4/4 Analytic number theory around torsion homology

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4/4 Analytic number theory around torsion homology
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usually when I have given several lecture series the attendance goes something like this I have so I'm happy that this time it's going to this so I remind you what would be risk of talk by regulators so the the new owners spend the 1st few minutes recalling the story from last week and then we'll talk about it so from last week situation was we had subgroups ordered an inside S O 2 of Knights recovered field and there's a focuses on matrices where where and divides even I remember everything all the coefficients in line side this spring here and there the conjecture that we were discussing is that when you take the million position at this and you just the finer parts that might have something part and you steal it by the volume you that this has indefinitely which is 1 of the 6 5 OK so you know
what I said last week the army in
the main tool and trying to understand this from an analytic standpoint site and heuristics for the main this story analytic portion and in particular the following formula so so the statement matter writers is literally true in Poland on compact case the size and this multiplied by regulator which for all divided by the volumes is equal to the some which you something to do with determinants of Russia OK so Pfizer last travel when I said in the previous lecture was 1st lady if you allow the pastor's plan and if you switch spread a different weight to the animal weight to view of some different ways we can prove the analog of this conjecture but for this conjecture so you yourself using the signed an accord in Formula One can improve the conjecture In the norm Compaq for leasing the complex case at least in this little now this close to being proven on Compaq is also war 1 conclude that if we know 2 things became the 1st few small eigenvalues so you understand you can understand this object rather well as long 1 newly issued this Delta 1 might have some values very close to 0 said Delta 1 has few small eigenvalues I wrote the precise meaning of that weeks rightist again this something we expect certainly heuristic leaks we strongly expect to be true but this is totally I have nothing to say about it you don't approve anything but the 2nd thing are is that you need to know so you want to understand the right-hand side of large and you can understand exactly how large you need to know that size comes from this finite group and not from are so you need to know that laws are still by the volume goes to 0 scientists discussed limit give you the definition are once more the move so this is what what I'll talk about today in particular say more about what the actual work so what interesting mothers are not just sits there trying to prove this but it it's actually very interesting and very OK that is it's it's related to to us many of the objects of number theory including all functions right so here's the definition of if you like than here so definition of R and I'll our justified in the case when the free part has ranked 1 otherwise similar definition would determine its went but as I said this is the hazardous last week broke all reminders of disappearing you have just as in the case of the thing I said very sorry this is lectures about classical whole Moffat forms disappearing between this and a space of modular forms OK so but this the average at best to be suggested so it kind of reminds you of the case of classical Oneill forms of but here in Iraq in our context this is the space of harmonic 1 forms on this come on and monitoring OK if you were working if you that the said make the same definition in the case the upper half blamed the harmonic 1 forms are sums of whole Moffat forms and their conjugates at the CDC and Konjic so this is really a panel exact analog here the space of whole Moffat wait to forms and pairing is the really is given by the same formula as in the 1st lecture so that is if you want to appear gamma and a former chuckle to integrate Omega on any path between W and Alan W. for any the OK so still other far-right Sosa now this yeah out of my way let's say this at some point it's like from switched among compact case but but you let me make the statements and in the complex case and no 1 has a slightly modified and Verizon so what do we do now so on weekends we can integrally normalized an element of flood of America in this as and by requiring that alright so I'll write it down it's like light-colored gamma North ,comma got to be 1 where gunmen naught and projects to a generator of the so in general you know and when you have so you should think of this as the space of 1 forms we have different ways to normalize the modular form of peso for analytic purposes when they did the 2 ,comma normalization that 1 sees it hitting a whole Norfolk forms is you can normalize Elton on to be 1 or you can normalize the 1st for coefficient to be 1 OK but in the context we're discussing this is an equally natural normalization you require that when you integrated against these cycles that you get 1 on a generator for this again once you've done that they need is defined as regulator to be on a in a product so maybe I should call this last I just have a notation for this let's go course subscript disease just as suggested this is integral normalization so the nominees the accident was also the site up to here were all things said last week and I'll try to say something about the nature of this most of the sting out of the question alright so there was a lot the only OK so anonymous data result from this paper with down over Shaun Alexander solar result from so that the address did result what life will be more of us a bit later about the proof but the main thing outright to communicators what the difficulty here again it's not that of maybe difficulty once so familiar with in the analytic theory of model for difficulties trying to sow speak find such an element of it so in other words despairing is not so hard to compute but if I just taking element you really have no idea where projected to win here OK it could you could have an element of very modest size right physical matrix here very modest-sized but get image here is very large begin that's the basic problem which 1 faces when you're trying to bound this from about that OK but this perhaps this result is actually go away more algebraic suggested that analytically so I suppose now this is really depends on suppose that and so really in non-contact it's OK so in other words that means at various points I should put in the word cost forms rather than all of them all forms of but this is
