 Thank you for the opportunity to present my work in this nice workshop, in this nice seminar. And yeah, also thanks for the participants for showing up. Yeah, I want to speak about equidistribution in our career theory and as an motivated being example, I want to discuss first equidistribution of zero sets of polynomials of integer polynomials. And to motivate against this, I want to recall below the equidistribution result on an integer polynomials. And to go this, let us first define the canonical height of an integer polynomial with leading coefficient a n and zeros alpha j. And classically, this is also the height of the zeros of alpha j, if p is not irreducible, if p is irreducible, sorry. And it's just defined as one over n of the logarithm of the absolute value of the leading term plus one over n of the sum of the logarithms of the absolute values of the zeros which have absolute value at least one. And then the theory in Babilo is that if you take a sequence of non-zero integer polynomials which such as the height is going to zero and the zeros of each pn are powers different. And if you take mu S1 to be the normed armature on the circle S1 and C, then for any test function from C to C, which is continuous to get the zeros for n to infinity is distributed like the armature on S1, meaning that if you evaluate F on the average of the zeros of pn, then for n to infinity, this is the same as the interval of S1 with respect to the armature. Yeah, and we can now ask some questions. First one is what happens if the canonical height is not tending to zero. And the other one would be, yeah, if we allow also that the height is not going to zero, can we reach something like equal distribution on the whole complex plane instead of only S1, which would be interesting. But to do this, yeah, we need some measure on C, which is finite volume. And then it's better to work with a compact space with all we choose the project line and there is a Fubini study metric. So I want to introduce it. So the perfect productive line defined over the integers is just p1z. And we take some homogeneous coordinates, z0 and z1 on the product line. Then we have the global sections of this product line of the line bundle on. It's just the homogeneous polynomials in the homogeneous coordinates of degree n. And we can identify them with this polynomial in here of degree n and the zero polynomial plus, for example, setting z1 to b1. And if you tend the complex numbers, we get the global sections of p1 over C is the same. And we can again identify with polynomials with complex coefficients. But now let me motivate the Fubini study metric where if you have a global section of ON over the complex numbers, the idea is to define a norm on this and it should be a function from p1 over C to the non-intuitive real numbers. And what one can do is, okay, just take the absolute value, but then it's not way defined just because it's depending on the, yeah, zero, one and zero is z zero and that one are only homogeneous coordinates. So we are only interested in the equivalence classes model. Multipleticists, so we have to divide out some factors. And if n is one, the canonical choice of such a factor is, takes the square of the absolute value of z zero and plus the square of the absolute value of z one. And on the tensor product of ON, the ON, we want that this is compatible with multiplication. And then we have to take the nth power of this. Okay, this is a good choice. And then we can compute the coverage of this metric on ON or Y or one. And if we try to compute it with respect to the coordinate z defined by z zero divided by z one on the complex numbers, which is isomorphic to the space where z one is not non-venishing, it's a series scale open subset of P1C then C1 of R1 is by definition just dd bar over two pi i the logarithm of the z one, the Fubini-Sulli metric of z one. And one can explicitly compute it in the coordinate z. Yeah, it's that word written there. And if one studies this one further and replacing d z by variable and imaginary part and checks directly that the i can go out and we get some positive form. So now the idea is that we want to study good heights for polynomials such that we get equal distribution with respect to this to this Fubini-Sulli measure C1 of R1. Yeah, that's what we're doing next. Next, we choose a non-zero section S of R1. And we also see it's a Fubini-Sulli height and the Bombieri height by the following integrals. The first one we're doing is the Fubini-Sulli height is just one over N of the integral of the logarithm of the metric, the Fubini-Sulli metric of S. And we add in half. So it will be clear later why we normalize it by adding a half. And for the Bombieri height, also the name of the Bombieri height will be clear later. It's just the other way around. We first take the logarithm of the integral of the, we first take the integral and then of this logarithm. So it's the logarithm of the L2 norm with respect to the Fubini-Sulli metric. And yeah, again, as I said, we can identify sections with polynomials and quite further. So if you have a polynomial P, then we can write it down and explicitly by coefficients. And we also see it's the sections SP having the same coefficients. And then we said that the Fubini-Sulli height of the polynomial, it's just the same as the Fubini-Sulli height of the section SP and the same for the Bombieri height. So now let me give you explicit formulas for these. If you have a polynomial with coefficients A j, it was a leading coefficient this A n and we