 Thank you very much. So I'd like first thank organizers for invitation to speak Here so it's a great honor to speak in a conference celebrating Arthur's birthday and I guess I should apologize at the beginning that I'm not Really, you know, I'm kind of new to this topic and probably I make mistakes so Experts should correct me at any time so like I guess a professor blog just said physics is difficult and I think periodic hot stir is also difficult for me So I just try to appreciate the box Okay, so Let's just to recall the following This sit up. So let's see start with okay over QP finite extension, so it's periodic field and and recall that So row a Piotic representation one of the Galois group to some vector space over QP is called Duran if If you Consider the following your first tense V over QP with the Fountain's pure ring pure ring B drum takes the Galois invariance Then its dimension is the same as the dimension of V okay, so this concept is This notion is important because We know that the ending piotic representations coming from geometry namely realized as a Action of the Galois group on the et alco homology of some some most projective algebraic varieties Satisfying this property. This is known as I guess as a fountain's conjecture and proved by various people like so The situation I'm interested very interesting. Oh, yeah, I should see this is a joint work with Okay, so the situation we're interested in is a falling. So suppose X over Over case it's a smooth It's a connected Rigid analytic variety So I could start with a drink right about the eventually I need to go to the analytical geometry So let's start right here at the beginning. So let me suppose L is Is a let's see QP or maybe let's see just for some place. Yeah, it's the same thing I guess let's see Zp local system et alco system and Suppose for so if you are given a finite extension Cape and a point on the variety. Let's see then you can consider the stock of this of This local system. So then the stock It's really can be Consider, I mean, you probably want to consider the geometric stock over geometric point and then the stock is going to be a representation of the Galois group Okay, so it's going to be a periodic representation. So Here's the theorem Maybe so So the students is falling if for Some point one point X L X bar is drum Then L Y bar is drum for every Point Let's see why in some finite extension. It could be some some other any over any finite Namely over any classic point the the corresponding Galois representation straw Okay, so you will see in fact the proof is really simple, but let me guess first make a few remarks and give an application before explain the idea so Let's start with remark The first remark is you can ask the same question if you replace the Drum by christening, but then of course, this is not correct Like there's different the family of right is at the most point that this has good reduction that at some point that there's the same semi-stable reduction for example hot you can also ask the hot state version I'm not sure. I guess it's a fourth, but I know I counter example okay the second is this is The situation you can think about the water I I don't know I Mean it should it should be correct, but let's I Just understand what is the beat around so far, so Okay, maybe later after I understand most sophisticated field of rings could be Okay, so the second is this this L you can think about it's a family of a periodic representations parameterized by a Right by the variety X. So usually it's called like a geometric family, but there's a another Notion Which is important in studying galore representations called as arithmetic family of periodic representations namely in that situation instead instead of having a top local system of writing you consider the the galore representation a Galore group acting on a vector bundle on some Rigid analytic space and in that situation, I think this the story is completely different I think it's like so probably this arithmetic family Then there's I think in this situation I if I know understand correctly it's proved by Bridget and the comb is that this is the drama locus is closed No Oh, yeah fix under some kind of it's close under some condition analytically closed so so it's very different from this geometric situation where you just required the representation to be drama at one point you get The drama is everywhere. So once I think one should really Compare this with this One should really compare this with this result of the name which equal to the principle B of the link because the title Which is a following? so if suppose you have a x over s is like let's say smooth proper of of map of complex varieties and Let's say s smooth connected complex at rate at it. So Then if you have a family the Family of hodge clock what's this I? Check the book which is the IE a global section. It's really a global section of some sheath on s There's a sentence which is should be horizontal with Gossip Manning connection then if It's a family of hodge class parametres by s what does it mean it's a global section of something I mean it's rated in the book in this way in the introduction. It's a little bit complicated I need to specify the wrong component at how component So I don't need to section about NF loss of the arm. Yeah, it's at least there's a drum component, but You may also want to add the at how component to Absolute host cycles. Yes, absolutely. Yeah, if