Vinberg's monoid and automorphic L-functions


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Vinberg's monoid and automorphic L-functions
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Bâo Chảu Ngô
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Institut des Hautes Études Scientifiques (IHÉS)
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Presentation of a group Multiplication sign 1 (number) Parameter (computer programming) Grothendieck topology Morphismus Thermodynamisches System Lecture/Conference Interface (chemistry) Integer Rational function Curve fitting Social class Physical system Area Focus (optics) Sigma-algebra Infinity Hecke operator Ring (mathematics) Limit (category theory) Functional (mathematics) Bilinear form Sequence Local Group Vector space Mathematics Duality (mathematics) Universe (mathematics) Phase transition Right angle Family
Axiom of choice Group action Multiplication sign Function (mathematics) Weight Mereology Food energy Group representation Mathematics Determinant Fiber (mathematics) Physical system Area Spacetime Theory of relativity Basis (linear algebra) Functional (mathematics) Bilinear form Measurement Sequence Substitute good Principle of maximum entropy Category of being Arithmetic mean Graph coloring Order (biology) Moving average Right angle Identical particles Point (geometry) Identifiability Presentation of a group Variety (linguistics) Student's t-test Surgery Rule of inference Theory Power (physics) Prime ideal Goodness of fit Lecture/Conference Natural number Condition number Series (mathematics) Addition Focus (optics) Physical law Division (mathematics) Mathematical model Local Group Equivalence relation Field extension Voting Spring (hydrology) Algebraic closure Universe (mathematics) Game theory Object (grammar) Film editing Family Local ring
Principal ideal Axiom of choice Group action State of matter Correspondence (mathematics) Sheaf (mathematics) Weight Mereology Food energy Proper map Subset Group representation Sign (mathematics) Length Fiber (mathematics) Physical system Spacetime Moment (mathematics) Sampling (statistics) Price index Functional (mathematics) Bilinear form Hand fan Wallpaper group Right angle Point (geometry) Computer programming Presentation of a group Similarity (geometry) Prime ideal Morphismus Lecture/Conference Well-formed formula Energy level Condition number Addition Series (mathematics) Statistical hypothesis testing Focus (optics) Physical law Mathematical analysis Equivalence relation Local Group Voting Network topology Universe (mathematics) Fiber bundle Local ring Spectrum (functional analysis)
Point (geometry) Equaliser (mathematics) Water vapor Complete metric space Mereology Morphismus Frequency Group representation Many-sorted logic Lecture/Conference Well-formed formula Subtraction Area Focus (optics) Spacetime Consistency Moment (mathematics) Basis (linear algebra) Incidence algebra Functional (mathematics) Bilinear form Subgroup Category of being Summation Funktor Network topology Quotient Equation Moving average Object (grammar) Fiber bundle Family Resultant
Musical ensemble Spacetime Variety (linguistics) Multiplication sign Moment (mathematics) Bilinear form Rule of inference Functional (mathematics) Vibration Frequency Goodness of fit Strategy game Lecture/Conference Meeting/Interview Term (mathematics) Right angle Family Local ring Resultant
Transformation (genetics) Variety (linguistics) Direction (geometry) Mereology Distance Event horizon Lecture/Conference Natural number Well-formed formula Subtraction Units of measurement Stability theory Area Series (mathematics) Spacetime Theory of relativity Conservation of energy Element (mathematics) Moment (mathematics) Infinity Volume (thermodynamics) Incidence algebra Bilinear form Functional (mathematics) Local Group Arithmetic mean Lattice (order) Angle Logic Right angle Figurate number Coefficient Family
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and a few wines at them when at this condition the dispute will you look at the role role in the end of the year so that the government present condition is if accrual to the central jail in become the added to commit from GM the central injury to the GM sentiment during the so the condition overall restitution and his identity Angier admits that his agenda said which meets so the glass if this is to be the case then when you could see manner see mental then and that the city to then you get the innovation to their spouse to In this situation clearly this condition could imply that was the siren Siam also said that could transform the the choice of the cement and power he's supported by support for and is included in the nature the injury if is such that evaluation of determinant of the infrequently it's all there to have these support so then we have no trouble 2 to make some of them please this sense as functions all while so it should be obvious to the left and right invented by the muscle of cobalt but it is not impossible support OK so that we have the the main topic should be try to understand this function sigh genetically and also he Anderton tend to EU didn't chase form so that the mantle of always would both these functions as best for the Transformers so that I could affect later so flustered money just now come back to the East many historically these funds from the the familiar