have zero alpha j, then the direct computation shows us that the Fubini-Sulli height of P is just this term. Again, one over N of the logarithm of the norm of the absolute value of the leading coefficient plus one over two N of the sum of, now it's the logarithm of one plus the absolute value of the zeros. And it's similar to the canonical height with the difference that we have not the maximum of one end. This alpha j, we just take the sum of both and that's the justification also for the name Fubini-Studioi because terms like these one plus alpha j absolute value squared is what you're usually getting if you're playing with a Fubini-Studio metric. And it's also justification for adding a half before because now the half is vanishing. We have only this clear formula. And for the Bombieri height, we get that it's one over two N of the logarithm of the average of the coefficients weighted by the binomial N over j. So that's what is classically called the Bombieri norm and that's why I'm calling this Bombieri height. And more generally, if P is just some real, at least one, we can define the P Bombieri height. Yeah, as usually we take in the sum, the sum into the power P and then we take the P through or after the logarithm we divide by P. And for P is infinity, we get the same with just the logarithm of the maximum of all these values of the absolute value a j weighted by the square roots of the binomials. And then we have some relate in between the Bombieri and the Fubini-Studio height because we have the Jensen's inequality. You remember that Fubini-Studio height was defined as the integral of the logarithm and the Bombieri height was defined as the logarithm of the integral. And the Jensen's inequality shows then that, yeah, the first one, okay, we had this normalization by a half. So if you take the Fubini-Studio height of P minus half, then it's really the integral of the logarithm is smaller than the Bombieri height. And we can again apply Jensen's inequality to show that it's also smaller equal to the Bombieri height with respect to this P, if P is at least two. Okay, and now let me come to my result for integer polynomials. So we again choose a P just as above and some test function. So F is from C to C, a continuous function such that the limit of F of that for that going to infinity is well defined and finite. That this meaning that F can be continued to a continuous function on P1, which is, yeah, which is bounded. So in the first assertion, we chosen a sequence of non-zero. Here we can even take complex polynomials that don't have to have integer coefficients and we assume that the limit of the difference of the Bombieri P height of Pn and the Fubini-Studio height Pn plus and half is going to zero. So we already know that this term, the bracket is always, yeah, non-relative and we are assuming that it's going to the smallest possible value in the limit. Then it holds that we get the equidistribution similar to Pilo's result but now we have equidistribution with respect to the Fubini-Studio measure. So meaning that the average of F in the zeroes of the polynomial Pn counted with multiplicity is going for N to infinity to the integral of F with respect to the Fubini-Studio measure. And the second assertion is that we now additionally take a sequence of reals which has smallest limit point which is positive and the highest limit point is still finite, so it's bounded. And we set Pn for a set of polynomials with integer coefficients of degree N and the P from here in norm is bounded by this sequence in. Then it holds first that, yeah, the difference of height we considered in A in the average is going to zero but this meaning because we have the sum of non-relative values going to zero meaning that it's going to zero for almost all whatever this means in the setup for almost all sequences in this Pn. And the second assertion is then we can apply the assertion of A to get that on average we have this actually distribution property for almost all sequences in these sets Pn, okay. Yeah, what I want to do is now I want to prove A and the proof of A will also show how you can get from B1 to B2, it's the same strategy but B1, I don't want to prove B1 directly but I want to show later in a more general setup of RK of theory how one can generalize B1 and then prove it also. So the proof of A and then also of B1 to B2 is that we can first rescale the polynomials such that of the other side of the section S Pn so the supremum norm is exactly one. So just multiply by some constant. So as I said, these are complex polynomials so it's nothing about integral numbers. And then we want to balance this, yeah, this difference, absolute value of this difference and first, yeah, we why did the first thing is that the first sum is just the integral over the divisor of this S Pn, the divisor of S Pn is the same as zero of Pn and the second integral is just the integral with respect to the C1 of R1, it's the Fubini's 2D measure. And we can put it together and integrate with respect to the delta, the RK measure of this divisor minus the C1 of On. Now it's On because we divide by, we multiply with one over N for both terms. But yeah, this form we're integrating with is exactly the same as the D-debar of the logarithm of the section S Pn of the Fubini's 2D metric. And now we can apply Stokes's theory to put the D-debar from the one side to the other side. So we get that this is equal to the integral, the absolute value of the integral of the logarithm S Pn with the Fubini's 2D metric times the D-debar of F. And