 So today I will talk about the local PRX Simpson correspondence and so originally the Simpson correspondence states some correspondence between the representation of fundamental groups and some Higgs bundles for projective smooth variety over a complex field and this analog is established by Houttings in 2005 in his paper and so there are two so there are local theory and also global theory and for global theory we need to assume that the variety is a curve and in this talk I will talk only about local theory and local means that we deal with certain affine open sub-scheme of variety and studies some relations between representation and Higgs bundles and graphically speaking the local theory compares the generalized small generalized representations and small Higgs bundles and I will explain a later more precisely the definition of these two objects and Houttings also constructs this factor in both sides from the right hand side to left hand side and also left to the right and actually so some so maybe some isn't so the argument written in that paper by Houttings has some difficulty to understand and this is pointed by this and so in this talks I will explain what's the problem and then how to resolve the problem yeah away I will explain how to solve that problem okay so to explain the theory I will introduce some notation okay so so that k denote complete discrete variation and mixed characters and sub-k denotes a ring of integers and little k denotes residue field okay and also pi is the uniformizer and k bar and algebraic approach okay as usual okay and and also for an okay module so let mk or mk bar and m hat denote just m tensor k and mk bar is just this one and so m hat is the higher completion or purely completion okay and so then throughout this talk I will consider the following algebra the semi-stable algebra over this and so this is it so this is a semi-stable algebra over okay and assumes the following so some to smallness condition first condition is that for any index set of this coordinate the close-up scheme defined by this index that is the ideal generated by PI and I contains I is is irreducible actually I don't use this explicitly I don't explain how this is useful I don't explain but I will be precise definition so speck of a I assume that also speck of a is connected and also the special fiber is so this is a condition this is a variety our or algebra we consider okay so then so yeah so I consider the geometric fundamental group of the generic fiber but I also need some more notation so so so this k denotes the field of fraction okay so curly k k bar is this algebraic closure containing the algebraic closure of the base base field and okay so then we we consider the finite sub extension so finite and an AL denotes the integral closure of A in L denotes A sub L so then I define the following set which which is L such that which consists of L such that this AL becomes a tau over A after inverting okay so we sometimes write this by the a trip so this means that we often add rock structure to a by this D coordinates and then this is a log trivial because of as a ring ring of look trivial because of speck of a and so this is the set and and that k unrampified is just a union of L okay so this is usual notation and so then I always note by delta the I'm sorry for a lot of notation so this is the same as some fundamental group of this log trivial with the base point given by this algebraic closure okay so now and so a bar of ah yes yes thank you so geometric fundamental yes or yeah yes yes and so then this is a union of AL so this is the one can also say that this is just in key for closure of A in in this field okay so now we can consider the definition we can define the small generative presentation the small way you are simplifying some of us because when you tens of escape bar so when you dress the field is not algebraically close but I'm sorry so yeah yeah yeah yeah you can have a ramified extension you're right you're right so I I just forgot to say thank you very much okay so okay so now we are ready to define maybe it's better to change okay so first I will explain the definition of this one so I define the category of generalized representation so and I denote it by this an object is simply the finally a finite projective A bar hat k module so this is I use the convention so hat means the pilot completion and sub k means just the instant k module is semi linear so delta is the color group of that and so delta naturally acts on a bar and also a bar hat and a bar hat k so the so this means that the action is semi linear with that action okay so the morphism is as usual so that's a bar a bar hat k linear map in the delta okay and so and I have to define a smallness so for for a rational number positive rational number we define okay so yes I'm sorry yeah I think otherwise you can't or maybe if you add some topology by using some a bar so then you can input anyway yeah okay and so for for such representation I define a b small or alpha small if exist and certain lattice that is finally generated a bar hat module such that the first condition is just if you tensor k so it gives v itself and the second condition yeah is that v bar v mod p to the alpha so this is some abuse of notation this is some fine maybe so so maybe this is some yeah element in in k bar which has some suitable variation or maybe some taking some roots of p and just you find a rational power yes and so then the second condition is that this is or maybe here I don't need so this is generated by delta invariant element so this definition is a little bit complicated because the reason is that yeah we are considering a semi-linear representation so it is not easy to define what is trivial representation so here is in something saying that module of p to the alpha this becomes a trivial representation yeah so this is a definition of