 Thank you very much for invitation and introduction. It's my great pleasure to speak in this conference. Before I begin my talk, I also want to express my gratitude to Professor Eluzie. I try to speak in French. As you can see, I'm Professor Eluzie, in 2004, when I was a student at the University of Tsinghua. At that time, Professor Eluzie gave us a course of Geometry Algorithm in Tsinghua. And then, recommended by Professor Fondin, Reno and Rossi Eluzie, I have the opportunity to proceed my studies in France in 2005. And then, first in Master's degree, and then a thesis with Laurent Pag. At the beginning of our knowledge, Professor Eluzie is always very kind to me in Mathematics and also in everyday life. In 2006, he was one of our parents when I moved to an apartment with Dillonton and Godonton-Opagonet. And in 2009, I participated in a group of work organized by Professor Eluzie and Alex Puti on Geometry Algorithm. Professor Eluzie took a lot of time to help me prepare my exposure to Crystalline. I remember very well that I did two repetitions before him before my public exposure. I took a lot of things from this unique experience. I sincerely thank you Professor Eluzie. Okay, now I start my talk about Newton's Diary and Weekly Atmospheric Curse in the Pierre-Dix-Rochtheory. It's a joint work with Dillonton. Okay, now first P is a prime number. And Fp is a finite field with P elements. Fp bar is the algebraic closure of Fp. Qp is their periodic fields. We denote by Qp-breath the weight vector ring with coefficient in Fp bar. The fraction field of the weight vector ring with coefficient in Fp bar. It can also be considered as the periodic completion of the maximal unrun field extension of Qp. We fix the P-double group X over Fp bar. We consider it's a rational deodorant module. So it's an isochrystal. For example, if the P-double group comes from abelian variety, so it's the p-entoscience of the abelian variety, then in this case the rational deodorant module is exactly the first crystalline cosmology group of X. The subscript Qp-breath means that we phase change to Qp-breath. Now we want to consider the definition of this fixed P-double group. Now we consider K, the field extension. Each one creates of A, it's not each one creates of X. Oh yes, yes, yes, it's a type of, thank you. If K is the field extension of Qp-breath, we denote by OK the ring of integers of K. And if we denote by X a P-double group over OK and dxK is the rational deodorant module of X, in fact it only depends on a spectral fiber and it has a hot filtration inside the rational deodorant module as a subspace of K-vector space. Again, if X comes from abelian variety, then in this case the rational deodorant module is exactly the first term of the cosmology group of the generic fiber of this abelian variety. And the hot filtration is exactly the hot filtration of this Dirac homology group. Now we have the pyramid mapping from the modularized basis of P-double groups that are isogenous to this fixed P-double group to the graph monion which parameterizes the d-dimensional subspaces inside this dx. So which maps X to its hot filtration? A priori the hot filtration is inside the crystal of this X. Now as our P-double group is isogenous to the fixed P-double group module P, so by the rigidity of crystals, these two isocrystals are isomorphic. So this case of vector space gives the d-dimensional case of vector space in the isoprystal determined by the fixed P-double group. I forgot to say that d is exactly the dimension of our fixed P-double group. Okay, in a word, what is the P-double mapping? It records the valuation of P, it records the valuation of hot structures for the family of P-double groups. So it's not very complicated, it's just the valuation of hot structures. So Grotendig asked the question in his ICM reports. He asked that, what's the image of this P-double mapping inside the graph monion? Here the image is exactly what we call the periodic period domain, or later we will call it admissible locus. In fact, in general, if we define the periodic period domain in this way, it's quite difficult to study the image because the periodic period mapping itself is already very complicated. And we can also define the periodic period mapping of more general set-up. Here we consider P-double groups, and now we can consider P-double groups with additional structures. That means we can replace the group GLN to an arbitrary-rejective group. So in the most general set-up, periodic period mapping is a term of them from local Schmura varieties to flood varieties. Both sides are as rigid analytic spaces over QP-breath, or as backward spaces. Okay. As before, this image is called the periodic period domain, or called admissible locus. And I will introduce what is local Schmura varieties. Roughly speaking, local Schmura varieties are generalizations of modular space of P-double groups with additional structures. In the most classical case, the P-o type, this is studied by Harper Poinsin. And in the most general case, this is