 Okay, so I'm very happy to have Professor Yan Xinyi from UC Berkeley who has taught me this hot index theorem for adding my models. So, that's why I come. My great pleasure to talk here. I'd like to thank the speakers from the models. From the three conflicts. I first give a sketch of data. I'm going to talk about. So, the first one is for preparatory points in algebra dynamics. So, this is in algebra dynamic systems. It is mainly application of the hot index theorem where I'm going to introduce this application. Then to introduce the hot index theorem I will recall the classic one. Since there are just so many versions for that. We consider the limit of line models. These are what we call a dynamic line model. You can view a dynamic line model like one of that limit of line models when you vary that into a model and you fix the general driver. After we introduce our method we introduce theory over finally generating a theorem. So, this overlapped method. Okay, we start with algebra dynamics which is much more accessible than derivative topics. I'm curious. I first introduce some basic terms in algebra dynamics. So, fk and if you look at the table. This is called a Polaris dynamic system. f will be the dynamic map and l will be the polarization. So, f overlapped with that variety. In the settings to study the behavior of the iteration of x and there is a one remark. This f is necessarily a finite map. The finite order. We can get a degree which is coming from q. Then here. Then we need iteration to take the limit and if it is an automorphism then we have convergence problem. So, we define x is trivial and that one means the iteration of f is finite. fA at c equals fB x for some... a not equal to b. This is just a subset of xk bar. We take k bar points since we don't have same k to be actually closed. This step is storage impedance. So, this is a basic theorem you can try to prove it. It's not that complicated. I'll give you some examples. The first example we care about actually which is the most important example is for a billion mirages. x, a billion mirage multiplication by 2 or by any here greater than 2. That's just use 2. We take L to be a symmetric line bundle. Symmetry meaning that if we use negative 1 to protect L you get L itself. So, then you might that if we use 2 to protect L you get L to the fourth r. The rq is just 4. In that case, this is pretty pf. It's just... and if there is basically a case or maybe just global conflict but then we can use case limiting argument to define canonical height. That's actually why we introduce this setting. This setting is like to define canonical height to use case argument. The second example x is gm so this f can be very arbitrary. Even if the space x is very simple but the map is very complicated and after iteration usually you don't have control at all. I want to single out here x is still m but then f is the square map. In that case we do have which is coming from the easy guess just take all coordinates to be 0 or root of unity and to be 0. So, these are our basic examples. How do you design example 1 and 3 and some exactly related to them? It is really hard to get a description for this set of periodic points. Now here I give you a remark this one is an electric actually here is a wide result in fact. In that case if I basically see an x as dimension 1 then the 2nd set is exactly that set of a communication point of this set in k equals c so you take the closure of the set of periodic points you get a close set the closure meaning closure under the counter problem since we already know that you service this now we do have some isolated points now we are taking them out then this is the Julia set some meaning we look at this example then this now this thing here is c then this 3df is just all the corner points then if we take closure counter closure then you just get all the corner points first so in that sense that case the Julia set is the whole set in the third example since all these x i are 0 or a loss of unity so if you take closure then you can imagine you can either 0 but really get the Julia set if the dimension is 1 and that's just q1 then the closure is just 0 infinity and the unit circle if dimension is 1 0 infinity and unit circle then isolated points are just 0 infinity and after you remove them you get the unit circle which is the Julia set if the dimension is higher there are different definitions of Julia set maybe not here for the first time so this is joint work with Shouya and Algebra Linux SimSense compares their set of set of three corner points so we have samex the samex different max and also maybe different polarizations intersect with previous star systems if they are equal if they intersect with star systems then they are equal the other direction is follows from the uploading result if they are equal then the intersect is just just the whole set so this result somehow indicates that this set you can just after you just remove some points and you want to recover some gallery system you can can't determine if x is just p1 say p1c and if g gx to x such that ppg contains infinitely many rules of unity ppg is exactly the set of rules of unity and zero and infinity so we use the canonical f and coordinate for g1 just take f to be just square if they intersect even then then equal and then you can actually get a classification which says this g should be a harm as what we can guess gx z to z to that so this is the result for pre-projected math paper also proved this result and it also was actually independently yes so instead of the power map take maybe the power introduce a coefficient which is a root of unity is it oh yes sorry thank you yes we need that here's the root of 1 