 Thank you very much for the invitation. I'm really happy to address the web seminar that has kept the number theory community together during the dark times of the pandemic. My subject today is going to be counting and equidistribution in hyperbolic space says whatever this means and I'm going to describe a number of results some of them going back to older times in the 50s all two very recent results of me and my collaborators who I'm going to talk later about. So let me start with a motivating example. I want to talk about the arithmetic function r of n which counts the number of ways of writing n as sums of two squares and consider its correlations. So I'm going to consider r of n multiplied with r of n plus four and I'm going to sum up those quantities up to x minus two but with some weight which may be confusing at the beginning I'll put weight one for n even and weight one half for n odd. The function r of n is almost multiplicative in the sense that r of n over four is multiplicative given by the character module of four as you see in this equation. It is also unbounded so correlations of these functions cannot be dealt with the theory of correlations of arithmetic functions which are bounded or have values plus minus one of things like that. The first step in trying to understand this correlation is to rewrite this equation in terms of more variables and eventually reduce it to something that has to do with automorphic forms. So since n is sum of two squares and n plus four is sum of two squares we write n as t square plus s square we write n plus four as r square plus u square and now we introduce four new variables a b c d where a is r plus s over two b is t plus u over two c is t minus u over two and d is r minus s over two. Then the condition that four which is n plus four minus n is the difference of sums of two squares can be rewritten with the new variables as the standard determinant condition a d minus b c is equal to one. Okay here now some parity considerations if we have that r and s have the same parity and t and u have the same parity then all the numbers a b c d are going to be integers. On the other hand if I give you integers a b c and d you could recover r u s and t by solving the linear system that I wrote before and here is the solution. Okay if n is even one can easily check that r s t and u have the same parity. If n is odd then r and u will not have the same parity and s and t will not have the same parity. So either r and s has the same parity or r and t has the same parity and that will cut down or counting by a factor of one half. This is why for n odd we introduce the weight one half in the previous slide. It doesn't matter whether you have followed all those parity considerations or not the upshot is that now we have four new variables which will be integers for us a b c and d. They will satisfy the determinant condition for a two by two matrix a d minus b c is equal to one and then by just substituting what is a b c and d in terms of r s t and u we say that this is n plus two which will be less than or equal to x because I was saying n is less than x minus two. So we have now re-rided our counting in more variables so we're looking at integer solutions of the determinant equation where the sum of the squares is less than or equal to x. Now that will push us very quickly to this realm of automorphic forms. So this is a very quick introduction to what happens in the hyperbolic upper half plane. We have complex numbers with positive imaginary part and we act on the hyperbolic upper half space by linear fractional transformations of the form a z plus b over c z plus d which we set into correspondence with two by two matrices a b c d with determinant one. So the group really that we are going to be interested here is sl2r and in particular discrete subgroups there were a b c and d r integers. This is the group that we call sl2z. Of similar importance in the whole arithmetic of automorphic forms elliptic curves we have Hekia congruent subgroups where we put the extra condition that n is dividing c if the level is n. The more general subgroups that one can consider and some of the theorems that I'm going to state are going to be stated in the more general framework those are subgroups which are co-finite or co-compact. This means that the quotient of the upper half plane by this group gamma is going to have finite hyperbolic area or be a compact space. We will also consider particular examples of co-compact subgroups coming out of quaternion algebra. Okay so what's now the geometry in the upper half space? We have something called the hyperbolic metric where we have dx square plus dy square over y square. This means that if I give you a curve x of t plus i of y of t in the upper half plane then you could use this formula here to calculate its length and geodesics in the upper half plane will be curves that locally minimize the distance between two points. It is a well-known result from hyperbolic geometry which you could find in let's say Ivan's books on automorphic forms and in many other places that the geodesics in the upper half plane are raised perpendicular to the real axis emanating from the real axis like the red line that you see here or semicircles centered on the real axis. Okay now we come to the further reduction of their initial problem to something that has to do with using hyperbolic distance. There is a fundamental point per invariant in the upper half plane that tells us how to measure the distance between z and w. So we take the absolute value