p-adic L-functions and modular forms.indico.ictp.it/.../58/contribution/34/material/0/0.pdf ·...
170
2056-2 Advanced School and Workshop on p-adic Analysis and Applications Alexei PANCHISHKIN 31 August - 18 September, 2009 Institut Fourier Universite' Grenoble-1 B.P. 74, 38402 St. Martin d'Heres FRANCE p-adic L-functions and modular forms
Transcript of p-adic L-functions and modular forms.indico.ictp.it/.../58/contribution/34/material/0/0.pdf ·...
2056-2
Advanced School and Workshop on p-adic Analysis and Applications
Alexei PANCHISHKIN
31 August - 18 September, 2009
Institut FourierUniversite' Grenoble-1
B.P. 74, 38402 St. Martin d'HeresFRANCE
p-adic L-functions and modular forms
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'2+<" ($"%")&'&/+) +, Z!
'% &!" ($+?"*&/<" 6/1/&
6/1
←−
Z/ Z
/& ,+66+7% &!'& ,+$ ! ("
, . . . , "
) ∈ Z!
["
, . . . , "
]3 &!" *+)8$0")*"%
! ("
, . . . , "
) ≡ @(1+.
)
'$" %+6<'26" ,+$ '66 # ≥ A /B &!" ";0'&/+)
! ("
, . . . , "
) = @
/% %+6<'26" /) -'./* /)&"8"$%9 C+60&/+)% /) Z!
*') 2" +2&'/)". 0%/)8 &!"
,+66+7/)8 -'./* <"$%/+) +, &!" $%&'()# * +, -.)# ,/0)12(-349
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!"#$"% &'"()"*+) ,"%%-.
!" # ($) ∈ Z
[$ ] %! & '()*+(,-&) -+ (+! .&/-&%)! $0 #
′($) ∈ Z
[$ ] -"1#(/,&) 2!/-.&"-.!0 &+2 13''(1! "4&" #(/ 1(,! α
∈ Z
"4! -+-"-&) 5(+2-"-(+
|# (α
)/# ′(α
)!|
< !"#$%
-1 1&"-16!27
84!+ "4!/! !$-1"1 & 3+-93! α ∈ Z
1354 "4&"
# (α) = &, |α− α
| < .
'( )*+,( -./0 12 /3456-/+3 50/37 -.( 0(85(36( +9 :0566(00/,(
;))*+</=;-/+30>?
α!
= α!−" −
# (α!−")
#
′(α!−")
.
@;A/37 /3-+ ;66+53- -.( 9+*=;B @;2B+* (<);30/+3 +9 # ($) ;- $ = α!−" +3(
0.+C0 -.;- -./0 0(85(36( /0 D;56.2E ;34 /-0 B/=/- α .;0 ;BB -.( 4(0/*(4
)*+)(*-/(0 !69# FGHIJKE FH(**(L&K%#
M+* (<;=)B(E /9 # ($) = $
−" − E -.(3 ;32 α
∈ { , ", . . . , ' − } 0;-/0N(0
-.( 6+34/-/+3 |# (α
)|
< O- -.( 0;=( -/=(
#
′(α
) = (' − )α −!
6≡ & =+4 'E .(36( -.( /3/-/;B 6+34/-/+3 !"#$% /0
0;-/0N(4# @.( *++- α 6+/36/4(0 -.(3 C/-. -.( 53/85(B2 4(N3(4 @(/6.=PBB(*
*()*(0(3-;-/,( +9 α
? α = ω(α
)# !"#"$ % &'()*(+)& ,-."/01!"2 3456$7 849:/7;$0/< 5/6 =06:!5. 90.=< )'>%? *"3;"=1".?@AAB CD E FG
!"#$%!
◦&' !"#$"%!%& '"( '"')*#$+ ,%"+#$!"& !-./ ' "!"01/+2$3.(.'" 4.)(
!" #! $ %&'(!) (*#+!&) ', "-! .$"! +!&) C
/ 0'1 $ (*#(!" ! ⊂ 2!
%'3(4)!1 %'3"43*'*( ,*3%"4'3( " :! → C
/ .-! ("$3)$1) !5$67&!( ',
%'3"43*'*( ,*3%"4'3( $1! 71'84)!) #9 7'&93'64$&(: #9 1$"4'3$& ,*3%"4'3( ;$"
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%'67$%" "-!3 ,'1 $39 %'3"43*'*( ,*3%"4'3 " :! → C
$3) ,'1 $39 ε > >
"-!1! !54("( $ 7'&93'64$& #($) ∈ C
[$ ] (*%- "-$" |" ($)− #($)|
< ε ,'1 $&&
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"-!3 #($) %$3 #! %-'(!3
(' "-$" #($) ∈ %[$ ] ;(!! ?@'#A>B: ?C$(-ADB</
=3"!1!("43E !5$67&!( ', %'3"43*'*( &F$)4% ,*3%"4'3( $1! 71'84)!) #9
43"!17'&$"4'3 ', ,*3%"4'3(: )!+3!) '3 %!1"$43 (*#(!"(: (*%- $( ! = Z '1 N
24"- = Q
/ !" " #! $39 ,*3%"4'3 '3 3'3F3!E$"48! 43"!E!1( 24"- 8$&*!( 43
Q
'1 43 ('6! ;%'67&!"!< Q
FG$3$%- (7$%!/ =3 '1)!1 "' !5"!3) " ($) "' $&&
$ ∈ Z
2! %$3 *(! "-! 43"!17'&$"4'3 7'&93'64$&(
($
'
)
=$($ − H) · · · ($ − ' + H)
'!.
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(!
"
)4( $ 7'&93'64$& ', )!E1!! ' ', $ : 2-4%- ,'1 $ ∈ Z:
$ ≥ > E48!( "-! #43'64$& %'!I%4!3"/ =, $ ∈ Z
"-!3 $ 4( %&'(! ;43 "-! &F$)4%
"'7'&'E9< "' $ 7'(4"48! 43"!E!1: -!3%! "-! 8$&*! ',
(!
"
)4( $&(' %&'(! "' $3
43"!E!1: "-!1!,'1!
(!
"
)∈ Z
/
!"#"$ % &'()*(+)& ,-."/01!"2 3456$7 849:/7;$0/< 5/6 =06:!5. 90.=< )'>%? *"3;"=1".?@AAB CD E FG
!"#$%&' (%)*$%)+,
!" #$%&&'#%$ (%!$")*& '+,")-.$%,'.+ ,!".)"/ &%0& ,!%, %+0 #.+,'+1.1&
21+#,'.+ : Z
→ Q
!" #$ %&'(($" '" ()$ *+&, -.$$ /0'1234 /5!.)67389
(!) =∞∑
!=
"
!
(!
#
)
, -:;<8
%'() "
!
→ = -$>!?' !@@A8 *+& # →∞; B+& ! *C" ('+" (!) ?$D"$? *+& ! ∈ Z4
! ≥ = +"$ !" %&'($ *+&,!@@A
(!) =
∞∑
!=
"
!
(!
#
)
,
%)$&$ ()$ +$E '$"(. !" #$ *+C"?$? *&+, ()$ .A.($, +* @'"$!& $FC!('+".
(#) =!∑
"=
"
"
(#
%
)
,
()!( '.
"
"
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(−2)"−#(
%
&
)
(&).
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!" #"$%"# &'$ (!) %# ()*(+# $",-.", /' ( 01%/" #-2 &'$ "(.! ! ∈ Z3 ! ≥ 45
6& "
→ 4 /!"1 /!%# #"$%"# %# .'17"$8"1/ &'$ ()) ! ∈ Z!
5 9# *(# 1'/%.",
(:'7"3 /!" %17"$#" #/(/"2"1/ %# ()#' 7()%, ;<#"$%&'() *'+,&'+-.=>5 6&
.'17"$8"1." '& "
/' ?"$' %# #' &(#/ /!(/ /!" #"$%"# ,"01%18 /!" .'"@.%"1/#
'& /!" !A"BC(1#%'1 '& (!) ()#' .'17"$8"3 /!"1 (!) .(1 :" "B/"1,", /' (1"."%/,+* 0.*,+-.5 D1&'$/-1(/")+3 &'$ (1 ($:%/$($+ #"E-"1." "
*%/! "
→ 4
/!" (//"2C/ /' -#" ;F5G> &'$ .'1/%1-(/%'1 '& (!) '-/ '& /!" #-:#"/ Z!