media because when Aegean D and roll is suspended presentation the dawn groups then side he didn't go out there is the function of GMD can tap with nature of the of 1st basically just a cut is he thought of interconnected so long and this is the costs of the honor this the single student days with us from the simple many another trader has been kind to related to classically had to do with GSP and new some standard interpretations that was to conclude by said that the and they had nearly Port Stephens In the welcome mat nor any other amendment on the face of the he's going to have to example they understand customer told me that and there is some missteps the conclusions the main entrance to the peoples of billion the In this case in the locomotive of colorful of something that is sure to multiply whenever OK not all of the 1st point to mention these injured only would like to understand what would play the role of these places of which was to construct misplaced in it for me to roll space that could too much of the CCS is going well this season 1 so there is someone with associated with the so this really constructive this area of the facility but he actually mentioned that the theory of mind and it would normally was not start mending the resistance right now that people that went the church and so it can be found annualized was vastly resists supports furious critical varieties and that had to do so song wanted to do is functions intimately but if if you do not understand the whole thing yet but let the committee could look at this stage ,comma and whether the the the the thing I would to use his authority to the altered so long in this series of Malik convenient tool not 250 but to Fiji Prime energy of the override so if it's something simple and forcibly entered as a substitute connected the prime any look good and look good consider all couples of G & amp lethargy a symbol of the new museum is an extension of the tallest lies 50 pride and enemies isn't it could vary amending this and not more qualified to provide a committee of the sea and that means outside on the left and the right to shield itself can be extended to have been under the assumption this up to the equivalent of the Grammys measure but more so than the left and right up to the extent of modification on which the some voters all this spring I think if goal if start of his not just big may be true the other 2 would speak Thomas this would not be in inappropriate Jews notices you know he gene-speak passengers picked on when these can be made that the practice picking his nose that significant regional and this to which I have tilted so long so there is something of technical but important east of what the meaning of calling the a billion as they shouldn't so we look at this matter G Nova castles and went to pay a key is selected the game some and so the regular phone from an M which is
unified on the left and the right to buy the submissive political where Smith and the model it's good to be said that the flat the list of billionaires relations if in his flat pleasure but reduced little fiber and the rest of us out into the city to assume this is going to look at this 1 week for which had been addition he's a very flat in this case have been booked full the basis of major union properly so we think that the intervals for me match that's not true to more complicated for him to which isn't it had been generally not complicated related to wonderful property fication Anderson simple and that I think the important point is when when after league called the fenced-off and thing the 1st of the new someone leaked it to make clear that he said I had to do with the Michelangelo OK so well construction so in this case of Gators sufficient could be in the because somebody's feet With Finland available only if there is no reason at this is for him to so after a support provided by the venue they have often of the deep and can be political find so some of them all at the time of the common Torre said of of of the Skagit eastern sky have mixed between the common political system of your pride and committed the book would writing to some kind of tricky common interests the focus so so that me consult 1 of the small into more than that of the universe someone so Liberty prime meat and not someone told us that he prime ncprize in the center of the prime the ticket for the press to be she prime time to buy more the I'm up taking that and so "quotation mark their heaviest this in-depth sequence to the to the plot to kill but the persist now is to provide more the prime In this become the borrowers about joining groups and entities that lead to the heart in this matter is even by supporters of his 6 more and have simple that the the identifier the muscle dollars abandoning hope the support of the year if so of course these you have about fire and fight extend to the people now you have no role by so little Miguel 1 on the part needle prominent in the weight of the prime and then there's this fundamental representations of the prime who went to jail the eyes of law Hi this weight and you understand this is all I always eat less and demeanor to try and end up the unique for only 1 way to do it to 2 brothers formed a look complicated mitigated and trousseau animal edition that too when he tried this wonderful so that this sort of right his so to let go this