Modulyarna forma golomorfna funkciya viznachena na verhnij kompleksnij pivploshini tobto mnozhini H x i y y gt 0 x y R displaystyle mathbb H x iy y gt 0 x y in mathbb R sho ye invariantnoyu shodo peretvoren modulyarnoyi grupi chi deyakoyi yiyi pidgrupi i zadovolnyaye umovi golomorfnosti v parabolichnih tochkah Modulyarni formi i modulyarni funkciyi shiroko vikoristovuyutsya v teoriyi chisel a takozh v algebrayichnij topologiyi i teoriyi strun Zmist 1 Viznachennya 1 1 Dopomizhni viznachennya 1 2 Modulyarna forma 1 3 Modulyarna funkciya 2 Vipadok grupi UNIQ postMath 00000020 QINU 2 1 Poyasnennya 3 Zagalnij vipadok 4 Prikladi 4 1 Poyasnennya 5 Kvadratichni formi 6 Operatori Geke 7 Zastosuvannya 8 Posilannya 9 LiteraturaViznachennya RedaguvatiDopomizhni viznachennya Redaguvati Nehaj g a b c d S L 2 Z displaystyle gamma begin pmatrix a amp b c amp d end pmatrix in SL 2 mathbf Z nbsp kvadratna matricya poryadku 2 z cilochiselnimi elementami i viznachnikom rivnim odinici Dlya deyakogo z H displaystyle z in mathbb H nbsp viznachimo funkciyu g z a z b c z d displaystyle gamma z left frac az b cz d right nbsp Takozh poznachimo G N a b c d S L 2 Z a 1 b 0 c 0 d 1 mod N displaystyle Gamma N left begin pmatrix a amp b c amp d end pmatrix in SL 2 mathbf Z a equiv 1 b equiv 0 c equiv 0 d equiv 1 pmod N right nbsp Dani grupi nazivayutsya golovnimi kongruentnimi pidgrupami rivnya N Takozh vikoristovuyetsya poznachennya G 1 S L 2 Z displaystyle Gamma 1 SL 2 mathbf Z nbsp Dovilna grupa G G N G G 1 displaystyle Gamma Gamma N subseteq Gamma subseteq Gamma 1 nbsp nazivayetsya kongruentnoyu Nehaj g G displaystyle gamma in Gamma nbsp deyakij element kongruentnoyi grupi Yaksho Tr g 2 displaystyle operatorname Tr gamma pm 2 nbsp de Tr displaystyle operatorname Tr cdot nbsp slid matrici to cej element nazivayetsya parabolichnim a vidpovidne peretvorennya parabolichnim Tochka s R displaystyle s in mathbb R cup infty nbsp nazivayetsya parabolichnoyu yaksho isnuye parabolichnij element g G g I I displaystyle gamma in Gamma gamma neq I I nbsp takij sho g s s displaystyle gamma s s nbsp Modulyarna forma Redaguvati Nehaj G displaystyle Gamma nbsp deyaka kongruentna grupa Funkciya f viznachena na H displaystyle mathbb H nbsp nazivayetsya modulyarnoyu formoyu stepenya vagi k dlya grupi G displaystyle Gamma nbsp yaksho vikonuyutsya umovi f g z c z d k f z g a b c d G displaystyle f gamma z cz d k f z forall gamma begin pmatrix a amp b c amp d end pmatrix in Gamma nbsp f z displaystyle f z nbsp golomorfna v H displaystyle mathbb H nbsp f z displaystyle f z nbsp golomorfna v parabolichnih tochkah grupi G displaystyle Gamma nbsp Modulyarna funkciya Redaguvati Nehaj G displaystyle Gamma nbsp deyaka kongruentna grupa Funkciya f viznachena na H displaystyle mathbb H nbsp nazivayetsya modulyarnoyu funkciyeyu dlya grupi G displaystyle Gamma nbsp yaksho vikonuyutsya umovi f z displaystyle f z nbsp ye invariantnoyu shodo diyi grupi G displaystyle Gamma nbsp tobto f g z f z g a b c d G displaystyle f gamma z f z forall gamma begin pmatrix a amp b c amp d end pmatrix in Gamma nbsp f z displaystyle f z nbsp meromorfna v H displaystyle mathbb