artin ell-function

ganithashaasthrathil, oru galois group "jii" yude oru rekheeya praathinidhyavumaayi bandhapetta oru tharam diriclet sreniyaanu artin "ell"-function. klaas feeld sidhaanthathekkurichulla thante gaveshanavumaayi bandhappett 1923 ll emil artin aanu eee pravartthanangal avatharippichathu. avayude adisthaana gunangal, prathyekichu chuvade vivarichirikkunna artin anumaanam, eluppamulla thelivine prathirodhikkunnuvennu thelinju. nirddishta noonn-abilin klaas feeld sidhaanthathinte lakshyangalilonn artin "ell"-functionukalude sankeernavum vishakalanaparavumaya swabhaavathe automorphic roopangalum langlands prograam nalkunnathupole oru valia chattakkoodilekku ulppeduthuka ennathaanu. ithuvare, atharamoru sidhaanthathinte oru cheria bhagam maathrame uracha adisthaanathil nalkiyittulloo.
nirvachanam.
oru parimitha-dimensional complexes vekttar space formula formula _ 2-nte praathinidhyamaaya formula _ 1 nalkiyirikkunna formula, formula _ 2 ennathu nambar feeldukalude parimitha ekstenshan formula _ 5-nte galois grouppaanu, artin formula _ 6 function formula _ 7 oru euler ulppannathaal nirvachikkappedunnu. formula 9-nte poornnasamkhyakalude valayathile oro prime ideal formulakkum, oru euler factor undu, ithu formula 8 formula 11-ll antaamifai cheytha saahacharyathil nirvachikkaan eluppamaanu (mikkavarum ellah formulakalkkum sariyaanu). anganeyaanengil, frobinius moolaka formula _ 13 formula _ 2 leeii oru konjugasi klasai nirvachikkappedunnu. athinaal, formula _ 15-nte swabhaavasavisheshatha polynomial nannayi nirvachikkappettittundu. formula _ 16-nte euler ghadakam swabhaavasavisheshathayulla polynomialinte cheria parishkkaranamaan, thulyamaayi nannayi nirvachikkappettathaan.
t" leeii yukthisahamaya functionaayi, formula _ 18 ll vilayiruthunnu, formula _ 19 saadhaarana riman seetta function notteshanile oru sankeernna variablaanu. (ivide "n" ennathu oru aidiyalinte feeld maanadandamaanu.)
soothravaakyam _ 16 ramify cheyyukayum "i" ennathu "g" yude upavibhagamaaya jadatwa grouppaayirikkukayum cheyyumbol, samanamaaya oru nirmmaanam prayogikkunnu, pakshe "v" nte upavibhaagathilekku "i" kondu sthiramaayi (pooyint thirichaayi).
eee gadakangalude ellah prime ideal formulakalude ananthamaaya ulppannamaanu artin ell-function formula _ 7. artin resiprosity kaanikkunnath pole, "g" oru abelian grooppaayirikkumbol eee "l"-functionukalkku randaamathe vivaranamundu ("k" yukthisahamaya samkhya feeld aayirikkumbol diriclet "l"-functionukal, pothuve heckey "l"-functionukal). noonn-abelian "jii" yum avayude praathinidyangalumaanu puthumayilekku varunnath.
oru prayogam deduckind seetta-functionukalude factorisationukal nalkuka ennathaanu, udaaharanathinu, yukthisahamaya samkhyakalkku mukalilulla oru nambar feeldinte kaaryatthil. pathiv praathinidhyam irreducible praathinidyangalaayi vighadippikkunnathinu anusrithamaayi, atharam oru seetta function "jii" yude oro irreducible praathinidhyathinum artin "ell"-functionukalude ulppannamaayi vibhajikkunnu. udaaharanathinu, "g" ennathu moonnu aksharangalile samamiti grooppaayirikkumbozhanu ettavum lalithamaaya cases. jii" kku digri 2-nte irreducible praathinidhyam ullathinaal, atharam oru praathinidhyathinulla oru artin "ell"-function, square, atharam oru nambar feeldinaayulla dedeckind seetta functionte factorisationil, riman seetta function (nissara praathinidhyathinaayi) signature praathinidhyathinaayi diriklettinte tharatthinte "ell"-function ulla oru uthppannathil sambhavikkunnu.
formula _ 5 nuu kooduthal krithyamaayi digri "n" yude galois ekstenshan, factorisation
athilninnu
saadhaarana praathinidhyatthile irreducible praathinidhyatthinte gunithamaan formula _ 29, "f" ennathu formula _ 30-nte kramavum "n" ennathu ramified primukalil "n/e" upayogichu maattisthaapikkappedunnu.
aksharangal klaas functionukalude oorthoonoormal adisthaanamaayathinaal, formula _ 31-nte chila vishakalana gunangal kaanichathinu sesham ganitha purogathikalekkurichulla dirichlettinte sidhaanthathinte saamaanyavalkkaranamaayi namukk chebotarev saandrata sidhaantham labikkum.
pravartthana samavaakyam.
artin ell-functionukal oru functional samavaakyam thripthippeduthunnu. function formula _ 31 athinte moolyangalil formula _ 33 maayi bandhappettirikkunnu, ivide formula _ 34 sankeernamaaya konjugettu praathinidhyathe soochippikkunnu. kooduthal krithyamaayi paranjaal "l" ennathinu pakaram formula _ 35 aanu, athaayath "l" neey chila gamma gadakangalaal gunichaal, thudarnnu meromorphic functionukalude oru samavaakyam undu.