and bluster and this rule blows and then you had this effect from Peru to most forms of Hitler's troops pulled out of the and for the local beauty and eye which consists of all this the embedded into some of the people of this morning would Madison surgery and blessed is the fight to be closure this this Easter and and and this is the closure of G plus carrying small uncharacteristic it may happen that impressed not long war in which case you want to normalize its that this is not a good thing that the people to under I so what being people for use of this season in Europe so Unisys system for every flat knowing it is can be "quotation mark didn't my major change the form which he left in place that means you have forfeited just aims of the head and less willing to play plus the the Upton upon up object E-Plus if you want to construct and another mourner it just look good to some did some good character some home from different to to person that can be extended to cleaner Moffett applauded politely for later plus the innovation and that the Easter Mississippi Pennsylvania look so now I want to take 1 minute explain why this had anything to do with it With I kind of pride and got on the composition because having to do because if you want that for example if you energy To be stating the land that day maybe emaciated wide conditions for if you want to look with the Union of fallen lavender pride in less than an hour by the in order you want to ride conditions and although the actual use on the fundamental way to division 2 to write down by the grave Genentech to will my and now if you want to let deciding what really want these days that role most of which is what to client under and you know use absolutely in the temple thanks to this condition become Canada's something become entangled .period the more to just recently that what achieve is how it could be
used conditions that complicated some level of course said Gandhi expressed and engine which increases intervals .period but that hesitate to do with the width the couldn't have known about it ,comma come back to the health functions sales function is to Medicaid to do with the farther you have this to mediate in the 2 Citigroup at the more I mean in Alfonsin Davies so correspond some it's a wonderful GM here it would put a white tend to see that the end of this had to be some but to the local characters of the blast in the furthest expand two-way wanted million the dominance of the System some Member Goldman Coker and call this the which 1 that the tools that have to this 1 to be the highest wit the so when you start a and will tested heaviest weight so it's splattered the construction to get them on and that is what we want in end this southeastern said this morning some distant and general so then become involved in this has been the best-selling paper brunt of my intention to my work what I just explained to me that before we see that the the subsiding this somewhat size subset of the support of society's visit my basic this equivalence is indeed to check the support of society is exactly that always addition couldn't contain always what calls is GFS Catholics and called for so the support of this function he took that it's a good point of some more this the construction OK now 1 this small at you can say what we have the support wanted the function so that is the she's fully competent at main numerically this could call these quiet was something called the complicated because nations follower and indicated comports policy which could complicate business well I mean I really want to win a complete 7 1 2 be overlooks offices so that put some there was about Gundotra that led dealt blows of them to be the Jets face and so solidified by the fact that point on this is the point in his value in the form of series on them character 1 day I was going on and we ask for the future that I see in this on a on the 1st and well he's can we decide that I and we don't know this summer but predicted to adjust by you go back destinations England and in winter when the duration of the program has missiles that .period of the this year solely for that reason displaced in locally Lisa pulled out of some defending national titles from 1 so it is that it is locally so whether call for a formal locally uses through formula was brought on financial and international but that it said in understands the pool of these as its history well there's a little softer natural visitors who might miss stop it appeared in the functioning in life and and focus Tokyo but it's OK so harmony as it is known to me this the DIC the function of the trace of wilderness these His son who did what so at least I mean I don't understand this little corner space mid-term and ionizing this is what I see as well as the clear to me yet but lags many sound something that they had to have has seen now Scioscia said I see at least on helpless 2 but think that it was OK so now is a good look at them World War moral of this story Brokers analysis CBO um small was collected because of all the financing and soda someone at 20 Prime Media principal energy on plans so what call what these more is the form just from additional impairment he and his party you mean nothing but a most section the global sections of and twist on the left Mike Keenan trees on the right by the try I think you could think of at the moment