important and you'll see why it's important OK now according to Langlands Program predicts OK so soul yeah I should have stayed in this setting here that we are witnessing the pulse of finite group India was in implicit in this but this is one-dimensional OK sizes 1 copy of the is the one-dimensional complex retrospect caseload just 1 that this is really Kansas State now over if you're ascending over in classical modular form those days a week 2 forms is 1 then this analytic curve associated this happens like that was be the 1st time it happens that level 11 over Cunanan is it's associate really become so the same thing is predicted although not proving here so Langdon's predicts that this 2004 but settled or harmonic 1 formative you right what that means it the mosque form of certain 1991 we became so associated this there is an elliptic curve the over the field when dealing with His squared minus the also conductor a man it's very other property his other matching between happy and values of this and .period counts it thinks it ourselves on the same stadium condition as overkill case of you look up the story forward to forms of duty that is now I should say this statement is actually not there a something which happens over an imaginary quadratic field which is that even when the dimensions spaces so over Cuba for example when the dimensions space model funds to you might have to elliptic curves but you could have an inbuilt in this could come from the BN service are in this setting something similar could happen even if the dimension is 1 that can come from a billion surface with "quotation mark turning on a multiplication of but which is sometimes called a fatal of Baker so I warned auditing nor that of I think exactly the same result should hold with the same proved innocent OK so amendments predicts this so rules supposed to be so it's true and also assumed virtually dire conjecture for unit's twist again when I get to exactly but I this is then this regulator are divided far-right so explicit fudge factor which I was so luminous quality family all the time the more important part of it is on the Net eh the miners to so this is some exclusive fudge factor explicit and involving only I that portion probability curve and the level and and and also the actually adjoint there that I enjoyed no values modular form but this is the part which is interesting this is the area all of the fundamental parallel parallelograms for the OK so In other words what you can do cricket take this elliptic curve take complex points and you can integrate over it but if you take an integrally and thereon In integrally normalized all 1 for which its conjugate because of the the absolute value of this it if this dominant class number 1 is slight issue defining this but again man most people without 1st simplicity OK now also what I wanted so so in fact I think that there should be an equality of this for so little the idea that I think that's what this is really under the quality so are his equal to some essentially explicit factor times that's been approved says that divide them and but I think in truth is equality now and let me say that another way to make it to try to say why this is an interesting thing this only needed the minus 1 is sent to the full things height on the curve Of the did so in other words if you right out explicitly in equation if you right right explosive any question as he is minimal equation which is why twitch if he is given by this then this is going to be approximately LLC what that means in 1 2nd it will measure the size of these coefficients the hope powers right this approximate signed me at this not bear it means the logarithm of the 2 sides is 1 of the ratio of the logarithms goes to 1 when something goes to infinity so while in other words if so this this or this statement cable what it says it this this thing here some measure the height of diesel to measure the odds many complexity of the and in the complexity of the model form 1 of the associated with me so at least it it came as a great surprise for me that these 2 things might be disseminated the cost both measure complexity of something but that the nature of the missile different that I can sort of ICC's approvals will be sort of very computational so that really explain why such a thing should be true moreover I think that this is a state this form might be true more generally OK that is to say that complexity of it in some normalized and successful things height of a variety should have something to do with this analog of regulated the Associated model for and that's great it's very speculative so OK anyway so so this is that I was somewhat interesting misses relation between on the height of the would occur now let's and let's see what that's that does for us in the context of our 1 abounded OK so so so so the point is that sold it to prove that goes to show that are Over the volume goes to 0 in this case and then you need only to show that the size of the coefficients of this would be curves are bounded I mean on a logarithmic scales basically my level and so this is a standard conjecture of diophantine geometry decided that size a coefficient of Olympic respond by level so this falls from From the meaning Standard conjectures for example that indicating for the free Spero conjecture these connections are much stronger than this city's conjectures about polynomial mounds it only once bonds and much much weaker but anyway the point is that yes also