now we can balance this by applying the triangle inequality for integrals to get the absolute value inside. And yeah, it's also multiplicative. And now the idea is that the absolute value of D-debar of F can be bound from above by the C1 of O1 because C1 O1 is positive and P1 is compact. So it says there has to be an constant such that we can balance the D-debar F by this constant times C1 O1. And also the absolute value of this logarithm of S Pn can be bound is just the same as minus the logarithm of S Pn because we assumed before that the supremum is exactly one. And so the Fubini's 2D metric of S Pn is always at most one, so the logarithm of it is always negative. Yeah, and this is exactly minus the Fubini's 2D height by definition of S Pn plus and half. And we can add just as we want the logarithm of the supremum of S Pn because we set it to be, yeah, the supremum norm is one, so the logarithm was zero. But the supremum norm is up to a linear term and N is the same as L2 norm that we get that this is the same as we can replace it by the P-Vomieri norm of Pn. And then we have to add some log N term. Yeah, but it's fine. And then, yeah, the first term here, so the bracket is exactly what we assume to go to zero and the second term, we have a log N divided by N. It's also going to zero for N2 infinity, so it's completely going to zero. This is showing that if we assuming that this bracket and the last line is going to zero, then the first term in this calculation has to go to zero. That's meaning that for these polynomials, we get this equity distribution. Now, let me go for the generalization to Hermitian line bundles on arithmetic variety. So it's really now our KELO theory. Let me recall some notions. First, an arithmetic variety is X over Z. It's an integral scheme, absolutely. And over that, it has to have nice properties like any varieties should be there, projective, flat, or finite, and separated. And we assume that the generic fiber, X tensor with Q should be smooth. So it says the advantage that also X tensor with C is smooth and then we can do complex analysis on the generic fiber. And for example, we can choose P1, Z, which we already studied before, and this would be the special case for our theorem with BZ and the theorem about the polynomials. Now, on this, we can have Hermitian line bundles. Hermitian line bundles are pairs of the line bundle on X and a system of Hermitian matrix HX on the fibers LX where X is running through all complex points of the arithmetic variety. And there's some assumption we're doing. It's not any system. We assume that if you have a section on some series key open subset U, then the norm of SX, which square is, which is just the self-product in the Hermitian metric at X should be smooth in the point X on the complex points of U. And we get also the curvature form, which is, as before, it's just the dd bar divided by two pi of the logarithm of the norm of SQM. So this is defined locally when S is invertible, but choosing another S, we can define it globally. One example is, yeah, just take the O1 on P1Z and equip it with a Fubini study metric, but in general, this will not be enough for us because we will see later, but what we can also do is multiply i as a metric on it by e to the minus epsilon. And this is written as, yeah, O1 bar and then by a shift by epsilon, where the metric on S is just the Fubini study metric on S times e to the minus epsilon. And then we come to small sections. The section is called small, respectively, strictly small. It's the supremum norm, which is the supremum of S over all complex points at most one, or it's strictly small if it's strictly smaller than one. And then the small sections form a finite set. So the global sections A0, XL are a discreet set and some real vector space. And if we bound some norm, we get the finite set. And supremum norm is this and more. And yeah, we can also say that this set is playing the role of global sections on the arithmetic, because what we are interested in in RK of theory is that to get an idea of the completion of the spectrum of that and there's not really completion. There's one point missing, the point at infinity, but in the set of schemes that does not make any sense. But what you can do is to think of it always by putting metrics on everything we're doing. And then one has to change notions a bit. For example, the global sections are then replaced by these small sections. And also for ampulness, we have some new notion which should be a thing of the parallel of the classical ampulness. So an emission line bundle is called as medically ample. If it's relatively ample on X over Z, so it's the vertical condition over each fiber. So over the fiber at infinity, so the hypothetical fiber should also be ample. This meaning just curvature form is positive. And then we need some horizontal condition to be ample. And that's doing by the following. For P high enough, the set of global sections of the tensor power of the P tensor power of L should be generated by strictly small sections. So as classically ample bundles are characterized by that for a high enough tensor power, it's generated by global sections. Yes. And an example is exactly the O1 bar shifted by epsilon is automatically ample for every positive epsilon. The first, of course, O1 is ample on P1Z. The C1 form of O1 bar shifted by epsilon is the same as the C1 of O1 because epsilon is a constant. If it takes the rhythm, it's an additive constant and then taking dd bar, it does not matter. And we already seen that the