 I would like to thank Alina, Michael and Philip for inviting me to give a talk in this seminar and I guess more than anything else for really organizing and running this seminar which has been a great resource for the number theory community. And also thank you all for coming here to spend part of your Thursday with us. So I'll speak on this beyond the spherical supernorm problem which is on work joint with my dear colleagues Valentin Blomer, Giergo Harzos and Peter Maga. So I promise the talk will end up being about automorphic forms and number theory but I'll start kind of with some amount of inspiration from analysis. So our setting here is that let's say we have a compact Riemannian manifold X and one of the ideas in global analysis is that you can encode the geometry of this manifold X with a special differential operator which is a second order and it's known as the Laplacian delta on this manifold X. So for example like the standard thing that we see in multivariable calculus is if you have some sort of a surface which is maybe a quotient of R2 you might be looking at the standard Euclidean Laplacian which is the thing we love from calculus del squared by del X squared plus del squared plus del Y squared. All right and so the building blocks of analysis on this manifold are the eigenfunctions of this Laplacian operator and so this is the satisfied equation that delta F is minus lambda F and I will always normalize my eigenfunctions to be L2 norm 1 in this stock. And so the super norm problem is really one of my favorite questions in math and it really asks if you normalize your eigenform to be L2 norm 1 so it has a certain mass and it's somehow distributed around your manifold and you could ask somehow how are these values distributed in particular how much can they concentrate let's say at a single point right and in that case you're really asking how large can be the point wise values of this function B and so you're really asking basically about the super norm about the maximum of this function over the whole manifold and you might want to ask what is the correct scale on which to ask this question so you can imagine that as this lambda this lambda is known in physics as the energy of your eigenstate as this energy becomes higher and higher you get these eigenfunctions that are somehow wavier and wavier and you could imagine that this maximum could actually start growing with lambda and it turns out that for the baseline question the correct scale is the power scale so you could be asking somehow how big does this super norm get as the power as a power of the Laplacian eigenvalue lambda all right so this problem has various sources of intuition where you can kind of start guessing what might be the right answer and in particular there is geometric intuition that comes from the fact that the very definition of the Laplacian involves averages over small balls like if you go back to the definition really what it does is it asks somehow if you look at the average of your function over small ball you're asking how much does that deviate from the central value right and because this involves averages over small balls in x that means that this operator somehow encodes the geometry the geometry of x right this is this leads to the famous question in spectral geometry which is like can you hear the shape of the drum like if you have information about if you have spectral information about the eigenfunctions can you recover geometric information about your manifold uh georgia there's a there's a hand raised in the chat christopher yeah yes um it's so you're asking about the soup norm so i assume that the so the eigenfunctions will necessarily be continuous yes probably c infinity okay yes thanks all right um and so again uh this this operator sees sees basically how the balls are expanding and so it basically encodes how the geodesic flow is running on your manifold so from the quantum mechanical perspective what this operator is doing is it's quantizing the geodesic flow its eigenfunctions are really eigenstates they're pure eigenstates of the quantized system and so we have this correspondence principle like you can ask physicists what this should do and basically the the little dictionary that i want to share here is that somehow in in the classical mechanics that's what's on the left side of this slide you can think about classical particles moving in space or following the geodesic flow and on the quantum side what corresponds to this is a discrete set of eigenstates you have a discrete set with increasing energy levels and then the idea is that the long-term evolution of your of your classical mechanical system should be should be encoded in the high energy eigenstates from lambda goes lambda goes to infinity really this corresponds to the semi-classical limit that physicists look at like where the parameter h bar is going to zero h you know what that means and then the idea is that also the geometry should be somehow reflected so for example if you have a stable closed orbit on your manifold then that ends up being reflected in a sequence of eigenstates which concentrate around your around your closed orbit if you look at high energy eigenstates you would be able to recognize some of the geometry of x right now on the other hand I said this last thing that I said was about stable closed orbits right but if you look at on a negatively curved manifold and you expect a more diffused picture for example the the classical geodesic flow is ergodic it's strongly mixing so you wouldn't really expect to see terribly pronounced features in the on the quantum on the quantum side and so for example if you look at the hyperbolic surfaces these are the surfaces that we really like in automorphic forms so these are the quotients of the hyperbolic upper half plane by let's say discrete subgroup gamma maybe finite co-volume