studied by Schoetser and Wänster. Or this local Schmura varieties, this is the local analog of Schmura varieties. And Schmura varieties is defined from Schmura data. Local Schmura varieties is also defined from local Schmura data. Here, local Schmura data, we have a triple GB mu. G is a rejected group of a QP. In our first example, then the group is exactly GON. B is an element in GQP-breath. In fact, this element, when G equal to GON, this element determines more precisely the sigma conjugacy class of this B determines the isoprystal. So in fact, B determines the isogenic class of a fixed P-double group of F-hebar. And mu is a minuscule co-character from the multiplicative group to G. In the GON case, B determines exactly mu. So this mu does not show up. For general G, we also have this mu. And we also require that B and mu should satisfy the Cartwitz condition. Roughly speaking, it means that the Newton polygon determined by B and the Hart polygon determined by mu should satisfy major inequality. And the F-flug variety attached to G-mu. When G is a quasi-split group, then this flug variety is defined by the quotient of G or by the parabolic subgroup determined by mu. For example, I'll still take GON as an example. If mu is of the form 1 appears G times and 0 appears n minus G times, then the flug variety is just to the Rathmian which parametrizes D-dimensional subspaces inside our n-dimensional vector space. This is flug variety. Okay. I will give you two examples about p-adic periodomas or two famous ones. One is the Lubinche case. In this case, G equal to GON and mu is of the form 1-0-0-0. B is basic. This corresponds to the fact that the isocrystal determined by B is isoclean. It already appeared in the talk of Tang Yunqing yesterday. And in this case, the p-adic period mapping is also called the Gross-Hopkins period mapping. It is studied by Gross and Hopkins. They give a very explicit formula about this p-adic period mapping. It's explicit, but very complicated. I learned this when I was in the second year or master M2. And they show that this p-adic period mapping pin, in fact, it is subjective. So in this case, our admissible locus, if you still remember, is the image of the p-adic period mapping. It's exactly the flug variety. In this case, the flug variety is the n minus 1-dimensional projective space. So in this case, the p-adic period domain is very clear. It's the n minus 1-dimensional projective space. Another case, the Genfield case, which is due to the Lubingtail case. In this case, the group we consider is a multiplicative group of the division algebra of invariant 1 over n over qp, and mu is due as before 1-0-0-0. Still B is basic. Then in this case, the p-adic period domain is called the Genfield up half space. It is the complements in the n minus 1-dimensional projective space of the union of all the qp-rational hyperplanes. You see, this is, this guy is already, and this Genfield up half space is already very complicated. And from this, you can also see that it does not have an algebraic geometry structure. It's only have a rigid geometry structure. Okay. Now, in order to understand the n-dimensional locus inside the flug variety, we want to, we need to introduce the algebraic approximation that we call it weakly n-dimensional locus. This is defining an algebraic way and should be a bit easier to understand. This is closely related to Fondant's definition of weakly n-dimensional filter isocrystals. Now I recall here. Let C be a complete field extension of qp-breath. I denote by few isocs, C over qp-breath, the category of filter isocrystals over C over qp-breath. That means the isocrystal is defined over qp-breath and the filtration is defined over C. And for a filter isocrystal over C over qp-breath, Fondant defines the notion of weakly admissibility. In fact, this is some kind of semi-stability. He introduced two invariants for filter isocrystals. One invariant is called the Newton invariants. Newton invariant is the pediatric valuation of the determinant of the Frobenius. And the Frobenius is the semi-linear. Its determinant is not well defined, but its pediatric valuation is well defined. This is the Newton points. Another invariant is called the Hodge invariants. Hodge invariant is defined from the filtration. So we have two, for a filter isocrystal, we have two invariants. One is the Newton invariant that is determined by the Frobenius. And the other is the Hodge invariant that is determined by the filtration. And Fondant calls a filter isocrystal weakly admissible if its Newton invariant equals to its Hodge invariant. And for any sub-objects of this filter isocrystal, we always have the Hodge invariant less than or equal to the Newton invariant. This is the definition of weakly admissibility. Now, how to define our weakly admissible locus? First, in the GON case. In the GON case, a point in the flug variety gives, in the GON case, the flug variety is always the grass mania. So a point gives a filtration. In fact, a filtration just means a sub-space in