of z minus w square and we divide with four times the imaginary part of z and the imaginary part of w and then the cos hyperbolic cosine of the distance between z and w is 2u of z comma w plus 1. Moreover we can evaluate what happens between the point i in the upper half plane and its orbit where we act on i by a linear fractional transformation a, b, c, d. The calculation gives you exactly the expression sum of squares a square plus b square plus c square plus d square minus two divided by four. And here is now what we are really asked to do in the correlation problem that I introduce as motivation. We want to count the matrices in sl2z so that 4u gamma i plus i plus two is less than or equal to x and this is the quantity that I call nxii. There is a more general problem where we consider w to be the center, we consider another point z and its orbit gamma z and we want to count the matrices in sl2z so that 4u gamma z comma w plus two is less than or equal to x. This is what we call the hyperbolic point problem. So here is a picture of Lattice point so we start with a point i. The points that we get on the same height are in fact acting on it by translations 1n01 and an integer and then points lower correspond to acting first with some other matrix in sl2z and then using horizontal translations. Those points seem to accumulate towards the real axis in our Euclidean i but actually in the hyperbolic sense they all go further and further away. In fact here is a picture of hyperbolic circle. I couldn't plot the whole of it because it was getting a little bit too big. So what we have is that we have center of the hyperbolic center of the point i and radius r. The hyperbolic center of the circle is lower than the Euclidean center of the circle but circles remain circles. Another way of thinking of the counting problem is that we draw the same hyperbolic circle and here in the tessellation of the upper half plane by the standard fundamental domains of sl2z we can count basically how many fundamental domains go inside this hyperbolic circle. This counting is difficult. Many of you may have seen the Gauss circle problem where we look at standard lattice points in z2 and how far they go from the origin inside the circle of radius r. This Gauss circle problem and at least Gauss's initial approach was based on the fact that the circumference or the length of the circle is 2 pi r which is essentially sort of like the square root of the area of the disk contained in it. Unfortunately this is no longer true in hyperbolic space and this makes counting difficult. So in particular we know that the area of a hyperbolic ball center that i with radius r is 4 sin r over 2 square and the length of the corresponding circle is 2 pi sin r and they both grow at the order e to the r. Okay now we go a little bit to the history. The first person to deal with this problem is Delsart in 1942. I want to pay particular attention to the work of Heinz Huber who was in Basel. He wrote a series of seminal papers in 59, 60 and 61 and we're going to come back to his work later on. So here is how the theorem looks like. The counting that we want basically we make w the center and we look at the orbit of gamma z up to some distance relating to x has a main term and an error term. I need to tell you what those are and I need to explain to you what we know about them. So the main term is a finite sum which is good and it contains expressions of the form x to the sj. Sj will be what we call spectral parameters. I'll tell you what they are and we multiply in front with uj of z, uj of w conjugate, uj of z is what we call a mass form. There are some extra gamma factors which you could safely ignore, some root of pi you could safely ignore and the sum is a finite sum over what we call the small eigenvalues of the Laplace operator. So what is happening here is that we have the hyperbolic Laplace operator and we consider the eigenvalue equation Laplacian uj plus sj1 minus sj uj is equal to zero and in the main term we take into account only those parameters sj that are between one half and one. There is always a main term because sj is equal to zero to one, well s zero is equal to one always occurs and that gives us what we would call the main part of the main term and the other terms corresponding to what we call small eigenvalues may or may not exist depending on the group and depending on whether we have proved the Selber eigenvalue conjecture. So this is how the main term looks like, it involves values of mass forms with eigenvalues less than a quarter or eigenvalue parameters between one half and one. Now this isn't a theorem until I tell you something about the error term, otherwise this does not make sense. So estimates for the error term look like big O x to the power a for some appropriate exponent a. Here is what we know about this exponent a. In 1975 St. Patterson approved that we can take the exponent a to be three quarters. The exponent two thirds which is better is known through a series of work of Selberg in 72, Anton Gott in 83 and Peter Goenther in 1980. Unfortunately Selberg never published his work on that, however hundred and notes of his with the exponent two third are available in the website of the Institute for Advanced Study. The exponent two third has never been improved since basically 72. Okay, so what is the conjecture about the error term? We believe that the error term grows at most like x to the half plus epsilon. And what I want to discuss now is evidence for that. The first evidence that I know is an old