 Thank you to all the organizers for inviting me here to give the talk. It's a really inspiring conference. Okay, so this is the title, explain joint project in collaboration with Alan Kohn. So everything I will say during this talk is obtaining collaboration with Alan Kohn. Here is a schematic overview of what I'm going to say in this talk. Let's say that our latest development of the study of the ideal class space of Q has brought us to determine a correspondence between a algebraic geometric space, which describes this quotient and the ideal class space, the original ideal class space with this analytic definition. Actually, for reasons which have to do with a connection with number tier and the description of the Riemann zeta function and the property of the Riemann zeta function, we take a quotient of the ideal class space of Q by the maximal compact subgroup of the ideal class cube. So this is the double quotient that is going to be put in relation with an algebraic geometric description of it. In fact, the first algebraic geometric description that we obtained last year is in terms of what we call arithmetic site. So arithmetic site is got to the topos over b and it provides the first link, the collection of points, the category of points of this site. It provides the first link with this double quotient. More recently we extend the scalar, so the first got to the topos, actually a semi ring topos. I will explain later what I mean by that. The first arithmetic site, the first topos have been recently extended by mean of extension of scalar from b. B is the same b as the previous talk, is the surface space that Gerjala has mentioned the first day. So b is the smallest semi field of characteristic one. It is the set that you saw before, 0, 1. It is a semi field and it is the smallest one. If we extend the scalar from this semi field to r plus max, in fact, we obtain a new version of the arithmetic site. It is like, as I will say later, it is like in parallel with what happened to an arithmetic curve over a finite field when you extend the scalar and you pass to an algebraic larger of fq. So this process produces a second description of this geometric space which then is the point of this scaling site. In fact, in one-to-one correspondence with this double quotient. All these two sites, these two gotenic sites are defined over a semi structure of characteristic one. So when I say characteristic one, I mean idempotent semi fields. So a structure for which x plus x equal x. And in order to link with the spectrum of the integers, we define a morphism of topoi which maps a prime in spet z to the orbit for Frobenius action defined over the points of these sites. Actually on these orbits we obtain very recently a Riemann Rock formula. So then these slides will disappear now on the main screen that will be left on the latter, on these two lateral screens so that along the way you can follow the various steps. I hope you can follow the various steps I will describe. The issue at some point was that the integers, there is no connection, no straight forward algebraic connection between the integer and this small, very small semi field of characteristic one idempotent structure. So in order to unify these constructions we had to think of a categorical approach which is through the theory of gamma sets and in particular through the definition of the sphere spectrum in the easiest possible approach via gamma sets. So in this way we obtain that both the integers and B they become S algebra once one uses a fully faithful fonder which embeds both Z and B and other instance of geometry construction of F1 as they all become S algebra. So S is the bottom, it's characteristic three so characteristic zero here, characteristic one here, S is characteristic three. It's a very basic yet fundamental structure which has already been developed in great detail by people in algebraic topology. But here we use the most easiest description of this sphere spectrum. So to do so then we have moved all after embedding Z and B into the category of S algebra we obtain what could be possible definition of absolute point or absolute algebra so this sphere spectrum. So then I start to develop the definition or recall the definition of the arithmetic site so this is a grotendiektopos. One start with actually is a very easy example of carrying, apparently carrying important information on the primes. So one start with the natural number and zero I add here the notation it will come up later and I denote by n cross the set of positive integers. So n cross is a small category with a single object which I denoted dot and the endomorphism of dot n cross dot. So we endow this category with what has been called the first day trivial topology in SGA4 is called the chaotic topology. So then together with this trivial topology we have a pre-sheaf of over this site and cross j got an excite and in fact because of the use of the chaotic topology any pre-sheaf is a sheaf. So here the two definition coincide because of the topology is rather irrelevant. So then any object here any sheaf over n cross is in fact nothing but a set which is the image via this map of the unique element unique object in the category endowed with an action of n cross. So the description of the sheaf reduces to as simple as possible as a set plus an action of n cross. Then I go rather quickly through the definition of points in the chaotic topology we saw the first day in the talks the definition in general but a point in this particular topos it's a morphism of topoi from sets so I denoted star it's a singleton so I denoted by sheaf over the singleton I mean sheaf of the singleton is nothing but the category of sets to n cross so which means nothing but the datum which we have when we provide to a joint fontor pi upper star and pi lower star pi upper star as written here is right a joint so it's left