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4/4 Automorphic forms in higher rank
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00:03
Hey Hey Jude and thanks for
00:17
coming so um and I announced last time be made application today which will basically feature all the ingredients that we had in the previous lectures will be the moment estimate for functions and in order to prepare for this I would like to recall states formula that we had last time or perhaps In the 2nd lecture so this is 1 of the few 11 few analytic results that we have on the hiring Whittaker function if I take the Whittaker function attached to the index of and multiplied by the Whittaker function attached To the index beta times the determinant of why the power and the harm measure on the group then this is the product of gamma functions that you would expect from rankings of the products G came from 1 to 10 gunmen are this class II of do mind is believed to have our cake and this has to be taken with a grain of salt I'd and Archer of the bars or correct but morally speaking this should be the case divided by gunmen are of in times the product 1 less than J strictly less than K this article than end the are of 1 plus Alfred J. my disqualification times gunmen are 1 I'd be dead just minus the 10 everything about OK I so we will use this a million times today and that what I would like to introduce knowledge Rankin said .period jail and also 1 of the things were 1 of the few things that worked exactly as the GAO too so for general and we consider the maximal degenerate I understand series the above that ,comma as I defined as in the case and equaled 3 and in the general case at the it's exactly the same thing as the sum over gamma modular the maximal parabolic subgroup determinant godmother to the power S and P is the and with this definition the same unfolding works as the G so if we consider the inner product of its time G bar against and I found serious the series then this is by definition the integral of age modulo gone they have gene got the of Dr. Coleman and
04:20
now we can do the usual unfolding and this is the sum over gamma Mort and this is an integral over age gunmen and by gunmen variance we can unfold this to an integral what .period but it also said Barros said determinant of to the power the and at this point I insert
04:59
the free expansion for 1 of these factors the the first one and you remember the expansion was a little complicated it had this summer Over lower outside a still in minus 1 module the UT potent subgroup and here again we can collapse integral and some and don't this becomes an integral over which marked the a subgroup of dimension in and then some over whether the piece of the indices of the Fourier coefficients AM that is the that divided by the normalization product and Jake the power and shake hands and chat times the Whittaker functions times the exponentially where these fixed jays are the 1st off diagonal the foreign expansion and this additional some collapsed with the integral and then I just copy the rest and at this point I can execute the interim end insert so if I integrators over X then by definition I just get back the expunged the free coefficients of Jews so I've performed the effects integration but I haven't before the integration over over X that's now to come so I integrator Rex and then I left only with the integral over a so this is the y integrations that that's the subgroup and what is left song everything is killed by orthogonal achieve characters except so be the Fourier coefficients divided by the normalizing products and now this Over to disappear because they couldn't twice from a and B. and get it Whittaker function attached to the Form Whittaker function attached to the 4 G so we call on was the diagonal matrices having the end and products thereof as entry the X doesn't contribute to the determinant so only the white contributes to the determinant and this is the HA measure on the group OK and here we are in good shape we can do a change of variables to remove this and then what we're left with is essentially the integral over to Whittaker function and this is precisely where States formulas so this sequels the reconciled Albert L. function of the times G bar at St divided by the 2 of them as times the Gummer ratio that can't comes from States formula where the rank somebody'll function is defined as follows by definition the and as this is needed to have a nice on a product and then divide by the 2 of them this year and as the usual case about you multiply by the 2 of 2 s the rankings of the girl function in order to get a nice on a product and then AM bars divided by everyone to the power and minus 1 and then all the way to end in minus 1 to the power was To the power so that the last coordinate comes with power 1 end of the quarter come with high although the rules theoretically this this unfolding is slightly more complicated but in principle at the exact same idea and give the exact same results OK so Rankin suburbs very robust that works for General so in particular but this
10:24