%1
C!
2(+ &(%)5 H'*"7"$3 %1 /!" #"E-") *" 2'#/)+ .'1#%,"$ (1)+/%. &-1./%'1#3
/!(/ ($" ,"01", (# #-2# '& C'*"$ #"$%"#5
!"#"$ % &'()*(+)& ,-."/01!"2 3456$7 849:/7;$0/< 5/6 =06:!5. 90.=< )'>%? *"3;"=1".?@AAB CD E FD
!"#$!%&#!'(" )(* +,)"&$,"-
!" #$ %&'$()!* + %&,,#"+"(-! +$$&%(+"(-! *('. / +' 0,&)#1! A +') +
2*&3'("! 4(5!5 %&,2+%" +') "&"+116 )($%&''!%"!)7 "&2&1&.(%+1 $2+%! ! 5 89!'
! ($ + 2*&:!%"(-! 1(,(" &; 3'("! $!"$<
! = 1(,
←−
!
!
=9!*! " ($ + 42+*"(+116 &*)!*!)7 (')#%"(-! $!" +') ;&* # ≥ $ / # / $ ∈ " "9!*! +*!$#*:!%"(-! 9&,&,&*29($,$ π
! ," : !!
→ !
"
=("9 "9! %&')("(&'
π! ," ◦ π
" ,# = π! ,# ;&* # ≥ $ ≥ % 5 89! (')#%"(-("6 &; " ,!+'$ "9+" ;&* +'6
# , $ ∈ " "9!*! !>($"$ % ∈ " =("9 "9! %&')("(&' % ≥ # / % ≥ $ 5 ?6 "9! #'(-!*$+1
2*&2!*"6 =! 9+-! "9+" ;&* !+%9 # ∈ " + #'(@#! ,+2 π!
: ! → !
!
($ )!3'!)/
=9(%9 $+"($3!$ "9! 2*&2!*"6 π! ," ◦ π
!
= π"
4;&* !+%9 # , $ ∈ " 75
!"#"$ % &'()*(+)& ,-."/01!"2 3456$7 849:/7;$0/< 5/6 =06:!5. 90.=< )'>%? *"3;"=1".?@AAB CD E FG
!" !"#( ,!) #! "$! !%&'()*! +',-.-".,/ '0 1** !%21*)!( *'+1**3 +',-"1,"
0),+".',- φ : → !4
!"#$%$&#
5 (.-"6.#)".', ', 7."$ 21*)!- ., 1 !%&'()*! A .- 1 !%*.,!16 $'&'&'68$.-&
µ : !"#( ,!) −→ A.
9'6 ϕ ∈ !"#( ,!) 7! )-! "$! ,'"1".',-
µ(ϕ) =
∫
ϕ$µ =
∫
ϕ(")$µ(").
:1+$ (.-"6.#)".', µ +1, #! (!;,!( #3 1 -3-"!& '0 0),+".',- µ(!) : !
→ A<
-1".-03.,/ "$! 0'**'7.,/ ;,."!%1((.".2."3 +',(.".',
µ(")(") =∑
#∈π− ,!
($)
µ(!)(#) (" ∈ "
, # ∈ !
). =>4?@
A, '6(!6 "' +',-"6)+" -)+$ 1 -3-"!& ." -)B+!- "' 8)"
µ(!)(#) = µ(δ! ,#) ∈ A (# ∈
!
),
7$!6! δ! ,# .- "$! +$161+"!6.".+ 0),+".', '0 "$! .,2!6-! .&1/! π−
!
(#) ⊂ 7."$
6!-8!+" "' "$! ,1")61* 86'C!+".', →
!
4 9'6 1, 16#."6163 0),+".',
ϕ"
: "
→ ! 1,( $ ≥ % 7! (!;,! "$! 0),+".',-
ϕ!
= ϕ"
◦ π! ," , ϕ = ϕ
"
◦ π"
, ϕ ∈ !"#( ,!), ϕ!
: !
π ,!−→
"
−→ !.
!"#$#% &!'()*+),*' -./#012"#3 4567%8 95:;08<%10= 607 >17;"6/ :1/>= *(?&@ +#4<#>2#/@ABBC D E FG
!"#$%#&%#' !(&'%(&"# ") '*% )+!' '*+' + ,-,'%. ") )/#!'&"#, µ( ) :
→ A
,+'&,0%, '*% 0#&'% +11&'&$&'- !"#1&'&"# 23456 2+#1 *%#!% &, +,,"!&+'%1 '"
,".% 1&,'(&7/'&"#6 &, 8&$%# 7- '*% )"99": !"#1&'&"# 2!"#$%&'(')'&*
!+'&,+'"-6; )"( +99 . ∈ / < +#1 ϕ!
: !
→ 0 '*% $+9/% ") '*% ,/.,
µ(ϕ) = µ( )(ϕ
) =∑
"∈#
ϕ
(*)µ( )(*), 234=6
&, %>%#1%#' ") ' )"( +99 9+(8% %#"/8* ' ≥ . 4 ?*%# /, 234=6< &' ,/@!%,
'" $%(&)- '*% !"#1&'&"# 234=6 )"( ,".% A7+,&!B ,-,'%. ") )/#!'&"#,4 C"(
%D+.>9%< &)
= 1 = 9&.
←−
1
&, + >("0#&'% +7%9&+# 8("/>< +#1 0 &, + 1".+&# !"#'+&# +99 (""', ") /#&'-
") '*% "(1%( 1&$&1 '*% "(1%( ") 2:*&!* &, + A,/>%(#+'/(+9 #/.7%(B6
'*%# &' ,/@!%, '" !*%!E '*% !"#1&'&"# 234=6 )"( +99 !*+(+!'%(, ") 0#&'% "(1%(
χ : 1 → 0< ,&#!% '*%&( 0 ⊗Q F9&#%+( ,>+# !"&#!&1%, :&'* '*% :*"9% (
!"#( ,0 ⊗Q) 7- '*% "('*"8"#+9&'- >(">%('&%, )"( !*+(+!'%(, ") + 0#&'%8("/> 2,%% GH+'I< GJKLM3I64
!"#"$ % &'()*(+)& ,-."/01!"2 3456$7 849:/7;$0/< 5/6 =06:!5. 90.=< )'>%? *"3;"=1".?@AAB CB D EF
!"#$%&' (&)*+,%%- .-/0)-1,0-+*/
!" #! $ %&'("()! (*"!+!,- ! : Z → C (' $ %!,(&.(/ 01*/"(&* 2("3 "3!
%!,(&. 4(5!5 ! (" + ) = ! (")- ! : Z/ Z → C65 73! +!*!,$8(9!.
:!,*&188( *1;#!, 4'!! <:=>?@ 6 #
,! (' .!A*!. $' $! "(;!' "3! /&!B/(!*"#C %
(* "3! !D%$*'(&* (* % &0 "3! 0&88&2(*+ ,$"(&*$8 E1&"(!*"
"− ∑
#=!
! (&)%'#$
'
"$ − F
,
"3$" ('-
∞∑
=!
#
,!
$!%
=
"− ∑
#=!
! (&)%'#$
'
"$ − F
. 4G5H6
!"#"$ % &'()*(+)& ,-."/01!"2 3456$7 849:/7;$0/< 5/6 =06:!5. 90.=< )'>%? *"3;"=1".?@AAB @A C DE
!" #$% &' (!)'*+$, %-$ .,!/)*%$ ,*)0
= Z
= #*1
←−
!
(Z/!Z)
("(!) ⊂ ")2 %-$ .,!3$(%*4$ #*1*% 5$*)0 %67$) !4$, %-$ '$% !8 6## .!'*%*4$*)%$0$,' ! "*%- '&..!,% "(!) *) 6 /9$+ /)*%$ '$% " !8 .,*1$ )&15$,':;-$) %-$ .$,*!+*( 8&)(%*!) # : Z/!Z → C "*%- "(!) ⊂ " 16< 5$ 4*$"$+
6' 6) $#$1$)% !8 !"#( , C): =$ (#6*1 %-6% %-$,$ $9*'%' 6 +*'%,*5&%*!)
$
"
: !"#( , C)→ C "-*(- *' &)*>&$#< +$%$,1*)$+ 5< %-$ (!)+*%*!)