exact sorry it's left a joint so it's right exact and we say that pi upper star shouldn't be shouldn't be only right exact but in fact it should be also left exact so pi upper star is exact as a fontor this point is in fact uniquely determined by a covariant fontor after using the unedan contravariant embedding of n cross in n cross so then you compose the contravariant unedan embedding with contravariant pi upper star and you get a covariant fontor for n cross to sets and the main fact is that this fontor is what has been called in several talks flat or filtering which is equivalent to flat this means that the category this category is filtering what does it mean means that you are given a set which is the image of the point of the dot point this set is not empty on this set there is an action of n cross so the category that I am alluding here is the category made whose objects are the objects of x and the morphism in the category are for example so are described by element in n cross which connects two points in x so for instance in this case this is a morphism which connects z to x so to give a flat or filtering covariant fontor it's the same as associating in this case in this particular top of case it's the same as associating a non empty set and the action of n cross which is given by with some property being filtering so the second step and the third step describes the meaning of filtering so for any two objects in x there exists a third object in x as for any filtering condition so that z is mapped to x and x prime through the action of two elements in n cross these are elements it's fancy notation but in fact it means simply that there are two elements in n cross one of which maps z to x and the second of which maps z to x prime so for any two points there is a point above z and z maps systematically it's mapped to x first and to x prime second the third condition for being filtering says that for x in x in this set and the kk prime in n cross so that x is connected so x is described so that the point kx or pkx pk prime x so then x is connects two points of the of the set capital X then if this case happen if this situation happen then necessarily the action must be I mean the element which produce the element of n cross which produce the action must be the same kk equal k prime so then in this set we will consider the finished of filtering filter in in SGA4 so the issue here is to characterize the points of this stop so then the first theorem of last year says that the category of points of n cross Včešte, da imaš q plus in izgleda se, da je to pozitivna del del q. In injektiv morovičnega grupa. Včešte, da, češte, da je počet, kaj je vzgleda v tem formu, da je vzgleda in cross in sets, potem na setu, kaj je, češte, češte, češte, češte, češte p dot, over which adds and cross. This is going to become a semi-group, actually the positive part of the totally order group, a subgroup of Q. And the addition in this H plus is provided by this formula. So if you take two points in H plus, the addition X prime is given by K plus K prime on Z for every Z so that these two conditions are satisfied. So this makes sense, because the category is filtered and satisfied the second condition that we saw in the previous slides. Then what happened is that H plus is the increasing union of subgroups, each of which is isomorphic to the additive semi-group of the integers. So by a process which is you done by taking the, what to say, the symmetrization, one obtain H. So H is the totally ordered group which this statement referred to. So in H there is a positive part which is a semi-group and the semi-group is described by, is provided to me by assigning a point, so assigning a set together with an action of n cross. In fact this H plus, because the statement is more precise says that non-trivial subgroup of Q, so the second, this line, explain how to interpret this abstract H, subgroup of Q in terms of unique embedding that is provided by this map. So notice that the K prime and K here refer exactly to this K prime above. If you choose X. Yes. Yes. May I ask you some question? Yes. Because this monoid n cross in just total of infinitely many commuting is equal to this, so prime numbers placed absolutely the same. No, but the structure will come later. So this kind of identification, somehow it's not symmetric? It's not symmetric, but the structure, the structure sheaf will give the additional structure. So yes, I will put in a remark the point when in fact I have to fix some structure sheaf on this top or so otherwise there are too many symmetries which are provided by the action of n cross. It will come up later, just a little bit later. The second part of the statement says that there is a canonical bijection of sets between points in n cross so isomorphic classes of point in the top and the double quotient of the finite adels modulo the action of q plus cross and z cross. How is it defined? Well, it is defined by associating to a point, like described here, an adel. So an adel, in fact, the adel a that I refer in this formula is obtained by, it's uniquely associated up to isomorphic, uniquely associated to one of these h in fact p is p h by the above statement but this p h is in fact h of a so this group subgroups of q each of them is associated to a unique adel and this adel Yes. Unique up to the at cross which is written here and the definition of the h is given as the collection of rational number so that a times q is in the at what that means, means that you should think about the adel at the place p say this adel is going to be zero if this h contains all the power of p otherwise, if it doesn't there will be a point where say p to the minus mu is still in h and p to the minus mu minus one is not anymore in h then I multiply so the adels at p at that point will have description p to the mu so then this is the definition of the adel which