TS is Absolute hodge at one point. It is every it is It's absolute hodge at every point. So this is what he called the principle be Which gives I guess such as a title of the talk Okay, so maybe Before again, let me explain the application of the theorem. In fact also the motivation of the of of the of This work. Let's start with Let's consider G a reductive group and the X is a family of like hodge structure with With G structure name needs basically Several copies serve her mission is a metric domains then attaching it to you have and let's see case open compact In the job finally add those then you get the Shimura varieties, which is Like at the level it's a quasi-projective variety whose set of complex point C points are just given by this double course post double quotient but The the theory of canonical model says that it's in fact It's a quasi-projective variety defined over some number of field called the reflex field Let me not go into detail, but just see there's such thing now if supp let me make the following assumption Let's assume like Let's see if the Q point of inner center is discrete I Mean this is technical something you can ignore in the Final adults so make this assumption Okay, suppose I have a V is a rational representation of G Then the then I get the in fact a Bady local system on this complex variety just as a social construction because Gq acts on this vector space you can take the social construction and in fact the theory test tells you In fact, what do you get this? It's a tall local system Okay For an MP you get the really a tall local system deep. You I mean a power is defined over C by this distance You are working with which kind of the townships Q and she's so Q Qp So fix P is this Qp. It's a Qp at all because it's fix a little P under Shimura right here P is Really the rational represent defined over Q just over Q. Yeah, rational Q rational Let's see and then you specify some P to talk about the etal local system You get something over over This is a the setup. It's general theory of the canonical model gives you such thing and then the color is color is in fact for every finite extension and a point F point Then the corresponding the stock It's a then it's galore representation of galore group for F Is drum at P? So this is going to be the calorie the proof is simple I mean this some observation of a Kaewon 9 that in fact because on the Shimura variety There are some very special point called special points. You can check the corresponding etal local system is drum there. So by the given gives you everything so I should see this Again, there's a remark This this term this color is not It's not new if the Shimura variety select is if it is some special kind of Shimura variety namely if this is this is known if I guess Gx is of Avian in type plus some conditions Because in this situation the it is known the corresponding Shimura variety just parametrized Parametrized certain Avian motives, namely those some motive is appearing in the Avian varieties and This local system or just the local system of their periodic realizations I mean this this is I mean how to prove the calorie I mean this balance from a suggestion of Kaewon 9 so so This is known for Shimura varieties of Avian in type But it's not known in general because for general Shimura varieties I mean it's it's expected. I think from the name that the Shimura varieties should should parametrized motives But we haven't in general. We don't know where to find the motive But nevertheless, we know like their periodic representation of periodic realization should be drum So maybe at one day, maybe we could one proofs from 10 meter conjecture before finding the motives we get something we need. Okay, so but Not known in general This wasn't known. Okay. All right. So okay. So anyways So Really, I said I need some condition. In fact, not every way V. It's some V really attached to Yeah, so I said that there's some condition not every Russian even in this situation you only for some V, you know, but now it's Your carol area gets for every V? Yeah, every V. But I want to make this assumption like the center Q point of the center is discrete one can relax this condition then it's not for every V, but I mean first you need to Descend the construct the local system on the Schmarovite that requires not for not for every representation you can construct it Okay, so anyway, so this is I mean this the whole idea is try to get to what every you would expect Like for example try to construct the stuka sound this kind of Schmarovite without finding the motive Okay, so I think that's The introductory part of the talk Present part so now I want to get to some the idea some idea of the proof which you will see kind of you see I'm sorry Does the theorem fault for QP local systems that don't have a ZP lattice? I Think it's okay because locally. It's a you see this is you see this is really a local Theorem it's a local theorem. So locally you do have ZP lattice, but Okay, so let me so the there are two ingredients needed in in this in this proof, so let me See the first one the first one is we need this recent like this pro et al Topology and on the rigid analytic Friday and the periodic shifts introduced by Peter Schultz. So So now let me assume X over K. So smooth may be curve Just I'm to prove the theorem I can assume axis curve, but you can just save the notation So then let me introduce a few more notations