H nbsp f z displaystyle f z nbsp meromorfna v parabolichnih tochkah grupi G displaystyle Gamma nbsp Vipadok grupi G 1 displaystyle Gamma 1 RedaguvatiModulyarna grupa G 1 I I displaystyle Gamma 1 I I nbsp porodzhuyetsya dvoma matricyami T 1 1 0 1 displaystyle T left begin array cc 1 amp 1 0 amp 1 end array right nbsp i S 0 1 1 0 displaystyle S left begin array cc 0 amp 1 1 amp 0 end array right nbsp Tozh dlya perevirki vikonannya pershih umov viznachen modulyarnih funkcij i form dostatno pereviriti vikonannya umov f z f z 1 displaystyle f z f z 1 nbsp i f 1 z z k f z displaystyle f 1 z z k f z nbsp Parabolichnimi tochkami danoyi grupi ye tochki Q displaystyle mathbb Q cup infty nbsp i vsi voni ye ekvivalentnimi tobto a b Q displaystyle forall a b in mathbb Q cup infty nbsp isnuye takij g G 1 displaystyle gamma in Gamma 1 nbsp sho g a b displaystyle gamma a b nbsp Tozh dostatno pereviriti golomorfnist chi meromorfnist lishe v odnij z cih tochok Najzruchnishe dlya cogo vzyati displaystyle infty nbsp Zavdyaki vlastivosti f z f z 1 displaystyle f z f z 1 nbsp funkciya f z mozhe buti zapisana cherez ryad Fur ye cherez q exp 2 p i z displaystyle q exp 2 pi iz nbsp Oskilki exp displaystyle exp nbsp na vsij kompleksnij ploshini ne rivnij nulyu to takozh q 0 displaystyle q neq 0 nbsp ale exp w 0 displaystyle exp w to 0 nbsp koli w displaystyle w to infty nbsp po vid yemnij dijsnij osi otzhe q 0 displaystyle q to 0 nbsp koli 2 p i z displaystyle 2 pi iz to infty nbsp tobto koli z i displaystyle z to i infty nbsp po dodatnij uyavnij osi Funkciya ye meromorfnoyu v bezmezhnosti yaksho f z n m c n exp 2 p i n z n m c n q n displaystyle f z sum n m infty c n exp 2 pi inz sum n m infty c n q n nbsp na vsomu vidkritomu odinichnomu kruzi Koeficiyenti c n displaystyle c n nbsp koeficiyenti Fur ye funkciyi f displaystyle f nbsp Yaksho c n 0 displaystyle c n 0 nbsp pri n lt 0 displaystyle n lt 0 nbsp na vsomu vidkritomu odinichnomu kruzi to funkciya ye golomorfnoyu v bezmezhnosti Poyasnennya Redaguvati Dlya G G 1 displaystyle Gamma Gamma 1 nbsp modulyarnu formu mozhna takozh oznachiti yak odnoridnu golomorfnu funkciyu F na mnozhini gratok v C displaystyle mathbb C nbsp Tut gratka ce pidgrupa L Z 2 displaystyle Lambda cong mathbb Z 2 nbsp v C displaystyle mathbb C nbsp porodzhena dvoma chislami w 1 displaystyle omega 1 nbsp w 2 displaystyle omega 2 nbsp yaki utvoryuyut bazu C displaystyle mathbb C nbsp nad R displaystyle mathbb R nbsp Odnoridnist F oznachaye sho isnuye cile k 0 displaystyle k geq 0 nbsp take sho F l L l k F L displaystyle F lambda Lambda lambda k F Lambda nbsp dlya vsih l C displaystyle lambda in mathbb C times nbsp i vsih gratok L displaystyle Lambda nbsp Dosit obmezhitis parnoyu vagoyu k inakshe F 0 displaystyle F equiv 0 nbsp Za dopomogoyu gomotetiyi l displaystyle lambda cdot nbsp mozhna zrobiti shob w 2 1 displaystyle omega 2 1 nbsp a w 1 H t C I m t gt 0 displaystyle omega 1 in mathbb H tau in mathbb C mid mathrm Im tau gt 0 nbsp bulo parametrom gratki Funkciya f H C displaystyle f mathbb H to mathbb C nbsp f t F Z t Z 1 displaystyle f tau F mathbb Z tau mathbb Z 1 nbsp maye avtomorfnu