this n-dimensional C vector space. So a point Bx from this pair Bx, we can define a filter isocrystal over C over Qp breath. Here B as before determines isocrystal and determine the Frobenius vector. So this gives the isocrystal, and this point X gives the filtration. So this is a filtered isocrystal. The pair Bx is called weakly admissible if the filter isocrystal is weakly admissible. Now, for general G, for general G, we use representation to reduce to the GON case, as you can imagine. A point is called weakly admissible if there exists a faithful representation from G to GLN, such that we get a pair Bx, such that we get a pair which is for this group GLN, such that the corresponding filter isocrystal is weakly admissible. Okay, now we are ready to define our weakly admissible locus. This is introduced by Harpo-Ponzing. So a point in the flood variety is weakly admissible if only the pair Bx is weakly admissible. Indeed, we can show that the admissible locus is contained in the weakly admissible locus. These two are both open subsets, open subsets in the flood variety. In fact, the definition of weakly admissible locus is elder brick. It is defined by removing the finite union of super varieties. It is also approximation of admissible locus in the following steps. Comments and fondant show that these two spaces have same K points. Here K is the finite extension of QP graph. That means that the admissible locus and the weakly admissible locus have same classical points. And Harpo shows that in general, these two spaces are not equal. Okay, now inside the flood variety we have two open subsets and the weakly admissible locus should be very prox to the admissible locus. It's natural to ask about the extreme cases when the weakly admissible locus coincides with the admissible locus and when the weakly admissible locus coincides with the weakly admissible locus. In the two examples that I have explained, in the lube intake case, the admissible locus equal to the weakly admissible locus equal to the flood variety. These two include both equality. And in the gym field case, the admissible locus equal to the weakly admissible locus. Okay, now we want to discuss about the extreme cases. We want to discuss about the extreme cases of the weakly admissible locus. We want to give a criteria about this. In fact, Harpo and Harpo will give a conjecture which describes when these two spaces are equal. I'll have to deal with the GLN case. In a joint work with Fargan and Shen, the basic case, in this case, admissible locus and the weakly admissible locus coincide if and only if the pair is fully Hodge-Newton decomposable. This fully Hodge-Newton decomposable, this is a purely group steritic condition that I will explain later. And I also consider the non-seq basic case. In the non-seq basic case, there is a similar criterion for a bit more complicated as we recall here. They introduce and systematically study the condition fully Hodge-Newton decomposability condition and classify all the possible pairs that are fully Hodge-Newton decomposable. This is one extreme case that weakly admissible locus is minimal if equal to the admissible locus. Another extreme case is studied by Harper Hall. He studied the case then when weakly admissible locus coincide with the whole flug variety. He also give a group steric criterion when these two spaces coincide and he called such typos weakly accessible. I do not plan to give the definition but I will explain the GLN case. Everything is very clear. Just one remark, if GB Mu is weakly accessible that means the weakly admissible locus coincide with the flug variety then B is basic hence it is determined by the pair G Mu so the weak accessibility only depends on the pair G Mu. So we can talk about when a pair is weakly accessible. First I would like to say what I want to do in this talk. I have just explained two extreme cases weakly admissible locus is minimal and weakly admissible locus is maximal equal to the whole flug variety. So the goal of this talk is to unify the two extreme cases in a way that the weakly admissible locus can always be considered maximal and give a group characteristic criterion for the cases that the weakly admissible locus are maximal. When the weakly admissible locus equal to the flug variety it is easy to imagine that this is already maximal it cannot be bigger. For the other extreme case when the weakly admissible locus coincide with the admissible locus it seems that it is minimal so what does it mean? How can I explain that to be maximal? For that I need to introduce the Newton's justification on flug variety and this also needs the frog fountain curve that I will explain later. Before I explain what does mean the weakly admissible locus is maximal first I would like to say that the extreme case is for GON now everything is very explicit now the group G is equal to GON that also suppose B is basic then as the weakly admissible locus coincide with the whole flug variety if and only if G mu is weakly accessible in this case it means that