theorem of Philips and Rudnik from 1994 that tells you that we can at least know that through a sequence of x's we are at least as big as x to the half. And in fact this is what we call omega results and they prove big omega of x to the half log of x to the exponent one quarter minus delta. The next result that I want to discuss is a result of Zamito from 1996 in his thesis. He proved what we call mean square results for the error term. So we square the error term and we integrate in a range of x's from x to 2x and we divide by x in front. And the result is consistent with the conjecture we believe because if the error term grows like x to the half then the square grows like x to the first power and we get some extra log square of x afterwards and nothing worse than that. So this is the result of Zamito that has motivated some of my work in subsequent problems. You may ask also whether we have numerical evidence for the conjecture. This goes back to the work. The first numerical investigation I know of but not for SL2Z is due to Philips and Rudnik. And for SL2Z here is some of the results that some of my master's students got in the past. We vary the x up to 10 to the 8. We do the exact counting nix and as you see it's about the last number here is about 6 times 10 to the 8 which is correct. The 6 is exactly 2pi over the volume of SL2Z which is the correct constant predicted by the theorem. The error term is listed here as you see the error term is growing and we also have in the third column what I would call the relative error term or normalized error term where we divide the error term by x to the half. And as you see the numbers we get in the third column are rather small. Let's now go to graph those things. So you could graph this normalized error term again up to 10 to the 8 and if you pay some attention at the numbers that we see here and here this is 5 and 5. So although on the x-axis we go up to 10 to the 8 on the y-axis we are between plus minus 8 really. So it seems based at least on the numerical investigation that the conjecture is true. However we are far away from proving it. I also do not believe that those numerical investigations can provide anything deeper in terms of proving theorems about these kind of things. Okay now I want to move to other counting problems in the hyperbolic space reminding people that know about automorphic forms that SL2R has three distinct types of subgroups. So up to conjugation we are interested into three types of subgroups. Elliptic subgroups which look like cosine theta, sine theta, minus sine theta, cosine theta, essentially S02. Hyperbolic subgroups which have two real eigenvalues with product one so they can be conjugated to look like M00M inverse and parabolic subgroups that shift to the right or to the left by the real number x. Now we will start with some discrete subgroup of SL2R, you could think of SL2Z and we can intersect the subgroup with some of those groups on the left and some on the right and form double cosets. Because H1 and H2 can be either elliptic hyperbolic or parabolic that leaves us with nine cases okay and the case that we have described so far is the case where we take H1 and H2 to be elliptic subgroups which essentially intersect with gamma would be a small finite group the stabilizer of points. Okay let's discuss now the situation about what we call the elliptic hyperbolic problem. This was discussed by Huber so Huber introduced it in 59 and also wrote his last paper in 98 about the same problem. So the problem is about a point in the upper half plane Z and its orbit let's say the point gamma Z or the various points gamma Z. We also put the hyperbolic subgroup in such a way that the corresponding closed geodesic on the Riemann surface H mode gamma lies on the imaginary axis and is represented by this dark second that you see here which I call L. So this same one goes let's say from the point i to m square i. Okay then we want to count the distance measure the distance from gamma Z to this imaginary axis and it's a very easy way and easy calculation in hyperbolic geometry that the cosine of the distance from Z to L is one over the standard cosine of this angle theta here. There is some change of variables to make the theorems look the same as in the previous case so we are counting the elements in this coset so that the distance from gamma Z to L is less than or equal to t and then what Huber studied and Anton got improved in 1983 is the following result here the counting has a main term c times x some subsidiary main terms here again corresponding to small eigenvalues mass forms and some error term which we call now ex zh. Huber for the error term he didn't get the best result what we know through Anton Good's work is x to the two thirds. Anton Good's work is basically a complicated monograph which I have here a local analysis of the Selberg's trace formula which is very difficult to read because out of those nine cases that I was mentioning before he discusses all of them at the same time with a unified approach without introducing any special functions and with complicated notation that makes the statements and the results difficult to read. So what you do in those cases is that essentially you sit down and try to redo any of those problems with your own means and understand with your own techniques and prove new theorems with your own techniques because frankly it's a very difficult book to read. Okay so what is the conjecture in the elliptic hyperbolic case we make the same conjecture that the error term should