in this way that it this gives at least assuming BSB and look very safe diophantine estimates of performing far weaker than what we had just than what we actually believe this
implies that the regulators indeed implies that slow growth of the writes about 1 and then I'll talk about whether of the proof of this comes from I would 1st have a little aside about this type of regulator some insight some slightly different context but I any questions about this yeah what no no no no nothing is is all transcendental and presumably yes Wednesday are divided the West I mean that artist is divided by the teacher yeah yeah you also see where that danger comes from a all all-white really should be 1 that certainly the proof of the users the results of the I use the full foreign BST in fact used when I say BSE-free E-Plus twists right I had a sewer use something is actually not surprisingly not written the literature anywhere which is that if you take an elliptic curve any twisted by directly character in uniform obvious differences but in end so that's what's being used is the author of the also explained that yes yet intact so I don't know how to prove this in the complex case OK in the Compaq case let us say you make the same assumptions would you get if that are as a rational multiples of the same thing OK so we're going to use that and does not compactness and some essential way for the poor I a lot of our minds was that it wouldn't make some use of modular symbols looking that but you don't have that so I would love to have such a statement in the in the Compaq case even the other questions where I'm going to have a 0 quota notices allows can make a slight aside about how this relates to some problems validity to that in the immediate remedial but may be all try to sketch the wall what goes into this kind of thing and then will come back so I'm in a sketch some of the ideas of proof and that this will sort of show power loss right so the basic point is you need in some way of course to be able to evaluate this spare rejected so there needs to be some class it be examples where you can evaluate so let's call so let's honor but called on while go plane lamented that to normalized form so abruptly itself as 1 so this is not going to be so rights over their crises have adjusted get this so take any so suppose we can produce some elements Delta In Canada and for which we can evaluate the such that we can explicitly evaluate you and by that this will mean in practice in terms of functions appears less the meaning of explicit evaluated Oh my God between directed the don't a so somehow we can explicitly workers of its school sorts this'll be practices from Udall functions right so that this Delta so let Delta and so you can take this and you can send it to the gunman on have which is as I said in the past something finite and selected ghost acute times generator a so the Delta goes to Q in this variable and the finite part doesn't matter at all we were doing this type of computation that into goals-against differential forms annihilate that it's there's no so Delta sub q ii adult Center keep times generator so then what this says in other words is that so this is cute times a generator on they go over the is 1 or in other words this tells you what the integrally normalize form is all magazine you get by by few crimes OK so as when as decision with 1 to explicitly evaluated just 1 valued the symbol you can expose explicitly you can understand what this interviewing normalized formers by the way I did say this this media this aside that I was on target with the end of this story will works very generally appear that is in
that setting all of the College of our many groups you can always ask about this Kennedy always integrally normal forms and it's always actually very subtle but it's very difficult to lose this infrastructure structure is very interesting and very sorry but the anyway that's still that this is integrally normalized form and so the regulator are it's going to be the interpreted this was itself which is Hughes quit over the square and the entire problems will be that we can bind is that we cannot we we cannot balance sheet of where the which are bound to you courses a I don't mean something to you In typical it it's more like at time you can get some naive down which is all the enormous size of 6 you need some OK so so may explain for example how you can try to use it in the pool complex case OK but as will the In the coal company I don't know how to solve this issue in the cold so I learned that there is no mean no for an answer at the end of this but it will just show how you explicitly so cousins have produced such don't get behind we have anything vote-rigging explosively computer that what the day to day do look at any questions don't you and this this is a good thing leader because the no no I just saying that we don't have the right to determine if the free part is enormous dimensional when you went to a separate another issue the free part is as large as the volume media so that the when you make this estimate you would have another problem of the free part of dimension of the very close to the full but other than that if it's significantly smaller any other questions who has the power to evolve other OK yeah on the free part here's what we know I just apology for example Nunavut bounded linearly in the volume if you think a little bit more and you can bounded by something which grows slower than the volume of volume over the logarithm the volume of that in and they're there to kind very interesting paper 1 by 1 him and 1 Marshall with some specific instances but by and large we we we don't have a robust