but it is up to demand that the ruling their victim and soulful if you have given someone to begin the final in animal feeds this is going to have to notify you bundles if you can get em have too often left and the right and twisted medical board it also someone would have been composed manner so always should look at some some fans but the such that genetically the the energy that is present the and as a law office of abundance then you want it shut down governing put on the school board I don't know of any of these groups based he's no this place of the pair of in the way he is a member of the city and the he's mn formed he did not to be so it genetically Olazabal I'm telling you not to be a long beyond achievement and this heavenly morphism too madam in isolation settled this matter to some of the same thing all fall EGM for a so when he gets eighties at at the space of prepares this man to the
space of the affairs of the failed then the the higher hits the fail In the end for the Ellison that bundle on Friday 1 section of the political so these amazing creatures union not 0 of C'mon C'mon protective devices on the coast so often and so on so that it could be CNN has some text possibly different ball into the King and extended it matches here and and it is good that we have a extended thinking there in the case of indeed contusion lending and feet attributed patient edition on July the 1st paper dramatically and that antithesis is concerned and so we based it it is not difficult to prove that the I C of the space between world was based from so OK so so if the largest again the case assignment for the AP I know see I this year .period the item of duplicity and then go home went to restrict that see effects to the fibers of me so I could call this Massachusetts since ascending path the AP minus 1 of the cases In Good to be could eye of sample books chief Dayananda of pain and I 6 Aesop overlooks chief correspondent presentations and that will be questioned Indiana into 1 and then you .period collectively submitted so this matter under this this fact histamine approved can be proven by the simple Franceville wallpaper so we this function must somehow I made at fault and maybe to totally stupid but I think that the guy who is the interesting point is if you even if you can identify mean reluctant either of the vaccine contigent locally this looks like a long that means you can obviously man X 2 states guarantee man locally Stewart said of all the most more than you be constructed to like to to the record this season and so will we see that the choice of on the high seas of effect the full direct Of the holdup function the scientists said the group would so that it gave me this for me indication that this would be the ICT admits it's exciting "quotation mark missed met this is not to this movement of this movement going in the form of a movie if you view of the adviser discussed reform .period here and in New York part of forms as disease and that this was only Goldstein is really easy and in compliance with all humility because the truth that for some things as about something we all have to do this is that the universe was Wallstreet device which was 1 point from his farm workers look at so well most of the songs on the CD does seem to have that he has a surprise is omitted interpretations so did there is what did he want to do with this function and so on I so they still at the end of sculpted proposal of the clemency so it now local Moffett representation fight to put the propriety of being this could tip preservation notable space to the beginning the 2 choices needed if it's a central character also assisted in the death of Phoenix would fit in the somewhat to just to make the space has her fine Ivorians and still pass on this 1 but it didn't patient home In and I'm afraid the way then you can write that down the face of a role pride in inventories of "quotation mark not upside you spawned by concern determined to the test so it's tempting to come around on to put the trainees to but the Chinese put up on the side of the pond according to his son and I would be a so up some of the function well did these consultants home when he didn't fall within the colorful I wonder if he seems to have been injured in the hope that the understood Spector has signed a suspect inside you understood the the the the the the right hand side is by and I plan the on right side the spectrum salary of medical blood B and finished understand that they can be can be proven that not converge on up to the vote could I of the chip would have presented the 1st wanted to find a percentage I was going to vote the company just some or all of the the world Paul but really but I don't know I expected on the on the domestic side if he sin mysterious but we do expect that the latest sign actually basic pay of the May similar construction began expanding the later in