then we have this generic bound and the generic bound ends up being lambda to one quarter okay so this this sees nothing except just the local geometry of your manifold and the idea is that because there's a strong mixing on the classical mechanical side you should not be able to concentrate quite this much and so the subconvexity problem in this setting asks for a bond that saves a little bit over one quarter in the in the exponent for the sup north all right so so if you don't know what to expect if you don't know what the truth is then you can always ask your computer to do some computations and to look at some eigenfunctions right the computer or other pieces of equipment so maybe I would just want to share a few pictures so this is this is an actual picture that was made under a low temperature microscope apparently there are 48 iron atoms that have been precisely arranged and then and then they were they were imaging this with the microscope and they color the picture so we can actually see what's going on this is this is an exhibit in alexandra and it was done in in ibm laboratories so this is fascinating like where you can actually see this you can see this concentration happening along a specific geometric shape these are known as whispering gallery effects or maybe more so in a in a more mathematical setting so people look at billiards so these are billiards in the in the usual plane and you you impose a boundary problem and you you ask for you ask for eigenfunctions so a computer can compute this for you and here your pictures of some of these eigenfunctions and the striking thing that you can't not see I mean you see this and kind of like there's this immediate view for example this closed orbit here so this is a periodic closed orbit of the geodesic flow if you hit the billiard ball it really will bounce around here and what you're seeing here is that is that eigenfunctions are picking up on that right so you can see eigenfunctions that pick up kind of more and more complicated orbits as you as you go into higher and higher energy levels all right or maybe another picture that I want to share is this is actually just the this is a different geometry is a geometry on the positively curved sphere and what I'm asking for here is these are really degree 11 spherical harmonics on this sphere there are 12 of them there are typical zonal spherical harmonics and what you can see is that there's a standard numbering schemes for these and you can actually see that there are these that concentrate just around the pole and then somehow as you go through the numbering scheme that kind of dissipates a little bit you can still see lots of concentration and by the time you get to the other end of the numbering scheme you get these eigenfunctions that concentrate around the equator of the sphere all right so eigenfunctions can look very very differently depending on the geometry that you're dealing with in the underlying right all right so I want to talk about maybe there's a question in chat regarding the locus of maximal modulus or versus the locus of zeros would that which one of those corresponds to the closed geodesics in these pictures in these pictures it's maximum modulus so all these pictures are about the size of the of the eigenfunction yeah all right so so yes these are not nodal these are not nodal lines what I was showing all right so um so this is a very difficult problem in general in analysis especially in the negatively curved case um and in the negatively curved case all non-trivial improvements beyond maybe we can save maybe a log or something like that um with analytic techniques but all power saving improvements that we have are for arithmetic manifolds and these are interesting because they give you access to to Kecker correspondences um and we look at eigenfunctions that are not just eigenfunctions of the Laplacian but really eigenfunctions of the whole family of the Laplacian and all the correspondences and those are basically automorphic forms um so for example if you look at mass forms on the on on arithmetic quotients of the upper half plane then we have this famous band of events in Sarnac which really started this subject and it saves from one quarter one quarter six over 24 and it saves down to five over 24 um in the exponent um and I would say more generally instead of thinking about this as quotients of the upper half plane you can think about this as like the upper half plane of course is the quotient of g mod k where g is s of two r and you can think about these estimates more generally with much more general groups g so there are results of Blohmert and Maga for SLNR um my own results with Blohmert and Herzsch and Maga that kind of cover the gl2 r setting and and there's a well known paper of Marshall that covers a wide class of league groups all right maybe as another example I just want to say this is also classical these are holomorphic forms of high weight um high weight k um and in that case of course f of z itself um somehow it really does have to be multiplied by y to k over two so that it becomes genuinely um genuinely invariant and when you look at the super norm of this it turns out that so k to one half is a trivial bound and you can actually get very very strong savings down to k to one quarter so you can get square root um the square root savings over the local geometric bound in this setting right so so these are classical what I want to say here is that um you can kind of I want to put the different spin on this problem um okay sorry before I say that I should say you can also there's many other questions you can ask you can ask these questions in the level aspect where you're looking at the covers of your surface or you can look for hybrid bonds etc so people have done this and I really want to kind of start thinking about this problem a little differently so so