vlastivist ekvivalentnu odnoridnosti F Golomorfnist F oznachaye golomorfnist f i polinomialnu obmezhenist rostu f poblizu mezhi H displaystyle mathbb H nbsp Z obmezhenosti viplivaye sho f x i y O 1 displaystyle f x iy O 1 nbsp pri y displaystyle y to infty nbsp i f x i y O y k displaystyle f x iy O y k nbsp pri y 0 displaystyle y to 0 nbsp Zagalnij vipadok RedaguvatiYaksho G displaystyle Gamma nbsp deyaka pidgrupa zi skinchennim indeksom grupi G 1 displaystyle Gamma 1 nbsp to mnozhina parabolichnih tochok tezh rivna Q displaystyle mathbb Q cup infty nbsp ale v comu vipadku voni mozhut ne buti ekvivalentnimi tozh umovi golomorfnosti i meromorfnosti slid pereviryati okremo dlya kozhnogo klasu ekvivalentnosti Dlya tochki displaystyle infty nbsp stabilizator porodzhuyetsya deyakoyu matriceyu T M 1 M 0 1 displaystyle T M left begin array cc 1 amp M 0 amp 1 end array right nbsp Oskilki f z invariantna vidnosno T M displaystyle T M nbsp to f z f z M displaystyle f z f z M nbsp Tomu yaksho viznachiti q exp 2 p i z M displaystyle q exp left frac 2 pi iz M right nbsp to mozhna dati oznaki meromorfnosti i golomorfnosti podibni do poperednih funkciya ye meromorfnoyu v bezmezhnosti yaksho f z n m c n q n displaystyle f z sum n m infty c n q n nbsp na vsomu vidkritomu odinichnomu kruzi Koeficiyenti c n displaystyle c n nbsp koeficiyenti Fur ye funkciyi f displaystyle f nbsp Yaksho c n 0 displaystyle c n 0 nbsp pri n lt 0 displaystyle n lt 0 nbsp na vsomu vidkritomu odinichnomu kruzi to funkciya ye golomorfnoyu v bezmezhnosti Yaksho tochka t Q displaystyle tau in mathbb Q nbsp ne ye ekvivalentna bezmezhnosti v grupi G displaystyle Gamma nbsp todi mozhna znajti takij g G 1 displaystyle gamma in Gamma 1 nbsp sho t g displaystyle tau gamma infty nbsp Todi funkciya F z f g z displaystyle F z f gamma z nbsp ye invariantnoyu shodo grupi g G g 1 G 1 displaystyle gamma Gamma gamma 1 subset Gamma 1 nbsp Todi f z displaystyle f z nbsp bude golomorfnoyu meromorfnoyu v tochci t Q displaystyle tau in mathbb Q nbsp yaksho F z displaystyle F z nbsp bude golomorfnoyu meromorfnoyu v bezmezhnosti Dlya G G N displaystyle Gamma Gamma N nbsp govorimo pro modulyarni formi rivnya N Modulyarni formi vagi k i rivnya G displaystyle Gamma nbsp utvoryuyut skinchennovimirnij prostir M k G displaystyle M k Gamma nbsp nulovij pri k lt 0 displaystyle k lt 0 nbsp i gradujovana algebra M G k 0 M k G displaystyle M Gamma oplus k geq 0 M k Gamma nbsp skinchennoporodzhena nad C displaystyle mathbb C nbsp Napriklad M k G 1 0 displaystyle M k Gamma 1 0 nbsp dlya neparnih k a dlya parnih k dim M k G 1 k 12 displaystyle dim M k Gamma 1 k 12 nbsp pri k 2 mod 12 displaystyle k equiv 2 pmod 12 nbsp i dim M k G 1 k 12 1 displaystyle dim M k Gamma 1 k 12 1 nbsp inakshe Bilsh zagalno yaksho G displaystyle Gamma nbsp diskretna pidgrupa S L 2 R displaystyle mathrm SL 2 mathbb R nbsp i G H displaystyle Gamma backslash mathbb H nbsp maye skinchennij giperbolichnij ob yem V stosovno 2 formi y 2 d x d y displaystyle y 2 dx dy nbsp to dim M k G k V 4 p 1 displaystyle dim M k Gamma leq kV 4 pi 1 nbsp dlya vsih k 0 displaystyle k geq 0 nbsp Zokrema dlya pidgrupi sho