is not written in these slides but in fact the formula here the statement here the requirement here makes the description of a rather visible Ok, so here I came to the point that was raised a few minutes ago in a way we want to restate the group of automorphism of the points of an at cross arising from the endomorphism of an at cross to avoid arbitrary permutation of the primes so then for this reason we introduce a sort of structure shift over this topos we introduce this what is a shift it's nothing but a set as I said with an action of an cross so in particular this is an example of a shift in the topos Z max these are the typical semi structure that people use in tropical geometry actually in tropical geometry people tend to use mostly R max this is Z max but R max will be the same by substituting Z with R so what one does is to change the addition in the classical description of Z so the addition becomes the max of two elements and the product becomes the addition this is a shift on the site n cross j and the action for example on minus infinity so minus infinity is fixed by the action of n cross and the maximum between n and minus infinity is n et cetera so then over this shift we define the action of n cross acts we state that n cross acts for the Frobenius endomorphism which are described here sending an element n cross k to the Frobenius action which is described by multiplying an integer by k actually just want to comment a second because it's not by chance that the algebra underline this topoi is in terms of semi-structure this semi-structure are important semi-structure so they are of characteristic one in the sense I said at the beginning x plus x equal x so the reason is that in the previous slides we had that point was uniquely up to isomorph is characterized by total ordered group of q so this total order is reflected here so in the structure which is idempotent so there is a kind of tight connection between the notion of idempotent semi-structure and partial orders so it's not a case that we the geometry which represents the ideal class space over in characteristic zero now is described at the level of the algebra by idempotent semi-structure the definition of the arithmetic site it's here it's like when one gives a scheme and a structure sheaf the scheme is substituted by the topos the structure sheaf is substituted by a selected sheaf on this topos which provides the restriction of the automorphism in view of the statement I made at the beginning so here we are in the description of the arithmetic site so then next what are the stocks of the structure sheaf so then the semi-ring structure is compatible with the action of n cross and now the stocks of the structure sheaf are following structure of semi-fields so if I look at the stock of the structure sheaf at the point of this topos which is connected, related one to one with this subgroup of Q, H abstract group yes this is a canonical isomorphism so the stock is canonical isomorphism to the semi-field H max it's like we consider in H the H as a abstract rank one group subgroup of Q in fact and then associating the corresponding semi-structure which is adding minus infinity make a variation of the addition rather than plus max and the multiplication becomes plus so what happen in particular for H equal z then the stock of the structure sheaf isomorphic of z max and the point associated to this order subgroup of Q, z corresponds to an Adele finite Adele which is in fact made by 111 at every place and the reason, if you remember the formula, the reason it's clear what we are considering at this point so the Adele's 111 everywhere corresponds to the point P attached, associated to z well then what happen about the rest this describes space which has a sort of classical point the remaining points which are associated to subgroup of Q subgroup of Q say H they sort of you might think of obtained by specialization on a veil so there is a specialization going from pH to PZ PZ is the you need a point and the rest are non-classic point which have some Adele's not of this form but more interesting description so not this particularly easy one so now in this setup the field, finite field of Q which is the basis of the geometry in characteristic P is played by this basic semi field B, 0, 1 in which 1 plus 1 is 1 because we mean plus as the maximum between two elements in this field 0 equal to 1 of the 0 different from 1 it's a multiplicative monoid with multiplicative identity different from the pointed, it's a pointed monoid two elements distinct we say that characteristic B is 1 meaning x plus x equal x so it's an idempotent condition which determine the definition of characteristic here B it's embedded so now B has no unlike the case of finite field when one consider finite towers finite field extension of a given finite field here there is no finite field extension the first one of B the first one the important one we are going to consider is R max plus which is the multiplicative version of R max so then because of the multiplicative version of R max is defined by 0 the set 0 infinity max as a sum and cross multiplication as the second operation so remember that R max would have plus here so here is the multiplicative version of R max this is suitable this version of R max is suitable for the description of the family of Frobenius automorphism of R max fixing B so B has no finite extension as I said then B also embed in Z max this is the usual version of Z max by sending 0 to minus infinity and 1 to 0 so if one think of try to go in parallel with the geometry say curve or finite field complete curve so if the curve is complete then there should be very few global sections only the constant so if one compute the global section of this topos global section for the structure I call structure shift of the shift we fixed Z max for this topos this is the general definition of