r is a dvr discrete evaluation ring we assume we have an arithmetic over r x over r is a projective smooth projective factor and instead dimension of the generic fiber x0 is the special fiber it is a divider we consider the secondary on x since the base is just a dvr not projective we can only consider the intersection of virtual divisors I think it's all but one divisor virtual divisors so it's just linear combination Q linear yes first consider the case it's dimension x if n equals 1 meaning that dimension of x is 2 then so this is the index theorem in that it is a multiple of x0 for some c in Q so this is very classical and you can write as linear of your useful components and then you here the intersection numbers you get a quadratic form and you work a little bit you can complete the square once you complete the square you get this one and then for the equality once you know it's the square sum of squares then you can get the condition of equality so this is very classical and I don't know who proved it first hard index theorem because it computes the index intersection index of the intersection in this vector space it is totally nappy but you need to multiply x0 so now if this n is greater than 1 then if n is greater than 1 then we we can just take m squared it is not a zero cycle so we don't get a number to get a number we need a lot of ampere numbers or just nine models to intersect it to get the correct dimension so what we do is we take up to lm-1 to intersect m squared to get zero cycles then you count the multiplicity you get a number intersect number then the result says just m squared dot and the equality holds if there is the same result the same property for us so it's similar still you try to complete the square so these are the classical results we try to take limit of this third the reason to take a limit is it's already coming from number three so I first write a little bit about multiplicity and the limit that's one example I think I will assume number three then we consider this one multiplicity by two now this is over okay if we want to do arithmetic intersection theory to study heights and recent points usually we want intramodals of A from here and the objective is left it's better to examine this is not everywhere defined then we can't find a good intramodel such that we can extend the map this is bad and what we do here is to vary the model in order to make this multiplicity by two define that idea not this A but once we love this A we get a different A then this one is different from that one and if you want to do iteration again then you need to love again so we have this A here to get multiplicity by two define here we need to love A front here we get a double prime and somehow you can imagine the limit of this sequence somehow gives you what I want here and also if you have a line model symmetric line model here if you extend here then of course you can extend here actually now what we do here is to you pull back I call it L front and then pull back again A double prime and then you consider the couple A double prime L double prime and you get a sequence do you get a front like arrow on the bottom there's no arrow so this construction this construction works for the study of canonical height input model so you study canonical height just in a second our kind of theory and then you take a derivative you get the guaranteed height so now I'm going to give you a daily diamond maybe first of all we still consider the second of the baby version just R is d by R R is d by R and that's the second field this time we consider all diamond models coming from integral models of X this is a direct limit which is essentially a unit you can pull back like models so then you get the transition maps and then you take the limit now we define this one the completion just pick half part we get this move so the completion requires us to follow we call it follows by effectivity it is as follows of radius we get R with the zero zero it's coming back the zero R the set of nine models M M is in that one same field as that one and then we write it's effected nine models are in parameters if M is real as that one just into a model we say M is in that pole it's like absolute value of M X zero X zero is there so then now this you define it using peak but do you want to take a peak 10s of the rationals or you want to take yeah yeah thank you so we take the rationals here even here but we can't use Z here this is one actually this is a complicated definition there is a an easier definition for that one so this is like easier I write it as interpretation really it's as follows we first look at this one we so in this pole this M can vary but dominated by this very good writer so then all nine models have created a generic fiber then that means after completion after completion it doesn't change the generic fiber so the generic fiber you can talk about the generic fiber of M is here and then now we integrate elements here as a nine model on the generic fiber with the metric actually this is a connection of metrics which is just a K bar space you put a metric on it and then we let X vary so at every fiber we get a metric maybe this RL this RX then if we take a point here we get X which is a small bit of K bar sorry our question from Tokyo so you really mean Pico device sorry so I don't understand the definition of effectivity so you look at high-smoke customer nine model just divisor divisor okay thank you yeah we should say divisor exactly or nine model with a distribution section which is just a section one on the generic fiber better to use divisor to understand that one and then we metric on each fiber now we vary the fiber we'll get a connection from