Modular Forms of One Variable Don Zagier Notes based on a course ...

1 downloads 53 Views 523KB Size Report
f or m s is m apped to itse lf b y a ll . ¬ etsУTpt u u| УН ( ~ Y di m S ¢Y r k b. L ¢ ) be a basis f or. L ¢ ove
      ! "#%$& '!(

)+*-,/.10324-05.167*9874;:1*9=?0/[email protected]#BF8HGI,5=?.1:KJL,1MONQPR=?BD8R@HSUT9T>S

V

 









Y



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

+



+ 1S  7S , S98 7S > 7S > 7S  , ,2 ,D ,  ,  2

S

2>S 2M, 2M, 28 2M

92 T 8M M8  M8 2 T8 > T8  T8  8ET > >2

>X8 > > >  >+



 "!#$"

%& ')( *,+-!" /. 102 *" 354""') #6 (

JR.0I

-* =/687R*Q6R % F * '  ,?JR.NRF =?05,3.R 4 PF . * ,?JR.A4 2 * CE. @  .18E,?BD*98R.16#P JR.18R* .18R*98#,?J 4 , ,?JR. . =/. . R BF0 ,?.18R:1. *  *(6R=?4 ,?BF:!F .CD6 & MI4 ! 4-BD=6 6R.1. P =?.105BD0/*-=?0 *  )4-8R6 I . J 4 CE' . 4-@E4-BF8 (S12) $ WR t t -u !ZF $&SX )^; D9T>S W")MW ,8ET) $ WE t t u !ZF &SX )^; d ,D8 DM>>, S 2D>7S  W")MW 2M,DM,) $ WE t t  $ !ZF - &SX )^; d W )5W +>SUT 2) WR t t -u !ZF / &SX )^; ,X8 +>S1D JR . !4-: , ,?J 4 , ,?JR. *9=?BF. = : *(G. J:1BF.18E,?0 *Q:1:1=/=?BD8R@4 =/. 4 /=?4 ,?BF*98 4 8Q<  2O. =?0 BF0 4 05P .1:1B 4  : 4-0/. * ,?JR.IP JR.18R* .18R*98 ,/J 4 C, ² ¢ BD8 @9. 8R. =K4 OBD0 05P4-8R8R.16 2 Z *-4=  0 I B ,?J=K4 ,?BD*98 4  *9=?BF. = : *(G. J:1BF.18E,?C0 ¡ , BF0+,?JRBF0+P JR.18R* .18R*98 I JRBF:KJ BD0 =/.105PO*98R05BD2 F. . *-= ,?JR. =?BF:KJR8R.10/0* ,?JR. 4 =?B ,?J . ,?BD: 4 PRP FBD: 4 ,?BD*98R0 * ,/JR.,?JR.1*-R= #* *(6R< F4 = *-= C0 JR. =?BD@9JLR, @ J 4-8R6 0/BF6R.I* & 2M)X4 =E. 0 0/. 8R0/. 4 F0/E * *-c= £; ,L& $ BF0 C. H(< 4  ,?* /F )&4-8RL6 I 6B /R2O. ) “” N S& A )@;EU $ š ¢ ] &PUVF[?\UQ$C)

&SA*; '&UVFW '&UQ$ u o&PUVF[?\UQ$9)Oq M)

2O. , I .1.18O B S, =?C. F4 ,?BF*9O8 & >) JR.*-,?JR. ? = 4-BD8 6R6B 5 . =?.18R: Z. I BD,/J ,?JR.: 4-0/.  ; S:1* .1E0 =?*  ,?JR.. RBF0 ,?.18R:1.* 0/X* @ : 4 /D.16 75* F6 *-4= G0 9µ¡ ( ! BF0 4 PR=/*-PO. = 6RBDC(BF0/*-=3*  M>,/JR.1G 8 ’M& ) BF034;052 @-=?*9P* ’M& !/) 4-8R6#. CE. =  *Q6R<  4 \ = *-=  ] &SX )+* %I .1BD@9JLQ, £ *988 ’M& ! ) BF0 - - 4 *Q6R<  4 \= *-=  *988 ’M& ) Z *-=/. @9.18R. =?4 /6EM ] &S²gX ) BF04 *Q6R<  4 N= *-4=  * I .1BF@9JEZ, £7*9l8 ’M& ): *-=A. 4-:KJ P *90/BD,/BDCE. 6RBDC(BF0/*-= ² * 

?





?















#

2@

8. 6".





? !

M 0/BF8R:1.