$
"
(# ) = %
",# 8!, 6## # ∈ !"#( , C). ?@:AB
C) !,+$, %! .,!4$ %-$ $9*'%$)($ !8 %-*' +*'%,*5&%*!) "$ &'$ %-$ 65!4$
(,*%$,*!) ?@:DB 6)+ (-$(7 %-6% 8!, $4$,< # ∈ !"#( , C) %-$ ,*0-% -6)+ '*+$*) ?@:AB ?*:$: %
",# B +!$' )!% +$.$)+ !) %-$ (-!*($ !8 6 .$,*!+ ! !8 %-$
8&)(%*!) # : C% 8!##!"' +*,$(%#< 8,!1 %-$ +$/)*%*!) ?@:EBF -!"$4$, "$ 0*4$
-$,$ 6 +*G$,$)% .,!!8 "-*(- *' 56'$+ !) 6) *)%$,.,$%6%*!) !8 %-$ )&15$,'
%
",# 6' ($,%6*) '.$(*6# 46#&$' !8 &H8&)(%*!)':
!"#"$ % &'()*(+)& ,-."/01!"2 3456$7 849:/7;$0/< 5/6 =06:!5. 90.=< )'>%? *"3;"=1".?@AAB @C D EF
!" # $%&'()!& : Z/!Z → C *+(
"(#, ) =
∞∑
=
($)$−!
,+ (-+ '!""+./!&0)&1 "2.+")+. 3-)'- ). #,.!*%(+*4 '!&5+"1+&( $!" #** # 3)(-
!(#) > 6 #&0 #07)(. #& #&#*4()' '!&()&%#()!& !5+" #** # ∈ C8 !" (-). .+")+.
3+ -#5+ (-#(
"(6− % , ) = −&
",#
%
. 9:8;<
!" +=#7/*+> )$ ≡ 6 ). (-+ '!&.(#&( $%&'()!& 3)(- (-+ /+")!0 ! = 6 (-+& 3+
-#5+ (-#(
ζ(6− %) = −&
"
%
,
∞∑
"=!
&
"
%!'
" ='
(
$ − 6
,
&
"
,+)&1 (-+ ?+"&!%**) &%7,+"8 @-+ $!"7%*# 9:8;< ). +.(#,*).-+0 ,4 7+#&.
!$ (-+ '!&(!%" )&(+1"#* 0).'!5+"+0 ,4 A)+7#&&8 $!"7%*# #//#"+&(*4 )7/*)+.
(-+ 0+.)"+0 )&0+/+&0+&'+ !$ &
",# !& (-+ '-!)'+ !$ !8 B+ &!(+ #*.! (-#( )$
) ⊂ C ). #& #",)("#"4 .%,C+*0> #&0 (* ) ⊂ ) (-+& 3+ -#5+ $"!7 (-+
$!"7%*# 9:8D< (-#( &
",# ∈ ) -+&'+ (-+ 0).("),%()!& +
"
). # ) 25#*%+0
0).("),%()!& !& * 8
!"#"$ % &'()*(+)& ,-."/01!"2 3456$7 849:/7;$0/< 5/6 =06:!5. 90.=< )'>%? *"3;"=1".?@AAB @@ C DE
!"#$%!#
!" #! $ "%&%'%()*$' +),(- $,. C(! , ) #! "/! "%&%'%()*$' 0%.1'! %2 $'' 34$'1!. 21,*")%,5 %, $ &+%6,)"! 5!" ! 7
!"#$%$&#
8 0!$51+! %, ! 9)"/ 4$'1!5 ), "/! "%&%'%()*$' 30%.1'! A )5 $ *%,"),1%15
/%0%0%+&/)50 %2 30%.1'!5
µ : C(! , ) −→ A.
:/! +!5"+)*")%, %2 µ "% "/! 351#0%.1'! !"#(! , ) ⊂ C(! , ) .!6,!5 $.)5"+)#1")%, 9/)*/ 9! .!,%"! #; "/! 5$0! '!""!+ µ- $,. "/! 0!$51+! µ )5
1,)<1!'; .!"!+0),!. #; "/! *%++!5&%,.),( .)5"+)#1")%, 5),*! "/! 351#0%.1'!
!"#(! , ) )5 .!,5! ), C(! , )7 :/! '$5" 5"$"!0!," !=&+!55!5 "/! 9!'' >,%9,2$*" $#%1" "/! 1,)2%+0 *%,"),1)"; %2 $ *%,"),1%15 21,*")%, %4!+ $ *%0&$*"
"%&%'%()*$' 5&$*!7
!"#"$ % &'()*(+)& ,-."/01!"2 3456$7 849:/7;$0/< 5/6 =06:!5. 90.=< )'>%? *"3;"=1".?@AAB @C D EF
!" "# $!%&'(#) *%+ $,!&#( &-.)'%/ !0 12# 3*1# 4#,( C
5 ⊂ C
5 *%( ,#1
A .# * $!67,#1# 86!(-,# "'12 1!7!,!/+ /'9#% .+ * %!)6 | · |A !% A
$!67*1'.,# "'12 12# %!)6 | · |
!% C
&! 12*1 12# 0!,,!"'%/ $!%('1'!%& *)#
&*1'&4#:
• 0!) ! ∈ A 12# #;-*,'1+ |! |A = < '& #;-'9*,#%1 1! ! = <5
• 0!) " ∈ 5 ! ∈ A: |"! |A = |"|
|! |A5• 0!) *,, ! , # ∈ A: |! + # |A < 6*=(|! |A, |# |A)>32#% 12# 0*$1 12*1 * ('&1)'.-1'!% ?* &+&1#6 !0 0-%$1'!%& µ(!) : $
!
→ A@ /'9#&
)'&# 1! * A89*,-#( 6#*&-)# !% $ '& #;-'9*,#%1 1! 12# $!%('1'!% 12*1 12#
&+&1#6 µ(!)'& .!-%(#(5 '>#> 0!) &!6# $!%&1*%1 % > < *%( 0!) *,, & ∈ ' 5 ! ∈ $
!
12# 0!,,!"'%/ -%'0!)6 #&1'6*1# 2!,(&:
|µ(!)(!)|A < % . ?A>B<@
32'& $)'1#)'!% '& *% #*&+ $!%&#;-#%$# !0 12# %!%8C)$2'6#(#*% 7)!7#)1+
|! + # |A ≤ 6*=(|! |A, |# |A)
!0 12# %!)6 | · |A ?&## DE*FGH5 DI'FJH@> K% 7*)1'$-,*) '0
A = = O
= {! ∈ C
| |! |
≤ B} '& 12# &-.)'%/ !0 '%1#/#)& '% 12#3*1# 4#,( C
12#% 12#  !0 O
89*,-#( ('&1)'.-1'!%& !% $ $!'%$'(#& "'12
O
89*,-#( 6#*&-)#& ?'% 0*$15 .!12 & *)# 8*,/#.)*& "'12 6-,1'7,'$*1'!%
(#4%#( .+ $!%9!,-1'!%>
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!"#$%!
◦&' !" #$%&'#(& )*++"' (,-.'*"-("% #-/ &!" 0#/1( 2"331-
&'#-%4,'+
5 *%"4*3 ('1&"'1,- 4,' &!" "61%&"-(" ,4 # +"#%*'" 71&! .18"- 9',9"'&1"% 1%:
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!"## $%&'()* +#' {,
} -# & "."'#/ 0, 102'32404" ,421'302" ,
∈ C(5 ,O!
) 32'6# 7328 C(5 ,O
!
) 0, &99 102'32404" ,421'302" 02 '6# 10/ &1' '0'&99.
:3"1022#1'#: 8704 5 ;3'6 <&94#" 32 '6# 7328 0, 32'#8#7" O!
0, C!
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)* +#' &9"0 {&
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(Z)) .$ .$%1%"6/.7 % /' 6%+9#%1.(+ ".#) %2 /'
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#
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!"#$%&'
!"#$$ %&#%
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!
"
! − 5
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=
!
#!
67,.!+-"#$ %#/$!8
= 5, !
= −5
9
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=5
:
, #
=
$
= · · · = ;, %
= −5
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=5
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= −>
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= −@
:
, !&
=<:5@
>5;
!'
= −=<A:@
@?A
, . . . ).
B(! �
ζ(#) = −(9π$)
9
,%
(&) =(9π$)
(# − 5)!