what I want to say is that you can look at mass forms and holomorphic forms for an arithmetic group gamma so maybe gamma is sl2z think about it like that um I can think about these really as arising from representations of sl2r and representations of sl2r come basically in two flavors there are principle series representations um they correspond to mass forms um and then they are discrete series representations and they give rise to um to holomorphic forms so the the first point I want to raise here is that actually both of these types of representations occur in literally the same decomposition of the same space of l2 of g mod gamma where g is now the whole group sl2r so so if you don't insist on on looking at g mod k mod gamma then you will pick up not just the spherical forms which are which correspond to principle series of representations but you will also pick up the discrete series of representations and they will all all occur in the same decomposition of l2 of g mod gamma so what we're trying to do in um in this paper is that we're trying to kind of switch the perspective on the soup norm problem um and to start thinking about it on the on the level of the group all right so so maybe for that um I want to start thinking about a similar picture where I look at the group quotient g mod gamma and the group g is maybe a little more has a more rich representation theory than just sl2r um and so we're going to be working with the group sl2c and I'll review that representation theory for you so first you have to look at the maximal compact the maximal compact is sl2 and the kind of the big new thing here is that this maximal compact is non-abelian uh so because it's non-abelian it's um so unlike so for sl2r the maximal compact is sl2 and so so the the irreps are just characters but here you're actually going to pick up kind of genuine finite dimensional representation um so for every for every half sorry for every half uh half integer l non-negative um you will pick up one uh two l plus one dimensional representation that we call tau sub l um and that's basically all that's that's the that's the full enumeration of irreducible representations of sc2 and then when you look at the representations of sl2c other than the trivial one um they are all actually principal series representations and they are induced from a character on the Borel subgroup so maybe you look at the standard Borel subgroup the elements of the diagonal are z and z inverse and what this character does is because z is a non-zero complex number it will have a modulus and it will have kind of the compact part the modulus you can raise to a continuous parameter new um and the compact part you can raise to um to a half integer um and from each of those characters you can induce to the full group and get a principal series representation I see that there's something happening in the chat or should we no okay good okay so so after I've enumerated um after I've enumerated all of these representations of sl2c then I can ask the basic question of automorphic forms which is that um I give myself uh maybe an arithmetic subgroup gamma I look at the space l2 of g mod gamma and I ask which representations occur there in terms of that only the unitary representations can occur I mean kind of like of course because it's the l2 space um and these unitary representations can be fully classified so there will be principle temporary principal series representations and this new must actually lie on this tempered axis IR there will be complementary series representations which occur only when p is zero so this is like the kind of like um okay there is no compact part really here and new must be between zero and one um so this is somewhat parallel to the complementary series representations for sl2r these are your forms that violate the amount of gen conjecture um the selberg robot engine conjecture and then finally um all of these representations pi new p that I just listed they are all irreducible and distinct except that there's this quote-unquote obvious um equivalence between pi new p and pi minus new minus p which is fairly obvious from just looking at what they're induced from all right so um so once I look at one of these representations what does it look like um so I have this representation space b new p this is induced representation space um and um and I can ask for it with respect to the action of of this compact subgroup k so it will decompose into a sum of irreducibles um and so what the way it ends up looking like remember these representations of k there are these tau sub l's they're indexed by this parameter l and what ends up happening is that only parameters l that start at absolute value of p and up uh keep occurring and basically for each of those um for each of those k types starting at p and up uh it will occur exactly once uh in this induced representation and and then in turn this v sub l is a two l plus one dimensional representation and it can be further decomposed with respect to the action of this compact diagonal subgroup uh so this m uh is is the compact part of the of the diagonal subgroup um and of course there the representations are just characters and you can decompose this two l plus one dimensional representation into a direct sum of one dimensional representations v l q so these v l q are one dimensional um so you can ask for what they are spanned by so these are the forms that we call phi l q and you can normalize them to be l to norm one so physicists call this the big nerve basis um of your um of your representation v l all right good so this is the representation theory of s l2c so this is what a principal series representation looks like um and it so basically it starts from these lowest weight vectors when l is equal to p and then you start going up this is similar to mass forms on s l2 r when you look at mass forms on s