mistit 1 G G 1 displaystyle Gamma subset Gamma 1 nbsp skinchennogo indeksu r dim M k G k r 12 1 displaystyle dim M k Gamma leq kr 12 1 nbsp Prikladi RedaguvatiOdnimi z najprostishih prikladiv modulyarnih form ye ryadi Ejzenshtejna vagi k displaystyle k nbsp sho viznachayutsya dlya parnogo k gt 2 displaystyle k gt 2 nbsp G k t 1 2 m n Z 2 0 0 1 m n t k displaystyle G k tau frac 1 2 sum m n in mathbb Z 2 backslash 0 0 frac 1 m n tau k nbsp de t H displaystyle tau in mathbb H nbsp Nehajg 2 60 m n 0 0 m n t 4 g 3 140 m n 0 0 m n t 6 displaystyle g 2 60 sum m n neq 0 0 m n tau 4 qquad g 3 140 sum m n neq 0 0 m n tau 6 nbsp modulyarni invarianti D g 2 3 27 g 3 2 displaystyle Delta g 2 3 27g 3 2 nbsp modulyarnij diskriminant Viznachimo takozh j t 1728 g 2 3 D displaystyle j tau 1728 g 2 3 over Delta nbsp osnovnij modulyarnij invariant j invariant Vikonuyutsya rivnosti g 2 t 1 g 2 t g 2 t 1 t 4 g 2 t displaystyle g 2 tau 1 g 2 tau g 2 tau 1 tau 4 g 2 tau nbsp D t 1 D t D t 1 t 12 D t displaystyle Delta tau 1 Delta tau Delta tau 1 tau 12 Delta tau nbsp Takozh dani funkciyi zadovolnyayut vidpovidni vlastivosti golomorfnosti Tobto g 2 displaystyle g 2 nbsp modulyarna forma vagi 4 D displaystyle Delta nbsp modulyarna forma vagi 12 Vidpovidno g 2 3 displaystyle g 2 3 nbsp modulyarna forma vagi 12 a j z displaystyle j z nbsp modulyarna funkciya Dani funkciyi mayut vazhlive zastosuvannya v teoriyi eliptichnih funkcij i eliptichnih krivih Poyasnennya Redaguvati Pri diyi grupi S L 2 R displaystyle mathrm SL 2 mathbb R nbsp z vagoyu k gt 0 displaystyle k gt 0 nbsp na golomorfnih funkciyah H C displaystyle mathbb H to mathbb C nbsp f f k g displaystyle f mapsto f k gamma nbsp g a b c d S L 2 R displaystyle gamma begin pmatrix a amp b c amp d end pmatrix in mathrm SL 2 mathbb R nbsp f k g t c t d k f a t b c t d displaystyle f k gamma tau c tau d k f left frac a tau b c tau d right nbsp stabilizator tochki 1 postijnoyi funkciyi pri parnomu k ce matrici z c 0 displaystyle c 0 nbsp a d 1 displaystyle a d pm 1 nbsp Pri diyi G 1 S L 2 Z displaystyle Gamma 1 mathrm SL 2 mathbb Z nbsp cej stabilizator ye G 1 n 0 1 n Z displaystyle Gamma infty pm begin pmatrix 1 amp n 0 amp 1 end pmatrix mid n in mathbb Z nbsp Mnozhina klasiv sumizhnosti G G 1 displaystyle Gamma infty backslash Gamma 1 nbsp perebuvaye v biyekciyi z c d Z 2 displaystyle c d in mathbb Z 2 mid nbsp nsd c d 1 1 displaystyle c d 1 pm 1 nbsp Ryad Ajzenshtajna E k t g G G 1 1 k g 1 2 c d Z c d 1 1 c t d k displaystyle E k tau sum gamma in Gamma infty backslash Gamma 1 1 k gamma frac 1 2 sum c d in mathbb Z c d 1 frac 1 c tau d k nbsp absolyutno zbigayetsya pri k gt 2 displaystyle k gt 2 nbsp i ye neruhomoyu tochkoyu diyi S L 2 Z displaystyle mathrm SL 2 mathbb Z nbsp tobto modulyarnoyu formoyu vagi k rivnya 1 Komutativne kilce M G 1 C E 4 E 6 displaystyle M Gamma 1 mathbb C E 4 E 6 nbsp Bezposeredno odnoridnu funkciyu vid gratki mozhna napisati yak G k L 1 2 l L 0 l k displaystyle G k Lambda 1 2 sum lambda in Lambda backslash 0 lambda k nbsp k gt 2 displaystyle k gt 2 nbsp Zvuzhennya yiyi na gratki L Z t Z 1 displaystyle Lambda mathbb Z tau mathbb Z 1 nbsp t