mu is central or it is of the form 0 appears n minus r tons such that r and n are co-prime to each other. This is the weakly accessible case and another extreme case weakly admissible locus coincide with the admissible locus in this case we know that these two coincide if and only if the pair is fully Hodge-Newton decomposable that means mu is central or one of the following three cases holds the first case mu is of the form 1, 1 only appears 1 tons and 0 appears n minus 1 tons all mu is of the form 1 appears n minus 1 tons and 0 appears only 1 tons and the third case, exceptional case n equal to 4 and mu is mu is of the form 1 1 0 0 I can show you in this example what is the main fully Hodge-Newton decomposable so in this last example n equal to 4 mu is of the form 1 1 0 0 then the Hodge polygon determined by mu is the upper polygon that are draw here it has two slopes 1 1 and two slopes 0 0 this is the Hodge polygon in equality all possible Newton polygons should lies under this Hodge polygon and they should have same n points moreover as a Newton polygon we should require that all the break points are inch grow points so for given a Hodge polygon there is only one basic Newton polygon basic means that all the slopes are equal so in this case we just connect the start points and the end points of the Hodge polygon this gives our Newton polygon in this case the basic Newton polygon is of slope 1 half, 1 half, 1 half, 1 half this is slope of our beam except this basic Newton polygon also are the possible Newton polygons we will show you some other possible Newton polygons for example we also have this blue polygon this is also a possible Newton polygon that satisfy major inequality with respect to this Hodge polygon you see this has slopes 1 1 half, 1 half, 0 and notice that this Newton polygon touch the Hodge polygon the first part and the last part it touch the Hodge polygon and two other examples of possible Newton polygons for example also this green polygon it has slope 2 thirds, 2 thirds, 2 thirds, 0 it also touch the Hodge polygon okay with this example in hand I think you can now understand what does mean Hodge-Newton decomposability Hodge-Newton appear Gmail you always consider Gmail in case for Gmail for a Hodge-Newton decomposable means that for any possible Newton polygon that are not basic it always touches the Hodge polygon so in this example you will see except this basic Newton polygon this basic Newton polygon does not touch the Hodge polygon for the other possible Newton polygon that satisfy major inequality always touch the Hodge polygon so this is the meaning of Hodge-Newton decomposability okay now I want to explain in order to explain the meaning of the weekly invincible locus maxima we need to introduce the Newton stratification on the flag right for that we need to use Farke-Fundan curve and reinterpret the admissible locus and weekly admissible locus as modifications of G bundles on the Farke-Fundan curve now we let C be a complete algebraic closed field over QP breath general by X of Farke-Fundan curve over QP equipped with close points infinity such that its residue field is equal to C I will not give the precise definition of Farke-Fundan curve but I will explain the most important property of Farke-Fundan curve that is that we need our final function shows that there is natural bijection between the isomorphism classes of isocrystals over QP breath and the isomorphism classes of vector bundles over the Farke-Fundan curve by generally mining classification we know that the category of isocrystals over QP breath is semi-simple with simple objects parameterized by rational numbers so translate on the vector bundle side we know that the vector bundles on the Farke-Fundan curve are always direct sum of stable bundles of the form of lambda, this lambda rational number but we should take care that it's not an equivalent of category the reason that on the isocrystal side for isoclean isocrystals of different slope there is no non-zero morphism between them but on the vector bundle side given to stable bundles or lambda and lambda prime if lambda is less than or equal to lambda prime then there is infinitely many morphism between all lambda and all lambda prime so the algebraic side and the isocrystal side can be considered to be the algebraic side the vector bundle side is the geometric side these two sides does not give the exact does not give same information this is also the reason that admissible locus does not coincide with the weekly admissible locus the weekly admissible locus reflects the information about the algebraic side and while the vector bundles reflects the information of the geometric side okay there is also a group sciatic version of this bijection again we replace GON by G which I know by BG the set of sigma conjugacy classes in G of QP breath when G equal to GON BG is bijection with the isomorphism classes of isocrystals of dimension N so in fact BG can be considered as the basis of isocrystals with G structure over QP breath as a group