metrics and then we want the metrics that connection to be continuous but not very concrete to get construction from here to here is very simple we first start with with model nine models nine models on integral models and then since nine models on integral models have integral structures they just induce metrics automatically so you extend supposedly you have a point you extend it to the integral model you get a fiber which is like okay bar and then this one in this one is a lattice ever you get a lattice once you get a lattice then you get a metric which makes this lattice that you need to pull and then you define the metric everywhere it is easy to check because the metric is continuous and now this is for pick mount for each pick mount you get a nine model with a metric now you can take a limit this limit process here this one somehow just means if you take the limit for metrics then the limit is uniform limit this one someone says if you have two different elements two different metrics in the sequence then you take the difference it is very small the quotient quotient is very small then it is uniform limit for metrics so furthermore if if this R is complete it is a DBR if it is complete then you can further get complete this metric has a metric on the book of inspects as a metric and the integral nine model over you can divide metrics metric on this one formally you go to the market now the definition is no I just briefly question when you wrote what you erase just now you wrote it as an equality but I think that in the right hand side you need to not to put just continuous for the usual topology on XK bar but you probably need the Berkovich space continuity otherwise you have too many disjoint small disks you cannot hope that it is true the quality is true when you just take the continuity forever for the topology on XK bar you're right so here is just continuous Berkovich topology here it's more than just continuous it's only some bounded property I just cheated so now if K is a number good then if X over K it's projective now we project X ma just do it now this time just pick up is this time this is the group of permission like model it is a pair consisting of a like model on the intro model and with a metric permission metric on the comments points in the setting of our category or maybe you just think this K as a global function built then you don't have to bring T it's just model over your projective curve so now pick X as a combination by a complication since we have even in many points on the base so there are just some requirements about that but let's ignore that so this is the identity line models we consider to be positive either vertical curves or horizontal curves chain form of the metric to be semi positive but if the sequence in each place have a have a look or pick the condition what's the they have a similar line you do have this in perpetrator yeah so so this is the setting we just consider the dvr case if we is not immediate if we is not immediate this is certainly the contents case so then you have a collection of metrics and every place you have a you have some capability so now the metric down to one down to un dv0 the generic So in the in the in the dvr this this That's all In the that we need one more assumption in the baby version. If furthermore, just one, means LI bar, epsilon, and it's net, and zero, such that. So pass them to a marker, but we want Lm is net. So then the possibility is like absolute value of n in some sense is bounded by LI. LI's equality, we mean m bar, should be a, so you can also, if I pick half case, you send it like limit of 0k, comes from the base. Pull back, the hot index center. They consider, in the second theory, directly surfaces. So they consider model case, meaning that you just, you only consider Hermitian line models without taking limit. At least more than what I stated, they actually proved that in that case, we don't have LIs, then they proved that this inner section, our kind of inner section, is negative of the narrated height period, defined using a different method. Then Moro-Wake extended there with us to, there with us to Hermitian line model case in general. So in our second, this theorem has two parts. In our second, the first part essentially follows from Moro-Wake's result, this is inequality, since every, and then in the second, you define by the same name of the process. So that's the reason that zero is obtained by the limited process. But the hard part, actually also the crucial part is this part, the equality, can give them equality. This part can be obtained by, not only by taking the limit, because if you take a limit, if you want to take it, then you even don't have less than one equal, then it is different. We proved it using a different method. Let us remark, that theorem works for finite generative fields. Then, as in the number of weekends, you extend K to, okay, in that case, you will feel it's finite generative. You first extend it to our family value over Z or a predicted value over Fp, this is your base, and then you take integral models, and then you take a limit. Eventually, you are defined as pick hat X, and then you can define positivity, and then you'll get a similar result. So one motivation to prove that for finite generative fields is to use our dramatic method to study as a result of any base field, just like the first example, the example of the dynamics. Your base field is arbitrary K, what we do is we first use deference principle to reduce it to finite generative fields, and then now we use integral models. Now we take a limit, we get a setting like this, then we can use arithmetic to study the probability of the original variety of generative fields. Thank