so I need Kn Which is K edge or into the second rules of one and the way choose it compatible and It gives you the element inside the K flat where K is I guess Just take the union of all of them take the periodic company Okay, so I also need new the map from the pro et al site mean maybe from the etal side Which appeared yesterday so a typical object object here, let me just draw the picture Is the following so I can I can consider you some open etal open subset and the take a Toric coordinate so episode stand the torus which is Inverse okay inverse and then I have this tower of Finite etal covers this Project limit formally I denote by a t infinity hat and I just take the fiber product so this is like this system in fact gives you a This is a etalon in the finite etal that gives you an object in the inner inner pro etal site Which I let me write it as Let's see spa a a plus and This is a okay, so a Infinity, let's see so a infinity So a infinity plus is You take the limit Complete and then the infinite is an infinity plus you invert p It's standard, but we want to consider the the Galois group of this tower so we get Gamma This Galois group the quotient is the arithmetic Galois group and then I have this geometric which is isomorphic to the p1 with a generator gamma Gamma axon I guess t 1 over p to the n that's Zeta p to the n t and here I also have the cyclotomic character. Okay, so this is a basic setup and I was so let me see All right, so what we need is really some period shifts on this space, so we have What we really need is Okay, so Let me think let me try write it. Okay. Let me write it. So I have integral shift o x plus Which is on the pro etal side, which is just full back of the integral pro sheet from x etal and it's a complete periodic complete version so What we really need is and Of course, I mean the rational version. It's just you invert p, but we also need this Important pure shift OB drama. I mean this appears before but I follow shots notation So it's defined as follows. I mean Yeah, maybe let me just give you the definition so I Have first of I have B ins which is Which is This a inf x appeared in yesterday's talk, I guess invert p so Then there's a OB Inf Which is all x Transferring this B which admits a map to O X Hat So, I mean because of this I'm I could start from edge break situation but here because this completion I Do have to go to the analytic situation. So and then OB drama plus. It's just the Completion with it with respect to this map. So and and then you OB drama This OB Drum plus Invert a T. Okay, it's usual. So I mean this is complicated, but Let me just Tell you what I really need. I mean, this is a I don't There's something quite complicated. So the what I need is the following some properties so first of all By definition OB drama is O X module Because OB inf is an O X module so the second properties there exists a filtration given by a the filtration by the Kind of theta so there's a filtration Such that if you Restrict I mean in general, it's it's very hard to describe this ship I think except if but if we pass to some object in the pro at a site like You infinity had indeed we do have some description so Grow I OB drama is gonna be O X hat when you restrict to You infinity then you adjoint a variable Here V is T inverse log of T flat I mean this is a The I really comes from the action of the arithmetic It's a tensor with cyclotomic character Yeah, so then that you if you just a chase in definition the god the element of gamma X on TS It maps T to tip. Oh, sorry X on V maps to Be goes to V plus one Yeah Okay, so another important thing I need is there exists a connection So, okay, so This is why so ah maybe so what does mean? I mean it means Okay, really it means that there exists connection means probably I should put it and write down the Meaning there's a Map from OB drama to Oh big drama Service Omega X. So Omega X, which is just the the pullback of the usual ship on the outside Okay Okay, so now Really the theorem comes from the following two propositions Proposition one if I consider E I which is direct push derived to push forward of This sheaf oh by the way if I have a at our local system on X I can pull back to X pro it out to get a local system there so I if I just do this push forward, this is a vector bundle and Proposition two If you consider its stock I mean, oh, sorry, it's fiber and one point Let's see if the X is the residue of X then this is exactly I think consider the H I of the Galois Cormolage of This L X bar turns you draw Okay, so Come to get these two propositions would imply the theorem Because I mean you get a vector bundle if it's a rank and one stock is the correct rank Rank is correct everywhere. So so in fact to Proof Proposition to I mean, I just need a zero in Proposition to but to prove it you need to use all the eyes in proposition one Just consider you zero is not enough But to prove so I mean to prove this is vector bundle The crucial observation is because it has a connection coming from the connection on OB drama so So it's really a vector bundle with the connection Integrable connection connection is the integral connection. So to prove it's a vector bundle You just need to prove it's a coherent shift. So that's I think the main observation One of the main observation. So it's a finance it's a finance problem to prove some shift Which is a probably huge. This is a huge ring and this pro etal covers