global section for in topos theory but 0 minus infinity so it's isomorphic to B so it looks like likewise what happened for a complete curve or finite field for which the group of global section is reduced to the constant Fq so in this case Fq as I said is replaced by B there is a second theorem now that put in what happen of this topos if we extend the scholars if we pass to R plus max if we look at the point of the topos better over R plus max well the points of the arithmetic side over R plus max the theorem states that is isomorphic to the double quotient which is the space that I recall as the space that has used in previous work modulo ziat cross and the action of R cross plus on this side corresponds to the action of the Frobenius how does it go this isomorphism so then a point as in algebraic geometry is characterized a point defined of some field in classical algebraic geometry as described here by saying I have a map between sets beta is the topos of sheaves over the singleton and R max plus to n cross o so then it's defined by assigning two component a component which is given by a point here and the second component in algebraic geometry we go to the reverse arrow so it's like looking at the stock of o at the point say which we know to be at h max for some h subgroup of q to R max plus and this is a local morphism like in the previous talk we heard so this is a local morphism in this case of semi fields so then in the description of these points will be characterized by looking at the image at the range of this f plus shrek whether first of all the range, one case is when the range of this local morphism is completely contained in B and these are the points in fact of the topos over B and these are in one to one correspondence in the previous theorem with a del class in the double quotient then there is the second case when the range of f p sharp is not totally contained, it's not contained in B so you go outside B and in this case what you get up to a parameterization of h, say h prime equal a log of u for u in the range different from zero you obtain not anymore a rank one subgroup of q but in fact a subgroup of r and the association here, this association will map a pair which is described by a del in the case of the finite sorry a is finite del and real number so a lambda h a which is in fact a subgroup of r if h a was a subgroup of q so then accordingly to the composition of the double quotient in two parts the double quotient here the composes in two parts by looking first at the finite ideals and then to the ideals which are not finite different from zero one obtain the distinction between subgroup of rank one subgroup which are containing q or subgroup of r not anymore in q the condition here is as we saw before so now I describe the arithmetic site with some detail and then I pass to the description of the map that I mentioned at the beginning so this is a particular important one because it connects a classical scheme classical stropos to this arithmetic site and it is provided by this uniquely determined by a filter in front so theta is the morphism of the point rather than n cross we consider n cross union zero this is at the beginning and later in the talk I will use n zero so in zero it's n union zero it's this non-negative multiplication this time yes so then a morphism a geometric morph of topoi is determined by a filter in front theta star from n zero cross from the category from the small category with a single object n zero cross z so it is given if I define what is this sheaf which is the image of the unique element dot in the category so theta to describe this sheaf s first of all one think what theta is explicit theta determines a map between the points of the topoi so a point here rational prime and so a point here in terms of the first theorem we saw is it's a set with an action of n cross so then the idea of the definition of theta comes by mapping p to to the positive part of a subgroup rank one subgroup of q which I denoted by hp and it's like here Alain says it's like localizing at p so taking a ratio of two integers at the denominator you see all possible powers of p and at the numerator you have integers so then the zero the generic point here is mapped to zero so hp in this particular example of rank one subgroup of q as before is going to be an h attached to an adel ap and the adel that turns out which hp is described here like I wrote before but the adels attached to this particular subgroup of q when you see only at one time all the power of p is given by saying at the place p the adels must banish because you have all possible power of p at the other places there is no issue and you put one so this is the corresponding adels in this case and therefore the definition of the stock in view of this setup is given by say at the stock of this sheaf at the point p because here we are at the level of sets with an action is described by semi group hp plus the positive part of hp the sections then you have to implement a good action of and cross on this sheaf on the section of this sheaf so the section are given by maps from you as usual in shift here to the coproduct for all p of hp plus so that the stock the germ of this section at p are different from zero only for finite remaining p in u and the action of n zero cross on this section is described by this formula so n adds some xi by acting on the germs of xi at p so this is the link between this classical topos the scheme spec z with the structure sheaf and the arithmetic site so this is we sort of link just leaving in two different words one is characteristic z or one is in characteristic one by this map of topoi and just add as a comment the theory of phisaggio but I would see later provides a unified framework for shifts of commutative rings like for instance the structure sheaf spec z and the preimage of z here is max so the preimage