the year of Dyson 1 less space that within what with so they won't want to do with this point and this so using tasteful followed booking was to the 1 function would like to have also in space of this function to put the trees from so sold I why don't we do suffice it I think there's a lot we want to order like to have some say the Spanish fast-paced depending on growing under all around me under of the place and this sort space have very specific in admitted that it could also cited the :colon that clique that would replace the pages of gays and you have this basis functions combat support a small quantity of and that this guy roller to 0 and then handed a copy of the form of a handful every rolled at passing still at its finest and as a brother and cousin problem this would be the Strokes based depending on goal With this magic from the water solid the Svoboda 0 procedural regional grow the space hallway and that all men of the West this summer camp short space couldn't that depending on roll all were doing by with which you always these before the completion of the new nor the rest of the attributed the detection and thank you and so did many function and that makes sense to me to make the rescue work I know that true the you that was more political than the 1 1 we and adjusting the dead man as just should have some kind of trace formula that applies depending on goal OK so here I think this attempted to the devoting up for to this push by a host of government cash now and then later as you push a button on a full who I would like to prove fossil equation of these analysts rookie my using the trades followed on Duma trees from a on list of this and so there is need to be so there no the resume at a exercise of some function in this place of Freeview which she Sequent oops I almost on place fetuses scrap some function and there are 1 0 like to to go to the tale of 2 1 man assess role and you would like to go to the space of S for the continent interval check so that there should be no it heard something Genoa forms and between the shock space and the equality of genetics I would look like opossum summation so from a single wall has given candidate for for the space defenders and quickly that is 1 proposal so there's another problem for is getting on the same day that was just that is what is to come so if you do that is not only a result of that is so pleased to be here for all that is bullish said Yukon is often the most of whom are you guessed right now are also in army yeah so pulled up over the inside the easier .period upset the form ozone and so on and just under the Fullerton following that the 2 days the the committee consistent OK so now are the of the new endoscope the institution known in Yamoussoukro she's is thought there he has a citadel originally not looking at the fossil equation he was more interested in the lobby of the that appears In this to our attention and we hope that all of them we know that the in the pulling functoriality East Molesey equivalent to partition Holkham offered representation in true into different backs back for the subgroup of the world that from the last part of it and they knew which is that you want to put the plantation depending on on those all of the ports getting room so controlled the poll that appeared on the right side would have to tell you what to put in different them so that these women they feed on what would like to do that and as always like to try to the genetic with these complicated functions to pull what to prove on the hospital said but of course dead he he said the intimidating because demand for the political consultants and Uganda made Nevada defied the space to start with as the moment to leave consistently lower to go to this when he said I think that this is a mental problem areas because list this year they have very selective interpretation of the lieutenant says and that it needs of this something that had been trying to do for them for a couple of years without much success to the fact that focus on now we had this if it poignancy and he also tools of American farmers here and to the center of the law so that you can twist can twist g bundled by the adviser went so innovative the bungalow can do cleaning of to just pull out the verdict about and twist by when incident OK so now the quotient this year most of years in the 82 dizzying you now corresponding to the category of of G bodyguard but then he decided the to that lead to the as amorphous and encourages jubilant adjusting mother convict abundantly can decide decided the he even as Moffitt and would be interested or you no when you look at the Duma to solve our article also used his Boyd the easy to guess at the to look at an amorphous of bundled over bundles so we look at aspects of em he's a space of so it so late he to be an object in this day and that the panel morphism of the bill payment or morphisms tolerated families the key of the command from the truly 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can decide to I couldn't stand to listen to music is very similar to the preparations this is mn I most the time OK so so 1 meant to used so the city Federation has come mn and to and they and using the program and joint cautioned that the moral and beautiful and all forms of 2 and pollution this women less brittle and Wilfredo and was the used and because they have to bear the costs same is this this cautioned to welcome on way parent and you want to start to study the Duma chase following a using the strategy Federation and the fullest .period 1 of the 1st issue Clinton the taste of it it would I them in the head of the list I see because it at will without Schieffelin got very very well but they want to be I see this family you just wear them I see this man would be the right function good it's not the same in the space of excess that