way of showing so Oh by the way I recently terms of what should be true away from this it in this setting the the free Parche dimension the free part grows of part of the volume Western 1 but we lacked very general ways of showing that I just wish finally there is the nation's only what it was called to form a yacht in the interest of the United States so yeah annoyed is in fact all states it's from this you find out about the walls forgery formula but in fact I mean not just the Wallstreet himself that that was his application or 1 of his applications in his original paper Welfel foreign for his formula was exactly in the setting to understand this off to rational OK so so that the story marks a lot it's going to be the most rigid fumble and back already approved by both retained the same paper where he shows that form so the outside as you asked at the the way 1 does this but this is not the main and I said the main problem will come later when we discuss how about cute but in terms of just producing such a Delta the answers given by the walls forge a formula which I think the league mentioned now at the end of his talk so it says so that is applied Is that automorphic representation 4 G L 2 over the field please Q squared minus the and Elise about extension "quotation mark OK then this fall and hence the victory in the space of 5 then walls bridges it is In this paper he gives an explicit formula for this there and
in fact he gives a formula for any metropolitan except very beautiful but anyway what is important for us is that up to some not fudge factors what we get is the essential function of pie Kansas central function of pie twisted by the already character social there "quotation mark grand during character associated Eleva overcame yeah Of that now allows also is what happens when when you when you make this work out explicitly what this says is that exactly computers some value of despair so in our case so we take Hi divvied up the report automorphic representation associated with God and will take Fiji beaten Becker imply that corresponds to the corresponding to Omega could be careful that cost and and then with the work out what this says so were you having roll over this idyllic space but when you project so the projection all this to gallon and montage is a union of 6 with a finite union of close 6 it's a cold the king and the I'm OK so that they may just right out of what you doesn't look like a piece of each of these close genetics this is just a complication pages have to unwind wall on the jails he projected onto here and you'll you'll see this beach unisex G. arises from some hyperbolic elements from Baltic woman cannot get in the following way so you take so I hyperbolic the space but this is gamma fixes so so alpha and beta kind fixes it it's exactly the same as if you know the story of an upper happened the same story gamma fixes to points on this plane at the bottom he joined them by a semicircle and what what happens is a Democratic moves you along the semicircle so if you take a point W. yeah and you keep applying gathered to it moves you further and further along again the way did duty GIA isn't just a project of this past the peace between pieces from W began the IWC this in projected to this question this otherworldly take 1 of these this piece here and you projected the caution is that of any other such fees should get the same close so if you OK so in other words when you integrate the former geologist all-for-one Eagle Ridge United at exactly the same as those paid by definition that's the same as what you get when you pair with honor God OK so that is the GI the definition .period exactly so this is true and so then what most bridges formula says but what it translates to is that you these over open 6 some of these pairings New Square it was it in this case it lacks the work to the wheels and square and you get the hell out of these funds factors of one-half pie we want tight times this quantification OK so in other words it gives you an explicit evaluation so this you can think of this as being carrying this summer he said the what is the product of all the Gamma Phi Gamma 1 of them is an explicit formula but the value produced explosive things for which you can compute that and as I said this is already done by walls version OK so he added to you statements really out of what he showed using that was he showed that the different held for different twists are related by rational number so this lament that led now by the way you also need to do something a little bit L half of pies 0 this useless to you but you can always find a quarter twist with non vanishing central that so and so this allows 1 the computer this regulator rational numbers In terms of elderly the appears but the initiated the contact among compact given this is this is useless for us many of the things it does tell you that this regulators something interesting because it has the same sort of its algebraic properties are the same as the central values functions but frosting parts subtleties the integral story that is how you bond askew so and I don't know how to balance so so no I I don't know how to say anything more more in the contract case which is of very this program problematic would you like to be in the history of the world with yeah that is yeah but even personas papers are are not hard to really use something like likely find many mention aquatic fields with Virginia wholly out yet another sign that the that's right now and that's sleep says that in the middle of this we say would in their similar problems even in more algebraic settings and yet even there is very far from obvious what it is so