huge cover But what do you get the something is finite? But anyway, what so if you look at the what do you really end up with if you from the Cube local system you get a vector bundle with connection. So It's it's kind of a remote Hilbert thing, but the only problem is that it may not have the correct rank But if it has correct rank at one point, it's correct. So it's basically the remote Hilbert Okay, so I what I really need to Move this No I guess no H2 right so, okay, so I really need to prove this the coherence shift That's not difficult, but Yeah, I mean if I have time I may be just commenting that but the key crucial thing is a proof Proposition one. So so yeah, probably like all B drama is much larger than B drama at the Beginning so you need to do something But okay, so so see to prove it's a coherence shift It's really a local statement. So I I can I can assume like axis. I Mean I can assume axis like this This is my X right now so So then the the starting point how do how to calculate some direct image so then the starting point is it's really this Business of like this perfect toys thing appears is you first the the The if you just the HR I The I derived push forward When you evaluate at you, which is there This is really can be calculated by some Galois combo, which Otherwise, you don't know how to calculate anything You just so I mean this really follows from I guess This is implied by a because there's no higher Comology when you restrict Restrict this shift to all infinite hand when you restrict But how do you know that the evaluation of you of our I Newell star Is the same as the ice come or so you could also consider the The the ice comology of you is coefficient in the drive on your star, which is more natural. So probably well so What I don't understand you can consider our new love star in the direct category Evaluate it on you and take the ice And this is seems to be more related to the right hand side Oh Sheifies the right-hand side So the right-hand side is a pressure for certification is our I Newell star and you have to prove that it is actually the shift It's something else is I think I at the end I prove the right-hand side the I think I just I guess I should prove the right-hand side is a shift directly. Okay, so Okay, so now I want to What I need to show, okay, so so really I want to show the following so this is a Hi will be drop Is a finite a module and the compatible with at our base compatible with space change a It's just Yes, this is what I this is what I mean. Yeah, so Alright, okay so What With a top is change if you have a top is change. It's a module. Then you turn service a over B turns over speed over a you get what? Okay So, uh Yeah, so I Yes Here is because I just Okay, so Right. So now I use the fact that there's a really a filtration on this ship So what I really need to show is I after I take the associate grade it I get what I want and the the Comergy for Associate if the let me just write it down. So what I really want is This guy. So it's a want Hi of Gamma on the grow I Be drawn maybe turn service well Grow J different This is a finite a module compatible with space change and Vanishes J is not enough Isn't this hot state? Hmm. This girl J. I'll be deep drama. It's in that. I think it's open Yeah, I think that should be over you said even without proof of theorem But I mean, I didn't know it's a you know, this guy wouldn't may not be a really a vector problem Griffith's transfer. Yeah, what do you get a different sense by Griffith's transfer said, okay, so This is what I want. So the problem is how to calculate this But let me just remind you let me just denote the let me write M infinity hat as I evaluate this this shift At you infinity hat, right, which Court according to this expression. So what do you really want to calculate? just it's a So what the really this guy It's really just a hi gamma head actually the one better okay, so So here comes the second K-ingredients or to kick J this yeah, of course, I need to This yes, of course Okay, so here's the main ingredients I need. Let me first formulate it in a In a in a way that Appearing the work of catalyze you but I later on I will explain the also in another way. So so here's the theorem So for sufficiently This is so for some sufficiently Large and there exists finite Projective a m ma sub module. So So I guess remind you a n is just the ring of functions on you and inside Such that stable under the action of gamma and induces this This is gamma equivalent isomorphism and The hi of gamma of the quotient is zero for any Okay, so this is a statement We use but let me explain it in a in a in another way which more Oh Why get the V so you you said you should get things which are like rank one back modules of a OX roof When you take this filtration, yes, yes, and now how So is this the filtration you look at or the filtration on yes, I just wrote Which relative to the powers of the kernel? Yes, yes, but when I take yeah, but when I you take a T inverse it becomes this It you it this V appears. I mean it's it it was in Peter's paper You invert You you take the filtration and be OB OB drum plus according to the kernel filtration Then you after you invert T you get something larger. Yeah. Yes Yes Yeah, that's something So, uh, all right, so let me I have both 10 minutes, I guess 12 minutes so Let me get to start the point is in fact this theorem this part is some how stringing the This