H displaystyle tau in mathbb H nbsp daye modulyarnu formu vagi k rivnya 1 G k t 1 2 m n Z m n 0 0 1 m t n k displaystyle G k tau frac 1 2 sum m n in mathbb Z m n neq 0 0 frac 1 m tau n k nbsp vtim G k t z k E k t displaystyle G k tau zeta k E k tau nbsp Vikoristovuyuchi she odnu normalizaciyu G k t k 1 2 p i k G k t displaystyle mathbb G k tau k 1 2 pi i k G k tau nbsp znahodimo rozvinennya yiyi v ryad Fur ye vid q e 2 p i t displaystyle q e 2 pi i tau nbsp G k t B k 2 k n 1 s k 1 n q n displaystyle mathbb G k tau B k 2k sum n 1 infty sigma k 1 n q n nbsp de B k displaystyle B k nbsp chislo Bernulli i s k 1 n d n d k 1 displaystyle sigma k 1 n sum d n d k 1 nbsp Kvadratichni formi RedaguvatiNehaj 8 t n Z exp p i n 2 t displaystyle theta tau sum n in mathbb Z exp pi in 2 tau nbsp teta funkciya Yakobi t H displaystyle tau in mathbb H nbsp Todi 8 2 displaystyle theta 2 nbsp modulyarna forma vagi 1 rivnya 4 Z odnovimirnosti pevnogo prostoru modulyarnih form viplivaye sho chislo predstavlen cilogo n gt 0 displaystyle n gt 0 nbsp yak sumi kvadrativ dvoh cilih chisel ye 4 d n d gt 0 d 2 1 1 d 1 2 displaystyle 4 sum d n d gt 0 d 2 1 1 d 1 2 nbsp Z togo sho 8 4 displaystyle theta 4 nbsp modulyarna forma vagi 2 rivnya 4 vivoditsya chislo predstavlen cilogo n gt 0 displaystyle n gt 0 nbsp yak sumi kvadrativ chotiroh cilih chisel ye 8 d n d gt 0 d 0 mod 4 d displaystyle 8 sum d n d gt 0 d not equiv 0 pmod 4 d nbsp Uzagalnyuyuchi rozglyanemo dodatno viznachenu kvadratichnu formu Q Z m Z displaystyle Q mathbb Z m to mathbb Z nbsp Q x x t A x 2 displaystyle Q x x t Ax 2 nbsp de A M a t m Z displaystyle A in mathrm Mat m mathbb Z nbsp simetrichna dodatno viznachena matricya z parnimi diagonalnimi elementami Z neyu asociyuyetsya teta ryad 8 Q t x 1 x m Z q Q x 1 x m n 0 R Q n q n displaystyle Theta Q tau sum x 1 dots x m in mathbb Z q Q x 1 dots x m sum n 0 infty R Q n q n nbsp de q e 2 p i t displaystyle q e 2 pi i tau nbsp i R Q n x Z m Q x n displaystyle R Q n x in mathbb Z m mid Q x n nbsp Nehaj N najmenshe dodatne cile take sho N A 1 M a t m Z displaystyle NA 1 in mathrm Mat m mathbb Z nbsp maye parni diagonalni elementi Todi dlya m 2 k displaystyle m 2k nbsp k Z gt 0 displaystyle k in mathbb Z gt 0 nbsp funkciya 8 Q displaystyle Theta Q nbsp ye modulyarnoyu formoyu vagi k rivnya N Zokrema dlya det A 1 displaystyle det A 1 nbsp 8 Q displaystyle Theta Q nbsp ye modulyarnoyu formoyu vagi k rivnya 1 Napriklad ce virno dlya gratki E 8 displaystyle E 8 nbsp m 8 displaystyle m 8 nbsp abo gratki Licha m 24 displaystyle m 24 nbsp Operatori Geke RedaguvatiNa prostori modulyarnih form vagi k rivnya 1 diye operator Geke T m displaystyle T m nbsp m 1 displaystyle m geq 1 nbsp Vin perevodit odnoridnu funkciyu F stepenya k vid gratki L C displaystyle Lambda subset mathbb C nbsp v sumu T m F L m k 1 F L displaystyle T m F Lambda m k 1 sum F Lambda nbsp de L L displaystyle Lambda subset Lambda nbsp probigaye pidgratki indeksu m Konstanta normalizaciyi vibrana tak shobi ryadi z cilimi koeficiyentami Fur ye perehodili v taki zh Skinchenna mnozhina gratok L L displaystyle Lambda subset Lambda nbsp indeksu m ototozhnyuyetsya