[
−
9#
+∞
∑
"=!
σ −!(')(
"
]
.
)
%
(&) = 5+ 9=;
∞∑
"=!
σ#
(')(" ∈M%
( !(9, Z)),
)
&
(&) = 5− >;=
∞∑
"=!
σ$
(')(" ∈M&
( !(9, Z)),
)
'
(&) = 5+ =A;
∞∑
"=!
σ(
(')(" ∈M'
( !(9, Z)),
)
!
(&) = 5− 9:=
∞∑
"=!
σ)
(')(" ∈M!
( !(9, Z)),
)
!"
(&) = 5+:>>9;
:?5
∞∑
"=!
σ!!
(')(" ∈M!"
( !(9, Z)),
)
!%
(&) = 5− 9=
∞∑
"=!
σ!#
(')(" ∈M!%
( !(9, Z)).
*+,,- 0!! -( CD!++!@;EF
!"#"$ % &'()*(+)& ,-."/01!"2 3456$7 849:/7;$0/< 5/6 =06:!5. 90.=< )'>%? *"3;"=1".?@AAB C@ D CE
!"# $%&'(#!#)%* %+ #,- .!&!*(/!* +(*$#)%*0
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:=
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!=
∑
" |!
!
− "
! =
∞∑
"=
!
− "
"
#− "
"
. $%& '()**+')( ,)'" +* "%)" ∆ = (# !
"
− #
#
$
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0%&1& #
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$
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7
!"#$%&'( )&%* +,-./0+ *&& !!!"#$% &'( )*+,-.) /012(3 4'(536 7674(18%
!!"#$$"%&'()%! (*+,-&./%*0(1&
:=∞
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2" 3 456*)7.589:;9.<=>?<.$78+?<.@AB7?<:;@@
2" 3 C56*)7.589:;9.<D>?<.$78+?<.@AB7?<:;@
2" 3 E/F!%5778A:C;> C@<D+78+=;C> 4@<:@$8G:H
! "#$ %" & "'"$ %( ! )#*"$ %# & #+(,$ %' ! -,#+$ %- ! )-*##$ %*
& +##+,$ %+ ! ))(-#($ %. ! ))'.",$ %), & '(#-)"$ %))
! (*,.##$ %)" ! '***(+$ %)( & #,)+'-$ %)# & )")*)-,$ %)'
& .+*)(-$ %)- ! -.,'.(#$ %)*& "*"*#("$ %)+ & ),--)#",$ %).
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!"#"$ % &'()*(+)& ,-."/01!"2 3456$7 849:/7;$0/< 5/6 =06:!5. 90.=< )'>%? *"3;"=1".?@AAB CD E CF
!"#$%&"'& !( )*+*"%,*"
τ( ) ≡∑
|!
!
!" #$% :
! " #$%&'()*+,-./0+
1+2 3 )4567, ) ,8/5674 ) /298567: ) 92/8/5678 ) 8,8422567/
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) ,80489804/567+4 ) 8;/442,/22567+: ) +,8+9;288/;567+8
) ,8:9+:/2:98567+/ ) :08098::::;567+9
) 042849/:/9+567+; ) +/;8;,+,:;22567+0 < =#67,2-
!"# $"!%"&'( !) *!'$+,-.% τ( ) /(## 012!&.#34
>?=@?AB
#BA@BA- B+,C3BDEF&(GHDGIJ#@(II(2#+-K +,-L '+C3M(JNJ#B+,-O,PL
>DQ%GR%GN%J#'+O+PK +22-L SD%TTNUN%V'J#W+-L
#>A?X- (#V-3NT#VY+K 2K !D&UD%TT#Z5%'(#Z<Z5=#Z7V--7,:K V--
#>A?X- ['(F#V-3NT#VY+K 2K !D&UD%TT#Z5#JFI#N3+K #J6G'NV'#;5V)9-<+-\,K
#)+-7N5#,5N)+-5Z7##N7,)N-.,-K =#Z7V---7;K V--L]
! " '(F#/0++-
14 3 )/+82+,9208+:94/24+:;;
! " ^^
555 &(J' G%JF&' UDI!F'%E NV 4K948 IJ_
!"#"$ % &'()*(+)& ,-."/01!"2 3456$7 849:/7;$0/< 5/6 =06:!5. 90.=< )'>%? *"3;"=1".?@AAB CD E CF
!"#$#% &!'()%*)+%'
!" Cp = Qp #!$%"! "&! '%()*!"+%$ %, -$ -*.!/0-+' '*%120! %, "&! 3!*#
Qp %, p4-#+' $2(/!015 6+7 - )%1+"+8! +$"!.!0 N 9 - :+0+'&*!" '&-0-'"!0
ψmodN -$# '%$1+#!0 "&! '%((2"-"+8! )0%3$+"! .0%2)
Y = YN,p = lim←−
m
(Z/NpmZ)∗
-$# +"1 .0%2) X = Homcont(Y,C×p ) %, ;'%$"+$2%2%1< p4-#+' '&-0-'"!01
;"&+1 +1 - Cp4-$-*="+' +! .0%2) -$-*%.%21 "% Homcont(R×+,C
×) ∼= C ;/=
s 7→ (y 7→ ys)<5 >&! .0%2) X +1 +1%(%0)&+' "% - 3$+"! 2$+%$ %, #+1'1
U = {z ∈ Cp | |z|p < 1}5>&! p4-#+' L4,2$'"+%$ L : X → Cp +1 - (!0%(%0)&+1 ,2$'"+%$ %$ X
'%(+$. ,0%( - p4-#+' (!-120! %$ Y 5
?5 !"#$%$&'"( )*%+&#
%, '%$1"02'"+$. "&!1! ,2$'"+%$1 +1 ,0%( "&! 1)!'+-* '0+"+'-* 8-*2!1 %,
'%()*!7 L4,2$'"+%$1 ;@&+'& -0! %,"!$ -*.!/0-+'9 -,"!0 - 12+"-/*! $%04
(-*+1-"+%$<5 !" 21 37 -$ !(/!##+$. ip : Q → Cp +$ %0#!0 "% '%$1+#!0
-*.!/0-+' $2(/!01 -1 p4-#+' $2(/!015
!"#$%& ?5?' ()& *+&#",, -&." /0,1.+2,'
ζ(s) =∏
l primes
(1− l−s)−1 =
∞∑
n=1
n−s (Re(s) > 1), ζ(1− k) = −Bk
k
,
@&!0! Bk -0! "&! A!0$%2**+ $2(/!01 .+8!$ /=
eBt =
∞∑
n=0
Bntn
n!=
t et
et − 1
.
B2" ,%0 c > 1 '%)0+(! "% p
ζ(c)(p)(−k) = (1− p
k)(1− ck+1)ζ(−k)
+*&!*) ?5C ;D2((!0<, 324 ",5 $2%5,2#+"% h(x) =∑n
i=0 αi xi ∈
Zp[x] 26&4 Zp 701) .)". x ∈ Zp =⇒ h(x) ∈ pmZp 2,& )"7
n∑
i=0
αi ζ(c)(p)(−i) ∈ pmZp (1.1)
>&+1 )0%)!0"= !7)0!11!1 "&! ,-'" "&-" "&! $2(/!01 ζ(c)(p)(−k) #!)!$#
'%$"+$2%21*= %$ k +$ "&! p4-#+' 1!$1!E
!"!##$"% &'(' 8&. k1, k2 ∈ N∗9 k1 ≡ k2 (mod(p− 1)pm−1)9 .)&,
ζ(c)(p)(−k1) ≡ ζ
(c)(p)(−k2) (modpm)
!"# $"%&'( ') *'!+%,-*%.!/ p0 (.* L0)-!*%.'!+ 1
!"##"$ ! "#$%&" !' !()& h(x) = xk1 − x
k2.
%&''( '( )*#'&#+ ,-, " *+, &- ./ !0& 1&,,2)3'13 4'5*#,(6
Sk(N) =N−1∑
n=1
nk =
1
k + 1[Bk+1(N)−Bk+1] (1.2)
3 10 %0 Bk(x) = (x + B)k =∑k
i=0
(ki
)Bix
k−i " !0& 7&53'#,, +',/3'2
* (,8
!"#$%$&# '()( ./ 0#1 A 2# . !'&+#" 1'3'4'567.4 &6!5 7'!1.6!6!5 Zp
.8 . 74'8#" 892&6!5 .!" 4#1 h 2# . 3'8616:# 6!1#5#&- ;'!86"#& 1*# A<
892+'"94# Ch(Y,A) '( .44 4'7.44= 3'4=!'+6.4 (9!716'!8 '( "#5&# < h '!