l2 r you start from lowest weight vectors these are weight zero vectors and then they keep going up when you look at holomorphic forms like that come from discrete series representation the lowest weight vectors are weight k and then they start going up all right um so so so after i gave you this kind of quick overview of the representation theory i want to say one particular type of representations are those when p is zero so when p is zero then you will see the lowest k type the k type when l equals zero zero will occur and that's exactly when your representation contains k invariant vectors and these k invariant vectors actually live on the quotient h3 mod gamma so these are the mass forms on arithmetic hyperbolic three manifolds right and for example you could look at gamma could be for example s l2 of z of i that's the group we take in our work and in that case this quotient this is just this is just the bianchi or bifold that corresponds to the gaussian integers all right um but beyond this and and maybe i should say for spherical mass forms the supnorm is well understood partly by kind of the same group of people so so blowmer hartzusch and i worked on that before but i do want to say that the supnorm problem makes sense not just on this quotient of the upper half space or if you will not just on the symmetric space but it makes sense on the level of the group quotient so you really can think about the supnorm problem just for eigenforms and g mod gamma and so so at that point you get a more general non-spherical supnorm problem which is to estimate the supnorm problem of these eigenforms phi sub lq which are normalized to l2 norm 1 or in general you can look at these vector valued forms um where you group um where you group a single k type together right and you can ask for the you can think about this phi sub l as the l2 norm um of that vector and you can ask for the supnorm of phi sub l over the whole manifold all right so uh so here because we're kind of opening a new problem so this is a new problem this is a problem that hasn't been discussed before it's a problem where you're trying to estimate the supnorm of things that don't live on the symmetric space they look they live on the group quotient and so somehow the new aspect here is to think about this the case when this p is large so p equals zero is the spherical case so we want it kind of a very clean situation we wanted a clean new situation and so so what we're gonna what we're doing here is we're taking a large p so we're taking something that's kind of that that has a really high dimensional lowest weight vector um and and and then we kind of then we keep the the continuous parameter about it right so we're not we're not interested in what happens when the continuous parameter is large which would be kind of like which used to be the large eigenvalue aspect we're kind of trying to minimize the impact of this and really ask about what happens when you when you increase the dimension of the k type in the supnorm problem okay and so just for comparison I want to say here are kind of the most optimistic baseline estimates here there's nothing obvious about these so nothing like this was known or I should say nothing like this was actually easy so I kind of hesitate to call these trivial bounds they were not trivial for us especially not the second one but basically um I will show you a little bit later what kind of machinery you can use to try to estimate these sup norms and you will see that if you don't input any any arithmetic input then these are kind of the baseline estimates so for the vector valued form the the best estimate is l to three halves and then for the scalar valued forms you can expect l right so again to be compared with the spherical super problem where this might be like some kind of a power of the eigenvalue or of the level right so here we're asking on the power scale from the dimension what's happening okay and so then I can actually go ahead and state our theorems um so here is the first theorem that was in our paper was that um again if you fix uh if you fix an interval for your continuous parameter can I ask a question yes absolutely hi hi all right would you call that I mean it's not trivial or not it is uh you can interpret it as a local bound right yes yes absolutely yeah all right that's all of us okay thanks I will say later it comes from the pre-transformal formalism uh so all right so uh so going back to here in this theorem again we're really kind of interested in what happens for the large dimension we're keeping the continuous parameter bounded the baseline bound was l to three halves and we managed to to knock it down to l to four thirds plus epsilon and here when we talk about super norms we are restricting this to a compact part of the domain so we are not we are not presently thinking about what happens high into the cusp where some sometimes analytic phenomena can happen high into the cusp and we're staying away from that all right um so the second theorem is um what happens when you actually look at these scalar valued forms this turns out to be much more delicate um somehow in a totally different universe of difficulty um and so we were able to prove that um if we look at these scalar valued forms in the vineyard basis uh pi lq um and we look at the largest one of them when q is up to l we were able to save from l down to l to 26 over 27 um the precise exponent there is nothing magical about it but the point is that there is savings over over l and then we actually uh tried tried hard to see uh what kinds of estimates can we get um for different special values of this parameter q because if you think back to the picture of the spherical harmonics that I showed you at the beginning um basically these this parameter q indicates um indicates along the compact how your form will behave um and the behavior