z mnozhinoyu G 1 M m displaystyle Gamma 1 backslash mathcal M m nbsp de M m M a t 2 Z displaystyle mathcal M m subset mathrm Mat 2 mathbb Z nbsp mnozhina matric g a b c d displaystyle gamma begin pmatrix a amp b c amp d end pmatrix nbsp z viznachnikom m Tomu T m f t m k 1 g G 1 M m c t d k f a t b c t d displaystyle T m f tau m k 1 sum gamma in Gamma 1 backslash mathcal M m c tau d k f left frac a tau b c tau d right nbsp Za predstavnikiv klasiv sumizhnosti mozhna obrati cilochiselni matrici a b 0 d displaystyle begin pmatrix a amp b 0 amp d end pmatrix nbsp z a d m displaystyle ad m nbsp 0 b lt d displaystyle 0 leq b lt d nbsp Tomu T m f t m k 1 a d m d gt 0 d k b mod d f a t b d displaystyle T m f tau m k 1 sum ad m d gt 0 d k sum b pmod d f a tau b d nbsp Vsi operatori T m displaystyle T m nbsp komutuyut i ye normalnimi vidnosno skalyarnogo dobutku Petersona tozh M k G 1 displaystyle M k Gamma 1 nbsp maye bazu spilnih vlasnih vektoriv Geke Ci vektori f mozhna normalizuvati umovoyu a 1 1 displaystyle a 1 1 nbsp dlya f n 0 a n q n displaystyle f sum n geq 0 a n q n nbsp i normalizovanij vlasnij bazis ye yedinim Prikladami normalizovanih vlasnih funkcij sluguyut D displaystyle Delta nbsp i G k displaystyle mathbb G k nbsp k 4 displaystyle k geq 4 nbsp Z kozhnoyu modulyarnoyu formoyu f n 0 a n q n displaystyle f sum n geq 0 a n q n nbsp vagi k pov yazuyetsya ryad Dirihle L f s n 1 a n n s displaystyle L f s sum n 1 infty a n n s nbsp Yaksho f normalizovana vlasna funkciya Geke to L f s 1 1 a p p s p k 1 2 s displaystyle L f s prod 1 1 a p p s p k 1 2s nbsp de p probigaye prosti chisla Dlya dovilnoyi modulyarnoyi formi f z a 0 0 displaystyle a 0 0 nbsp ryad Dirihle prodovzhuyetsya do ciloyi funkciyi vid s i zadovolnyaye funkcionalnomu rivnyannyu L f k s 1 k 2 L f s displaystyle L f k s 1 k 2 L f s nbsp de L f s 2 p s G s L f s displaystyle L f s 2 pi s Gamma s L f s nbsp tezh cila funkciya Zastosuvannya RedaguvatiZ gipotezi Shimuri Taniyami Vejlya dovedenoyi Vajlsom Tejlorom Brejlem Konradom Dajmondom naprikinci dvadcyatogo stolittya kozhna eliptichna kriva nad Q displaystyle mathbb Q nbsp mozhe buti parametrizovana modulyarnimi funkciyami viplivaye Ribet velika teorema Ferma dlya n gt 2 displaystyle n gt 2 nbsp ne isnuye dodatnih cilih a b c z a n b n c n displaystyle a n b n c n nbsp Posilannya RedaguvatiJ S Milne Modular functions and modular forms kurs lekcij D Zagier Elliptic modular forms and their applications The 1 2 3 of modular forms Universitext Springer Berlin 2008 pp 1 103 Literatura RedaguvatiSarnak P Modulyarnye formy i ih prilozheniya M FAZIS 1998 ISBN 5 70364029 4 Tom M Apostol Modular functions and Dirichlet Series in Number Theory 1990 Springer Verlag New York ISBN 0 387 97127 0 Robert A Rankin Modular forms and functions 1977 Cambridge University Press Cambridge ISBN 0 521 21212 X D Mumford Tata lectures on theta I Progress in Mathematics vol 28 Birkhauser Boston MA 1983 Yu I Manin A A Panchishkin Vvedenie v sovremennuyu teoriyu chisel Moskva MCNMO 2009 Enciklopediya Suchasnoyi Ukrayini Otrimano z https uk wikipedia org w index php title Modulyarna forma amp oldid 34396132