sciatic version of the previous bijection we have a bijection between the set BG and the set of the isomorphism classes of G bundles over the fact from that curve which maps our sigma conjugacy class of B to EB the corresponding G bundle two points in the flood variety this defines the modification of the G bundle EB that we denote by EB X in fact all sides of the points infinity are the G bundle EB X is isomorphic to EB and for neighborhood of infinity we take the trivial bundle and the grueling data is given by this point X roughly speaking it's like that's the modification for simplicity you can always imagine the GON case then the G bundle is exactly that bundle there are points in the flood variety it's called admissible that means it's admissible occurs if and only if the modification is semi-stable so you'll see this is very easy to verify as we also know that the degree of this vector bundle is zero so semi-stable also means this is a trivial bundle and a point is in the weekly admissible occurs if and only if or the modification is a weekly semi-stable for semi-stability we know that we need to test for all the sub-vector bundles and for the weekly semi-stability we do not need to test all the sub-vector bundles but only test all the sub-vector bundles arising from sub-isopristals so we have much less conditions to verify from this definition you can easily see that the admissible occurs is contained in the weekly because for the weekly semi-stability condition we have less conditions now I can introduce the Newton stratification on the flood variety in fact the Newton stratification parameterizes the isomorphism classes of the modifications more precisely a point is in the Newton stratifier corresponding to B' if and only if the modification of EB at x is isomorphic to the G bundle EB' in particular we know that admissible occurs is exactly the strata corresponding to 1 so this corresponds to the chiral bundle so this is strata corresponding to 1 this is the open strata in this Newton stratification then it's a natural question to ask which Newton stratifier contains weekly admissible points as we know that admissible occurs is contained in the weekly admissible occurs and we would like to know which Newton stratifier infects with the weekly admissible occurs in our joint work with Faganation we prove that in fact this result is not written explicitly but it is written in the proof of our main results we show that the weekly admissible occurs does not infect with all the Newton stratifier it only infects with the Newton stratifier that a hot Newton indecomposable with respect to new B mu minus 1 what does mean here in fact associated to new B mu minus 1 we also consider it as a hot polygon and to B prime we also have a Newton polygon these two guys are called hot Newton indecomposable if and only if these two polygon does not touch each other this is the so-called hot Newton indecomposable in particular when G mu is fully hot Newton decomposable if you still remember the definition for hot Newton fully hot Newton decomposable it means that in this case on the right side there is only one Newton stratifier corresponding to the admissible occurs so in this case we see that the weekly admissible occurs equals to the admissible occurs so in fact this argument shows that fully hot Newton decomposable the equality implies admissible occurs equals to weekly admissible occurs and very recently FEMA also shows that in fact the weekly admissible occurs infects with each of the Newton stratifier appears on the right then we say the weekly admissible occurs is maximal if the equality holds this includes the equality now we can see that for example when G mu is fully hot Newton decomposable in this case we know that on the right hand side we have nothing but just the admissible occurs so this equality holds so we know that in the extreme case when the admissible occurs equal to the weekly admissible occurs or the weekly admissible occurs is maximal by definition there are several natural questions to ask first is that in which cases the weekly admissible occurs is maximal this is the first question and a more precise question that for each of these Newton stratifier for which stratifier is completely contained in the weekly admissible occurs these are the two questions that we hope to answer in our drawing work and in this talk I only plan to explain I only plan to give a complete result for the first question for the second question I just mentioned words for the second question I think we are able to give a criterion just for the GON case for GON when a given Newton stratifier is completely contained in the weekly admissible occurs in fact this is called a criterion but maybe it's better to call it a logarithm it is defining an inductive way that we can compute whether a given Newton stratifier is contained in the weekly admissible occurs and for the moment we can only deal with the GON case the reason is that to deal with this kind of problem we need to determine given two vector bundles