
langlands-shahidi reethi

ganithashaasthrathil, langlands-shahidi reethi oru nambar feeldil connectucheytha reductive grouppukalumaayi undaakunna pala kesukalilum automorphic ell-functionukal nirvachikkunnathinulla marggangal nalkunnu. pothuvaaya rekheeya grouppukalude cuspidal automorphic praathinidyangalkkaayulla rankin-selberg uthppannangal ithil ulppedunnu. eee reethi praadeshika gunakam enna sidhaantham vikasippikkunnu, ithu aisanstain seerees vazhi aagola sidhaanthavumaayi bandhippikkunnu. thathfalamaayundaakunna "l"-functionukal oru pradhana functional samavaakyam ulppede niravadhi vishakalana gunangal thripthippeduthunnu.
praadeshika gunakam.
oru praadeshika feeldil "f" ennu nirvachichirikkunna oru levi subgroup "m"-noppam bandhippicha ardha-vibhajana reductive group "g"-nte saamaanyathayilaanu krameekaranam. udaaharanathinu, "g" = "gl" ennathu "l" raankinte oru classical grouppaanengil, athinte paramavadhi levi upavibhagangal "gl" ("m") × "gn" roopathilullavayaanu, ivide "gn" ennathu "n" raankinte oru classical group "gl", "l" = "m" + "n" ennathinte athe tharathilumullathaanu. eph. m (f)" ennathinte irreducible generic praathinidyangalkkaayi praadeshika gunakam enna sidhaantham shahidi vikasippikkunnu. pothuvaaya praathinidyangalil ninnu parabolic induction vazhi labhikkuna praathinidyangalkkaayi intertwining oopparettarmaarude sidhaanthavumaayi jodiyakkiya vittekkar modalukalude savisheshatha property vazhiyaan praadeshika gunakam nirvachikkappedunnathu.
laamglaandsinte ainstain seereesinte sidhaanthathinte pravartthana samavaakyathil prathyakshappedunna aagola intertwining oopparettare praadeshika intertwining oopparettarmaarude ulppannamaayi vighadippikkaan kazhiyum. m" oru maximal levi subgroup aayirikkumbol, uchithamayi thiranjeduttha aisanstain shreniyude forrier kofifishyantukalil ninnu praadeshika coffificientukal undaakukayum bhagika "l"-functionukalude gunakam ulppedunna oru crude functional samavaakyam thripthippeduthukayum cheiyunnu.
praadeshika gadakangalum pravartthana samavaakyavum.
oru induction ghattam aagolathalathil pothuvaaya oru cuspidal automorphic praathinidhya formulayude crude functional samavaakyathe bhagika "l"-functionukaludeyum γ-gadakangaludeyum vyakthigatha functional samavaakyangalilekku parishkarikkunnu
vishadaamshangal saankethikamaanu "s" oru sankeernamaaya variable "," s "" s "nuu purathulla" v "ennathinu nirakkaatha formula _ 3 ulla parimithamaaya sthalangalude oru koottam (adisthaana aagola feeldinte), koodaathe formula _ 4 ennathu laamglaandukalude eratta grouppaaya" g "yude oru prathyeka upavibhaagathinte sankeernamaaya lye algibrayile" m "-nte anubandha pravarthanamaanu. g" prathyeka rekheeya grouppaaya slide (2) aayirikkumbol, "m" = "t" diagonal maatriksukalude paramavadhi torres aayirikkumbol, π oru grassenturector anubandha γ-gadakangalum tettinte theesisinte praadeshika gadakangalaanu.
pravartthana samavaakyathile avayude pankum parabolic inductionumaayi bandhappett gunanakshamatha ulppedeyulla praadeshika gunangalude oru patikayum γ-gadakangalude savishesha savisheshathayaanu. v" oru archemedian local feeld nalkumbozho "v" archemedian allaathathum formula _ 3 "m (f)" enna ann-raphid principal seerees praathinidhyatthinte gadakavumaayirikkumbozho artin ell-functionukalum artin root nambarukalum ulppedunna oru bandham avaa thripthippeduthunnu. praadeshika "l"-functionukalum root nambarukalum εformula _ 6 umm formula _ 7 ulppede ellah sthalangalilum "p"-adic grouppukalkkulla langlands vargeekaranathiloode nirvachikkappedunnu. pravartthana samavaakyam roopam edukkunnu
ivide formula _ 9 umm formula _ 10 umm poortthiyaaya aagola "l"-function root nambarumaanu.
automorphic "l"-functionukalude udaaharanagal.
langlands-shahidi ell-functionukalude oru poornna pattika ardha-vibhajana grouppaaya "jii", maximal levi subgroup "emm" ennivaye aasrayichirikkunnu. kooduthal vyakthamaayi paranjaal, adjayint aakshan formula _ 4-nte vighadanathe dainkin diagramukal upayogichu tharamthirikkaam. ainstain seerees sidhaanthathiloode automorphic "ell"-functionukalekkurichulla aadya patanam laamglaandukalude euler uthppannangalil kaanam, automorphic praathinidyangal ellaayidathum nirathillaathathaanu enna anumaanathinu keezhil. langlands-shahidi reethi nalkunnathu "l"-functionukaludeyum root nambarukaludeyum nirvachanamaan, oru vittekkar modalinte nilanilpp aavasyappedukayallaathe "m" prathinidheekarikkunnathil mattoru vyavasthayumilla.
l"-functionukalude vishakalana gunangal.
aagola "ell"-pravartthanangal thripthikaramaanenkil avaa nallathaanennu parayappedunnu
langlands-shahidi "ell"-functionukal functional samavaakyathe thripthippeduthunnu. lambamaya strippukalile athirthi nirnnayathilekkulla purogathi s-ll cheythu. ezz. jelbart eph. shahidi. koodaathe, valare vishalamaaya kathaapaathrangalaal valachodikkalukal ulppeduthiya sesham, langlands-shahidi "ell"-pravartthanangal poornnamaakunnu.
l"-functionukal aprathyakshamaakathirikku ennathaanu mattoru phalam. pothuvaaya rekheeya grouppukalude rankin-selberg uthppannangalkk oro yadhaartha samkhyaykkum 't' enna soothravaakyam poojyamallennu ithu prasthaavikkunnu.
fangtoriyaalittiyilekkum "pii"-edic grouppukalude praathinidhya sidhaanthathilekkum prayogangal.
ivide, unitory parabolic induction vazhi formula _ 22 labikkum