
smith-minkovsky-seagal maas formula

ganithashaasthrathil, oru janussile laattisukalude (quadratic roopangal) bhaarangalude aakethuka, avayude automorphism grouppukalude kramangalude parasparafalangalaal bharamullah oru soothravaakyamaanu smith-minkovsky-seagal maas formula (allengil minkovsky-seagal maas formula). pinda soothravaakyam palappozhum avibhaajya chathura roopangalkkaayi nalkappedunnu, ennirunnaalum eathu beejaganita samkhya feeldilum ithu chathura roopangalilekku saamaanyavalkkarikkam.
0, 1 alavukalil pinda soothravaakyam nissaaramaan, 2 alavukalil ithu adisthaanaparamaayi saangalppika chathura mandalangalkkayulla diriklettinte klaas nambar soothravaakyangalkku thulyamaanu, 3 alavukalil chila bhagika phalangal gothold aisanstain nalki. 
uyarnna alavilulla pinda soothravaakyam aadyamaayi nalkiyathu addehamaanu, ennirunnaalum addehathinte phalangal varshangalolam marannu.
ithu veendum kandethukayum minkovsky pepparile oru pishak kandethukayum thiruthukayum cheythu.
maas formulayude prasidheekaricha pala pathippukalilum pishakukalund; prathyekichum 2-adic saandrathakal sariyaakkaan prayaasamaan, koodaathe 0,1 alavukalude nissara kesukal kuranjathu 2 alavukalude kesukalil ninnu vyathyasthamaanennu chilappol marakkunnu. 
 integral quadratic roopangalkkaayulla maas formulayude krithyamaaya prasthavana oru exposity akkoundum nalkuka, ithu vishwasaneeyamaanu, kaaranam avar ithu dhaaraalam vyakthamaaya kesukalil parishodhikkunnu.
pinda soothravaakyathinte sameepakaala thelivukalkkayi kaanuka.
smith-minkovsky-seagal pinda soothravaakyam adisthaanaparamaayi weyl-seagal soothravaakyathinte sthiramaaya padamaanu.
pinda soothravaakyathinte prasthavana.
f" oru "n"-dimensional positteev dephinit integral quadratic roopamaanengil (allengil laattis) appol pindam
athinte januss ingane nirvachikkappedunnu
f" enna athe janussile ellah samagra asamathva roopangalilum thuka ullidathu, oottoo (λ) ennathu λ-nte automorphism grouppaanu. 
n" ≤2 nuu pindam nalkunnathu ingane ennu paranjukondu nalkia pinda soothravaakyathinte roopam
ivide "m" p "(" f ") ennathu" f "-nte" p "-maas aanu,
mathiyaya valia "r"-nuu, "p" s "ennathu" f "-nte nirnnaayakatthe vibhajikkunna" p "-nte ettavum uyarnna sakthiyaanu. n" ("p" r ") enna samkhya" n "bye" n "maatriksukalude samkhyayaanu.
"x" poornnasamkhyakalulla gunakangal modaaya "p" r 
ivide "a" ennathu "f"-nte gram matricks aanu, allengil mattoru vidhathil paranjaal automorphism grouppinte kramam mooad "p" r "kuraykkunnu.
chila rachayithakkal pinda soothravaakyam "pii"-adic saandrathayude adisthaanathil prasthaavikkunnu.
pakaram "pii"-maas. p"-maas "f" enna punarnirmmaanatthinu keezhil maattamillaathathaanu, ennaal "p"-saandrata anganeyalla.
0 allengil 1 enna alavilulla (nissara) kesukalil pinda soothravaakyathinu chila mattangal aavasyamaanu. munnilulla 2 enna ghadakam prathyeka oorthoogoonal grouppinte tamagawa nambarine prathinidheekarikkunnu, athu 0,1 alavukalil 1 maathramaan. koodaathe "m" p "(" f ") nuu munnilulla 2 enna ghadakam oorthoogoonal groupile prathyeka oorthoogoonal grouppinte soochikaye prathinidheekarikkunnu, ithu 0 alavukalil 1 maathramaan.
pindatthinte vilayiruthal.
pinda soothravaakyam ellah primukalkkum mukalilulla ananthamaaya ulppannamaayi pindam nalkunnu. ithu inipparayunna reethiyil oru parimitha ulppannamaayi mattiyezhuthaam. parimithamaaya samkhyakal ozhike (2 ditt ("f") vibhajikkaathava) ellaavarkkum "p"-maas "m" "p" ("f") staanderd p-maas stad "p" ("f") nuu thulyamaanu,
p" 2 ditt ("f") yeei harikkukayaanenkil randaamathe variyile legender chinnathe 0 ennu vyaakhyaanikkunnu.
ellah "p"-pindangalkkum avayude staanderd moolyam undengil, motham pindam
staanderd maas
evide
oru eratta poornnasamkhyakalude riman seetta functionte moolyangal bernauli samkhyakalude adisthaanathil nalkiyirikkunnu
athinaal "f" yude pindam yukthisahamaya samkhyakalude parimithamaaya gunanafalamaayi nalkunnu
pii"-maassinte vilayiruthal.
eph" enna roopathinu pii-aadik jordan vighadanamundengil
q" ennathu "p" enna shakthikaliloodeyum "f" q "ennathinu" p "enna nirnnayaka prime" n "(" q ") enna alavum ullidathu, 
appol "pii"-maas nalkunnu
ivide "n" (ii) ennathu type 2, "p" = 2 ennivayude ellah jordan gadakangaludeyum alavukalude koottamaanu, koodaathe "n" (i, i) ennathu aduthulla gadakangalaaya "f" q "," f "2" q "ennivayude motham jodikalude ennamaanu.