series has a role as a poet as equals 1 with constant residue and so if I take the residue of this is as good as far as equals 1 and take if equals G than just get them the Peters along since the residue that is equal to 1 of the items that series constant we have appeal is right the this equals 1 at that the at what kind of some time to make sure the Gamma factors from aviation should write this so the completed function my notation and the completed functions include the appropriate time yes fun and questions OK so then let's move on to the application the for example moments of melted so this is now a real analytic number theory flavor on and is based on a paper of mine that was published perhaps 2 years ago so as a woman that's indicates any boasted that's the case we all familiar with OK and I want to prove the following theorem take a classical homomorphic cost form of weight case and safer level and then we look at the following 2nd moments with some over a basis of cost forms Of this space so be carries the basis of this case the rankings of functions held onehalf every time she doesn't matter whether put a bar here or not because in 1 everything real but this we restrict guide OK and so this summer has K terms roughly and on Lindelof we would assume that the whole thing is of size skate and this is what the theory says there is a bond of sigh skated want Simpson so this is as strong as Linda on average and it's on the edge of subsequent accepted the Subcommittee so they the the convexity estimate the square root of care and so on if I dropped all but 1 term just get square of that that's sits on the edge of the complexity OK so how How would you proved well if you're trained as a classical analytic number theorist that the 1st thing you do in this situation is wrapped up in approximate function equation opened the square and before the summer overdue using the Peters but if you do this then the following happens approximates functional equations plus interest formula this leads to a double some coming From the square open the squares so you get some of and em both are roughly half the size scale and you have land that involved in land that and and then there is the sum of a sea coming from the From the Peterson ,comma
15:39
turns out that this some overseas especially bounded so that we can just to see to be 1 of the the Klosterman from vanishes and only the best of functions and you
15:52
get something like this would all of the people to understand the advantages from the formula for them and it disappears because that's what yes but so this is the
16:07
beast you would have to estimate are offside scarier to sums of length came so square or cancelation would give you the desired care but if you look at this doesn't look very nice because both variables are size case the arguments of size can be in excess of size scale so you're in the fragile transitional range of specified he knows something about Bessel functions if the argument is the same as the index that the disaster so probably you can push through if you if you're a good analytic number theory but that's probably very hot so it's probably hard instead I would like to use a very different approach and in addition this approach would never work and so I use a very different approach so instead we do the following go to fix notation that may removed the if you guys just landed times in to the came was won over 2 times and that and I defined the L infinity factor of if terms G that's the usual infinity factor that you get in Rankin celebrated scum of as of a prior to this time from got out of this plus came minds wander over fold higher to the claimants 1 and as you can see that doesn't depend on average only depends on the common way so and the weight is something that we don't change during the arguments that we just don't know this by and and then we have seen before in month in much greater generality that the square of the 2 norm it is well it's essentially L. infinity of 1 times the finite function but it turns out that the residue on the final day of functions bounded above and below so it's of size K the power plus minus epsilon This is my shorthand notation for an upper and lower bond at the same time it's abounded about biking to the excellent bounded below back into the minds and now let's run the following arguments so we are interested in the sewage L 1 half their times you square was the 1st thing we do is we work with the completion of malfunction of the completed functions easier to handle that but the final part of the a function this is some overdue onehalf at times just squared divided by an infinity of 1 square OK
19:25
so far I haven't done anything OK
19:35
I 40 to keep track of what we're doing so becomes the 1st step I use rankandfile Burke to ride this as but in a product so this is 1 of the island fileted onehalf square times some over inner product of the best OK you're a writer different it doesn't matter if Jeter and I got home I went right that's ranking sobered unfolding the but yes the existing 1 and once is banking OK we want to prepare for spectral analysis I would like to have a sum so this is the some of it he forms but I would like to have an L to normalize version of this so I would like to have G be up to normalize to artificially belt to normalize its it belt infinity 1 square times some overdue the new product in he called . com on squared and now I divided by the 2 nor often GE squared
21:15