grow as big O of x to the half plus epsilon and in 2016 with Hadzakos we proved the mean square result consistent with the conjecture similar to the result of Tsamiso the error term in mean square grows like big O of x to the first power log square x and in 2019 he also proved a big omega result and in fact a lot of big omega results that are complicated and I will just leave them as big omega of x to the half. The next problem that I want to discuss is what we call the hyperbolic hyperbolic counting problem in this case I've plotted here the Riemann surface that is the quotient h mod gamma and in this case we have two hyperbolic subgroups giving us two closed geodesics on the quotient which are represented here by those green things. Okay this problem has been started by Anton God but also people in dynamical systems in particular Parconen and Pologne people in automorphic forms Kimball, Martin, McGee and Wambach there is also an unpublished letter of Suzuki and work of me and my students Lekas and Voskou. Okay for the simplified situation we are allowed to take the two geodesics to be the same but let's look first of all how what we're going to measure geometrically in this case so what we are going to do is that we're going to draw geodesic segments that start on the one of the green geodesics and go towards the other geodesic also to meet it perpendicularly so this is the first geodesic segment here but the geodesic segment could wrap around quite a lot on the Riemann surface like this one here that goes around the hole as well so those geodesic segments are going to have lengths those lengths will form a discrete set increasing to infinity and we want to count the length of those segments that's the geometric problem. Okay let's move now to the situation in the upper half plane where we might visualize things easier I'm assuming here now that the two geodesics are the same I represented both of them here on the imaginary axis from I to M square I and I look at various images of the imaginary axis under linear fractional transformations those are the semicircles that you see and the dark part of the semicircles is the image of the geodesic segment here corresponding to the closed geodesic on the quotient and here is the geodesic segments of which we are counting the length they are represented by these green arcs here meeting perpendicularly the oops the semicircles and the imaginary axis so here's what we know about the situation we can see how the group element gamma gets involved into this or more precisely how the matrix entries ABCD are helping us count the lengths of those hyperbolic segments this is achieved by an elementary lemma that if you look on the imaginary axis IY2 and you look at the image of the imaginary axis under gamma the smallest length which is essentially what the fundamental point pairing variant does is going to be twice the absolute value of AD plus BC if the product of the entries is positive and zero otherwise and here is the theorem which I attribute to Anton God in 83 but with a new proof by me and Leca in 2023 we want to count the number of elements in the double coset so that AD plus BC is less than or equal to X and we have a simtotics for that there is a main term subsidiary main terms having to do with small eigenvalues and an error term what do we know about those things the constant in front of the main term is twice the length square divided by pi times the volume in the subsidiary main terms we don't see the values any longer we see the integral of the uj over this geodesic segment square and for the error term the result is still of the form x to the two thirds the conjectures x to the half plus epsilon and we have a mean square result which we proved in 2024 that is consistent with the conjecture there is also a theorem a big omega result of oscun that it is big o of x to the half as expected so all those theorems so far go in analogy let's now move to an arithmetic application of this result we consider a particular quaternion algebra and it's embedding into s2r and i'm not going to write it down for you right now and we consider a particular hyperbolic matrix m00m inverse where m is the square of a fundamental unit in q adjoining root 2 and then we look at how this matrix acts on the imaginary axis and it produces a geodesic segment l so it is the imaginary axis divided by the group generated by h we also fix some prime p congruent to 5 modulo 8 and some issues with quaternion algebra as orders which are not necessarily maximal etc that i don't want to go into but here is the new arithmetic function that plays a role in this counting problem we want to count ideals in z square root of 2 that have norm equal to k and then a theorem that goes back to hedge hall in 78 gives us correlations of this arithmetic function so the number of ideals of norm k multiplied with the number of ideals of norm pk plus one summed up to capital x plus what happens if you have number of ideals here of norm pk minus one has asymptotics and those are the asymptotics that we have there is a subsidiary main term and a big o of x to the two thirds this result was first proved by hedge hall for p is equal to 5 and generalized last year by my student was cool actually hedge hall proved a slightly more precise result for p is equal to 5 he didn't have the sum of the two summands so he had them separately and so the natural question is can we separate those two summands and get half in the main term the answer is yes if we notice that actually there's some hidden parameters