kevala moolyam 1 ulla oru prathyeka sankeernna samkhyayaaya "w" (ρ) ulla ithu artin root nambaraanu. rund tharatthilulla gunangalekkurichu ithu aazhathil padikkappettittund. onnamathaayi, roobert laamglaandukalum piyari dillign langlands-dilligne local sthiraangangalaayi oru factorisation sthaapichu; automorphic praathinidyangalumaayulla anumana bandhangalumayi bandhappett ithu pradhaanamaan. ρ, ρ * enniva thulyamaaya praathinidhyangalude kaaryatthilum, pravartthana samavaakyathin oro vashathum ore l-function ulla onnaanu. beejaganithaparamaayi paranjaal, ρ oru yadhaartha praathinidhyamo quaternionic praathinidhyamo aayirikkumbozhaanu ithu sambhavikkunnath. appol artin root nambar + 1 allengil −1 aanu. eathu adayaalam sambhavikkunnu enna chodyam galois module sidhaanthavumaayi bandhappettirikkunnu.
artin anumaanam.
artin ell-functionukalekkurichulla artin anumaanam parayunnath, oru nissaaramallaattha irreducible praathinidhyatthinte artin ell-function formula _ 31 muzhuvan sankeernna thalatthilum vishakalanaparamaan ennaanu.
ithu ekamaana praathinidyangalkku perukettathaanu, ell-functionukal pinneedu heck pratheekangalumaayi bandhappettirikkunnu-prathyekichu diriclet ell-functionukalkku. oru dimensional praathinidyangalil ninnu prachodippikkappedunna ellah praathinidyangalkkum artin anumaanam sariyaanennu kooduthal pothuve artin kaanichu. galois group supersolvable allengil kooduthal saadhaaranayaayi monomial aanenkil, ellah praathinidyangalum eee roopathilullathinaal artin anumaanam nilanilkkunnu.
function feeldukalude kaaryatthil andrey weyl artin anumaanam theliyichu.
dvimaana praathinidhyangale imej subgroupinte swabhaavam anusarichu tharamthirikkunnu athu chaakrikamo dihedral, tetrahedral, octahedral allengil icosahedral aakam. chaakrika allengil dihedral kesinulla artin anumaanam erichu hekkinte srishtiyil ninnu eluppathil pinthudarunnu. tetrahedral cases theliyikkaan langlands bays chenchu lifting upayogichu, jerold tunnel octahedral cases ulkkollaan thante jolie vipuleekarichu; andrew wiles modularity anumaanathinte thelivaayi eee kesukal upayogichu. richard taylor mattullavarum (pariharikkaanaavaattha) icosahedral kesil kurachu purogathi kaivarichittundu; ithu gaveshanathinte sajeeva mekhalayaanu. projective imej subgroup pariganikkathe, serinte modularity anumaanatthinte thelivukalil ninnaan vichithravum irreducible aaya dvimaana praathinidyangalkkaayulla artin anumaanam pinthudarunnathu.
induced pratheekangalekkurichulla brouverinte sidhaantham soochippikkunnathu ellah artin ell-functionukalum heckey ell-functionukalude positteev, negatteevu avibhaajya shakthikalude uthppannangalaanennum athinaal sankeernamaaya thalatthil meromorphic aanennum soochippikkunnu.
 ellah formulakalkkum gl (enn)-nulla automorphic praathinidyangalumaayi bandhapetta ell-functionukalumaayi bandhapetta langlands thathwashaasthrathil ninnulla sakthamaaya phalangalil ninnaan artin anumaanam pinthudarunnathennu choondikkaatti. kooduthal krithyamaayi paranjaal, langlands anumanangal adalik group gln ("eiii" kyu)-nte automorphic praathinidhyathe galois grouppinte oro "enn"-dimensional irreducible praathinidhyavumaayi bandhappeduthunnu, ithu galois praathinidhyam irreducible aanenkil oru kuspidal praathinidyamaanu, athaayath galois praathinidhyatthinte artin ell-function automorphic praathinidhyatthinte automorphic ell-functionu thulyamaanu. cuspidal automorphic praathinidhyangalude ell-functionukal holomorphic aanenna ariyappedunna vasthuthayil ninnu artin anumaanam udan pinthudarunnu. laamglaandukalude pravarthanatthinulla pradhana prachodanangalilonnaayirunnu ithu.
anumaanathinte ardham.
durbalamaaya oru anumaanam (chilappol dedeckind anumaanam ennariyappedunnu) ingane parayunnu
m"/"k" ennathu nambar feeldukalude vipuleekaranamaanenkil, appol koshyant
avarude dedeckind seetta functionukalude formula _ 39 poornnamaanu.
m"/"k" galois aanenkil eee anumaanam nilanilkkumennu aramatta-brower sidhaantham prasthaavikkunnu.
kooduthal pothuve, "k" nuu mukalilulla "m" nte galois closser "n" ayirikkatte,
koodaathe "g" ennathu "n"/"k" enna galois grouppaanu.
koshyant formula _ 39 thulyamaanu
swaabhaavika praathinidhyavumaayi bandhapetta artin ell-functionukal
k"-inverients complexes "m" ulcherkkunnathil "g" yude pravartthanam. athinaal artin anumaanam dedeckined anumaanathe soochippikkunnu.
jii" ennathu pariharikkaavunna oru grooppaayirikkumbol, koji uchida aarum swathanthramaayi eee anumaanam theliyichu. dablyu. 1975ll vaan derr vaal.