C1 O1 is positive. And the third is a bit the most non-trivial part. So the Z zero to the J times the one to the P minus J are the basis of the global forms of the P tensor power of L. If it computes a supremum norm, it's just, yes, we have to multiply by E to the minus two epsilon times P because it's a P tensor power. And then the supremum of this Fubini study metric and one can check that this supremum is at most one. And then we get it smaller equal to E minus two epsilon P but it can be one. And that's why we really need this epsilon to be strictly bigger than zero. So it's not mathematically empty if you only take the Fubini study metric. And now we come to my theory. So if you take an arithmetic variety X of dimension D, at least two, it's absolute dimension, so the spec Z has already one dimension and the relative dimension of X over spec Z is then one less, it's D minus one. And let L bar be an arosmetically ample Hermitian line on X. So then for every D minus two, D minus two continuous test form phi on X, C on the complex points of X, it holds the following distribution result that again, if we take the average over the small sections of the P tensor power of L of the difference, absolute value of the difference of the integral of phi over the divisor of S, yeah, norm by one over P and the integral over the complex points of phi wedge with C one of L bar, then the limit for P to infinity, this is zero. So it's again, I'm searching about almost all sequences because we have a sum of non-digitive values getting zero in the limit. Yeah, so for almost all sequences of small sections for the P tensor powers of L, P is going to infinity, we have an equidistribution of the divisor of the section S with respect to the C one of L. And if you check, it's very similar, it's not really exactly the same as in the case of polynomials if you apply it to the case P one Z and the line one O one epsilon. So let me give you an overview of the proof. So the first thing I want to do is to get rid of the four five as talk theory, but similar to the proof for the polynomials I gave you was that the advantage that we have not to play around with this form five, which is very arbitrary though. Second thing is, there is already some result about equidistribution of sections, but it's about random holomorphic sections and complex analysis worked out by Barakhtar, Kovman, and Marinescu. And it's with respect to certain probability measures on the complex closed sections. And the problem is that these probability measures are not discrete, so we cannot directly apply it on just a subset of small sections, which is just finite. But what we can do is to reduce it to apply it to the real subspace of real global sections in the complex global sections for the probability measures supported on symmetric compact and convex subsets. So these symmetric compact and convex subsets, we call Kp living in the real global sections. And the main example in mind is just takes the real global sections which has supremum norm at most one. So if you balance the norm, then you get something symmetric compact and convex. So we want to go now from the real global sections to the lattice points in it. And yeah, the integer global sections are just the lattice and the real global sections. And we can decompose the real global sections into boxes, as I said, to every lattice point. And a result by Moriwaki shows us that then these boxes are getting exponentially fast, small for P is going to infinity. And the other thing we have to see that is, if you have, that's the integral we are interested in is going to zero for P is going to infinity for a sequence of section as P with, yeah, subnorm at most one, which will be always the case for us. And then the same holds true for another sequence, which the property that the difference of both sequences is going exponentially fast to zero. So the supreme is a limb so for P to infinity of the norm of the difference of SP and S prime P and then taking the P's route is strictly smaller than one. So I will give more details on this later, it's just an overview. And the last thing is then reducing from almost all sequences in this set KP, which we've done in step three to almost all sequences in the lattice points in KP, which will be just the small sections because at least in our example where KP is given by the bounding the subnorm by one. And this can be done by, yeah, just step four and five we know that we have these boxes, as I said, it's the lattice points and we know that the box is getting smaller and if we have some sequence for sections lying in the boxes we can reduce it to the base points of the lattice boxes, which are the lattice points. So first let me speak about getting rid of the form phi by Stokes' theorem. The Stokes' theorem and the positivity of C1 allows us to compute to the difference of, yeah, if you take the integral of phi over the divisor of S and norm by one over P and the integral of phi wedged with C1L over the complex points of X, this is, yeah, similar to the proof for the polynomials we get that this is just phi wedged with the dd bar of the logarithm of S and again we can use Stokes' theorem to put the dd bar on the other side. Look, we can now put by the triangle inequality for integrals, the absolute value inside and now again, the absolute value of dd bar of phi can be bounded by the compactness of X and by the positivity of C1 of L by some constant times the constant can depend on this L and on the phi times the C1 of L bar to the d minus one. And thus