of z max which is a sheaf over this topos now I pass to the description of the scaling site as I said the scaling site well the previous theorem one of the previous theorem says that after if I consider the point of r plus max of this arithmetic site then I get one to one correspondence between this double quotient and the arithmetic site but just passing to the consider in a point of r plus max is not sufficient at least is not one we were not happy about this process so we were looking for a topos whose collection of point right away without extending passing to r plus max would provide a setup in which the collection of points are immediately in one to one correspondence with this double quotient this is the scaling site the scaling site is not as easy is another grotenic topo but is not as easy as the previous one in the sense that the topology is not anymore the basic topology we used before so the goal is provide a geometric analytic structure over this double quotient so we had to define a site first and then go on so now the category we used to define the site it's denoted c here and they are the object of the category are made by bounded open interval possibly empty inside zero infinity and the morphism between two objects are described and cross the action of n cross should be there in some form so this is given by all elements in n cross which maps first open interval omega to the second one if omega is not empty otherwise it's the single term so the on this zero infinity this is the Euclidean topology that we use so there is a jump in deepness with use of the Euclidean topology and so for every object in the category then we consider the collection of all ordinary covers so this is for the definition of the sieve of omega so these are ordinary covers so the union of omega i is omega but omega i are open in omega and the proposition states that k in fact define a growth in the topology on the category c first statement and the second statement says that the sheve the sheve over this site so the topos is canonical isomorphic to the category zero infinity similar to the red product n cross so of n equivarian n cross equivarian sheves on zero infinity so then you see immediately ordinary sense because over zero infinity there is the Euclidean topology there is a big jump of deepness at this point because the topology is very refined and before was the trivial topology so then this big jump on one side would bring more interesting and somewhat more difficult interpretation of the points in the topos but on the other side you should pay back to give us right away the structure that we had to work over in the previous site by passing two points define over a plus max so the scaling site by definition is the site c with the growth in the topology j, which I just described and the scaling topos is the category of sheves over this side or equivalently in view of the previous proposition n cross equivarian action on zero infinity so the theorem of this idea says that in fact then as before one look at one study the points of these topos the category of points of these topos with support at zero and then there will be a statement when the support is not zero so the first case with support zero is the same as the category of point of n cross which is not in bad the points of the first of the arithmetic side define over b point with support different from zero which is canonically equivalent to the category of rank 1 subgroup of r so the points that we saw in the previous statement for the arithmetic site up to passing to those points of a define over a plus max so in view of this theorem the goal was achieved in the sense that immediately we have that the collection of points is in one to one correspondence with the double quotient here and as a consequence there are canonical bijection of sets between the isomorphism of points of these topos the points of the arithmetic topos define over a plus max after extending and the double quotient so the, as I said we pay a price because the topos got any topos is more sophisticated and more difficult to understand from the point of view of the study of points the pay is that the result is automatic as before there will be a structure shift attached to this topos but if we simply extend the structure shift z max to our plus max then and then for technical reason pass to the reduce structure attached to the tensile product then this is not quite a shift anymore in this second topos because it's not the case that any price shift is a shift so what happen is that the definition of the shift is obtained starting with this reduced semi ring which is simply the extension of scalars from z max to up plus max up to a reduced form and then localize so we have to localize this gadget and this will be the stock of the shift so the Legend transform allow one to describe this semi ring in details what is it given in fact more in general given around one subgroup of r h in r an element in the tensile product here is z max but we can do this definition, the description in general where I take h max this is a Newton polygon and as a Newton polygon in r2 is the convex hull of the union of finitely many quadrants xj, yj minus q where xj, yj it's a point, the pair describe a point in h cross r and so the Newton polygon in fact is uniquely determined by the function max lambda xj plus plus yj where lambda is an element in r plus so now let's look at to describe the stock and let's look at the consider the ring of convex piecewise affine continuous function on zero infinity with slope in h and finitely many singularities pointwise operation on this function then the proposition says that this reduced tensile product called r it's in fact isomorphic by mean of the association sending the Newton polygon to the corresponding affine function ln to the semi rings rh of convex piecewise affine continuous function etc