Y >'( 1*# :.&6.24# yp : Y → Z∗p$ 1*# 7.!'!67.4 3&'?#716'!/@ 6! 3.&16794.&$
1*# A<892+'"94# C1(Y,A) 7'!86818 '( 4'7.44= 7'!81.!1 (9!716'!8 '! Y
A61* :.49#8 6! A- ( A = Cp 1*#!
∀h > 1 Ch(Y,A) ⊂ Cloc− an(Y,A) = {ϕ : Y → A | ϕ 4'7.44= .!.4=167} ⊂ C(Y,A),A*#&# C(Y,A) = {ϕ : Y → A | ϕ 7'!16!9'98}-
2/ B "681&62916'! Φ '! Y A61* :.49#8 6! . !'&+#" A<+'"94# V 68 .!
A<46!#.& +.3 Φ : C1(Y,A)→ V $ ϕ 7→∫
Yϕ dµ-
7/ B +#.89&# Φ : C1(Y,A)→ V 68 . 2'9!"#" "681&62916'!C |Φ(ϕ)|p <C|ϕ|p A*#&# C "'#8 !'1 "#3#!" '! φ-
"/ 0#1 h ∈ N∗- B! h<."+688624# +#.89&# '! Y A61* :.49#8 6! V 68
.! A<46!#.& +.3 Φ : Ch(Y,A)→ V A61* 1*# ('44'A6!5 5&'A1* 7'!"616'!C
('& .44 t = 0, 1, . . . , h− 1,∣∣∣∣∫
a+(Npm)
(yp − ap)tdΦ
∣∣∣∣p
= o
(p
m(h−t)).
('& m→∞-
94 A = Cp !0&3 (%%'5- 3: !' ;<=> (3- ;= ?@>A "#%0 ( *(+ Φ %(3
.& #3 B#&,/ &C!&3-&- !' !0& A2*'-#,& Cloc− an(Y,A) '4 ,'%(,,/ (3(,/! %
4#3%! '3" '3Y '4 !0& +(5(*&!&5 yp : Y → Z×p 8
!"#$"% D8E FG(H#5I& )*#&# #D6818 . 9!6E9# >2'9!"#"/ +#.89&# µ(c)
'! Z×p A61* :.49#8 6! Zp 897* 1*.1
∫
Z×
p
xkdµ
(c) = ζ(c)(p)(−k), k ≥ 0
F#+.&G- J0&'5&* D8K " &B# L(,&3! !' J0&'5&* D8E F./ 3!&:5(! '3 '4
h (:( 3"! µ(c))8
93 !0& +5&"&3! +(+&5 1& %'3"!5#%! p2(- % - "!5 .#! '3" '3 Y 1 !0
L(,#&" 3 Cp "!(5! 3: 45'* - "!5 .#! '3" 1 !0 L(,#&" 3 "+(%&" '4 *'-#,(5
4'5*"8
!"#$#% &!'()%*)+%'
!" p#$%&' L#()*'+&,* ,( -).,+$#/",0,1%+ ! "#$ %$&'%'&(# ) *+,)" ',
Lp : X → C×p - .$, /0
Lp(x) :=
∫Yxdµ
(c)
1− x(c)c, x ∈ X (1.3)
12 "# 3 ! ,-4$ ! %(4$ ('4$ 3" x = y−1p 56 3,7 "#$ *+,)" ', 189:5 !
,7$($,7$," '* 3 )#' )$ '* c; *'& 344 < & )#4$" )#3&3)"$&! χ mod pm6
χ : Z×p → Q× → C×p ',$ #3!
Lp(χykp) = (1− χ(p)pk)L(−k, χ) ∈ ip(Qab).
=, -$,$&34 $.$&0 7 !"& /+" ', ', Y 2 "# .34+$! , Zp 7$>,$! 3 p?37 )
L?*+,)" ', 1"#$ ,',?@&)# %$7 3, A$44 , "&3,!*'&% '* µ5;
Lµ : X → Cp, µ(x) =
∫
Y
x(y) dµ.
=* D(s, χ) =∑
n≥1 χ(n)cnn−s
! 3, 3& "#%$" )340 7$>,$7 )'%(4$B L?
*+,)" ', "2 !"$7 2 "# 3 < & )#4$" )#3&3)"$& χ 2 "# "#$ (&'($&"0 D∗(−k, χ) ∈Q *'& 3, ,>, "$ !$" '* )'+(4$! k, χ 13,7 2 "# 3 ,'&%34 C3" ', D∗(s, χ)'/"3 ,$7 /0 %+4" (40 ,- D(s, χ) 2 "# )$&"3 , $4$%$,"3&0 *3)"'&!56 ',$
)',!"&+)"! +!+340 "#$ )'&&$!(',7 ,- p?37 ) L?*+,)" ', L = LµD !"3&"?
,- *&'% "#$ 34-$/&3 ) !($) 34 .34+$! D∗(−k, χ) , !+)# 3 230 "#3"
LµD(χykp) =
∫
Y
χykp dµD = ip(D∗(−k, χ)),
3,7 "#$ $B !"$,)$ '* !+)# 3 %$3!+&$ ! $D+ .34$," "' -$,$&34 C$7 E+%?
%$& )',-&+$,)$! *'& "#$ !($) 34 .34+$! D∗(−k, χ)9 F'&%+43! *'& "#$!$
.34+$! )'+47 /$ D+ "$ )'%(4 )3"$7 3,7 ',$ +!$! .3& '+! %$"#'7! ,
'&7$& "' '/"3 , !+)# )',-&+$,)$! 14 G$ "#$ *'&%+43! '* "0($ 189H5 ,
"#$ (&''* '* "#$ I#$'&$% 859 F'& %'7+43& *'&%! ',$ +!$! -$'%$"& )
"''4! 4 G$ %'7+43& !0%/'4!6 )'," ,+'+! *&3)" ',!6 "#$ J3,G ,?K$4/$&-
%$"#'7 $")96 1.' & LA3,M:N6 LJ3OHN6 LA3,?P3N6 LPQRAN59
S$ (&'('!$ 3 ,$2 %$"#'7 2# )# (&'7+)$! 3 *3% 40 '* p?37 ) %$3?
!+&$! !"3&" ,- *&'% 3 7 !"& /+" ', Φ ', Y 2 "# .34+$! , 3 !+ "3/4$
.$)"'& !(3)$ M =⋃
m≥0M(Npm) '* %'7+43& *'&%!T "# ! *3% 40 '*
p?37 ) %$3!+&$! µα,Φ,f ! (3&3%$"& C$7 /0 ,',?C$&' $ -$,.34+$! α ∈Cp 6= 06 '* "#$ '($&3"'& U '* @"G ,?Q$#,$& ', M6 3,7 /0 3 (& % " .$
)+!( $ -$,*'&% f 2 "# 3, 3!!') 3"$7 $ -$,.34+$ α 6= 0 1', 3, $3!0
%'7 >)3" ', f0 '* f 3! 3, $ -$,*+,)" ',59 U,$ !30! "#3" 3 (& % " .$
)+!( $ -$,*'&% f =∑
n≥1 a(n, f)e(nz) ∈ Sk(Γ0(N), ψ) ⊂ M 12#$&$
(e(nz) = exp(2πinz))5 ! 3!!') 3"$7 "' 3, $ -$,.34+$ α * "#$&$ $B !"! 3
)+!( *'&% f0 = f0,α =∑
n≥1 a(n, f0)e(nz) !+)# "#3" f0 | U = αf0 3,7
f0 | T (ℓ) = a(ℓ)f0 *'& 344 (& %$ ,+%/$&! ℓ ∤ Np5
=, "#$ '&7 ,3&0 )3!$ |α|p = 1 3 )',!"&+)" ', '* !+)# %$3!+&$! )'+47
/$ '/"3 ,$7 *&'% V 73W! 7$%('"$," e = limr→∞ U(p)r!