in this problem and all those issues that I'm trying to explain are easy to explain in graphs rather than writing down formulas which eventually one has to do so again in the picture we have the imaginary axis and the geodesic represented by this segment from i to m square i and we look at its image its image will fall either to the left or to the right so that distinguishes already two cases but also the way we are traversing the image plays a role so if we look for instance on the right here for the element gamma 3 gamma 3 of 0 is here to the right of gamma 3 of infinity so that moves in this direction while gamma 4 of 0 is here and gamma 4 of infinity is here and you're moving to the right direction so that gives you an extra parameter another plus or minus so what happens and was in fact understood by god is that the four subcases in the hyperbolic hyperbolic case and one aims to separate them this is done in a theorem that was understood in some contra-indue papers of H. Halin 78 with no details of proofs also in Anton God's book which is unreadable and then my student in 2023 has written down a complete proof that we believe is understandable so each of the four cases gets its fair share of asymptotics one quarter of what I was describing in the previous slides as a result we could also separate in the arithmetic counting the two summands as we expect so what goes into this separating the four cases so here I have the same picture but I've also plotted a ray emanating from zero which is not the geodesic because it is not vertical it goes through an angle theta with the imaginary axis the image of this ray is not going to be a geodesic either but we notice that in the case of gamma three where you are moving from right towards left the image of this ray is in fact included inside while in the case of gamma four when you move from gamma four zero to gamma four infinity to the right we've got that the image of this ray is outside and playing with this inside outside and left and right and using some sort of perturbation or let's say differentiation in the theta parameter we can in fact separate the four cases and get the full version and understanding of Anton Good's theorem from 83 okay let me now move to equidistribution in the classical hyperbolic point problem we've got the center z and the orbit gamma w and the point is at a certain distance from z so it puts it on some hyperbolic circle that I have plotted here to specify the exact location of the point gamma w we need also to specify an angle which we get as the angle between the vertical line and the tangent at the point z of the hyperbolic radius these angles are well known to be equidistributed it's an old theorem of selberg, Anton God and Nichols if you want more precise results on equidistribution for instance at which rate can we control the discrepancy in the equidistribution this result of my collaborator Morton is there and his student Thuelsen let's now see what happens in the elliptic hyperbolic case in this case we have the orbit of the point z so let's say gamma z is here and then we observe the geodesic segment that starts at that point and goes towards the imaginary axis and meet the imaginary axis at the point p of gamma you could ask what happens with the points p of gamma we have to reduce them by the hyperbolic element to be inside this segment from i to m square i and one expects that they also become equidistributed this follows from the work of good but also from all the work of Herrmann if you want more precise results this can be done this is work in progress of my student that means discrepancy with error there and applying the error to write inequality as in all previous cases in the case of hyperbolic hyperbolic then this is what the image looks like we've got the imaginary axis and the semicircle and we are looking at the geodesic segment in green from this point to this point the points p of gamma are in this segment the point q of gamma is on this ray you could move it back and get another point and in this case we also have joint equidistribution that means we count all double cosets so that the distance between l and gamma l is less than or equal to r and that goes in the denominator in the numerator we put all those elements such that p of gamma and q of gamma belong to the let's say the interval 1 m square to the square in fact the correct way to do it is to use a logarithmic measure and then in the end we can prove this equidistribution result for error terms and discrepancy this is work in progress of my student okay now i want to go a little bit more quickly about other cases that i have not discussed that have occupied the research community in let's say analytic number theory and automorphic forms for a while the parabolic elliptic case so you have a parabolic subgroup on the one side and an elliptic case on the other for the elliptic subgroup we can start for instance with the negative discriminant and consider corresponding Hegner points in the upper half plane for instance you could take the point i then the image of i under sl2z is going to be a complex number which will have a real part rational mu over m and an imaginary part i over m and the interesting thing is that if you follow all the identifications mu is not random mu is going to satisfy a congruence condition mu square congruent to minus one modulo m it's a result of vis air and rudnik from 2009 that the real parts those rational numbers mu over m equidistribute modulo one of course it also follows from the work of good but