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| In the theory of [[automorphic form]]s, an area of mathematics, the '''fundamental lemma''' relates orbital integrals on a [[reductive group]] over a [[local field]] to stable orbital integrals on its [[endoscopic group]]s. It was conjectured by {{harvs|txt|authorlink=Robert Langlands|last=Langlands|year=1983}} in the course of developing the [[Langlands program]]. The fundamental lemma was proved by [[Gérard Laumon]] and [[Ngô Bảo Châu]] in the case of [[unitary group]]s and then by Ngô for general reductive groups, building on a series of important reductions made by [[Jean-Loup Waldspurger]] to the case of [[Lie algebras]]. ''[[Time (magazine)|Time]]'' magazine placed Ngô's proof on the list of the "Top 10 scientific discoveries of 2009".<ref>[http://www.time.com/time/specials/packages/article/0,28804,1945379_1944416_1944435,00.html Top 10 Scientific Discoveries of 2009], ''Time''</ref> In 2010 Ngô was awarded the [[Fields medal]] for this proof.
| | Adequate has been talked about meals that purportedly sets our taste buds on fire. Primarily based mostly on cutting performance and handle comfort and balance, it also things in corrosion resistance, sharpness as received, and other attributes of the 4 standard knives in the set-chef's, paring, slicing, and utility-and a matching santoku (when obtainable). In addition to finding Ratings for the most up-to-date models, you can now shop on-line working with an ad-free interface exactly where you can invest in kitchen knives in a safe-purchasing environment. The blades of these knives will have double bevel edges.<br><br><br><br>Anyway, I utilised to sharpen my high-quality kitchen knives applying sharpening stone. Quite a few persons believe that forged knives are substantially better than stamped knives. The manufacturing course of action of stamped knives has enhanced so considerably that they are now viewed as to be equal to forged knives by most experts. The paring knife is one of the most valuable of the Japanese chefs' knives that you could have in your kitchen. This knife is quite light and not difficult to use at all.<br><br>Messermeister knives, like the name sounds, are rooted in Germany—the Meridian Elite line becoming forged in the extremely very same German town as the preceding knives from the Big Two. Kitchen Knife Fundamentals ($7.95) has got all the core material from the KitchenKnifeGuru web-site, but in an quick-to-read format that only an eBook can give. Most of Global's knives are not forged, but created of a higher-good quality steel that has Best Kitchen Knives Blog been tempered and heat treated to new levels of sophistication.<br><br>I can't consider attempting to prepare meals without a decent set of knives. They use 18/ten stainless steel for their merchandise and present a lifetime warranty on all of their goods as properly. Right here are what I find to be the best ten most impressive kitchen tools produced by RÃSLE. These tongs also have a patented, simple to use locking method that enables you to open and lock them applying one hand. It mashes all kinds of vegetables and fruits, and fits completely in your hand for safe use.<br><br>But the set is economical, gets wonderful testimonials on Amazon (4.eight stars) and the bread knife is a definitely good length (the blade also has that slight curve that McDermott recommended). This set would be an great choice for a higher-site visitors kitchen, such as in a roommate predicament, or if you just don't want to spend hundreds of dollars for knives. If you adored this article so you would like to obtain more info about [http://www.thebestkitchenknivesreviews.com/ Best Kitchen Knives Made In Usa] kindly visit the web site. General, the Wüsthof Classic knives performed properly and the set appears terrific on the counter in its wood block. So that's who he sells knives to the most.<br><br>You really only have to have a few critical ones for most kitchen tasks, and the additional knives that come in sets often go unused. They make good gifts, particularly for weddings, college graduations or for any other occasion when a person may well be setting up a new kitchen. Knife sets, which tend to come with a storage block and honing Best Kitchen Knives Of 2012 steel, also look good, regardless of whether you are giving them as a gift or you want a uniform set on your counter. Global knives have a lifetime assure.<br><br>These knives are produced from one strong piece of steel, with the steel extending the entire length of the knife like into the handle. I do attempt to give fair and honest assessments when I do solution reviews so it is only correct you get the good and the bad - that mentioned, I have found it hard to obtain anything truly negative about this knife set. The good quality of the knife you use is also significant.<br><br>I wouldn't let a four-year-old youngster use a chopping knife, for instance. The earlier they understand, the superior they cooks they will be when they are grown. Getting in the kitchen can be a actual education with finding out Math Abilities and significantly far more. There are numerous excellent cooks amongst the male members of society and some handle all the cooking requires of the family. The high quality is initially class and will last a lifetime, Choose from specialty knives to 13 or 15 piece chef sets.<br><br>Individuals have different preferences, various cutting styles, different cutting duties at function - so diverse knives perform for unique folks. Considering the fact that I use these knives, Im forced to carry my old wusthof about to use on jobs that are rough on a blade. I normally use my knife mainly for just chopping so I never devote a lot on a chef knife. Major restaurant kitchen may possibly be a harsh place for a Takeda or Watanabe. Both knives have its personal pros and cons. |
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| == Motivation and history ==
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| [[Robert Langlands]] outlined a strategy for proving local and global [[Langlands conjectures]] using the [[Arthur–Selberg trace formula]], but in order for this approach to work, the geometric sides of the trace formula for different groups must be related in a particular way. This relationship takes the form of identities between [[orbital integral]]s on [[reductive group]]s ''G'' and ''H'' over a nonarchimedean [[local field]] ''F'', where the group ''H'', called an [[endoscopic group]] of ''G'', is constructed from ''G'' and some additional data.