it's enough to prove, yeah, the same result not for the absolute value of this difference but for the integral at the end without the constant. So the constant is not important because we are interested in showing that this is zero. So we have to show that the limit of the average over the small sections of this integral tends to zero. Okay, and we have to take this integral good in mind because it will be now the very important term for the rest. So let me tell you about the execution result in complex analysis we use. So Bayrakti, Coleman and Marinescu introduced the following condition on probability measures and proved the following theory. Conditioners, they call it condition B, there's also condition A, but it's not important for us because it only matters for, yeah, varieties with singularities and if the metric has singularities it's not the case for us. So we take a sequence, the sigma P of probability measures on the space of complex global sections and we say that these sequence of probability measures satisfy B. If for all P, there exists a constant Cp such that the following integral is bounded by Cp. So we take the integral over the complex sections with respect to the probability measures of the absolute value of the logarithm of the inner product of S and U. So S is a complex section and U is also a complex section living in the unit sphere and you have to say that on the complex sections we take the L2 product induced by the metric on L bar. So we have, yeah, so classically the L2 metric by the integral of this H of S and U and then we get some product on this complex sections and yeah, then it also makes sense to speak about the norm of U to be one in this L2 norm. And the theorem they gave is if we have such a sequence of probability measures satisfying B and the limit of this constant divided by P is going to zero then we have said yeah, here's this integral we are interested in is integrated over over the complex sections with respect to the probability measures is going to zero. And now one sees, okay, one is it's possible to apply it if our probability measure is discreet, for example, like in all setting where we are interested in the small sections which are just finally many then we have the problem that this cannot be true the condition can be true because then for this S we have only finding any S in the integral will be a sum and so there will always be a U which is orthogonal to the S and then the logarithm of zero it's infinite. Yeah, it's not a good idea to apply it to the small sections instead but we can instead apply so we choose a sequence of symmetry compact and convex subsets Kp in the real sections which are also living as a real subspace in the complex sections and we may assume that the Kp are bounded by balls from above and below and the radius of the balls satisfying the following condition that the P's root of them is going to run for P's going to infinity and an example is for before we take Kp to be the real sections with bounded supremum norm by one and then one can show that the probability measure defined by taking the Lebesgue measure restricting to the set Kp and normalizing such that it has volume one in total it's setting p and that is the property that the limit of the Cp the constant in the condition be divided by P is going to zero so we can apply the theorem and we get that the integral we are interested in is going to zero on average over these Kp in the space of real sections but that's not really enough you want to go to the small sections which are just lattice points in the real sections so this is the step where I wanted to do this decomposition into boxes so we had already considered the real sections and in there we have the integral sections and these are just the lattice so let's show it's just one is to check that the integral sections are torsion free what follows from algebraic geometry in these lattice we can choose a basis and we choose a basis sp1 to spdp dp should be the dimension of this lattice of minimal norms and then it's a result by Moriwaki that the norms are then so small that even the the maximum of all the norms of the base vectors is bounded by a constant times p to the d-1 times the constant to the p where this constant v is strictly smaller than one so it's meaning that if p is going to infinity the norms of the base vectors are going to be exponentially fast to zero and now for every lattice point x we define a box qx just starting in the lattice point x and then going in every direction of the base vectors by coefficients from zero to one but not including one just looking like the half open cube in the dp dimensional space and the theory by Moriwaki shows that this box qx is going exponentially fast, small for p going to infinity and this box is the property that we get in this joint decomposition of the space of real sections the real sections are just the disjoint union of the qx where x is running over all the integral sections okay and now the idea is if we know the distribution for the real sections we know that it's for almost all sequences the case that we get distribution and then these for if you take one sequence the sections lying and at some point in these boxes qx and we should reduce it to the base point x of the box qx and that's what we're doing in the next slide if you have two near sequences of sections we can do it in more general that sp and sp prime be two sequences of sections of complex sections and we assume the following the subnorm is again bound by one we always have this and