1!$$ V 73
LV X:N5 3)" ,- ', p?37 ) %'7+43& *'&%!T "#$ %3-$ '* e ! )',"3 ,$7 ,
!"# $"%&'( ') *'!+%,-*%.!/ p0 (.* L0)-!*%.'!+ 1
!"#!$ %& Mord ⊂ M '( )*+,& -+.&*!+'* /0,1& '2-+* 23 $ 2, '( M45
61+%1 +! 7*'6* ,' #& 8&*&2 ,&- #3 %&2, +* %9 !!+% 9 .'-"9 2 ('2.!:
;*& '#, +*! µα,Φ,f = ℓf (eΦ) !" # $%&'#()* )&+*#" !", ℓf ∈ Mord ∗- .+
# ,!"* /*+*"#) 0#$* 12*+ α 6= 0 !+* 0!%)3 &,&'#'* '2&$ ,*'2!3 %$&+/
&+$'*#3 ! Mord'2* 4"&,#"5 602#"#0'*"&$'&07 $%($4#0* Mα ⊂ M ! U
612&02 &$ #)$! ! 8+&'* 3&,*+$&!+7-
9- !"#$!%&#!'(" )!#* +,-&." !( /'0&-,$ 1'$/"
:*' A (* #+ #)/*("#&0 *;'*+$&!+ K ! Qp !" &'$ "&+/ ! &+'*/*"$
OK - :*' %$ 8; #+ *,(*33&+/ ip : Q → Cp #+3 )*' Mk(Γ1(N);A)<Mk(Γ0(N)ψ;A) (* '2* $%(,!3%)*$ ! A[[q]] /*+*"#'*3 (5 '2* q=*;4#+$&!+$
f =∑
n≥0 an(f)qn ∈ Mk(Γ1(Np
m),Q) ! 0)#$$&0#) ,!3%)#" !",$
1&'2 #)/*("#&0 >!%"&*" 0!*?0&*+'$ an(f) ∈ Q &+ i−1p (A)- @+* 4%'$
M = ∪m≥0M(Npm)< 12*"* M(Npm) = Mk(Γ1(Npm);A)< #+3 S =
∪m≥0S(Npm) '2* A=$%(,!3%)* ! 0%$4 !",$-
!"#$%&' () *+',-+./,+(0' 1+,2 3"%/&' +0 M4
:*' Φ : C1(Y,Cp)→M (* # 3&$'"&(%'&!+ !+ A 1&'2 B#)%*$ &+ M-
#7 +'&0',&+0 *+',-+./,+(0'4 >!" # 0!,4)*; +%,(*" s ∈ C #+3 a, bmodN4%' 6(5 #+#)5'&0 0!+'&+%#'&!+7C
Eℓ,N(z, s; a, b) =∑
(cz+ d)−ℓ|cz+ d|−2s (0 6= (c, d) ≡ (a, b)modN) .
D'#"'&+/ "!, '2&$ $*"&*$ < !+* !('#&+$ '2* !))!1&+/ E&$*+$'*&+ 3&$'"&=
(%'&!+$C 4%' s = −r< 0 ≤ r ≤ ℓ− 1<
Er,ℓ,N(a, b) :=N
ℓ−2r−1Γ(ℓ− r)
(−2πi)ℓ−2r(−4πy)r∑
amodN
e(−ax/N)Eℓ,N(Nz,−r;x, b)
= εr,ℓ,N(a, b) + (4πy)−r∑
0<dd′,d≡a,d′≡bmodN
sgn d· dℓ−2r−1W (4πdd′y, ℓ− r,−r)
· e(dd′z) ∈Mr′,ℓ(N), 12*"* r′ = max(r, ℓ− r − 1) ,
W (y, ℓ−r,−r) =r∑
j=0
(−1)j(r
j
)Γ(ℓ− r)
Γ(ℓ− r − j)y
r−j,
ζ(s; a,N) =∑
n≥1n≡a(mod N)
n−s,
εr,ℓ,N(a, b) =(−4πy)sΓ(ℓ+ s)
Γ(ℓ+ 2s)δ
(b
N
)ζ(1− ℓ− 2s; a,N)
∣∣∣s=−r
+Γ(ℓ+ 2s− 1)
(4πy)ℓ+s−1Γ(s)δ
(a
N
)[ζ(ℓ+ 2s− 1; b,N) + (−1)ℓ+2sζ(ℓ+ 2s− 1;−b,N)
]∣∣∣s=−r
F2*$* $*"&*$ #"* &+ /*+*"#) +*#")5 2!)!,!"42&0 ,!3%)#" !",$ 6$** D*0=
'&!+ G7 &+ $4#0*$ Mr′,ℓ(N2)< 12*"* r
′ = max(r, ℓ− r− 1) (%' &+ 0*"'#&+
!"#$#% &!'()%*)+%'
!"#" $%#& !'# %()(*('+%, - #./. ℓ ≥ 3- r = 0 (' r = l − 10 !12 $%#"#
"#',#" +'(23 # 2,"$',43$,(1" (1 Y × Y 5,$% 6!)3#" ,1 M7
Er,ℓ((a+ (Npm)× (b+ (Npm)) := Er,ℓ,Npm(a, b) ∈Ml(N2p2m).
40 !"#$!% &'()%!" *'"&+, 8(' !1& f =∑
n≥0 an(f)qn ∈ Mk(Γ1(Np
m)(1# +3$"
Φf (a+ (Npm)) :=∑
n≥0n≡a( mod pm)
an(f)qn ∈Mk(Np
2m)
0 !"#$!% #-.#! +."$.+ 9!)"( 5,$% ! "+%#', !) +()&1(*,!)0- "## :;,<=>.
/.&!"0+, ,0 8(' !1& ?,', %)#$ %!'! $#' χ mod pm6,#5#2 !" ! @31 $,(1
(1 Y 5,$% 6!)3#" ,1 ip(Qab)- $%# ,1$#/'!)
∫
Y
χ(y) dΦf = Φf (χ) =∑
n≥0
χ(n)an(f)qn ∈Mk(N
2p2m)
(,1 ,2#" $%#1 5,$% $%# $5,"$#2 *(23)!' @('* fχ.
,,0 A%# 2,"$',43$,(1" !0- 40- 0 !'# 4(312#2 9!@$#' ! '#/3)!',"!$,(1 (@
$%# (1"$!1$ $#'* ,1 !00 5,$% '#"+# $ $( $%# pB!2, 1('* (1 M =∪pmMk(Γ1(Np
m), A) ⊂ A[[q]] /,6#1 4& |g|p = supn |a(n, g)|p @(' g =∑n≥0 a(n, g)g
n ∈Mk(Γ1(N), A) 0.
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N f =∑
γ∈Γ0(Npm)\Γ0(N)
f |k γ.
K1# (4$!,1" !@$#' ! 1('*!),"!$,(1 ! +'(H# $('
π(f) = [Γ0(N) : Γ0(Npm)]−1TrNpm
N f
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3!($! )$*# %& M )&' %& S 45 g | U =∑
n≥0 a(pn, g)qn6 3!"-" g =∑
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7
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M(N0pm)
πα,m//
Um
��
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≀(Uα)m
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n≥1
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⋂
n≥1
Im(U − αI)n ∩M(N0) =⋂
n≥1
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@"# g =∑
n≥0 a(n, g)gn ∈ Mk(Γ1(N), A) #%"* |g|p = supn |a(n, g)|p
)/ 0 :"332'"1*"' p20')( *$&- $*
M = ∪m≥0Mk(Γ1(Npm), A) ⊂ A[[q]].
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∫YϕΦα := (Uα)−m
πα,0
(( ∫Yϕ dΦ
)|
Um)∈Mα
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Φα(a+ (Npm)) = (Uα)−m′[πα,0(Φ(a+ (Npm)) | Um′
]. (5.1)
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∫
a+(N0pm)
ytp dΦ ∈M(N0p
κm) 1.&"# h = [ordp α] + 16 (6.1)
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′'+?;&$%"0< 0(-:$ 1.&"#
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∫
a+(N0pm)
ytpdΦ
α = (Uα)−m′πα,0
((∫
a+(N0pm)
ytpdΦ
)| Um′
).
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tpdΦ ∈ M(N0p
mκ) 4'- &11 <+-+8/1#$ 8/&-&8$#-, χmodN0pm
=.+$/
0&1(#, +) A> %#8&(,#
∫
a+(N0pm)
ytdΦ =
1
ϕ(N0pm)
∑
χmodN0pm
χ−1(a)
∫
Y
χytp dΦ.