smallness but assume that the thing is nice that like a free module no I don't assume so that's the actual difficulty in in the theory the difficulty comes from that yeah of course if we can replace this with such thing so maybe my lecture is not necessary to solve the problem actually so anyway so the we so the smallness is just defined by so for exist alpha which is bigger than two over p minus one such that so this is the the definition of the left hand side I wrote okay it's easier to do you know it's okay but it's that was old but okay but maybe it's better to use okay so now uh so I so so he expand expand on the side and so I also to state the theorem precisely I miss also need to introduce some notation so omega is just just this is easy so this is the logarithmic differential module so this is just a free module with generator p log t i but here I actually I made a mistake so maybe so I should remove one coordinate so maybe I should start with two so it's a little bit okay okay so then I introduce the and that's that's of course a kind of period rings for p i for history this case to define precisely yes usually this is to know it by a inf but I omit in yeah so this is the use this is the usually the starting point of the various period rings or in period history and so and because I denote this one so this is an element in here and so and p under bar is just without the compatible system the reduction mode pi of the p to the n through of p and n so this is an element in here and so bracket means the time you know it and so then uh it is known that this is isomorphic or c or c is just the completion of the okay bar okay bar is the integral in the ring of k bar and and the second graded question is the pre-oc module so this is well known and and I introduce the following notation xi inverse omega is just the home of o c of a okay bar module of xi a okay bar to omega one so this is free a module so this is just uh so if you fix the base so then this is just the same as omega one but I just avoid to choose some anyway actually this has a basic xi so if you fix then this is the same as omega uh omega so now uh we are ready to define the category of fix bundle and so this is uh no no I needed to introduce more notation a zero so this is notation come from counting so this is the base change of okay to okay bar yeah and fix bundle is defined on the general the purely completion maybe precisely so yeah so the purely completion and in breaking period and so this is the object is just uh finite regenerated uh a hat k module uh project and uh plus uh so I didn't take completion so here I think I could go with a k bar so uh so this is uh linear uh such that this uh kind of integrity condition so this is a use just more reason this maybe it's uh as usual just a linear map uh compatible is uh okay so then okay so and then uh I similar ideas uh generalized representation I explained before I want to define the smallness uh the smallness is uh oh here okay so for fix bundle for on a zero hat k uh I define uh m theta is up and upper is supposed to be right number uh upper small if if similarly so it so exists a finitely generated a hat k module such that so this is contained yeah and uh I'm sorry so so this generated m okay so now we can state a theorem ah thank you so the same as uh no not same all right thank you uh so this is small if uh exist alpha uh which is bigger than one over p minus one such that uh and and and now our theorem states that this one thing I correct some argument so I didn't introduce the notation for small but I just write small here so exist an equivalence of category between the two categories omega and uh actually so it depends so actually so this depends dog smooth lifting of a zero hat over a two a two is a okay bar when you're on the square uh yes yes uh yes yes you're right okay so now so I will explain what's the problem yeah in the construction of the center from right to a zero to c yes extended to o c instead of okay ah yes yes without completion just a extension okay all those things depend only on the form of completion along the special fiber so you should yes maybe so it's enough to okay so now I have two remarks or maybe votes are related so one remark is that here I work with rational coefficients so but also gp theory for finite free ever had representation so just replace everything ever had just remove some in some sense remove the subscript k and also the smallness in this case is defined uh just using some taking uh no we don't need to take a lattice just yeah just the assumptions yeah yeah yeah uh you one can take some good basis such as that model of p to the r for it becomes action becomes true but then it could be that there is a that it is even in the integral theory you can always find the general projective modules so of course the risky locally they are free but even when you define this locally trivial so of course look there is another but uh so maybe the difference is not so big but still you could have a case in the integral case where the definition of small are not exactly the same no no no right right right right yes it's different so maybe so that's the main difficulty yeah actually yeah so yeah actually so starting to find a few p theory by actually considering uh something but uh it's not some exact sequence such that uh three zero and v one are uh this one so finite ever representation of delta and uh it seems that he claimed that one can show that v zero and v one also both are small but yeah it's easy to make v zero small but for we can see we can see if we can see if so that's a very simple problem so yes the idea yeah his idea is just take uh yeah this kind of lattice and take a generator and then lift it to give v zero and then uh consider a subjective map and here we have a zp theory and and he I think he I'm not sure if whether he explicitly write down