patek Simpson correspondence first to consider by faultings, but also Abes and the growth so so what let me just understand in that content. So see I Cannot because This question is local so I can assume Let me assume L This is I mean I start with a Zp local system. That's good. So this is small Which means L module of some p to alpha is a trivial for Alpha big there Is this Bunt Okay, so in this situation what the in fact what I mean I just really need a very simple part of that correspondence Simpsons Simpson correspondence, which is in fact that there exists a unique Mk finite projected a k module with a Linear action of the geometric fundamental group such that I'm having infinity. It's isomorphic to mk tensor a k of What infinity Gamma geometric Affirmative and then in addition the the common the common knowledge The tri of gamma geometric Mk is the same thing as a tri gamma M infinity this is just a very Very I guess Very basic version I need so basically if you just take and just mention if you take the log of theta That gives you the hex field and that's I guess I let me ignore the tape twist. So that's a that's a Simpson correspondence and this statement says that the the hexco homology Hexco module also calculate to the Galois homology here. So this is a something you need so Therefore, what do you so So then you Reduce then one can show if you just consider the hi of gamma geometric Jim on m hat infinity of this V So now I can ignore the psychotomic character for a moment. This is gonna be there's a map natural map of mk V And this map is gonna be isomorphic this one can show this easily So really I want to calculate the hi of this guy, which is it's good It's becomes smaller this it's much smaller so uh Okay, so now the geometric fundamental group Gamma Jim is really simple. It's just a topological group generated by one element of gamma. So You you just do the following lemma You just do the following lemma For a linear algebra, which is good exercise so so let m be a q vector space with Gamma is a automorphism then you have this let me write m Generized invariant subspace to be just those m such that the gamma minus one And it's zero for some a large enough. Just a generalized eigen space Just one one then you see What do you get is the following then Ah, yeah, and let me consider define an action Gamma act on m actual into variable V as just as before m MV Goes to gamma m V plus one. Let's see hi Hi Okay, so it's just the part you tensor those two actions when it's on M and once on Q V with gamma X by shift So basically the the algebra acts as derivation then right the Gamma invariance of this guy is just a m generalized invariant and the if M equals m Generized invariant So if it's all the space of generating eigen space the gamma coin variant is If I remember correctly, it's also It's zero Okay, so the linear Little in exercising linear algebra. So now what I really need is the following simple but crucial observation and MK Generized invariant This equals MK So you can now you can apply therefore you can apply this lemma to the situation this lemma tells you this guy would be just MK If I is zero zero if I bigger than zero then you Reduce to calculate certain Galois co-module for the arithmetic Galois group Which is kind of I mean the finalness is kind of easy there. So this lemma I think this is interesting It's really equipment to see the hex field Interesting. It's new potent Really get the new potent hex field, but this is not sub not really quite surprising namely original the periodic simple something correspondence is a local system and gives you a hex field But now my local system L is not just a local system on X capital K It's a really a local system defined over this X over little K. So there's the action of arithmetic Galois group So that for in fact the fourth is the the hex field is new potent. Basically, you just check because of Let me write a generator here as delta Delta because of this delta when you base change to K is isomorphic to L Based change to K. It comes from arithmetic fundamental group local system on X little K. This would force this Emma K with this hex field It's isomorphic to MK of chi Delta theta maybe my plus or minus I can't remember What? then That means the characteristic polynomial of the hex field must be I mean all the coefficients of the characteristic polynomial Hicks field must vanish. So it's new potent. So this is exactly the Classic proof by like Simpson when he proves if a local system support a complex variation of heart structure, right? It It at his end it's a Why but you are proving that theta is an important thing as you said I'm sorry to correspond to variation of our structure. Yes this case it corresponds also to correspond Really? Okay. The whole state of the local system. Is it the run? I don't it's drama No, but the whole system is there The whole local system is drama See, oh, maybe I don't know Yeah, maybe I don't know but I mean I just want I think this music fact that If the local system really comes from arithmetic it has a hex field is new potent It's argument for the local monty. Oh, yeah, I said pulling that the breakfast and the characters Yes, yeah, okay, so okay, I guess from here you get this then it's The wrist argument is it's not. Yeah, I think I probably stop here I Mean it's not really assume see in this theorem of collage do they pass to some a n which is really Go to something make it small