never expressed this way it's a much harder result and perhaps a lot more interesting to look at what Duke Friedlander and Ivanietz proved in 1995 they consider not just for the point i but also for other irreducible quadratic polynomials x square plus d the fractions mu over p where now you consider the equation mu square congruent to d modulo p only over the primes and the result is that actually those fractions also equidistribute modulo one this is a much harder theorem moving to the parabolic hyperbolic case in this case you divide on the left by parabolic subgroup and on the right by hyperbolic subgroup and all the thing that changes versus the previous situation is that now the discriminant becomes positive and now we are interested in solving the congruence equation mu square congruent to d modulo m but for positive d an old result of who he says that the fractions mu over m equidistribute modulo one and Arpat Toth in his thesis in 2000 proved that if we restrict only to prime numbers p they also get equidistributed more recently in 2023 Jens Marklouff and Welsh computed the percorrelation of those fractions mu over m and i'm not going to go into that okay here is some other recent work that is rather exciting in 1999 Erdos and Hall refined a lot the original result of Gauss about counting points in the circle actually the result of Gauss can be slightly refined to count points inside sectors but Erdos and Hall prove that we have equidistribution on most circles representing sums of squares on the circle themselves so here for a particular circle we have all the solutions to the equation x square plus y square is equal to the given integer and for most circles we are going to get equidistribution it's natural to ask whether a similar result holds in the hyperbolic plane in 2021 Headsakos Krubek Leicester and Wigman proved that this is true for most hyperbolic circles and for the group SL2z slightly later Kerubini and Fasari generalized that by looking at some other situations that deal with other Hegner points rather than just the point i what is interesting to notice is that when we started with the original problem we were dealing with r of n r of n plus 4 so we're really considering two distinct circles one with a radius square root of n and another one with radius square root of n plus 4 and essentially the considerations we had mean that we're looking at integers that lie on integer points that lie on those two circles so what I have plotted here is a bunch of circles in pairs according to color that go along odd numbers that we consider here so the radius square should be four apart so we have one and root five root three and root seven root five and root nine which is three now as you see in this picture in one case the circle seems to be sort of darker somehow confused in color this is because in the outer of the annular that we have in black we also have the inner circle for the annular that appears in red so what happens is that the equidistribution of angles or equidistribution of points now can be understood through some clever arguments by moving back to the Euclidean case studying Gaussian integers okay there's another interesting theorem of Friedlander and Ivanets in 2009 which they call the hyperbolic prime number theorem so again we consider matrix a pcd again we measure how far we go by the sum of squares a square plus b square plus c square plus d square but now we demand that the result is a prime number and you could ask how many matrices do we have like that up to some of the four squares less than or equal to x it is not an unconditional theorem that we have uh Friedlander and Ivanets need fairly strong conjectures the simplest one or more where no one would be the alias Halberstam conjecture or some result which may be slightly weaker like level of distribution somehow close to one and the result is that we know that this counting grows like x over log x upper and lower bound we don't have proven asymptotics in this case this problem is supposed to be almost equally hard as the twin prime conjecture okay so let's go back to our sl2r uh in sl2r we have the cartandi composition which is the kak k is so2 and that provides us for every matrix in sl2r with two angles for each one of the sl2s to theta's theta1 and theta2 which we consider to belong to zero pi square and the result from 2024 with uh more than reserve is that we have equidistribution of the pairs of angles if we vary over primes p which are written as sums of four squares subject of course to the determinant condition ad minus bc is equal to one we could even do something similar for the hyperbolic hyperbolic case the decomposition here is probably not as well known as the cartandi composition but here it is we could write a matrix in sl2r as a matrix cos v since v since v cos v and on the left a hyperbolic matrix and on the right another hyperbolic matrix with parameters y1 and y2 and in this case we consider a particular quaternion algebra embedded into sl2r relating to q root 2 so this is what uh matrices will include x0 plus x1 root 2 root 5 x2 plus x3 root 2 and similar things on the second row it is the same thing that we're considering in the earlier arithmetic application m is going to be 1 plus root 2 square and the way to count in this case is bc divided by 5 this is the product of those entries divided by 5 which is x2 square minus 2x3 square and here is the result from 2024 if we look at log y1 and log y2 and we normalize with this quantity here which is fixed beforehand those numbers that