so this is the this is the tensile product description of in terms of the Newton polygon and then the stretcher sheaf of over this topos affine by localizing this rz because in fact now rather than h one look at z because we are extending scalar from the stretcher sheaf of the arithmetic side to r plus max so then for any bounded describe the section of this sheaf so for any open bounded interval a collection is on omega maps from omega to r max and this section are convex piecewise affine continuous function on omega with slopes in z the action is provided by this formula and then through this definition I look at the stock of also the theorem says that if I consider a point in this topos associated to a rank group of r h the stock at this point p equal p h is the semi ring r h of germs now of r max value convex piecewise affine continuous function with slope in h and yeah so in the case of the abstract rank ranks of group of r h we we consider the associate point with support in 0 so let h be an abstract rank for group of r and let p h 0 be associate point with support in 0 then the stock at this point is described by the semi ring h cross r max associate to the max plus construction so this theorem gives the description of the stock under the attention paying the attention that is not enough to consider the extension of scalar of the above of the shift z max for the arithmetic site one has to be more careful because in this setup shifts are not finite sets with an action most sophisticated so why do you take germs because it's the stock so yes this is the corresponds to the theorem before that when we look at the stock for each this is ok so then the definition of the scaling site is therefore the scaling topos over which we fix this structure shift that I just described and in this case denoted a hat this is view as a relative topos of r max plus the structure shifts here locally are structures over r plus max the theorem which gives the final viewpoint on this on the points of this scaling site says that even if we look at the point of the scaling site over r plus max this is not about the collection of points of the of the site so this is a bijection of sets and then there is a canonical projection from the scaling site to the arithmetic site sort of by forgetting zero infinity and just retain and cross and one derives therefore the final bijection of the points of say the scaling site over r plus max or the points of the arithmetic site over r plus max and the double quotient that we we were looking to study using algebraic geometric gadgets then the remaining part here it's about a definition of Cartier divisor theory for this scaling site so then the structure shift has a reduced structure and immediately you can look at the embedding so there is an embedding of the structure shift for any fixed omega there is an embedding of the section in omega with the semi-field of fraction of omega as I said that this is in view of the reduced structure that we imposed for technical reasons over the structure shift of the arithmetic of the scaling site technical reasons if you wonder what is behind this technical reason is because this is in order that the Frobenius acts suitably on a tensor product in this framework of semi-structure so then this embedding embeds convex piecewise affine continuous function in the semi-field of piecewise affine continuous function so we lose here the convexity is not holding anymore and the natural action of n cross on this shift of fractions defines a shift of semi-field on the scaling tops so from now on it's like a sort of generalizing the theory of Cartier divisors because now you have a shift of function shift of fractions in this context and so then we have a point on this topos which corresponds to subgroup of r then look at the germ of this of an element in the stock say f the germ at h then there is a definition of order of this germ as h plus minus h minus where h plus and minus it's described here by a limit this limit that you have to have in mind the Newton polygon point of view so these are affine piecewise continuous function you look at one point in which you sort of the slopes are different and so then h plus and h minus gives you the slopes from the right and the slope from the left and the two slopes are different because there is a singularity at the point at that point the function vanishes and the order of vanishing is described by this difference ok so now this part it's about the statement I made at the beginning that that the image of prime determines a orbit for the Frobenius action in either one of these of these sites so then this is something that we don't see because in spec z we see only p we don't see the orbit of the Frobenius so again in pursuing this approach with the sites arithmetic and scaling is that now you can see how the action of the Frobenius how the Frobenius on these sites act on points so if suppose p is a rational integer and consider in the subspace of points of the scaling site cp it's a collection of points with the property that each of these points is one to one correspondence with subgroup with rank one abstract group which is isomorphic to one of these hp I described before in the process of giving the definition of h p is a prime so hp are those subgroup of q the ideal of which at p is zero is one otherwise so schematically if you take all of these cp all these orbits and these orbits embeds so map to the image of spec z via the map theta here and this collection of orbits in fact it's a subspace of the points of the scaling site and the points of the scaling sign maps to the point of the arithmetic site so this diagram is Cartesian and so this collection of the orbits of the Frobenius in a at cross defines the extension of scalar define an explicit wave to see the extension of scalar in a at so this is in analogy to the scheme that one obtained