this v one but he yes so and then but here we don't have a zp we cannot apply zp theory so that's a simple problem and uh so in in our book with Ahmed and Michelle Grower yeah we so yeah it's discussed from that approach but we cannot solve from by this method yeah we have some equivalence equivalent uh yeah yeah condition to for for even to be small but it's not easy so can you say so in your book uh so the I did not look exactly but what is the result which is so we don't go through smallness we go through another condition which is for us this either don't go or it's an admissibility condition and it's enough to build the theory but to have that this admissibility is equivalent to small is equivalent to exactly this statement because of this you have a okay so there is one application which is trivial and the other application is a consequence of what the application is explaining to you so you go through the admissibility yeah so yeah in that terminology the result is that the admissibility is equivalent to the smallest yeah yeah yeah admissibility with the right representation are the same as the small representation that's okay so now I will try to explain how to yeah prove this theorem yes and to prove that I use a generalized sensory and I fix some notation and the idea so I fix some an underbar so this is a set of uh some uh top of positive integers and I think no I don't need six and I introduce a infinity a and bar so this is integral closer a in the following subfield of k and ramified so this is defined by this and so this is abuse of notation this is joining all p p power roots of ti so this is uh adjoining all uh roots of ti uh by uh some p power some p power multiplied by ni and then I define gamma n bar as the automorphism algebra over a zero so you are 10 bigger than zero dc no d minus c uh d minus c okay so let me see I have to ah yeah so I'm sorry I also need to introduce more so maybe ah I made a mistake so yeah so I also need some finite level for this so here I an m m bar is m1 m2 ah I'm sorry so notation is not good but I have to start this too and so here so I also needed a finite level version so this is for so this is replaced by so minus in minus infinity is replaced by minus mi and also this is I'm sorry for and then I need the following ring which is the union of the periodic completion of this of this finite level ring and this is contained in the periodic completion of this one this is a bit delicate this is a completing completion on the finite level and take union so this is the completion of the actual taking union so this is okay so this is the case uh so this is this must consider this must be and no no horizontal divisor so d is equal to c so so this actually so n n bar does not appear in there because this is the n n bar comes from the horizontal device so the first statement is that so for so this generalized representation so there exists n bar in under bar as above such and there exists a similar representation of gamma n bar with coefficients in this tilder ring such that after expansion of scalar so we get so the proof of this is just first first we use the farthing's almost set up theory to to distance to some kind of a infinity but actually if you have a so for some divisor along a horizontal divisor horizontal structure so we have to add some some roots of coordinate for some number also prime to be also to to kill their amplification so here that's because n under bar remains and I don't know maybe it's not true without n we need to take some rules of the horizontal coordinates of horizontal divisor this is the first claim and the second claim is that this we can further descent to the case n bar is zero if g is small so this is for alpha small so here I introduce notation infinity is just an infinity zero under bar zero under bar is just a no one one is okay and so and gamma is gamma and large and the statement is if it's this term for this integer such that every this alpha small then the infinity which is just defined by the fixed bar of the scalar group is a representation of gamma in a tilde so a into the infinity is also defined similarly they're just tilde and we can further descent so but this is for any alpha so alpha if so this is so we don't need alpha is bigger than two of two over p minus one so this is if you weaker assumption is enough for this part okay so and remark is that of course I have to remind you that the extension is not is not better unless so so this course is a trouble to this course but this distance because so this is not I forgot to write you down so if you leave this is a tar so then it's quite easy just a color descent but if this so this is not a tar so this ramifies along the horizontal divisor so we have a trouble and but if you have a small as you can kill the ramification along the device so is it just prime to peer or ramification or also the p so the for for the for the prime yeah so of course it can have some but we can't take so we already take up so here we I have to take the people who so yeah okay so you're just doing just use that yeah trying to keep up yeah so this but this looks like a time problem does the plan to yes yes yes so okay so it's just yeah it's not difficult yeah we need some argument and also okay I hope also just in so this gamma is of course asymptomatic to this uh free go uh group so this is a free group over over zp uh free free module over free module over yes and so now so this representation is much easier to handle than the original v so not the for the you may zp to the zp to the d i don't look at the zp one to the d without the product is a product or yes I know I I yeah I also should say here that in essence uh the alpha is alpha prime small for alpha right okay so for then I get the same same I I can't okay so then uh we also have an action of week so the action is in something the analytic so