defied earlier but this Christensen to be the results my resume or merge them into faggots OK so for now 1 the tho again this well well and then of course you but there must have speculation because even vibration when you when you start to always Templeton call all had the words space is a lot of problems but but you can never happen would part has been studied extensively and image sold by a local rule moment should from political venison is not is not known but it takes at least expect expect that the this is a completely new terms unless course others in the fight this criminal myself filling from of this is covered a bit of variety then the say that the answer is very complicated meet Genevieve Simard is simply to provide a sense of form that it is invitation of modal me on that this 1 of the new underwriting has so you would think that there a global rosier and and that somehow what they're interested in the it is tasteful ESA somehow Kieth should be the locomotive political communicators snowplows the scope and so that is what
happens and of course there's not a whole
fully and look beyond endoscopic undertook pre-eminence Gandhi and translating the question of stability composure bridge that had a lot the producers so the richest start muted and there seemed to be 1 the thinking please maybe it is question and this is you Over the years the was 1 of the matter is that he is looking again on it so Vermont and a into 2 and a half and look at that distance To do this and an open so the man from and these the basic and international getting the right to put a thing you can think about is that this the family of home To covering a projected Gulf British firms like great so so that is the demand a divided on this occasion devised then something better but when when the device becomes of them to proceed but the before that then divided the committed pitifully you have nice Emilia covering to have bid varieties Joakim bids and have a temptation of the Parliament of group on the issue 1 of that abilities so that very well-known the percentage of the bread hopefully the vocal on the issue I'm a bit of abilities and in that case at the unknown pulled some steps stability stories about women go to infinity but but it up developed of Bostick enough there should be some whistleblowing was in a situation the figures on holiday resort was and is on the phone if I were you out of a the unit volume grew you do you told if you want to do give you had to do with me while I was reading this is the Fed formula on this side of the then last man and and what kind of something that that could you will have the during the year before her home in there is always an we extend my whole thing is this the move saying In the area of the world the EU is whole at a I mean is that they believe that there is still a very easily relations between them Honda and analysts itself I mean In the telescope eateries anti-sodomy this discovery and the wonderful grace and you have met between this short space and then the edges has popped to close a letter to alone in generally seem to be rather know that transforms here as the transforms that if you if you know the Indigo note I under just 1 element of the event 1 that of the Hall of Fame at meetings in traffic they which falls in but in the this and no underscore P up .period that with his size and extended their lead she's the piece is not the moment to go on above the hole in the side it's new and more more precisely some constituting into the behaviors so it is it's not enemies not deleted didn't want to try to have any beside and this core moral G that how angle to infinity and different companies have been too different and this is his precise gate can give his answers fall don't thank you for of that we in so far only the people of the world that he knew he so far as to say that calls for an answer from all parts of the jungle if you're into his vehicle rolled recently with Morgen logical wishes influence answer the it seems on nose and healthful from the consignor none of involved Health ocean the faulty machines the boy and his gaffes in particular coefficient tried to some of that mn is a mean the simplest case like a truncated the series to some of the and you end up to to some which and how how it ghost season from other people motherboards and that what we want to continue to compare different will and that should be blessed incidents precise value the question is on course you would like to make your own motives this certification is so Of course if you know anything about smaller local groups normally in the 1st 2 months of this workstation here so for example who has the courage this is what for the internationalization of the same political transformation little natural form this or similar and so well here you only need to know about the fate from of 1 of the functions and which is public principally with the some of someplace had put other functions that I would have concessional returned to the court directions at the end of the year the performance of the congestion of and had been at least about 2 and a friend of conservative forms it started important to the ruble information and they're going to have a mechanic from the Tennessee but I think followed the board question is not relinquished much of we were questioned and


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