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| The first case considered was ''G'' = ''SL''<sub>2</sub> {{harv|Labesse|Langlands|1979}}. {{harvs|txt|last1=Langlands|author2-link=Diana Shelstad|last2=Shelstad|year=1987}} then developed the general framework for the theory of endoscopic transfer and formulated specific conjectures. However, during the next two decades only partial progress was made towards proving the fundamental lemma.<ref>Kottwitz and Rogawski for ''U''<sub>3</sub>, Wadspurger for ''SL''<sub>''n''</sub>, Hales and Weissauer for ''Sp''<sub>4</sub>.</ref><ref>[http://www.newton.ac.uk/programmes/ALT/seminars/051316301.pdf Fundamental Lemma and Hitchin Fibration], Gérard Laumon, May 13, 2009</ref> Harris called it a "bottleneck limiting progress on a host of arithmetic questions".<ref>[http://www.institut.math.jussieu.fr/projets/fa/bpFiles/Introduction.pdf INTRODUCTION TO “THE STABLE TRACE FORMULA, SHIMURA VARIETIES, AND ARITHMETIC APPLICATIONS”], p. 1., Michael Harris</ref> Langlands himself, writing on the origins of endoscopy, commented: | |
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| {{cquote|... it is not the fundamental lemma as such that is critical for the analytic theory of automorphic forms and for the arithmetic of [[Shimura varieties]]; it is the stabilized (or stable) trace formula, the reduction of the trace formula itself to the stable trace formula for a group and its endoscopic groups, and the stabilization of the [[Grothendieck–Lefschetz formula]]. None of these are possible without the fundamental lemma and its absence rendered progress almost impossible for more than twenty years.<ref>[http://publications.ias.edu/rpl/series.php?series=55 publications.ias.edu<!-- Bot generated title -->]</ref>}}
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| ==Statement==
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| The fundamental lemma states that an orbital integral ''O'' for a group ''G'' is equal to a stable orbital integral ''SO'' for an endoscopic group ''H'', up to a transfer factor Δ {{harv|Nadler|2012}}:
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| :<math>SO_{\gamma_H}(1_{K_H}) = \Delta(\gamma_H,\gamma_G)O^\kappa_{\gamma_G}(1_{K_G})</math>
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| where
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| *''F'' is a local field
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| *''G'' is an unramified group defined over ''F'', in other words a quasi-split reductive group defined over ''F'' that splits over an unramified extension of ''F''
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| *''H'' is an unramified endoscopic group of ''G'' associated to κ
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| *''K''<sub>''G''</sub> and ''K''<sub>''H''</sub> are hyperspecial maximal compact subgroups of ''G'' and ''H'', which means roughly that they are the subgroups of points with coefficients in the ring of integers of ''F''.
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| *1<sub>''K''<sub>''G''</sub></sub> and 1<sub>''K''<sub>''H''</sub></sub> are the characteristic functions of ''K''<sub>''G''</sub> and ''K''<sub>''H''</sub>.
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| *Δ(γ<sub>''H''</sub>,γ<sub>''G''</sub>) is a transfer factor, a certain elementary expression depending on γ<sub>''H''</sub> and γ<sub>''G''</sub>
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| *γ<sub>''H''</sub> and γ<sub>''G''</sub> are elements of ''G'' and ''H'' representing stable conjugacy classes, such that the stable conjugacy class of ''G'' is the transfer of the stable conjugacy class of ''H''.
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| *κ is a character of the group of conjugacy classes in the stable conjugacy class of γ<sub>''G''</sub>
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| *''SO'' and ''O'' are stable orbital integrals and orbital integrals depending on their parameters.
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| == Approaches ==
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| {{harvtxt|Shelstad|1982}} proved the fundamental lemma for Archimedean fields.