they're getting exponentially fast near to each other so the the norm of sp minus sp prime and taking then the piece root gives a sequence which has yeah this is the biggest limit point strictly smaller than one so if we put the piece root on the other side we have set it smaller than some v to the p as Moriwaki resolved and the third thing is that the sequence sp should satisfy that for the integral we're interested in the limit is going to zero and then the question is is this also satisfied for the other sequence of section and indeed one can show that also for sp prime p the integral we are interested in taking the limit for p to infinity goes to zero and let me give the idea of the proof first is prove it just by complex analysis so we can prove it for smooth productive complex varieties by induction on the dimension so the case of the dimension zero is just the base case this is easy it's not completely trivial but easy enough to do it one should say that the varieties have not to be connected so in the case dimension zero we have finally many points which are yeah we have to discreet topology on them and a norm of a section is then just association of positive real numbers on these points and now for the induction step we use dog's to reduce to divisor of a section of a very ample line yeah that's what one can always choose by the theory we can always choose if you take a very ample line by the divisor which is smooth in general the case we can also take this divisor in general such that the sp and sp prime I again have only zeros in co-dimension but that's not enough one gets an error term but this can be made smaller than any epsilon bigger than zero by choosing the section using again the equation theory by Raktar and Marinesco was the second part and then one can make the error term small and arbitrary small one gets that this integral for sp prime p is smaller than epsilon for every epsilon then it has to be zero so let's conclude the proof we have to use also some geometry of numbers as in the title of this slide saying though the idea starts with subsets of positive density so let's look at the convex sets kp and intersect them with the lattice points of integral sections and in there we just take tp to be subsets and positive density means that the smallest limit point of the ratio of the number of points in this tp and the number of points in the of lattice points in this kp the number of points of integral sections in kp should be positive and then it follows that if we look again at the decomposition of the real sections that if we take the volume of the of the boxes associated to the points in the tp and divide by the volume of all kp so we also intersect the boxes with kp then this is also positive so first one thing that directly follow but it's a problem because one has to be careful with the points intersecting the boundary where the boxes intersecting the boundary of kp by geometry of numbers one shows that the boxes intersecting the boundary of kp does not do not matter let's let's do some work it's not directly following and then we can take the third step of our discussion of the proof to show that there is a sequence sp where sp lying in this disjoint union of these boxes intersected with the kp where the x is from the tp such that the integral we are interested in is going to zero so this is true because we know in the average over this kp this integral is going to zero and if we restrict to some subset such that the volume is positive in ratio to the volume of kp it also has to be almost everywhere to be zero and in particular we can choose some sp such that this is going to zero and now we can use the fourth and the fifth step to conclude that there is also a sequence sp prime where the sp prime is in tp which is just sp in the disjoint union of this qx with kp and this meaning this is lying in some qx and we just take this base point x of the box where it is lying in and this is then the sp prime it is just the x where sp is lying in qx and then we know by the result by Moriwaki that the difference of sp and sp prime is getting small exponentially fast and then by the fifth step we know that because we have set the integral, the limit of the integral for sp is going to zero we know it also for the limit of the integral of sp prime and now we should come back for the average sum and what we can use is that this integral in general is bounded by our Kiloff theory but at least in this case we are interested in sections of subnorm at most one it is bounded by our Kiloff degree of the arithmetic self-intersection number of the line bundle L and this is not depending on p or on the section and because this is bounded we get that on average over the lattice points in Kp the integral is going to zero because if this would be positive then for a subset of this Kp intersected with H0 of positive density it should be everywhere positive and this cannot be this case because such a subset is looking like the Tp at the beginning of this slide and we showed that in every such as Tp we can choose one section for which the integral is going to zero so in the end we conclude that also this average of the integral is going to zero and by the example that Kp is given by rounding the sub-norm by one we know that this is exactly the small sections if we intersect with the integral sections and by the first step in the proof we know that this gives again our distribution result when we put in again the test form file ok that's all for the proof and I thank you for your attention