over on the fact from that curve we want to determine which kind of vector bundles can be realized as the extension of these two vector bundles and we can give our answer to this question in GON but just in an inductive way that's why our criterion is just a logarithm and for the other groups other than GON we don't know whether we can generalize to other groups this is still a working progress we need to work on that okay so in this talk we only focus on the first question in which case the weekly admissible occurs is maximal these are our main results or we can give a group search criterion or when the weekly admissible occurs is maximal that the weekly admissible occurs is maximal if and only if the pair g mu is we call it weekly for hot Newton to composable what does it mean before I give a definition about the weekly for hot Newton to composability I recall the minute criterion for the fully hot Newton to composability proved by Gois He and Nie they show that the pair g mu is fully hot Newton to composable if and only if the pairing of mu with any relative fundamental weights is less than or equal to 1 for any relative fundamental weights now we have a similar minute criterion for the weekly for hot Newton to composability you see it's very similar you can also consider this as a definition for the weekly for hot Newton to composability in the process split case we say a pair g mu is weekly for hot Newton to composable if and only if the pairing of mu with the relative fundamental weights if it's an integer then it's less than or equal to 1 there is also a similar criterion for the non-process split case but in this talk for simplicity I just give the process split case so it's clear that fully hot Newton to composability implies the weekly for hot Newton to composability but maybe it's still not that clear what does mean I will also show it to you in the GeoN case in the GeoN case everything is very clear okay for GeoN a pair is weekly for hot Newton to composable if one of the following occurs the first case is the weekly accessible case weekly accessible means is equivalent to say the weekly items for locals equals to the whole flag right in this case mu is central or mu is of the form 1 appears r times and 0 appears n minus r times such that r and n are co-prime to each other the second case is the fully hot Newton to composable case this is the case and the third case is the the weekly items for locals coincide with the items for locals in this case that we have mu is central or mu is of the form 1 0 appears n minus 1 times and 1 appears n minus 1 times 0 or n equal to 4 and mu is of the form 1 1 0 0 and last line is the new cases 1 appears 2 times 0 appears n minus 2 times or 1 appears n minus 2 times 0 appears 2 times or exceptional case that n equal to 6 and mu is of the form 1 1 1 0 0 0 you see it looks very parallel to the fully hot Newton to composable case before I give a sketch of the proof to persuade you that this kind of theorem should be true ok I still take this exceptional case as our example the case that mu is a g is g o 6 and mu is of the form 1 1 1 0 0 0 in this case the basic Newton polygon is of the form or 1 half appears 6 times this means the corresponding iso crystal is the direct sum of 3 copies of simple iso crystal of slope 1 half and mu b mu minus 1 is of the form 1 half appears 3 times and also minus 1 half appears 3 times so the hot polygon given by this mu b mu minus 1 appears 3 times so we have 1 half appears 3 times and minus 1 half appears 3 times so for all the Newton's drawer appears in this flood variety it satisfies its Newton polygon with respect to the polygon determined by mu b mu minus 1 that means all the possible Newton polygon should lies under this polygon and have the same end points as this hot polygon so for example the only basic polygon has slope 0 this corresponds to their admissible locus this is one Newton polygon this is another possible Newton polygon it has slopes 1 3rd 1 3rd 1 3rd minus 1 3rd minus 1 3rd minus 1 3rd so these are the two Newton polygons that have hot Newton intercomposable with respect to mu b mu minus 1 I recall that a Newton polygon is called hot Newton intercomposable means that the two polygon does not touch each other so for these possible Newton polygons they do not touch the hot polygon these are the only two Newton polygons that are hot Newton intercomposable with respect to our hot polygon I can show you other possible Newton polygons they are not hot Newton intercomposable for example this green polygon it has slope 1 half 1 half minus 1 fourth appears 4 times and we have 5 other hot Newton decomposable Newton star and by our result we know that the weekly admissible locus should be contained in their Newton star corresponding to hot Newton intercomposable Newton polygons that means the weekly admissible locus should contain in the admissible locus union with the Newton star corresponding to this B' so there is only two Newton star so in order to show that the weekly admissible locus is