we distribute in zero one square or modular one square and this is true if we restrict that the counting works over the prime numbers okay i think that my time will be finishing soon so let me just say some quick things about what goes into the proof we need to consider periods of mass forms of weight zero and of weight two so a period is basically integrating our mass form of weight zero that's what zero symbolizes here over the geodesic segment i m square i we act on the mass form of weight zero by a raising operator and then we consider its period as well huber proved a mean square estimate for the periods of weight zero and my student was could generalize that for weight one and higher weights the next ingredient that we need is a relative trace formulae that looks like a daunting slide so i don't want you to try to memorize it at all i just want to tell you a little bit what relative trace formulae look when you work with hyperbolic subgroups rather than parabolic which is more well known so in those relative trace formulae there is a geometric side on the left and there is a speck flow side on the right let's start with the geometric side there is a quantity relating to the problem we've seen this quantity before ad plus bc there is a test function f and some simple transform of it g0 and g1 and we evaluate f g0 and g1 over the various double cosets that we are involved and on the right hand side there are the periods square of the mass forms of weight zero or of weight two and in front of them we've got some other transform of the function f of the test function those transforms are complicated the first one we call the hubert transform and they're given by specific integrals versus hypergeometric functions and this is a process similar to the selva kaishan datansform that one normally meets in the analytic theory of automorphic forms in fact it's a process that can be deduced from the selva kaishan datansform as well and the last ingredient that one needs is large sieve inequalities i will state them here for the case that we have essentially co-contacts a group and the eigenvalues are discrete parameterized by tj so we have some finite sequence a j and the l norm l2 norm square will be the summation of the mod aj square where we extend up to the height in the spectrum t moreover we fix some point in the upper half plane or in the modular surface and find a number of points x1 x2 up to xr in a big interval x2 to r those points need to be well spaced by at least the distance delta and then the theorem which i list here the large sieve inequalities that have appeared before or used before is by tsameso lekas and boscu so they always look the same we try to exploit the oscillation of x to the i tj using weights either the values of the mass form or the periods of it or the periods of the mass form of way two in the right hand side we see the l2 norm of a in all cases and something that has to do with the average growth of those quantities here in the case of values the local vi log gives you t square in the case of periods we just get t okay i think my time is running out very soon so let me now go to the open questions um there are obviously huge number of problems and results that i did not mention um let me say that my number one interest would be to improve the error term from big o to x to the two thirds to anything really better than x to the two thirds it's a problem that has been left open for 50 years we don't know any improvement for any group arithmetic or not and for any points uh z and w and certainly not for any interesting points like kegner points the second problem is refined statistics for the various lattice counting problems for instance per correlation in all the cases the per correlation has been computed in a bunch of cases but not in all of them then there is the question of arithmetic applications uh there've been some other applications that i haven't described in h3 but we need to write down more because they can be actually fairly concrete then there is the issue of higher dimensional hyperbolic spaces and equidistribution there we have results in higher dimensions already in Fricker's thesis in basal um but um somehow the large cv inequalities that we have in three dimensions seem to be worse and they do not produce the expected mean square results so we have even less evidence of what is happening concerning our conjectures in higher dimensional hyperbolic space the next item is higher rank in other situations it goes without saying that the problems that i described are only in a particular flavor applied to hyperbolic space a much bigger project has been uh originated in the work of Duke, Rudnik and Sarmat in 93 and the paper was called density of integer points on a fine homogeneous varieties and in that paper they introduced some of the problems also that i've discussed but various other problems in higher rank that seem to be a lot harder in particular the case of sl and z seems to be the hardest well one of the most interesting and hard ones in 2023 Blommer and Lutzko provided a first i would say good error term for the counting problem for sl and z so the main term grows like x to the power nn minus one and they can save something from the main term in the error term and what they save behaves better and better as the dimension of sl and z increases so this constant delta n behaves like one plus one over two plus we go one over n it is highly desirable that other cases of a file homogeneous varieties are studied thank you very much for your attention