you very much. We will start with the pairs. Any questions from pairs? So when you work with a finite generative field, so you said that you, let us say, you write it as the field of fractions of some model, but then you cannot get things to be flat over it unless you change the model. So do you work with a fixed model of the finite generative field or all the tower of all models? That's a good question. We don't need to vary models. We use the way of flattening theorem to get, always get relatively less confidence. And we, to write it down, I'll write a little bit. We need to vary both the base and the total space. So if K is finite generative over, let's pick the case of finite fields. Just like our density over, we call it little k. Then we first extend to be over K, projective, and with function field K. Now, if it actually is over K, then we extend to completion. It's a simple example of, say, to polarize the dynamical system where you know in advance that the intersection of the pre-periodic points or the ischidians, but you don't know that they are the same. If it's possible to construct an example like this. So the examples are, if we know, are coming from the group, group loss. For example, if you take a billion varieties, then the set of pre-periodic ones is always the whole torsion group. And you can take two actually very independent morphisms, but they have the same pre-p set. Yeah, but you know it in advance. You know it in advance without using your theorem. I mean, I want to understand if you know a situation where you don't know the conclusion of a theorem before. Then we don't know. Yeah, it's very hard to check. I think that's all for my list. Okay, so any questions from Tokyo? So you made a assumption that something is zero on the z-x variable. So you just need a numerically zero or a rationally zero? Sorry? So really, we are assumption in this number field case. So you said this entire set of morphisms is zero on the z-x variable. So you just ask, it's a numerically zero? Or you need a rationally zero? You mean this one? Yeah, this assumption, yes. Yeah. So you just need a numerically zero? Intersection number is zero? Equal to zero. Yeah, so this is just as a number? As a number, right. Numerically. Thank you. As a marker? So I didn't get to you. So your assumption is in the limit case. What was the new ingredient in the limit case? In the limit case, what's your question? Yeah, so you knew this model case. But you have proved this limit case. So what is the new ingredient in the proof? The new ingredient in the proof of the limit case. We use a trick. We call it a variational method. Last year, it was a complex geometry. He proved, actually he obtained a simple proof of the unique parts of the Calabia theorem in the complex settings. His method can be used here. Variational method. So the method somehow says the following. If this m minus one part equals zero, as in the third, then we can get the following. We get m dot l one prime. So it's just we... If we have that one, then we change the metric. Then his method, there is a very nice trick. But somehow we managed to prove that if we change the metric, then this is still true. Then it's like we get a lot of conditions, extra conditions. And then somehow... Thank you. Okay. Another question? Thank you. That's from Tokyo. Thank you. Okay, so any questions from Beijing? One question. So when you explained the definition of peak your completion, you said that you would take all models of, let's say, for a finitely generated field of Fp. So you wrote an inverse limit, but you probably meant a direct limit. Somewhere very, right? Yeah, but... Is that in X over V? Yeah, do you write that as a project limit? Using a bit gently. Oh, sorry, it's really a direct limit. And then to define this completion, you have to... It is not clear how one extends your definition, because you looked before just at one fiber. And now you have many prime divisors. And also when you change the models, you get more prime divisors. So it is not clear what is the estimate that one uses to define this completion. Yeah, so it's like for, like infinitely many types of completions. So for each type, we fix an infected divisor on the base. And then for this divisor, we define the models. We can get a completion. Now we enlarge the divisor on the base to get different types of completions. Eventually we get all of them. So maybe... Let me write here. You take a one model and divisor... We have X over K. K over K. Now we get X over V over K. Now we take... We take this D. Inside this V, defective. Then we get a completion. Using the activity by key, completion of S8. Now. Now then take union over all V. In the number field case, I suppose that you allow yourself to not work with a fixed divisor but to make some... to use infinitely many primes. Is this the case? No. We only have 599 primes. So in every step, if this is the number field case, then this is okay. And this D is just like 599 primes. So every step, we only need 599 primes. Okay. Only this completion. Not the... Yes, I understand. Yeah. First take a completion, then take a union. This is by... I have a good question for the idea about this athletic line model. No. Actually, I lose the complicates in no kind of way. Here are the compatibility conditions. It's completely... What's the compatibility condition? You put it on the metric. So in the number field case, it's just like, if you take a line model from each model, and you get model metrics, then you want your athletic metric to be... to agree with the model metric at almost all places. So any other questions? Any comments? From Paris? Well, from Tokyo? No? Okay, so... Let's sign this figure again.