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| {{harvtxt|Waldspurger|1991}} verified the fundamental lemma for general linear groups.
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| {{harvtxt|Kottwitz|1992}} and {{harvtxt|Blasius|Rogawski|1992}} verified some cases of the fundamental lemma for 3-dimensional unitary groups.
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| {{harvtxt|Hales|1997}} and {{harvtxt|Weissauer|2009}} verified the fundamental lemma for the symplectic and general symplectic groups Sp<sub>4</sub>, GSp<sub>4</sub>.
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| A paper of [[George Lusztig]] and [[David Kazhdan]] pointed out that orbital integrals could be interpreted as counting points on certain algebraic varieties over finite fields. Further, the integrals in question can be computed in a way that depends only on the residue field of ''F''; and the issue can be reduced to the Lie algebra version of the orbital integrals. Then the problem was restated in terms of the [[Springer fiber]] of algebraic groups.<ref>[http://www.claymath.org/research_award/Laumon-Ngo/laumon.pdf The Fundamental Lemma for Unitary Groups], at p. 12., Gérard Laumon</ref> The circle of ideas was connected to a [[purity conjecture]]; Laumon gave a conditional proof based on such a conjecture, for unitary groups. {{harvs|txt|last1=Laumon|last2=Ngô|year=2008}} then proved the fundamental lemma for unitary groups, using [[Hitchin fibration]] introduced by {{harvs|txt|last=Ngô|year=2006}}, which is an abstract geometric analogue of the [[Hitchin system]] of complex algebraic geometry.
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| {{harvtxt|Waldspurger|2006}} showed for Lie algebras that the function field case implies the fundamental lemma over all local fields, and {{harvtxt|Waldspurger|2008}} showed that the fundamental lemma for Lie algebras implies the fundamental lemma for groups.
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| ==Notes==
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| {{reflist}}
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| == References ==
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| *{{Citation | last1=Blasius | first1=Don | last2=Rogawski | first2=Jonathan D. | editor1-last=Langlands | editor1-first=Robert P. | editor2-last=Ramakrishnan | editor2-first=Dinakar | title=The zeta functions of Picard modular surfaces | publisher=Univ. Montréal | location=Montreal, QC | isbn=978-2-921120-08-1 | mr=1155234 | year=1992 | chapter=Fundamental lemmas for U(3) and related groups | pages=363–394}}
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| *{{citation|last=Casselman|first=W.|title=Langlands' Fundamental Lemma for SL(2)|url=http://www.math.ubc.ca/~cass/research/pdf/SL2.pdf|year=2009}}
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| *{{citation|first=Jean-François|last=Dat|url=http://www.math.univ-paris13.fr/~dat/publis/lf.pdf|title=Lemme fondamental et endoscopie, une approche géométrique, d'après Gérard Laumon et Ngô Bao Châu|publisher=[[Bourbaki seminar|Séminaire Bourbaki]], no 940|date=November 2004}}
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| *{{Citation | last1=Hales | first1=Thomas C. | title=The fundamental lemma for Sp(4) | doi=10.1090/S0002-9939-97-03546-6 | mr=1346977 | year=1997 | journal=[[Proceedings of the American Mathematical Society]] | issn=0002-9939 | volume=125 | issue=1 | pages=301–308}}
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| *{{citation|url=http://www.institut.math.jussieu.fr/projets/fa/bp0.html|editor-last=Harris|editor-first=M.|title=Stabilisation de la formule des traces, variétés de Shimura, et applications arithmétiques}}
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| *{{Citation | last1=Kazhdan | first1=David | last2=Lusztig | first2=George | title=Fixed point varieties on affine flag manifolds | doi=10.1007/BF02787119 | mr=947819 | year=1988 | journal=Israel Journal of Mathematics | issn=0021-2172 | volume=62 | issue=2 | pages=129–168}}
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| *{{Citation | last1=Kottwitz | first1=Robert E. | authorlink1=Robert Kottwitz | editor1-last=Langlands | editor1-first=Robert P. | editor2-last=Ramakrishnan | editor2-first=Dinakar | title=The zeta functions of Picard modular surfaces | publisher=Univ. Montréal | location=Montreal, QC | isbn=978-2-921120-08-1 | mr=1155233 | year=1992 | chapter=Calculation of some orbital integrals | pages=349–362}}
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| *{{Citation | last1=Labesse | first1=Jean-Pierre | last2=Langlands | first2=R. P. | title=L-indistinguishability for SL(2) | doi=10.4153/CJM-1979-070-3 | mr=540902 | year=1979 | journal=[[Canadian Journal of Mathematics]] | issn=0008-414X | volume=31 | issue=4 | pages=726–785}}