along the compact is going to be very very different depending on what value of q you choose and so what we found is that uh for q equal to zero we were able to improve our estimate to l to seven over eight that's that's about three times better savings than what we get for a generic q and then actually we get a very very strong saving when q is plus minus l and there we were able to save a square root so so so our upper bound is l to one half uh up to an epsilon all right so those are the three main results um I'll pause for questions okay so seeing none um I'm sorry is is it what are the assumptions um HECA yes okay yes and gamma is s l to z of i we just stuck to that subgroup uh I have a question yes you said use a pre-trace formula right so no spectrum other than the bottom one right is it involved um when you say bottom I mean you you cannot take I'm taking you in a in a in a short interval let's say length one I mean so you essentially use use group action pre-trace without special resolution of laplacium well it's coming up I'll I'll explain yeah I mean this of course you have to use arithmetic eventually yeah yeah but it's it's involved already in the structure of your find itself right so you don't need to probably say it twice I don't know whatever let's see okay good okay so um so the basic the basic setup here is somewhat classical and I will just write it um mostly in classical language so the way we get the way we get a handle on these point-wise values is by this pre-trace formula so what you do is you feed it let's say a compactly supported function f or something closely really maybe almost compactly supported function f uh on the group um and you automorphize it by by summing overall um you automorphize this this kernel um by summing overall uh gamma and gamma and what ends up happening is that this will um this will be this will be gamma invariant so it will have um an expansion uh in terms of an orthonormal basis of the l2 space on g mod gamma and uh and after you kind of go through the motions you find that you will get the point-wise values of phi at this point g the point g is the point g that's appearing on the geometric side um so you're going to get the handle on those point-wise values and they will be weighted by a certain spherical transform of this function f that you feed into the pre-trace formula so this is the spherical transform f hat of phi and your classical spherical superdome problem this will be the selberg-Karisch-Changran transform that we all know and love all right so the first setup um the first step in this setup if you want to do anything is that you have to understand this spherical uh spherical transformer spherical inversion so there's this transform that takes a function f and it sends out this this function that I might start calling h of phi because the way you want to use this formula is that you want to prescribe your h of phi so it's to emphasize your chosen form phi you're trying to get a handle on phi of g squared so you want the h of phi to be large around your chosen form um and then once you kind of emphasize your form phi as much as you can then you look at the geometric side and you kind of want to understand according to this function f um what's happening on the geometric side and so it best what you can hope to achieve in this kind of a setup is that you can try to have on the geometric side you can try to have that only the identity term dominates um and then on the spectral side what you can hope is that you can isolate your form phi um where you can isolate this discrete parameter p maybe to be um at plus minus l or close to that um and you can hope to isolate new in an interval of size roughly one and you can't do better than that because of the principle basically if you try to isolate your phi better you will pay a huge price on the geometric side uh and so uh if you are able to do all of that then this leads this yields the local geometric bound and that's exactly the bound that i was that i was mentioning earlier basically there are approximately l squared representations in this in this scenario um that's the plancher l density the plancher l density is l squared the intervals of size one um and so you will pick up l squared representations and then you by the time you take the square root that's the l that you get for the scalar valued functions if you are able to execute all of this all right um now as i said in order to improve on this you actually need some arithmetic and this is this the first idea that goes back to even in ccarnate is that you want to use what we call the amplified pre-transformula so the amplifier you introduce an amplifier that also sees the operators that are combined with certain coefficients x sub n um and um maybe you want to first look at the geometric side so on the geometric side now the sum over gamma is not just going to be over your subgroup gamma but it's going to be over let's say matrices of determinant n or something similar to that weighted with these coefficients xn and on the spectral side you will get a transform not just of your archimedean test function f but also of this sequence xn and basically you can think about this as the non archimedean um non archimedean test function that you're putting into your into your pre-transform right so we get both of those transforms um and so then the idea is to pick uh not just f but to pick both f and the amplifier x to additionally emphasize your chosen form phi and what happens if if you're successful you will have better kind of combined localization on the spectral side then on the geometric side you will get a counting problem and it involves kekker correspondences and these kekker correspondences you you see them according to the size of this function f so i keep repeating how you really really have to understand the size of this function f that you're