
kaalathe anumaanam

samkhyaasampradaayathilum beejaganita jyaamithiyilum, tettu anumaanam joonn tettinte 1963 leeii oru anumaanamaan, athu kooduthal kanakkukoottaavunna maattamillaatha, ettal cohomology galois praathinidhyatthinte adisthaanathil vaividhyathilulla beejaganita chakrangale vivarikkum. beejaganita chakrangalude sidhaanthathile oru kendra prasnamaanu anumaanam. hodge anumaanathinte ganitha analogaayi ithine kanakkakkaam.
anumaanatthinte prasthavana.
vii" ennathu oru feeld "keey" nuu mukalilulla minusamarnna projective inamaayirikkatte, athu athinte prime feeldil parimithamayi srishtikkappedunnu. k" kall "k" yude verthirikkaavunna closser ayirikkatte, koodaathe "g" ennathu "k" yude kevala galois group gall ("k" s/"k") ayirikkatte. k" yill invertable aaya oru prime nambar l shariyaakkuka. v" muthal "k" vare adisthaana vipuleekaranathinte ell-aadik cohomology grouppukal (ell-aadik poornnasamkhyakalile gunakangal, scalerukal pinneedu ell-aadik samkhyakal ql lake vyaapippichirikkunnu) pariganikkuka; eee grouppukal "g" yude praathinidyangalaanu. eathenkilum "i" ≤0 nuu, "v"-nte oru codimension-"i" upavibhagam ("k"-ll nirvachichathaayi manassilakkunnu) cohomology grouppinte oru ghadakam nirnnayikkunnu.
ithu "jii" kondu nischayikkappedunnu. ivide ql ("i") "i" th tettu twistine soochippikkunnu, athaayath galois grouppaaya "g"-nte eee praathinidhyam cyclotomic pratheekatthinte "i" th power upayogichu tensor cheythirikkunnu.
galois group "jii" nischayicha "dablyu" yude "dablyu" "jii" enna subpace "vii" yude codimension-"aii" upavibhaagangalude klaasukal kondu oru ql-vector space aayi vyaapichukidakkunnave tettu conjecture prasthaavikkunnu. oru beejaganita chakram ennaal upavarggangalude parimithamaaya rekheeya samyojanam ennaanu ardhamaakkunnathu; athinaal "w" g "-nte oro gadakavum ql gunakangalulla" v "-leeii oru beejaganita chakratthinte klaasaanu ennathaanu thulyamaaya oru prasthavana.
ariyappedunna kesukal.
devicerukalkkulla tettu conjecture (codimension 1-nte beejaganita chakrangal) oru pradhana thuranna prasnamaanu. udaaharanathinu, "f": "x" "c" ennathu oru parimithamaaya feeldinu mukalilulla minusamarnna projective uparithalathil ninnu minusamarnna projective vakrathilekkulla oru roopeekaranamaayirikkatte. k" ("c") enna function feeldinu mukalilulla oru vakramaaya "f" nte pothuvaaya fiber "f" "k" ("c") nuu mukalilulla minusamaarnnathaanennu karuthuka. appol "x" ennathil vibhajakangalkkulla tettu anumaanam "f" enna jacobian inathinte birch, swinnardan-dier anumaanathinu thulyamaanu. ithinu vipareethamaayi, eathenkilum minusamarnna sankeernamaaya projective inathe vibhajichavarkkulla hodge anumaanam ariyappedunnu (lefshetts (1,1)-sidhaantham).
abelian enangalile vibhaajakangalkk tettu anumaanam sariyaanu ennathaanu orupakshe ariyappedunna ettavum pradhanappetta cases. parimithamaaya feeldukalilulla abelian enangalkkulla tettinteyum nambar feeldukalilulla abelian enangalkkulla falittintinteyum sidhaanthamaanithu, ithu mordell anumaanatthinte faalittins parihaaratthinte bhagamaanu. sarhin eee phalangal parimithamayi srishtikkappetta eathu adisthaana mekhalayilekkum vyapippichu. abelian enangalekkurichulla bhaajakangalude tettu anumaanam soochippikkunnathu "c" 1 × vakrangalude eathenkilum gunanafalathil bhaajakangalude tettu anumaanamaan... × "sii" enn.
abelian enangale vibhajichirikkunnavarkkula (ariyappedunna) tettu anumaanam abelian enangal thammilulla homomorphysangalekkurillu sakthamaaya prasthaavanaykk thulyamaanu. athaayath, parimithamayi srishtikkappetta "k" feeldil "a", "b" ennee abelian enangalkku, prakrithi bhoopadam
ithu oru isomorphisamaanu. prathyekichum, oru abelian inam "eiii" ennathu athinte tettu module "echu" 1 ("eiii" "keey" s, isal) leeii galois praathinidhyam vazhi aisojeni vare nirnnayikkappedunnu.
2 allatha savisheshathakalulla parimithamayi srishtikkappetta feeldukalil k3 uparithalangalkkum tettu anumaanam baadhakamaan. (oru uparithalathil, anumaanathinte nissaaramallaattha bhagam divaisarukalekkurichaanu.) swabhaavasavisheshathakalaaya poojyathil, keey 3 uparithalangalude tettu anumaanam andrey tankivum theliyichu. 2 allatha parimithamaaya feeldukalilulla k3 uparithalangalkk, nigard, oogas, chaals, madapusi peraa, moulik ennivar tettu anumaanam theliyichu.
 sarvekalkku ariyappedunna tettu anumaanathinte kesukal.
bandhapetta oohaapohangal.
parimithamayi srishtikkappetta "k" feeldil "x" oru minusamarnna projective inamaayirikkatte. x"-nte ell-aadik cohomology galois group "g" = gal ("k" s/"k")-nte praathinidhyam ardha lalithamaanennu (athaayath, irreducible praathinidhyangalude nerittulla thuka) ardha lalithamaanennu ardha lalithatha anumaanam pravachikkunnu. 0 enna swabhaavathinte "k" ennathinu, tettu anumaanam (mukalil paranjathupole) athinte ardha lalithathaye soochippikkunnuvennu kaanichu.
k" enna kramatthinte parimithamaaya "q" ennathinu, tettu anumaanavum ardha-lalithamaaya anumaanavum sakthamaaya tettu anumaanathe soochippikkumennu tettu kaanichu, athaayath "t" = "q" − "j" ennathil seetta function "z" ("x", "t") nte dhruvathinte kramam "j" modulo samkhyaa thulyathayude beejaganita chakrangalude groupinte rankinu thulyamaanu.
hodge anumaanam pole, tettu anumaanam beejaganita chakrangalekkurichulla grothendeekkinte staanderd anumanangalil bhooribhaagatheyum soochippikkum. athaayath, ithu lefshetts staanderd anumaanathe soochippikkum (lefshetts isomorphisathinte vipareethathe oru beejaganita correspondence nirvachikkunnu); diagonalinte kunneth gadakangal beejaganithamaanu; beejaganita chakrangalude samkhyaa thulyathayum homological thulyathayum onnuthanneyaanu.