‰ y} ~ z Ž k•’M& )

‰ }ˆ?y ² z ² ~ Ž k•’M& ! ) W z ;I] ‰ y&P²¨X )Wz²  ] ‰ ² yp}XsXjW* ~ Ž &}v?² )C&P²¨X )W~ Ž ; &}XjW~ ) ¢ ] &S²gX )  JR. 0/2 05P4-:1. * >² ¢ &P’(M& )) 05P4-8R8R.162T46 *-=0] &S²gX ) I B ,?J€]k€² ¢ &S’& ! )) Mb²  ! m  M  ! ;   MBF0A: 4/D.16 ,?JR.05P4-:1.*      & JRBF0 6R.F 8RBD,?BD*98 ;   #L&)^; $F    -  + +   - 6R. 7, & )@;S +! - !  .0/ + j² " ³ $ & " $& '*)) +  + -  JR. P *9BD0/0/*98 0/< [4 ,?BF*98 *-4=  <  4 4-06RBD0/:1›‘a  " ” = BD0;4-T8  0 *(*-,?J 0 =K4 P BF6  [ *-s= m Vm  4 =/@9.I4-8R6 ]Q&)@;  ] & )e˜ ~ 6R.18R*-,/.10 ,?JR. @ $œ6R› /B .1/ 8R0/ BF*98 4  *9=?BF. = ,/"=K4-$ 8RR0 *-$œ4= ›  / *   ] % R. =?:1BF05. D>M F,?JR.  *9=/BF. =;,/=K4-8R0 *-=  * ] &)Q; ˜/š  & ‡q M) BDF 0 ³ ˜ š  %M I JR. =/. # &) ; $    BD0,?JR. HQ< 4-6>=K4 ,?BD: *-4=  4-0/0/*(: B 4 ,?.16 ,/*  ;  š BF0 4-8R6 JR.18R:1.-M T2  4-8 4 6Q,?BF: :1*98E,?BF8L< 4 ,?BD*98 M *-=4 6OX BD8 ,?JR.#PRPO. "=$J 4 / /@ P  4-8R.  , . / &SX ) BD0 *-2LC>BD*9BD*9=* ,/JR. I JR* F.H: * /F. : ,?BF*98 *

 BF05,?0A4 P =?.G6 4D @9. 2R=K4-BD:;PR=?*Q* ,?J 4 , ,?JR. =K4-8 = * 4 PO*90/B ,?BDCE. 6R.G F 8RB ,?.A. CE.18HBF05BD2F . 2T >+ M 2 , BD, BF0(H ¥ Q P9K  EK  

EK  (  ´ 8R. : 4-8 @9.18R. =K4D B«1J . ,?JR. ,K4 0/. =/BF.10 2  BF8R:GF

? >

>

N(JR*7I ,?J 4 , B/  BF0;4-8 . CE.18 0R [. ,/=?BD: ,6£ $ ,6£ 4 ,/=?B OM,?JR.18 $ ; &ed S7) ¢ 6R. , & )BD0 :1*98R@-=?



0 (

. 8



/(

4 .  . 

$( " (/!6'&)(6

#%:

( $. 

.  4 !$#% 

 ) (

. " 4  !



#%

(

(

':

(" (6:

!/("  45(

( 1 (6: ( 7.

8.

(6: . 8

#,8 : !6". :



#%

6'9  (, .  (

8 -:;!6 . : #%





. 8

. 8*

 45(



/(

( 25#) .%:

 % ( 



$. . 8



 

























=)>





 ´+8,?JR.&*-,?JR. = J 4-8R6 M,/JR.P *9BD0/0/*98 0/;  $ d

S $ ‰ 6 1? S5d  n  W"  Ž





 / &|u  )>;

 ¢ &|u  ) &(8* d  $ ) W ,S $   4 -hx~ u "?/~r ¢ š F   % - S $  &|u  ) &(8* d  $ )Ld S $ BF8-hx~ u "?/~r ¢ š F  &  )^;¨d ¢ , ,  % ¢ 

$

=)>

 =)>

?

?



$

 :



-



-

d 2$  W  $

& £  ,)u & £  8T) 

JR. =?.ABD0 4-8H4-8 4F*9@9*9BF8R@ ,?JRBD0 2  ¢ š ˜ š - &•; & dG] )w?X8T)&4-8R6 BF8E,?.1@-=?4 ,?BF8R@ * CE. =   


















 "

 



&R)œ˜ š $œ›‘ " ~ •˜ $œ›‘ "  u

=)>

JR. =?. &)&BD0 4-8M :1*98L,/BF8LP4-8R6RBD8R@ ,?JR. =?BF@9JE, J 4-8R670/BF6R.* 

?









*%

š

#

?

y "% š

?

*#&



?

?

& ,?JR.Z *-=4 M"H $; $F M $j; d µF); $F 2  *-=4 BD8R@ 4M) &  ) 6R. P .18R6R0I*98 # *98   *Q6Ru  )&q S‘ :1) %& >F  $9)^; & >F ) %& $G)* *-=I4/ >Ftu  $k '  ¡ , BF0 . 4-0/B66 0/.1. 8 ,?J 4 , =K4 ,/BF:7=?.1: BDPR=?*Q:1BDS,  0?94 >0,/J 4 , *-= *(6R6 PR=?/B .10  4-8R6 ) ,?JR€. ¬ .1@9. 8R6>=?. R0 ; 2 * FE 0 &  ) 4-8R6 &  )34 =/.AC. HQ< 4  6B  *-= );BF0 :1*98R@-=?