we have an action of if this so this is actually that's all for the sense of operation okay so then I will try to explain how to how to use this theorem to prove the for the original theorem okay so now yeah in some sense what we did is descent from this one or maybe an idea is just a trivialized action if you kill the action of the gamma so then the action of gamma uh continues with respect to the discrete apology so I guess we just then apply the kind of descent but actually this this is has again a trouble because we have ramification along the horizontal divisor so and actually if you allow horizontal divisor we need to have some time to explain okay so then uh what is the period ring actually for our correspondence we have a period ring over the a bar hat k actually so we have uh exist and certain period ring so our is uh or sequence of your dreams I mean so here we have so this is an a bar hat algebra with an action of uh here we have an action of uh delta and also we have a theta here so this is and uh and this has a following property so this is a period ring for local since some correspondence so so the invariant part is just this algebra the a bar hat k which is a coefficient ring of no I'm sorry so this is from coefficient ring of this field and uh if you take the theta zero part you get the coefficient ring of the generalized representation and then uh actually factor is constructed is actually using this ring uh and uh so just if you're given the small expand also just the corresponding representation is given by cancelling uh m and takes the theta equal to zero part so this is as usual a construction in the period closed theory and so r should be sufficiently big and uh if you write this b then if you write this representation as b so then uh so so the m can be reconstructed using this b uh and uh here the ring so the invariant hat so so the problem is that in some sense if you're given a small representation then this model is large enough yes this is the first one is that it's a not is balanced the coefficients is n the source of d is now the map d is so it starts from a not oh thank you yes and so then uh uh so my strategy is to be given a small representation and construct a candidate of m uh directly using uh sensory so okay so then uh here's the fourth position and so if you are given a representation of gamma representation of gamma a okay then uh okay so i yeah i think i should say so i fix uh basis of this one uh gp and basic by one so i i fix the basis and then uh the gamma is again isomac this is and here's so this is a sense of data and uh frame is that if b is small then exist some alpha such that this can be the smallest can be interpreted in terms of so the action of the b group so this is convergence of the action of the group so this is uh and this convergence actually allows us to trivialize trivialize the action of the alpha and uh so already is that uh uh for b define b infinity b to be the alpha i should and here i actually introduce another ring sub sub ring of this one but i don't explain here i'm sorry so what the idea is that introduce some sub ring and uh some pure ring and uh groups that if you take the the trivial part of or for the action of b algebra so then uh if r is big enough so this recovered uh the representation and then uh the purple and so now i the proof is something i said b is again the as the infinity i'm sorry i should right ah no i'm sorry i so this is the infinity then i start with small and then uh by uh by two we get small representation of gamma in here and then uh corollary four we get uh this one uh so infinity d r d infinity r so in my paper i own tundra puzzle uh so r is we get this one and so here action is uh discrete and then we can descent actually but actually so we need some argument for horizontal divisor but anyway so we the infinity infinity and so delta for for this one uh we have a nice one isn't like this so this is a outline so maybe so we use descent so here also we use some kind of smallness but after small enough and then we using some pure ring uh we cube the action of b algebra so then you get a discrete action but this is a uh module so this is a module projective module over a cube the infinity okay and so this is the union of finite algebra so we can do if you if this is a tau so it it's easily the descent but for for the multiplication we need some argument and argument and then we get this and so then this is a half to zero module and we are done this is it so only one direction was close the other direction is actually relatively easy yeah but because yeah one can describe the action of uh so action on a kind of yeah yeah first yeah you you can construct uh the infinity dire directly from the he expand using the spirit ring and the action is it's just given by a kind of exponential of the hex here so then i just control the ending the definition of small you use different exponents in both cases sometimes bigger than one of the p minus one and sometimes bigger than two of the minus one yes the explosion and this is a so when you go through the proofs if matches that is i don't know exactly i'm sorry i didn't explain that so this it's made in order to match it's made exactly in order to match okay and in this step that you require that you explain here yeah for do you need the do you need alpha small or you need really small for for and let me see uh for for this step yeah for here for here yes for here are you prove the convergence of uh of this reaction and then it implies that this module is big enough yeah that's the argument but i didn't explain this at all so maybe it's not easy to okay yeah so here we really need the smallness yeah and also here uh actually for here i don't need uh so far if you have started with alpha small then maybe some alpha prime bigger than alpha one can prove this okay so first thing the speaker