and to compensate for this we know what the 2 Norm squared is it's essentially elephants and 1 not to a small error term so the right catered 1 year so that this equality is really an upper and lower OK so what do we need for this step 2 is again rankings several classes we need to good upper bound for the residue at St equals 1 set up a bond for the residue at St Paul's 1 L profit there's times can spot now this looks very much like applying possibile moved this out of some the negative case political 1 eligibility had 1 balancing its at 1 per squared and bypass about well maybe it's specials inequality but this is so this thing year square at times he had 1 half right this summer the spectral expansion of the square OK so at this point it's not really clear how to containing a nice thing would be to apply cautious shots at this point unfortunately quarter Schwab not easily available and I would like to estimated by the 2 normal at times to normal of the but the 2 of you doesn't exist so I cannot easily apply kosher shots but I I have something what we can do something similar and I will explain this later this is the sort of regularization the suspects for 4 needs some regular so that this was real so trees possible Morris some regularization that I will explain in a moment it turns out that this it is bounded by cater the little 1 divided by Bill onehalf squared and electricity 1 times being a product of its time and daughter ,comma 1 plus excellent so basically what is happening he you trade off 2 copies of the onehalf against 1 copy Of the at work OK I will explain the step later let's believe that for the moment now we can use rankings Albrecht again I so this is a mail functions so this gives came to the little 1 electricity 1 divided by electing to have onehalf square times Bell Infinity at 1 Pleasants and again I have to uh major residue years so 55 is the same as that too we need Rankin cell broken up about 4 the residue 1 and that's it now we only have to collect the scum of actors and you see precisely what happens you get a half power of Kate 4 1 against half and you can another have powerful 1 against her and you end up with To the 1 excellent so this States formula In the jail to cases that gives a precise description Of these gunmen tactics so in this case it's just dumb of Katyn scum of plus excellent divided by gunmen of K minus a half pence ,comma came a half in this case twice a half of OK so as you can see we did basically nothing I mean we never looked at for a coefficient of which it's a it's a very soft on wouldn't do anything the only not completely trivial steps if this and I'm going to explain this won't so
27:04
concerning so there is the at onehalf square is by definition the integral always a fundamental domain of H. Montgomery all of the squared times the square and here I estimate so you pick a special fundamental domain name pick the usual the so I mean that I'm only interested upper bounds but it doesn't really matter so so I take fundamental domain with sufficiently large white coordinator so either this or even the corresponding Siegel said and the like simply used before it expired June 2 boundaries by the White onehalf square right now so that the constant term the 4 expansions dominating everything else is rapidly decaying and now you see Hollywood tradeoffs 2 versions of 1 half against 1 version of 1 of the reasons why to the want less Epsilon 1 plus 2 and this in turn is bounded about from the Eisenstein's series of 1 excellent again for expansion just so as to the identity terms and you just add the others artificially and given up about this is grounded by interval over it from the square at the end of the line plans Epsilon Is this contains as a constant term y to the oneplus Epsilon and entered quoted by positivity the other terms for real argument all nonnegative so you can artificially at them if you want precisely what OK so freely told me that this argument also plays a role in his newspaper with action yet so in any case this is how you regularize this the idea that serious 1 so that's all and that's that's complained that completes the probe what is another's was the warmup the at any equals to cases and well let's do the same thing for GM so now the sale on Jalan and we have some good chances to 2 succeed because as I said we basically we did nothing I mean we use rankings Alberta we used the state's formula we use possible OK and we have to do something about this regularization Of course we do cheerful must forms the oval Moffat forms and so let's have the same on Geo and philosophy but it's obvious the theorem this is a really bad news for the I uniformly in a neighborhood that's true because here I'm evaluating it and want it OK fine so what would have yet but I certainly can bond the residue then you can also bonded in Epsilon away but theorem that there ,comma GB "quotation mark simplicity temper so so satisfying the Roman conjecture that spherical glass forms for the group's they sell and then With respect of spectral parameters new and new but in I'm not consistent with what I said earlier in 1 of the previous lectures said that my unitary accesses the real access knowledge the Nigeria axes but I think you can call it that so they purely imaginary by which I mean that they satisfy the Roman which conjecture OK then if so
32:22