going to be seeing um and so for example in the spherical super norm problem this is the setup um this is the classical setup what happens with this function f is the dysfunction f ends up concentrating close to the compact uh and so what you end up needing to estimate um is this diafantin count so this is where number theory really ends up entering is that you need to estimate the number of these kekker correspondences gamma as such that the distance um of this point that you're seeing on the screen to the compact is what is is at most delta right so you so you need to count how many how many correspondences gamma have this property that by the time you apply this correspondence you land fairly close to the compact and fairly close is controlled by some parameter delta which could actually be as large as one but somehow you need to be able to control this both for delta close to one and then for delta very small all right so the first um i want to kind of talk about this spherical inversion a little bit because this is really a kind of a big a big difference that happens here right so uh so when you fix this parameter l um and you take a function f uh in the tau l isotypical subspace so these are these are functions that are um let's say compactly supported um and they behave according to tau l relative to the k action on both sides so we're not going to be taking functions f that are that are by k invariant that's not useful if you take a function f this by k invariant you're going to pick up only the spherical forms so we need to look at functions f that are in the tau l isotypical subspace and that's exactly what allows us to pick up to pick up the forms that we are interested in so for those um for those test functions f you're going to be seeing a spherical transform that is given by this formula really what you're doing is you're integrating the function f of g um against this spherical function and this spherical function i denoted by phi new p l so this l is the is comes from the tau l type and this new p comes from the representation type where you're trying to compute your spherical transform and this function phi new p l of g that is literally like a special function like like you could literally you know if grating rigid worked very hard it would be in grating rigid like like it's a special function um and um it's known as a spherical trace function um this is how it's given um so it's the it's the it's the trace of what happens after you apply your representation and so you have this map that takes a function f in the tau l isotypical subspace and it sends it to this to this spherical transform f hat of new p and this turns out to be an isomorphism of Hilbert spaces and there is an explicit inversion of this isomorphism this is due to Gelfand and Neymar in 1947 it's a it's famous work of theirs that applies kind of more generally but i'm talking about an isomorphism of Hilbert spaces here all right now here's the interesting twist here is that uh for analytical purposes you don't want a function f that's like just some random l2 function i mean you need to have good control on this function f uh you want this function f to decay reasonably you want it maybe to be reasonably continuous etc so so if you want a function f for example to be compactly supported and smooth on g um then this function f hat of new p uh is not just a random function of new it's an entire function of new because of course it's some sort of a transform of of a compactly supported function i mean just think about the Fourier transform of a cc function is going to be an entire function and so in particular this function f hat is going to see all of the non-unitary spectrum as well so this function is not only defined for new on the temperate axis it's defined on all of complex numbers all right um and so what you end up seeing here is there is this unexpected symmetry and when i say unexpected i'm just going to see it was unexpected to us um so there is a symmetry in these spherical trace functions that this spherical trace function phi new p l has a symmetry which relates new p to p new which which kind of like initially you're like this makes no sense why would you ever switch new and p new is a continuous parameter p is a discrete parameter um so this symmetry only makes sense when new also happens to be um uh like a half integer right so new also has to be a half integer and it has to be super far off the temperate axis um all right um so so the uh question uh yeah go ahead Christopher Lloyd why don't you just unmute and ask away you're muted thank you in in the previous integral defining f hat you integrated over all of g not just the compact right yes okay but of course if f is compactly supported then it's okay okay so they're probably related just by a multiplicative factor as in the as in the volume of the of the maximum torus or something yes yes okay all right so anyway so there is an explanation like when i saw this i kind of couldn't really accept it until i had a good explanation for this and the explanation is that pi new p and pi p new they're not isomorphic to each other but they actually agree up to a small finite dimensional factor or sub quotient so one of them is irreducible and the other one is reducible and and it has a small finite dimensional piece which ends up contributing nothing to the spherical trace function and so the spherical trace functions end up having the symmetry but you see once the spherical trace functions have the symmetry then your spherical transform is going to have the symmetry of any function f right so you're seeing this function and and so at that point this kind of becomes really really interesting because now the question of somehow how do you go back i mean you can only go back from functions that satisfy this extra symmetry and you would maybe want to ask can i even go back