A-**! *! =#$*-$3 B7C7 7$ ,(?8#, $' ,/'. $/&$ 4'- & 8'),$&)$ C > 0&)* 4'- &11 $/# '@#) ,(%,#$, '4 $A@# a + (Npm) ⊂ Y ')# /&, |Φα(a +(Npm))|p ≤ C! BA '(- &,,(9@$+') $/#-# #C+,$, m
′ ∈ N ,(8/ $/&$
Φ(a+ (Npm)) = Φ(χ(a+(Npm))) ∈M(Npm′+1),
$/#) $/# α3@-+9&-A @&-$ Φα'4 Φ +, 2+0#) %A =D!5>E
Φα(a+ (Npm)) = (Uα)−m′[πα,0(Φ(a+ (Npm)) | Um′
].
F) $/# α3@-+9&-A ,(%,@&8# Mα ⊂ M ')# /&, Uα = αI + Z 4'-
& )+1@'$#)$ p3+)$#2-&1 '@#-&$'- ZE 4'- &11 g =∑
n≥0 a(n, g)qn ∈ MG
g | U =∑
n≥0 a(pn, g)qnG |g | U |p ≤ |g|p &)* |g | Z|p ≤ |g|p!
H#C$ &11 $/# 4()8$+'), Φα(a+(Npm)) = α−m(α(Uα)−1)m [πα,0(Φ(a+ (Npm)) | Um]
&-# %'()*#* %#8&(,# |α−1|p = 1 &)*
(α(Uα)−1)m = (α−1Uα)−m = (I + α−1Z)−m =
n−1∑
j=0
(−mj
)α−jZ
j.
(6.2)
! "#$%$& '"()*&+*,&(
!""# "# $%&"!&' ()*) !" ordp α > 0 #$% &!" h = [ordp α] + 1' (!$)!*$! +#, "* -*.$%
∫
a+(N0pm)
(yp−ap)tdΦα = α
−mκ(α(Uα)−1)mκ
[πα,0
(∫
a+(N0pm)
(yp − ap)tdΦ
)| Uα
].
/+! $*01, *2 "+! *3!0#"*0,
(α(Uα)−1)κm = (α−1Uα)−κm =n−1∑
j=0
(−κm
j
)α−jZ
j
#0! .$42*01&5 -*.$%!% -5 C1 > 0 #, n = dimMα%*!, $*" %!3!$% *$
m' (!$)! 2*0 #&& t = 0, 1, . . . , hκ − 1 *$! +#,
∣∣∣∫
a+(N0pm)
(yp − ap)tdΦα
∣∣∣p≤ C1· |α|−mκ
p ·Maxy∈a+(N0pm) |yp − ap|t· |Φ|p
≤ C1·C2|pm|t−ordp α·κ = o(pm(hκ−t)), h > ordp α
#, m→∞ 6-!)#.,! |Φ|p ≤ C2 #$% |α|−mp = p
m ordp α7'
8' !"#$% &'$'('#)&*+ p,"-*+ ('-.$"# /'#(0 '/ 1%)! r ≥ 0
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#-&!, 6*?!0 "+! )*13&!< $.1-!0, q = e(z) = e2πiz
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Pr(A) ={g =
∑rj=0 ω
−jgj :4"+ gj =
∑n≥0 a(j, n, g)q
n ∈ A[[q]]}' B*$>
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+*&*1*03+4) 2.$)"4*$, 6,!! C(4DEF7 Qr,a ={∑r
j=0 Im(4πz)−jgj(z) :4"+
gj =∑
n≥0 a(j, n, g)e(nz/a)}#$% 3." Qr = ∪a≥1Qr,a'
2&!'#!( 8'G3 +, $%& A-'"./0& Mr,k(Γ1(N), A) ⊂ A[[q1]][ω−1] "#
'"./0+! #"!'1 "# 234& r ≥ 0 +5. "# 6&78%2 k ≥ 1 #"! Γ1(N) 71 8&5&!+2&.93 2%& 1&!7&1 g ∈Mr,k 672% +08&9!+7: :"&;:7&521 a(j, n, g) ∈ ip(Q) 1/:%2%+2 2%& :"!!&14"5.&52 :"'40&< 1&!7&1 =.&5"2&. +01" g,
g = i−1p (g) =
r∑
j=0
Im(4πz)−j∑
n≥0
i−1p (a(j, n, g))e(nz) ∈ Qr,1
1+271#3 2" 26" :"5.727"51> ∀γ ∈ Γ1(N)? g|kγ = g +5. ∀γ ∈ SL2(Z)?g|kγ ∈ Qr)
9, /2Mr,k =Mr,k(N, p) = ∪m≥0Mr,k(Npm)? 6%&!&Mr,k(Np
m) =Mr,k(Γ1(Np
m), A))
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! !"#$%&' (&"% ")"%*+,- ,.&).+&,) .&(,&. /0 1&,23+
ℓ > 0
"#$$ %&'()*+, -./ s ∈ C '01 a, bmodN 1$20$ "'# 30 4$563.0 7+ 68$
93#$0#6$30 #$/3$#:
Eℓ,N(z, s; a, b) =∑
(cz+ d)−ℓ|cz+ d|−2s (0 6= (c, d) ≡ (a, b)modN) .
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>B63.0 @368 C'?B$# 30 0$'/?D 8.?.=./E835 <./=# "<./ '?? s = −r @368
0 ≤ r ≤ ℓ− 1+:
Er,ℓ,N(a, b) :=N
ℓ−2r−1Γ(ℓ− r)
(−2πi)ℓ−2r(−4πy)r∑
amodN
e(−ax/N)Eℓ,N(Nz,−r;x, b)
= εr,ℓ,N(a, b) + (4πy)−r∑
0<dd′,d≡a
d′≡b mod N
sgn d· dℓ−2r−1W (4πdd′y, ℓ− r,−r)
· e(dd′z) ∈Mr′,ℓ(N2),
" !F+
@8$/$ r′ = max(r, ℓ− r − 1)G
W (y, ℓ−r,−r) =r∑
j=0
(−1)j(r
j
)Γ(ℓ− r)
Γ(ℓ− r − j)y
r−j, ζ(s; a,N) =
∑
n≥1n≡a(mod N)
n−s,
εr,ℓ,N(a, b) =(−4πy)sΓ(ℓ+ s)
Γ(ℓ+ 2s)δ
(b
N
)ζ(1− ℓ− 2s; a,N)
∣∣∣s=−r
+Γ(ℓ+ 2s− 1)
(4πy)ℓ+s−1Γ(s)δ
(a
N
)[ζ(ℓ+ 2s− 1; b,N) + (−1)ℓ+2sζ(ℓ+ 2s− 1;−b,N)
]∣∣∣s=
H! 4,.+(,56+,/). 1,+3 7"%6&. ,) )&"(%* 3/%/#/($3,- p8"9,-
#/96%"( 0/(#. /0 +*$& r ≥ 0
I$6 g =∑r
j=0 ω−j
∑n≥0 a(j, n, g)q
n ∈Mr,k ⊂ A[[q]][ω−1]! JB6 |g|p =supn,j |a(j, n, g)|p! K0$ 8'# ' pA'135 0./= .0 Mr,k!
I$6 α >$ '0 $3;$0C'?B$ .< U .0Mr,kG g|Um = pm(k/2−1)
∑u mod pm g|
(1 u0 pm
)=
p−m
∑umod pm g
(z+upm
)G
Im(z + u
pm
)=Im z
pm
=⇒ g|Um =k∑
j=0
ω−jp
mj∑
n≥0
a(j, pmn, g)qn
"'0 306$;/'? .E$/'6./ .0 Mr,k+!
!"#$%$&# '()( ! M(α)r,k = Ker(U − αI)"
#! Mαr,k =
⋃n≥1Ker(U − αI)n"
$! Mαr,k(Np
m) =Mr,k(Npm) ∩Mα
r,k%
! "#$%$& '"()*&+*,&(
!"#$"% !"& ! "#$ %% m ≥ 1 &'( A)*#+,%(Mαr,k(N0p
m) =Mαr,k(N0)
-. #/ 01-&( $ 12 1+ -& +#(. 1#& +(3(1+ #1 m4
5! 6'( /#%%#7-18 +- 8$ * -. 9#**,& &-:( ;/#$ A = Qp!