G With so is fixed No . 1 it is fixed and I'm summing over G I so I'm something of a spectral parameters that our in both 1 neighborhood of the given spectral parameter you that is the same as sort of taking the same way if you take the same weight then you spectral parameters also off 1 neighborhood and it's on the knows the same thing but the next 1 is this off 1 as the dismissal 1 so I knew when you are at a distance also won L onehalf its times G square and I claim that that that the boundaries of of the cardinality of this 6 so what's the cat analogy of the said well that's a spectral density that new and that this is measured by the harsh and receive function so this is the density and you have to give it careful if I'm very close to the walls that the measure can do something funny so let me write the till the end but by this I mean so by definition this is the product that a you may want to hear it products 1 less than Jerry let them came less than an 1 plus when you jail minus so the measure itself may behave very strangely say 1 of the 2 of us so there is a pair of parameters of equal and so I have to adjust this a bit by adding 1 plus years yeah but anyway and add to the 1 but it's lot so that I lose its alone on the way all the buttons bearing the same theorem your averaging rankings functions by taking a fixed form of very large spectral parameter and then you average over off 1 ball and the cardinality so that the upper bound of the continuity of but how do we proved as well we proved in the exact same version say if we have all the ingredients available OK so ranking Albert available but we need an upper bound for the L function at 1 end in a neighborhood of 1 and for the residue for much of it certainly was 1 of the things you have to give it all up probably needed I think it was just laziness because if if if the manager and conjectures not satisfied that maybe you have to be modified that you have to figure out what exactly states formula says the 2 conjugate or do you take the negative in so that in can certain it's not necessary it's just convenient so that you don't have to bother about exceptional item OK so as I said we need to upper about and that's in the literature the pellets at times due this close to 1 for an apt to fall business proved Professor Harold Brown and then for general and by John who was probably in the audits although what else do we need so this is amenable parsimonious Alice always available famous through that we need states formula OK so states formula tells us so that the end game will be that we need to estimate L Infinity would it at 1 square divided by an infinite and how often states formula tells us what this is the terms of a gammaray and we just have to verify that it coincides with the spectral density and that's what it does so this is of the order of magnitude of 1 oversee children square OK so what remains a step for we need to do a similar trick for general I've science series in order to make the transition from 1 half to 1 focused world a little lemon the OK so remember up there the important thing was to balance the eyes fences and a fundamental domain by basically the 1st for a coffee and we do the exact same thing he had to the Z 1 half is bounded by determinant of to the onehalf just less epsilon plus dual determinant of Z 2 onehalf cluster saloon where there still there W that minus transposed W and nobody is the long vitamins and then what you do is you bring this back into canonical us out the composition and take the determinant so in canonical form and then that has to be a suitable said I can't be too close To Bottom of the upper house space so is that it is in his zeal men that is something why and age such that saved the expired chase unbounded and the wide jails all not too small the constant squared of 3 of which were placed but it's a valid concept OK and assuming that this limit true we get is a corollary that the integral Over the fundamental demand which contain which is contained in it will that squared times he observed onehalf square the of we came to terms with the 2nd term we make a change of variables to get back to you and the price we have to pace that we get killed for the for the past form so this is bounded by integral it that squared plus there is the killer but this is just the bull lost determinant of visits to the 1 excellent can run the same argument with the form itself and with the dual form and you get the same result but little sketch the proof of this land that is not very exciting it's essentially the same idea we use the foreign expunged all of this the generalized sense we use a special form of the so 1st we right so 1 of the main features of the maximal general Eisenstein serious is the fact that it's really extends which so if you slightly we normalize things the chairman of that the power is divided by the Zito in this at times as at Epstein's ETA function they transport wide transposed Y X where X Y is merits at that point in it's over too where is that of a matrix M and a complex number role is the usual extends the functions onehalf times some all no 1 0 victory no 1 with the corresponding quadratic for schools and to the poor so you have the values Of the quadratic form if in this positive definite but this guy he is certainly positive definite the classical objects to this extent so OK I so the foreign expansion of this extends the