for every function that satisfies this extra symmetry and this is effectively the palerina problem like if you think about like for classical Fourier transform there's a question of somehow if i start from cc functions what kinds of functions can i get as a Fourier transform of that right and the cc infinity palerina problem was solved by wang um in actually a paper that goes back to 74 um and and so the basic problem of this symmetry is that it makes it very very hard to localize to p equal to plus minus l we remember our our goal in the pre-transform is to localize to our favorite representation as much as we can so we would like to localize to one value of p but the problem is that if you want all the other values of p to get zeros then you're also going to have to you get a lot of points off the tempered axis where your function has to vanish and now all of a sudden making a function that satisfies this it actually becomes very very hard analytically so um so one thing that we that we did in our paper and i just want to advertise this because it might be useful to other people is that um we eventually decided that trying to do this within the cc class uh it was just very hard we spent many months and we were not able to do it um and so we ended up proving a schwarz class palerina theorem so this is a palerina theorem that classifies if you start from functions that are that are tau l isotypical on both sides and they're smooth and have all rapidly decreasing um derivatives you want to ask what kinds of functions can you get sorry you can ask what kinds of functions can you get there's this vertical transform of this and the answer is that you you need to get functions that are rapidly decreasing in vertical strips um and that satisfy this this extra symmetry in addition to the analytic continuation that i already commented on and so with this theorem once we were able to prove this schwarz class palerina theorem we were able to take basically a variation of the gaussian um for our uh for our spherical um for a spherical kind of function so this is not exactly uh this is not this ends up being not exactly concentrated on p equal to plus minus l but but it decays very very rapidly for other values of p like already for p equal to l minus one uh you get kind of like an exponential savings and so these other values are not a problem for us all right so so the last thing i want to talk about in terms of mathematics that we encountered is um is i want to talk sorry i said last thing second last thing um i want to talk about the the concentration on the geometric side and so if you want to understand the concentration on the geometric side uh you need to understand the concentration of these spherical trace functions and in fact you need these generalized spherical trace functions where you have an extra parameter q which controls like which vigner basis element you're trying to you're trying to to pick up um and so this ends up being an analysis of of special functions as i said this is like a special function that you know you're actually in the rigid was much thicker it would be there right and so you actually have to estimate these intervals these are like six two-pole intervals and you just estimate them with a lot of stationary phase analysis um and so we proved that they concentrate close to various sets i'm just going to tell you what kinds of sets we've got because it's really interesting so some of them concentrate close to the identity matrix some of them concentrate close to the centralizer of a in k and this ends up being the set of diagonal and off diagonal matrices in k like it it's a union of two one-dimensional sets um it can concentrate to the set of diagonal matrices in g which is a two-dimensional variety um they can concentrate close to the compact this is similar to the to the spherical supernorm problem and this is a three-dimensional variety here um you can concentrate to this funny set of matrices a b c d where the modulus of a and d agrees and the modulus of b and c agrees so interestingly this this contains the compact k but it's actually four-dimensional there's a four-dimensional sub manifold um and and then you end up being left uh after the pretrace formula you end up being left with a number theoretic problem where you're trying to count matrices where this where this quantity g inverse gamma over root 10 g um is close to one of these manifolds and so in the spherical supernorm problem this was always how close are you to the compact and here we get kind of a a variety of different counting problems and so because they're different counting problems you kind of have to solve them differently right um so maybe i want to kind of give you a so this leads to to these counting problems in c4 or c8 depending on which problem you're in and it's intersections of you know small balls they can be centered randomly not necessarily at rational points they can be centered in thin wafers sort of like a like a like a purette cookie uh or they can be uh oh sorry those are thin cylinders or they can be in thin wafers so sort of like a like a like a flat plane that's a little thick and then of course it generally it lies in a general position so it doesn't have to be like rationally sloped or anything like that so these are very interesting counting problems and these are the parts of this these supernorm problems that we really like to work on because they really have arithmetic flavor to them um so maybe just to kind of i said how these functions concentrate close to these things and i just want to kind of show you what those estimates look like so that you understand what i'm talking about so for example for this um for this spherical function that doesn't have a parameter q so i'm not trying to pick just one element of the vineyard basis um i can take this matrix g um i can conjugate it to an upper