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| *{{Citation | last1=Langlands | first1=Robert P. | title=Les débuts d'une formule des traces stable | url=http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/endoscopy.html#debuts | publisher=Université de Paris VII U.E.R. de Mathématiques | location=Paris | series=Publications Mathématiques de l'Université Paris VII [Mathematical Publications of the University of Paris VII] | mr=697567 | year=1983 | volume=13}}
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| *{{Citation | last1=Langlands | first1=Robert P. | last2=Shelstad | first2=Diana | title=On the definition of transfer factors | doi=10.1007/BF01458070 | mr=909227 | year=1987 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=278 | issue=1 | pages=219–271}}
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| *{{Citation | last1=Laumon | first1=Gérard | title=International Congress of Mathematicians. Vol. II | url=http://mathunion.org/ICM/ICM2006.2/ | publisher=Eur. Math. Soc., Zürich | mr=2275603 | year=2006 | chapter=Aspects géométriques du Lemme Fondamental de Langlands-Shelstad | pages=401–419}}
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| *{{Citation | last1=Laumon | first1=Gérard | last2=Ngô | first2=Bao Châu | title=Le lemme fondamental pour les groupes unitaires | doi=10.4007/annals.2008.168.477 | mr=2434884 | year=2008 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=168 | issue=2 | pages=477–573}}
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| *{{Citation | last1=Nadler | first1=David | title=The geometric nature of the fundamental lemma | doi=10.1090/S0273-0979-2011-01342-8 | year=2012 | journal=[[Bulletin of the American Mathematical Society]] | issn=0002-9904 | volume=49 | pages= 1–50 }}
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| *{{Citation | last1=Ngô | first1=Bao Châu | title=Fibration de Hitchin et endoscopie | doi=10.1007/s00222-005-0483-7 | mr=2218781 | year=2006 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=164 | issue=2 | pages=399–453}}
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| *{{Citation | last1=Ngô | first1=Bao Châu | title=Le lemme fondamental pour les algèbres de Lie | doi=10.1007/s10240-010-0026-7 | mr=2653248 | year=2010 | journal=Institut des Hautes Études Scientifiques. Publications Mathématiques | issn=0073-8301 | volume=111 | pages=1–169}}
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| *{{Citation | last1=Shelstad | first1=Diana | title=L-indistinguishability for real groups | doi=10.1007/BF01456950 | mr=661206 | year=1982 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=259 | issue=3 | pages=385–430}}
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| *{{Citation | last1=Waldspurger | first1=Jean-Loup | title=Sur les intégrales orbitales tordues pour les groupes linéaires: un lemme fondamental | doi=10.4153/CJM-1991-049-5 | mr=1127034 | year=1991 | journal=[[Canadian Journal of Mathematics]] | issn=0008-414X | volume=43 | issue=4 | pages=852–896}}
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| *{{Citation | last1=Waldspurger | first1=Jean-Loup | title=Endoscopie et changement de caractéristique | doi=10.1017/S1474748006000041 | mr=2241929 | year=2006 | journal=Journal of the Institute of Mathematics of Jussieu. JIMJ. Journal de l'Institut de Mathématiques de Jussieu | issn=1474-7480 | volume=5 | issue=3 | pages=423–525}}
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| *{{Citation | last1=Waldspurger | first1=Jean-Loup | title=L'endoscopie tordue n'est pas si tordue | url=http://www.math.jussieu.fr/~waldspur/endoscopietordue.pdf | publisher=[[American Mathematical Society]] | location=Providence, R.I. | isbn=978-0-8218-4469-4 | mr=2418405 | year=2008 | journal=Memoirs of the American Mathematical Society | issn=0065-9266 | volume=194 | issue=908 | pages=261}}
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| *{{Citation | last1=Weissauer | first1=Rainer | title=Endoscopy for GSp(4) and the cohomology of Siegel modular threefolds | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-89305-9 | doi=10.1007/978-3-540-89306-6 | mr=2498783 | year=2009 | volume=1968}}
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| == External links ==
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| * [http://www.fields.utoronto.ca/audio/02-03/shimura/laumon/ Gerard Laumon lecture on the fundamental lemma for unitary groups]
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| * {{cite journal |last=Basken |first=Paul |url=http://chronicle.com/article/Understanding-the-Langlands/124368/ |title=Understanding the Langlands Fundamental Lemma |journal=[[The Chronicle of Higher Education]] |date=September 12, 2010}}
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| [[Category:Algebraic groups]]
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| [[Category:Automorphic forms]]
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| [[Category:Theorems in algebra]]
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| [[Category:Theorems in number theory]]
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| [[Category:Langlands program]]
| |
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