function can be found in the literature the so computer inductively for a spot promulgate ordered terrorists there's a matrix S is given the following form identity identity Q transposed time the plan following her notation this is now a matrix S and then I hear and this is the end 1 block and this is the end to block so this has anyone rose and this has been roles and told us we have it 1 class and 2 then gunmen are off to that the move role can be written in terms of the extends function associated with the smaller matrix S 2 and then you can inductively move on so this is guy Motorola ,comma are 2 role as a cases 2 we and this is a smaller matrix plus a matrix of frenzied a function for tea are to Rome writers and 2 divided by determinant of St 2 square 2 of the world tour among a complicated terms it that takes care of the cross products and this is the final terms is dying from them we need term for the cross but this is rapidly decaying because now comes the 4 expired in terms of Special K functions and the bestlooking function decays rapidly so it will not contribute to much so there is a son of a and B paying visits To the N 1 nonzero and beating Z to the end to nonzero France proved 1 quarter in 2 months .period from and there is no technical transport as to be onehalf largest reported to times between principles here so here the crust this 1 how and to minors or 2 pi squared transposed the transposed was to indeed and that's it OK and that it's an exercise to boundless so you this and actively reducing the dimension step by step and this can be estimated trivially using the rapid decay of the best function and then you continue with this too which is onedimensional lets you apply the same argument and you keep doing and then eventually you end up let me continue here so eventually you end up with a sound J from 1 to end some of the easy terms that may contribute some polls but otherwise they're easier and then is that 1 up to Jade to the miners onehalf said Jay the GAO over to landless rural uniformly in matrices tests of the formed extransport was that eggs where capital there is a diagonal matrix Is it ended up to said 1 and they order such that is the page and that 1 of the smallest and none of this means more other energy plant decision so this is the bound for the expands function based on the foreign expansion if you plug this into the above formula and you get a bound for the island that series and if you combine everything when you get them so this requires a bit of work a casebycase analysis but the idea is fairly straightforward and so this proves the result the moral of the story is that in higher rank it's often useful to use of techniques and not to try to you things like approximate function equations and then you end up with a total mess that you cannot handle the soft techniques to generalize OK Well I guess that's the end of what I would like to teach hope you enjoy this and that nobody will be too angry for entertainment early but maybe your question In good question would question them and I should also mean I said that soft techniques more easily generalisable but at the same time of course they are not strong enough to prove something really likes convexity they do give something highly nontrivial them namely the best possible moment estimate but it's not strong enough to prove some convexity which is morally equivalent to an asymptotic formula with powersaving heir to what we can we can look at this Provencio where we failed and I mean basically we fail almost everywhere together to get an asymptotic formula so that the 1st small cheaper here than in the 2nd is when we apply Bessler the inequality that have because there is more than just saying this is our business Over the whole space of our dimorphic forms including forms of weight can so the whole amorphous forms of weight K are of course the picture that contributes but there also must forms of weight that also contributes to here is inequality and then again we have the small fluctuation in the residue of I on scrap paper I worked out a version where I replace K to the epsilon by a but I don't know how to make this into an asymptotic formula let alone any character the In chances of the all but will potentially is but that's a totally different story but of course you also have rankings I and perhaps a bit sloppy that so this this is rankings of 1 version of rankings Virginia jail intends you can during conservative all possible combinations you intend Jilin and you're asking for Jia lanterns dealing might as well and this thing is of course a totally different story because of the period for there is a period formula but it's it's a very different periods and will involve a summation over the border dimensional GLA minus 1 spectrum and there's a very beautiful work by Matthew young in this direction he he is computed several moments for the G L 3 times jails
00:00
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Titel  4/4 Automorphic forms in higher rank 
Serientitel  Summer school Analytic Number Theory 
Anzahl der Teile  36 
Autor 
Blomer, Valentin

Lizenz 
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DOI  10.5446/16455 
Herausgeber  Institut des Hautes Études Scientifiques (IHÉS) 
Erscheinungsjahr  2014 
Sprache  Englisch 
Inhaltliche Metadaten
Fachgebiet  Mathematik 