triangular matrix with a compact matrix um and i get an estimate that looks like this on the right hand side um and so how do you read an estimate like this i don't want to talk about the proof i just want to say how do you read this estimate well you say okay so the baseline estimate is l but then i look at the second term and the second term tells me well i actually get much better than l unless z is very close to plus minus one uh and then i look at the third estimate and i say well i still get much better than l um unless u is very close to zero so i get a much better estimate than l unless z is close to one and u is close to zero but if z is close to one and u is close to zero then my g is close to the identity and so what this says is that this spherical function in a certain soft sense it's concentrated like it could be pretty pretty big close to the identity but then otherwise uh it starts to decay right um and then the other ones are kind of similar flavor so i'm not going to go into details but like once i have a value of q right then um i can see for example here i get savings unless g is close to k and g is close to that set of diagonal matrices in a certain uh precise quantitative sense um and again for q equal to zero and q equal to plus minus l we got uh upper bounds in terms of distances to some um to some other um so so here you can see these sets s and n and here you can again see the sub manifold of all that matrices all right so this is what those estimates look like after analysis and this in a sense is a lot of our hard analytic work and our paper is to prove these concentration estimates and so in the end i wanted to say like a couple of new features that i'm really excited about um in this line of work so first of all what i really like is that there's a mix of positive and negative curvature aspect so we're really kind of we're really not throwing away the compact we're really interested in the compact um so there is a part of this picture that is positively curved and there's a part of this picture that is negatively curved and you actually when you're doing analysis you start seeing both of those coming together right um the other thing that i that we didn't necessarily expect and that i think is really interesting is that the analysis of these spherical functions is highly sensitive to this parameter q um and the case when q is plus minus l is very very interesting and interestingly it is reminiscent of holomorphic forms it's reminiscent of what happens when you do the super norm problem for holomorphic forms um so um so basically when q is l or maybe very very close to l we're thinking maybe maybe about square root of l away from l uh you you see basically the holomorphic form super norm problem and then when q is in the bulk when q is not within square root of l of l um then you then your problem is much more reminiscent of the mass form super norm problem um so in this case when q is plus minus l we uh the reason why we get much stronger estimates is that we adapt the method of coyote and steiner and we write it in a we execute it in a different way we don't actually use the theta correspondence we combine the Voronoi summation formula and we get essentially our hands on the on the fourth moment estimate of our point wise values and we can only do this somehow in order to get our hands all the way to the fourth moment uh we actually have to be able to take our expected correspondences with n as big as the as the approximate functional equation because we're trying to collect um we're trying to we're trying to we're trying to collect precisely the value of phi so we really have to go as far as the approximate functional equation goes and so we need very very strong localization for q equal to plus minus l so that we can take the heck of correspondence is very far um and so maybe um so so i just want to say this was some somehow interesting when i gave a talk on this um in the budapest conference on automorphic forms uh we talked to jack budcain a little bit about this work and jack pointed out that um actually you can um you can bootstrap this scalar scalar form estimate um so we have the scalar form supernorm estimate which is l to one half and you can bootstrap it to the vector vector value supernorm estimate which is l to one which is actually much stronger um than what i previously announced here um and so so it's really interesting this is something that was not what i could initially anticipate it is that you would think that the vector valid forms are easier to um are easier to bound because your spectral your spectral average is is richer and you're trying to you're trying to isolate less so it looks like this should be an easier problem it should lead to more localization but that's not really how it works it turns out that these forms when q is plus minus l they seem to play a very special role and it seems like trying to estimate those is the correct way to estimate uh also the vector value right so that was somehow something unexpected and i thought very interesting um and finally i was maybe even my primary motivation in this work was that i realized that unlike the spherical super problem the concentration in the non-spherical super problem is not just along the compact so you get different you get different um different counting problems and you encounter new difficulties but maybe you don't encounter some old difficulties along the way and finally i would like to say that the spherical inversion for sl2c seems to be underutilized i'm not seeing use much by automorphic forms people on the level of l2 spaces it is classical and we now have both the cc and schwarz class spherical inversion available so this bailey reiner theorem of ours i hope would be a useful tool for other people who would want to work on this i mean that i would like to thank you for your attention