maximal that means we need to show any points in this Newton star corresponding to B' every point is weekly admissible we need to show this but on the other side a point in this Newton star if it contradicts with the weekly admissibility then the problem should come from these breakpoints but by the weekly 4 hot Newton decomposability this tells us that the X coordinate of the breakpoints cannot be the dimension of the sub-isocrystal B in this case the X coordinate of this breakpoints this is 3 but for in our case the isocrystal is direct sum of sub-isocrystals that are always of slope 1 half so all the sub-isocrystals should have dimension even so this breakpoint cannot come from a sub-isocrystal this tells us that this breakpoint cannot come from a sub-isocrystal so it cannot be non-weekly admissibility so this means that any points in this Newton star should contain in the weekly admissibility locus in fact we can also use this criterion we can also give another definition about the weekly 4 hot Newton decomposibility we recall that a pair G mu is called fully hot Newton decomposable means that for any Newton strutter that satisfies major inequality if it's not basic then it touches the hot polygon and for the weak fully hot Newton decomposibility it means that for all the Newton strutters that satisfy major inequality with respect to the hot polygon if it's not basic and if it does not touch the hot polygon then it cannot come from a sub-isocrystal this can also be considered as a definition for the weekly 4 hot Newton decomposibility okay okay finally I will give an outline on the proof in fact this is exactly a group from the relation of what is explained in this example the proof is quite direct the first step is that we explain the pairing mu the relative fundamental weights is the integer means that the isocrystal corresponding to B can be decomposed into direct sum of sub-isocrystals with some given shape this given shape is related to this relative fundamental weights so I recall that why I have this because this appears in our minor criterion I explain that this this pairing the integer means that our yes it means that our isocrystal can be decomposed in a given form roughly speaking like this the second step we show that a pure G mu is weekly for hot Newton decomposable if and only if the pure G mu B mu minus 1 this is similar in the fully hot Newton compatible case our criterion is about G mu but in fact it's more practical to deal with G mu B mu B mu minus 1 in fact in our example I say that the weekly for hot Newton decomposability implies as coordinates of the break points of something actually the weekly for hot Newton decomposability for the pair G mu B mu minus 1 you'll see as this polygon is mu B mu minus 1 but not the polygon mu so this condition is more easy to use so we need to prove that these two conditions are in fact equivalent so by this step one and step two or step one plus step two we can show that weekly for hot Newton decomposability implies that the weekly admissible occurs is maximum and for the other side we need step three if a pair G mu is not weekly for hot Newton decomposable then in this case we can construct very explicit points that are not weekly admissible and this contradicts with the mathematicality of the weekly admissible occurs so this proves the necessity okay I think I stop you okay thank you thank you are there any questions so just historical remark in the older days it gets inequality, it gets conjecture and inequality so it was Newton above Arch but of course for the reason of course it's much better now to have the reverse inclusion but sometimes I always don't realize the other side it depends on whether we draw the polygon in a convex way or in a concave way yes yes last question probably I'm not it's hard for me to understand that but how do you construct those on the last slide you said that you are able to construct some you can construct points that are not weekly admissible such a construction must be very difficult no no not very difficult in fact in fact it's in fact in this situation I think indeed we need to construct some extensions of vagabond, extensions of vagabond but in the case that we needed in fact the extensions are always the far front and curl you mean the bundle over the far front and curl yes the modification of vagabond over the far front and curl and how do you do that or in fact or in fact we first how to say that maybe just in this example no this is not an example or maybe in more complicated examples if there exists we want to construct points that in this example we do not exist this con points if it exists con points I think we can construct the points from the Newton polygon where it's from the break point that comes from sub-isocrystals we can we can construct some extensions of vagabond in fact they are split and satisfy the non-weekly Hodge Newton incompatibility condition okay thank you are there other questions from the room are there any questions from the room that's in the speaker again