by extending the scalar of a scheme like spec fqt to the closure of fq so as I said this description in spec z was not visible in spec z you have only p you see only p it's only when you move to an extension of q that you would see the action Frobenius action the orbit but in this case is provided by this space so now what I said at the beginning is that each of these cp they are like they corresponds to what classically would be an elliptic curve and so then the next statements it's about the it's about a setup which is definition of rimarok formula I'm getting confused I just have to approach more so then what I'm up to it's a statement which define rimarok formula on these curves we don't claim that these are tropical curves by the way they are curves so defined over semi-structure and we claim that they are they corresponds to an elliptic curves in the classical context because an elliptic curve in a classical context is described by a quotient of c cross modulo q to the z in this case the curve would be rather than c cross you would have r plus cross modulo p to the z so then the rimarok that we will see it's different from the classical one because the degree of the divisor is not anymore an integer in this context is a real number so then the map which send in r cross plus to cp induces a topological isomorphism from the quotient r cross plus modulo pz to cp then what we do is cp after all it's a subspace of the scaling in which there is a structure shift so we pull back the structure shift restricted to cp to this quotient so we call it OP so this is the shift on this quotient whose section are piecewise affine convex function with slope in HP 5 minutes and so then through the definition of the shift of quotient defined before kp then one can prove that a global section has ordered zero and the order uses this definition so it's the order I defined before pull it back to the quotient and the divisor like in the classical context cartiel divisor is a section from cp to say h of the bundle of pair h which is a subgroup of r and small h and h are isomorphic abstracts to HP and h, small h is an element in h, this is in parallel with the theory of cartiel divisors for curves and the degree of divisor on cp is this finite sum I'm going a little bit faster so I will put up the theorem so the theorem says that if I pick up a divisor on these orbits of Frobenius actions then the degree being zero and also k of the divisor being zero, k is the map which is defined by looking at all possible so all these h are lambda HP so then there is a map, evident map sending h to HP modulo p minus 1 HP so then if the degree of a divisor is zero and it's also the image of k of d by k is zero then this principle so then there is a notion of Jacobian for the curve and the isomorphism of the Jacobian of this curve with z modulo p minus 1 z then in fact there is a definition for what classically we can see that the shift of global section attached to a divisor over the curve of Cp this is in parallel with the classical definition of OXD in algebraic geometry and for each of these for any section here this denotes the norm at p denotes in fact the max of the periodic norm of each lambda is in HP so then with this setup one consider in fact a suitable notion of dimension for these modules, the difference between this case and the classical case here is visible again because h0d in the classical context will be a vector space say over a field in this case there is no vector space anymore, this is a modules semi module in fact so the one has to implement a correct definition for dimension in this case we took this possible definition provided that in fact one proves this is the topological dimension classical topological dimension provides that one proves that this limit is finite indeed it is finite it coincide with the degree of the divisor for any divisor which is not positive divisor degrees is not negative and the formula that I alluded before the rimarok formula is as this description so you see it's like an elliptic case because the canonical divisor is not present so you have this formula so then just to collect together I don't have the time to go into the S algebra machine in details but I want to say just a few words here so as I mentioned at the beginning these constructions are in characteristic one so they are in overriding important semi structure spec z, it's a gadget in characteristic zero likewise the ideal class space of the double quotient so there is no algebraic connection between z and b so in order to gain a unique bottom structure characteristic 3 then we thought to use a certain construction which are very useful and known and we refer to the construction of S algebra so any of these categories in particular semi rings and so rings certain constructions in are a kelov geometry that Durav provided in the past, in the recent past the theory of monoids they all each of these structure, algebraic structure embeds faithfully here I consider for example the monoids and the semi rings embeds faithfully as subcategory of the category of S algebra so these hyper rings are there because in our project we also met hyper rings which are more sort of exotic form of algebraic structure than the one I talk about so then with this setup when we consider structure sheaf for example over spec z and one is able to reinterpret structure sheaf structure sheaf of S algebra so in this way this is a unified point of view in order to be characteristic 3 so in order to avoid to talk about characteristic p characteristic 0 or characteristic 1 but to have a unique source categorical structure at the bottom which is characteristic 3 and which provides the possible definition of absolute algebra or absolute point I think I stop here, I'm sorry I don't have time to go into the details but this is the life thanks Maybe we can take extra time for questions I