is betting on local systems really given by representation of pie 1 and a better just corresponds to the defining representation all this just a two-dimensional representation on the more precise about this later on so and then skipped shootaround might actually watered out bits it's actually very interesting more or less done in the work of Lebanon Rossendale where they lived quite a chunk of actually and I will write it down later on but I'll skip that pledges to Hodge what you can do is you can take a each of better intensive with all of In 1 1 analysts and so this will be I'll call the age for this this is a vector bundle services the ages or locally constant chief this is a vector bundle and got a canonical flight connections this is the and I'll write this down next time but this polarized very of hard structure this is the most canonical 1 what's called a cell to all that luckily homogeneous so I will describe incomplete detailed completely elementary it's also a good way if you don't know theory of the polarized just referred to to the effect the new products this is actually a flat in a project going from Geneva and so it proved the the hue of minus 1 price it it's just this is just a couple of polarization just means basically it means that this car at a new product that satisfies the ring nonviolent relations and we want you need to lean on that and told them good to know it's there in the general theory you need a polarization so this is a and let me just described it like this but fiber a state where he is to find a cure is high 1 of the over acute 4 0 tents with here -minus just we down before like itself to the queued around the area later on but I should say more about this place .period lots of reasons it's convenient to work with the based .period I wrote down on the Q the right let's say more about the basic .period so the 1st thing you is that since every said we a base .period it's non-zero at every price rise so if we talked so you might think I K let's take some of the defined as the euro over Q whatever in M 1 1 Q now but they'll be some prime at which this has been a reduction sir and that Prime we can no longer use this as a base point because it's no longer an aspect to if forced to use them productions somewhere after you so there's only been on various ways to draw this modular space you can draw cost picture out here With this is basically where infinity doesn't say I the rule road Cuban of him on and helped you discuss them here so this is really the the cutest down out here but if we choose to do here that works because the takeover so the takeover thank God the answer is L series in Q but this if you just want peace everywhere it's still and so on and you know the analog of the tension vector will be the pull it back to said the queue the 2 1 In pulls back here so this is going to be held at the base .period DBQ and this guy is non-zero proprietor from the based .period which is not 0 at all times and so the only 2 possible base .period Wikinews opossum minus the it's also very natural when you work in the upper half .period like explained over here it's more or less topologically using the upper half playing the projection from the upper house .period abuses .period and and distinguishing imaginary act so minutes for what may be 1 of the Watson sensible place to stay yelling and make 1 point here and then stopped everybody wants to win so this is the point that most confusing as well here right now and future of the week this and this is why I have 2 different locations for weight filtration so if you have a 1st of all the citizens group if you have a variety you've got a variation on structural reforms .period here to local systems it won't have a fiber of India by method of all review all this because I expect most people didn't know it but you have this local systems for example this might be out at out punctured cutest Kansas local system age and if you look at it the 1st thing you can do is you can take the extent of you and you get the flat vector bundle this guy here it has a natural extension to the entire this get which is called the liens canonical expansion and you get a fiber that canonical extension of it here and to every change invective here you get a mixed Hodge structure on the TV if you started out with a pure heart structure get a mixed taught structure on the fiber of the canonical extension of of the origin of the disease so what happened and this is a very elementary calculation older next time completely explain is what's the limit next Todd structures the limit action sales of the makes taught structure depends on the actual tainted effectively refused different engine directors give different results some of the land that makes Todd structure even this was pure white 1 of the few I'm going to by age this is new denied that Foggo by Stevie Hugh actually turns out to be true why but queue of minus 1 and go this week because this was pure the variation age pure white this is pure 1 sir but what I want to do is this 1 define well we I should say we have wait 0 of age equal to 0 and wait 1 of of each equal to 8 that's because age was a variation of what 1 because H 1 of an elliptical as of white 1 as it's 1 of is a white 1 in every theory every interpretations but he was getting something next so we have to give this way infiltration different names to this is way too this got waves era should the average of these as 1 answer we're going to get away infiltration here so we also have wait filtration which is called the money driving weight filtration which is going to be all right I don't this guy on this limit and extort structure which is going to be a M 0 contained and minus 1 which is equal to 0 this is going to be equal to 1 is contained in them too "quotation mark it said that to enhance your great questions this 1 here and this 1 here and this is Q minus 1 and security and so In the same thing applies in Galois theory views this if you look at the wheel of the takeover when but if you look at the takeover of this guy it's a H 1 splits Askew pustules minus 1 the In a mobile phone .period just got 1 minute 40 seconds according to what please stop so sir 2 did you it equal to Q plus a queue of minus 1 and you will realizations but because it's a direct sound all these assertions that these are realizations here plus a few of minus 1 which is a mix tape mode so this is going to be the starting point this is why denoting the weight filtration and because so In a selects elliptic motives are going to be extensions of cement take twists of symmetric palace of this variation this limit makes Todd structures and make statements is the realization that make statement and his forces the things to be genuine motives and makes the story very rigid and then you can ask all perhaps there are no interesting examples and what I'll explain next time is there are interesting examples if you look at an elliptical and non-zero attention vector you can tell you pardon fundamental group of the -minus 0 With this is the battle in the car you've removed the identities of the free fundamental group and got this victory here and this guy here I and you look at the variation of maybe I should put fully algebra here so this gives you a local system actually over and 1 vector-one and this is this is a universal mixed elliptic motive and its fiber to limit makes taut structure which is the 1 associated to the takeover there is a genuine next so I try to explain all that I will stop here questions should be asked for and


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