carefully mentioned this pieces of Kartik persona which is dealing with quit turning algebras over Q and already you there the contact that these questions are difficult here it's much worse because nothing is alright so now what what we see non-combat Casey we have available fortunately streak of modular symbols gathering any other questions so here what we can do evaluated pairing :colon select actually take With so let's choose that use an element and gamelan on which projects to build a generators 1 in focus over and that additional lyrics accuse 1 so in other words regulator will be given by if you look at that formula over there was just 1 over despairing Delta but only just right out here we have the fall under the following thing we can do so to evaluate integral from WTO gamma W ago you can take this point W to infinity OK so this this idea respectively from London we didn't use any point W. Anderson as long as Omega comes from a cost forms of decays rapidly you can replace this integral so what happens when W. goes infinity this gap as a W is going off to infinity gamma W's down here and this becomes a geodesic we can take this to be a geodesic so II integrated coming of when it passed from infinity so little what will out will write this in terms of malfunctions and no bills will be a problem there which all discussed because of this alone doesn't these are In terms of the union can see this as their the special case in fact of this wall specially formula retake held to be displayed Quarterdeck extension but it's also so this and that and the have another feature which will be useful all right so I'll let me discuss what these intervals look like adjust to keep it a more familiar out of discussed at the end of the story for the upper half .period abatement everything works exactly the same sold 4 million Monday welcome to compute the analogous story for the upper half of it is really works are exactly the same Nevada right out here and that is here cannot exposes for expansions were here would have a Bessel functions so that so imagine you have a we to cost form and as Elmendorf don't public favor a CD what you want to do you join infinity so don't applied to infinity is the point in so you joint infinity anyone integrated CDC over this it was the the but again this is it this will be easy enough to do that the the issue delving into gravity problems and so how do we do this limit 1st taken very simple case of it let's say a overseas actually 0 of maybe actually 0 computers then it's also free with right for expansion as some of the didn't and what we have is just in general From infinity 2 0 some of the end give the money to buy and why the wife and this just gives you 1 over to pie times is the standard the central function talking about the central allocation so let them use it for me now would it go Of course this this and this so if we're actually in this case the then you get the central functions then you can apply dire and merchandise is the central value for all the L function for someone just ignore that and 1 of the it's product several factors to teach Ephraim which group the elliptic curves a total dollar number divided by the portion part square kind of period as on in that match recorded case it's this complex area so in this on realistically simple case where it was he works out to 0 ii but in this case we would get this this pairing works out to be this so some integer which comes size of the public teacher fringe groups divided by size of portions squared times only and then this regulator comes out to be the reciprocal of so cautioned the 4th times even minus taxes the Commission said what you owe something to cheer OK so that's something that was that's the nature of the list of those state that I made the regulator this divides leader orders injuries but anyway it the regulated divides of fudge factors which is poor sizes portion group which can be very rebounded times this is the interesting part of this thing is the height of Fulton's Heidi although this however that's the way this argument is meant to work that day this this this the analog of this essentially never happen but there's always a overseas non-zero then you have to do something more complicated but your entire warrior is in fact that why I chose an element Delta projecting the 1 and the entire worry isn't this element Delta is extremely complicated because of the old you warriors in fact is a receipt has some very large height so in that case was
but hoops it said yeah so it's a worse if you're not so lucky what happens then you can still write this is a combination of malfunctions so we need to know that we have no other source of understanding the interval properties of things besides birds and became so if you don't have and you completely lost In OK so we have no we need to go through L. functions because in this in in there and we talk about modular curves you can make direct arguments sometimes using algebraic geometry right here there's nothing there's no algebraic structures said the only that that's of everything must be Roselle functions because there's no other way no other handily have stoked what happens here the integral from indefinitely of receipt of the disease AS as saying that a similar thing if you work this out you get 1 of few buyers I initiated this is what the sum I write in the sense of analytic continuation In that then you have this factor you to so this is gonna function but you can write it as so this can be written as as some linear combination so bad for analysis you can rewrite this in terms of directly characters so he can be written as a linear combination also things like Bellows 1 half past half times skies were Kaiser directly character of conducted dividing so this is sort of some 80 Ky His directly character of conductor dividing sea OK right now you can hope it added the same thing you can apply Bergeron .period understand the kind of interval property of this now you have other problems today Kaiser talking to professor and backed up this year you know you can kind of make analogs of of this kind of story also and in the Compaq case but you run into similar problems a Kaiser not integral because you know when you bite out this for example this city is primed