have two questions one is about the scaling of course because when you have all these identifications there is zero infinite semidarek how do you define in these terms morphism in this scale what is it but I just put a formula at the board I put a formula at the board so you can see Voila the other thing was when you discuss Riemann-Rohr theorem 4mg complex function and you said you embedded into semifil of fractions but is it really unimbeding for some semirings in general not if you would consider the tensor product which you can define for two semi-structure will be in general it wouldn't be and so that's why we pass to reduce structure is like moving from a ring to the reduced ring and then when you have a reduced structure but this thing is there it helps not only at that point but as I said also at the moment when you want to define for example we have a theory of correspondence I didn't talk about this on the self-product of the arithmetic site and so this correspondence in particular Frobenius correspondence in this for this type on the self-product they do not behave well they are not endomorphism if because they are semi-structure so that's why we use systematically in that in that development about the Frobenius correspondence also a reduced structure so the reduced structure is like to avoid badly behavior of Frobenius on the self-product because even the polynomial ring in one variable of B for this canonical mapping to the fraction semi-factor fractions is not injective but here you mean it for this or for me again it is because we sort of kill we sort of kill the other part you did 0.5 all this is a fine I'm interested in the last thing where the compactified spec Z over S how do you see the real prime? this is like in the construction of Durov so spec Z bar we are against the wrong answer for him G and N of Z of the integer that the real prime is a finite group preserving one polytope and also a pogonite group in our case what we have is that when we take spec Z it's no longer true that Z a tensor product over S with Z this is not isomorphic that's fine in Durov theory it does reduce to the diagonal yes but Durov gets the wrong answer it's the real prime and the real N we have the correct pick up group so what we prove is that at the real prime the pick up group gives you exactly a plus star and which Durov doesn't ok but it's the L1 metric not the L2 metric it's a finite group sa group of GLNR you get a bad log of GLN VP in GLN QP it's a orthogonal group fitting in GLNR ok but we don't care about GLN at the moment we care about the pick up group of spec Z bar and what we have found is that with our construction we have one parameter family of invertible shoes and this is exactly what Durov was missing because he was getting too close which is not the right answer so we get exactly the right angle of answer and we have a right arcane of answer but you want the L2 to know at the real prime and you get the L1 know once you start like with gamma sets with sets and you get the right extension no but you could have if you wanted at the real prime if you want you can take any norm I mean if we stay satisfied with the condition normally we work any convex set we work any convex set works so you can take M2 if you wish but I mean this was not the point the point was to understand that when you hide a convex inequality at the Archimelian place in its rational coefficients who create poles coming from the denominators at other primes and this is perfectly made in the state now what the real thing that I didn't have time to make go into the scientific homology slide you see the real important thing the real important thing is that we would be in the dark if we didn't have a very powerful homology theory which is present and thanks to the work of Matt Senn, Esselot and all these people they have developed psychic homology over Archimelian's OS so this means that there is an immensely powerful machine which is waiting to be used and has been used extremely successfully already by Esselot and Matt Senn for instance there is a recent result by Esselot which says that when you take and when you look at it as an Esselot if you look at the psychism you obtain exactly the crystalline convolution and moreover you obtain the fountain that's there when you do filtration in the correct way this means that there is this enormous machine which is available and it's available in the largest possible context which is the context of Esselot so this is why we believe that this is how it works but if you start with Esselot you are of monad you are obliged to use the L1 norm not the L2 norm no, not in the module in the module you can use the L2 norm you can perfectly use the L2 norm in the module because it's convex combinations that you have in modules yes but the local ring so it's big because I think it's always you have these important tricks it's about non Archimelian fields you really miss Archimelian but over us we don't have the idempotent when we have over us we don't have an idempotent to do the picture that you should have is that spec Z is still characteristic 0 and then we modify it to something in characteristic 1 but it's not the same thing as talking about the Archimelian paste the Archimelian paste is not treated in characteristic 1 by no means by no means it's the same idea as the idea of shorts and all these people in a fountain are passing from one characteristic to another so this is the general idea but it's not the discussion of the Archimelian paste we didn't discuss it at all technical question related to the question why do you lose convexity when you go to embed into the field of fractional it takes a difference it takes a difference field of fractions the product is addition so when you pass to the fractions you have to look at my master you have to take a look you have to take a look at my master secondary with titan