m" p "(" f "q") enna gadakathe diagonal ghadakam ennu vilikkunnu, ithu "p" gadakangalulla feeldinu mukalilulla oru prathyeka oorthoogoonal grouppinte kramatthinte "p" madangu sakthiyaanu.
otta "p" nuu athinte moolyam nalkunnathu
n" vichithramaayirikkumbol, allengil
n" erattayum (−1) "n"/2 "d" q "oru chathuraakrithiyilulla avashishtavum aakumbol, allengil
n" erattayum (−1) "n"/2 "d" q "oru quadratic noonn residuve aakumbol.
p" = 2 ennathinu diagonal ghadakam "m" p "(" f "q") kanakkaakkunnath kuprasidhamaanu. (eee notteshan "f" q "neey mathramalla," f "2" q "," f "q"/2 ennivayeyum aashrayichirikkunnathil thettidharippikkunnathu.) 
appol diagonal ghadakam "m" p "(" f "q") thaazhepparayunna reethiyil nalkiyirikkunnu.
phoom bandhikkappettirikkum okden moolyam + 2 allengil −2 mooad 8 allengil
phoom swathanthravum okden moolyam −1 allengil 0 allengil 1 mooad 8 allengil
phoom swathanthravum okden moolyam −3 allengil 3 allengil 4 mooad 8 ullathum aayirikkumbol.
ζ "d" ("s") nte vilayiruthal.
diriclet seerees ζ "d" ("s") nte aavashyamaaya moolyangal inipparayunna reethiyil vilayiruthaam. m" erattayaanengil 0 kondu nalkunna χ ("m") enna diriclet pratheekatthinu nammal χ ezhuthunnu, koodaathe "m" vichithramaanenkil jacoby chihna soothravaakyam _ 21. eee pratheekatthinte modulasinu "k" ennum athinte kandakdarinu "k" 1 ennum ezhuthukayum χ = χ1si idukayum cheiyunnu, avide χ1 pradhana pratheekamaaya mooad "k" umm si oru aadima pratheekamaaya mooad "k" 1 umm aanu.
ell-seereesinte pravartthana samavaakyam ithaanu
evide "g" ennathu gasses thukayaanu
s" oru positteev poornnasamkhyayaanenkil
ivide "b" s "(" x ") oru bernauli polynomialaanu.
udaaharanagal.
n"> 0 enna alavilulla unimodular laattisukalude kaaryatthil pinda soothravaakyam 8 kondu harikkavunna λ aanu
ivide "b" "k" oru bernauli samkhyayaanu.
alavu "n" = 0.
mukalilulla soothravaakyam "n" = 0 nuu paraajayappedunnu, alavu paramavadhi 1 aayirikkumbol pothuve maas formula parishkkarikkendathundu. "n" = 0 nuu oru laattis maathrameyulloo, poojya laattis, bharam 1 aanu, athinaal motham pindam 1 aanu.
alavu "n" = 8.
pinda soothravaakyam motham pindam nalkunnu
alavu 8-nte oru unimodular laattis, iii8 laattis undu, athinte automorphism group 696729,600 kramatthile "iii" 8-nte weyl grouppaanu, athinaal ithu eee saahacharyathil maas formula parishodhikkunnu. 
pindam poojyamallenna vastutha upayogichu 8-aamm alavilulla oru unimodular laattisinte nilanilppin smith yadhaarthathil oru nirmmaanarahithamaaya thelivu nalki.
alavu "n" = 16.
pinda soothravaakyam motham pindam nalkunnu
16-aamm alavilulla rund unimodular laattisukalund, onnu root sisttam "iii" 82 aanu.
automorphism group oof oordar 2 × 6967296002 = 970864271032320000, root sisttam "dii" 16 ulla onnu, automorphism group oof oordar 21516! = 685597979049984000.
athinaal pinda soothravaakyam ithaanu
alavu "n" = 24.
24 enna alavilulla 24 unimodular laattisukalund, avaye neemiyar lattisukal ennu vilikkunnu. avayude maas formula check inn cheiyunnu.
alavu "n" = 32.
eee kesil pindam valuthaanu, 40 dashalakshathiladhikam. ithu soochippikkunnathu 80 dashalakshathiladhikam thulyarundu ennaanu.
32-aamm alavilulla unimodular lattisukal, oronninum kuranjathu 2 automorphism groupund, athinaal pindatthinu paramavadhi randilonnu sambhavana cheiyunnu. eee vaadam parishkarichukondu, atharam oru bilyaniladhikam lattisukal undennu kaanichu. uyarnna alavukalil pindove athinaal laattisukalude ennavum valare vegathil vardhikkunnu.
saamaanyavalkkaranangal.
chila janussukalile roopangal anusarichu oru chathuraakrithiyilulla roopathinte praathinidhyangalude bharamullah ennam kanakkaakkunna kooduthal pothuvaaya oru soothravaakyam seagal nalki; oru roopam poojyamaayirikkumbol smith-minkovsky-seagal pinda soothravaakyam prathyeka kesaanu.
pinda soothravaakyam tamagawa nambar enna prasthaavanaykk thulyamaanennu tamagawa kaanichu. 
oorthoogoonal group 2 aanu, ithu spin grouppinte lalithamaayi bandhippicha kavarinte tamagawa nambar 1 aanennu parayunnathinu thulyamaanu. lalithamaayi bandhippichirikkunna eathenkilum semi simple grouppinte tamagawa nambar 1 aanennu well kooduthal pothuve anumaanichu, eee anumaanam 1988 ll coatvits theliyichu.
 verukalillaatha (allengil nalkiyirikkunna root sisttam upayogichu) unimodular laattisukalkk oru maas formula nalki.