for their use and and what will you do some expansion of this year some of kind .period gal sums look something like this but you have a divide that will divide by C minus 1 which comes from when you do for but in this event is non-zero might you have something like this so you get the denominator so you get denominators related to and this is again for this is that you're there but now what you can do this so this is that this is where this sort integral difficulty gets ends up with these denominators but now you can play the following game so to avoid these the denominators you take your original Delta we right Delta his Delta you right as the product of 2 other elements adult honor is Delta 1 Omega Delta to Omega 3 if you you can do exactly the play exactly the same game for these of what you do find a few carefully is you can get a different set of denominators for the right hand side OK so can arrange to get different films the and then again and then so dusty denominators which potentially occurred here I couldn't actually occur in the 1st place OK so just as final this is a similar story in the Compaq for the nominators become related to class numbers but I don't know that I don't know how to play this kind of game effectively because of this last step is the point where we can avoided the kind of win something the reason that the that concludes the for 1 Biden here so right so regulated divides some explicit multiple some explicit and as you dance with its exposed at times all made it in the minds of Gaza's Saudis explicit things really just came in from the portion the size portion elliptic curve so let me just say 1 word about why you expect this to be equality the the failure this to be equality comes from this image the that endanger traced back it is that in that case which group of this elliptic curves now the sort of thing which would need to happen for this to be a it is 1 of the qualities that you would need to have not just of the Jefferies Group B nontrivial that there be some the divide all the orders of all the teacher fringe groups of Old West a case of this kind of thing is is that although it's very hard to prove their conjecture the colored bargaining others suggesting that that kind of phenomenon should never have OK so so I would say to show the quality we need to know another way to say this is something like non vanishing more P results a case which you could think of being about the teacher training being about they'll function but I'd like to think for that reason I think this is really equality and he also 1 more time I think this this general phenomena this might be a general phenomenon regulators Arlington Heights and if so that's quite and there's quite interesting because somehow the 2 sides are of different natures what Minnesota the
questions that have this is the only the revival of is what it is through the use follow should it exited donated to that have eroded down on us so you can do you can run this argument so to speak elliptic curves that contribute to it you can run this argument for each 1 individually and then they'll be another factor about conferences between those 2 forms which I think should be so is that it has to do think you can it's export I never so you know and I have never the that argument problems but has not been mentioned was a formed by former you have some sovereignty because the integral structure doesn't break up formed by former nonetheless I think it's easy to what would I say that without having faltered very carefully when the rented freebie Delaware it went start to go that I guess that it seems to me it should be fine even if directed the free part was on sort of volume to a powerless and 1 but I haven't really thought very carefully about system here I so I said you were also is interested in the only thing this was not related to this fall all I think this what was so yeah for example in paper as part of pieces right he doesn't have a winning an analog of this we're in the walls Pejic context that you when you for analysis on the on this kind of basically an opening for Nelson class groups and you end up dividing by class yeah the Union of soldiers were injured in the school leaders who were not only is of the view is so already none of you lose you don't have much control over things like that thank you for example here in this element where Delta now this which generates levels typically peak in homage corresponds to a generator you don't have much control of weather corresponds to a maximal ordering you know the key and not really shy and I don't while I don't know if so that would have been although accuse of the Europe of the the this is not the only I have to I haven't had much success with the of the world I don't know because of the size of the injured were the 2 of you know what you have to see what was important here in a way that you could not this C-minus 1 you can think of is coming from a class number of the general quarters where and it what is important in this denominator avoidance argument that you have such an explosive understanding of of those some of language the move is part of a move to you don't have none OK I go 6 doing anything you yeah but you would like to we wouldn't be where it is needed and what isn't I'm sorry that I have a hard is the mediated by the leader of the of the year year .period divides this year and I really like way you don't you don't think this is enough to abounded yet because the artist is divided by so it's less than this so this is fine for the purpose of bounding but like I would like to know what's true and I think the taxes should be qualities yeah but for the strictly speaking for the application this is fine yeah the you know regional nations of on which this you know I I like that it already has an interesting incarnation for Geel to over but I don't know from jail 1 2 I think it really might not have gotten explain why the afterward because GL 1 a billion the law and mn