Mr,k(N0pm)
πα,m//
Um
��
Mαr,k(N0p
m)
≀(Uα)m
��
M(N0) πα,0
// Mαr,k(N0)
;πα,m -. &'( 3$#<(9&#$ #1&# &'( α)3$-* $= .,5.3 9( 7-&' &'( 2($1(%⋂n≥1 Im(U − αI)n ∩Mr,k(N0) ;(>, % &# &'( +-$(9& .,* #/ %% &'( #&'($
3$-* $= .,5.3 9(.? πα,m = Rα,m(U) 7-&' .,-& 5%( 3#%=1#*- % Rα,m ∈A[x] .(( @$#3#.-&-#1 A4B!4
!"#$%$&# '()( C(& α ∈ A 5( 1#1)D($# (-8(1: %,( #/ &'( #3($ &#$ U
#1 Mr,k 1+ Φ : C1(Y,A) →Mr,k +-.&$-5,&-#14 6'( α)3$-* $= 3 $&
Φα : C1(Y,A)→Mαr,k #/ Φ -. 8-:(1 5=
Φα(ϕ) = U−m
[πα,0U
m(Φ(ϕ)]∈Mα
r,k
/#$ %% m .,E9-(1&%= % $8( ; :(9 Φ(ϕ) ∈Mr,k(N0pm)!4
#$% &%'()*)+( ), )(&%-%(&%(* +. *$% /$+)/% +. m 01,,23%& ,24/)%(*56
5178%9 :6 ;7+-! !"< :9!
!"#$"% !=& C(& |α|p = 1 1+ Φ 5#,1+(+ +-.&$-5,&-#1 7-&' : %,(. -1
&'( A)*#+,%( Mr,k #/ p) +-9 1( $%= '#%#*#$3'-9 *#+,% $ /#$*.4 6'(1
&'( +-.&$-5,&-#1 Φα+(01(+ -1 F4B -. %.# 5#,1+(+4
@$##/4 >* ), )&%(*)/15 *+ *$1* +. #$%+7%3 ?!@ 0)( A$)/$ *$% $+5+3+7-$)/
/1,% A1, *7%1*%&9!
@B! h'()%*++*,-" )*+.$*,/.*#0+&
C%* Φj : C1(Y,A) → Mr,k :% 1 .13)56 +. &),*7):2*)+(, 0(+( (%/D
%,,17)56 :+2(&%&< j = 0, 1, . . . , r∗< r∗ ≥ 1)! E+7 1(6 +-%( ,2:,%*
a+ (Npm) ⊂ Y -2* Φj(a+ (Npm)) = Φj(χa+(Npm))!
!"#$"% @B!@& C(& 0 < |α|p < 1 1+ h = [ordp α] + 14 G,33#.( &' &
&'($( (H-.&. κ ∈ N∗.,9' &' & /#$ %% j = 0, 1, . . . , hκ − 1 1+ /#$ %%
m ≥ 1 #1( ' . Φj(a + (N0pm)) ∈ Mr,k(N0p
mκ) ;&'( %(:(% 9#1+-&-#1!4
G,33#.( 1(H& &' & /#$ t = 0, 1, . . . , hκ − 1 1+ /#$ %% a + (Npm) ⊂ Y
#1( ' .
|Uκm
t∑
j=0
(t
j
)(−ap)
t−jΦj(a+ (Npm))|p ≤ C|pm|tp = Cp−mt (10.1)
7-&' .,-& 5%( 9#1.& 1& C > 0 ;&'( +-:-.-5-%-&= 9#1+-&-#1!4 6'(1 &'($(
(H-.&. 1 hκ) +*-..-5%( *( .,$( Φα : Chκ(Y,A) →Mαr,k ⊂ Mr,k .,9'
&' &
∫a+(N0pm)
yjpdΦ
α = Φαj (a+ (N0p
m)) ;/#$ j = 0, 1, . . . , hκ − 1!4
!"# $"%&'( ') *'!+%,-*%.!/ p0 (.* L0)-!*%.'!+ 12
!""#$ ! "#$%&" !' (&)*+, !-& %'./*!*'. '+ 0)'1!- 2343 /55 +') Φαj (a+
(N0pm)) ∈ Mr,k(N0p
mκ)4 6.& -7" U = αI + Z8 Zn = 0 '. Mα
r,k(N0)8n = rkAMα
r,k(N0pm;A)4
6. !-& '!-&) -7./ 9, !-& %'./*!*'." '+ !-& !-&')&: 1& -7(&
∫
a+(N0pm)
(yp − ap)tdΦα =
t∑
j=0
(t
j
)(−ap)
t−jΦαj (a+ (N0p
m))
= α−mκ
αmκ(Uα)−mκ
[πα,0U
mκ
(t∑
j=0
(t
j
)(−ap)
t−jΦj(a+ (N0pm))
)].
;-& '<&)7!')" α−1(Uα)−mκ =
∑n−1i=0
(−mκ
i
)(α−1Z)i 7)& #.*+'):=, 9'#./&/
9, 7 %'."!7.! C1 > 0 -&.%& !-& %'./*!*'. 23>435 *:<=*&"
∣∣∣∣∫
a+(N0pm)
(yp − ap)tdΦα
∣∣∣∣p
≤ C·C1|α|−mκ
p |pm|tp = o(pm(hκ−t))
1-&. m → ∞ 9&%7#"& |α|p = |p|ordp α8 ordp α < h8 |pm|−κ ordp α
p =o(pmκh)4
334 !" #"$%&' #(")* +)),-$+(-%& %. + #/-(+0," ,-&"+1
.%12
?&! α ∈ Q ⊂ C 9& 7 .'.@A&)' &*0&.(7=#& '+ U '. Mr,k(C) 7""'%*7!&/
1*!- 7 <)*:*!*(& %#"< &*0&.+'): f ∈ Sk(Γ0(N), ψ) 7./ =&! f0 = f0,α 9&
7 %'))&"<'./*.0 &*0&.+#.%!*'. (f0|U = αf0)8 =&! #" /&B.& f0 = f
ρ0 |WN0 8
fρ0 =
∑n≥1 an(f0)q
n8 WN0 =
(0 −1N0 0
)4
!"#"$%&%"' (()() %& U∗ = W
−1N0UWN0 '( )*+ *+!,')'%( -+.)"! /0%.+
Sr,k(Γ1(N0),C)1 )*+ %23"'() "0+!%)"! 4')* !+/0+.) )" )*+ +)+!//"( /.%5%!0!"26.)$
7& 8(+ *%/ f0|U∗ = αf
01 %(2 #"! %55 9:""29 0!',+ (6,7+!/ l ∤ Np
"(+ *%/ Tlf0 = al(f)f
0$
.& ;*+ 5'(+%! #"!, ℓf,α : g 7→ 〈f 0, g〉 "( Mr,k(Γ1(N0),C) -%(<'/*+/ "( Ker πα,01 4*+!+ πα,0 : Mr,k(Γ1(N0),C) → Mα
r,k(Γ1(N0),C)=)*+ 0!"3+.)"! "()" )*+ α<0!',%!> /67/0%.+ 4')* )*+ ?+!(+5 Ker πα,0 =Im(U − αI)n) *+(.+
〈f 0, g〉 = 〈f 0, πα,0(g)〉.
2& @# g ∈M(Npm+1,Q) =M(N0p
m,Q) +) α 6= 01 "(+ *%/
〈f 0, gα〉 = α−m〈f 0, g | Um〉
4*+!+
gα = (Uα)−m
πα,0(g | Um) ∈Mα(Np)
'/ )*+ α<0!',%!> 0%!) "# g$
! "#$%$& '"()*&+*,&(
! "# $%&'
Lf,α(g) =〈f 0, α−m
g|Um〉N0
〈f 0, f0〉N0
,
( #) Lf,α :M(Npm+1;Q)→ Q *&( +,# -. /0.1 Lf,α '%.Mr,k(Γ1(Np),C)
,' 2 3# 2 04 . Q! -#2 &( . 5,'&' - %#,6% A7+,# -. /0.1 ℓf,α ∈Mα
r,k(N0)∗'%)( &(-&
i−1p (ℓf,α(g
α)) =〈f 0, α−m
Um(g)〉N0
〈f 0, f0〉N0
*∀g 8,&( )0 9), #&' ,# i−1p (A)!:
;.00/ 0/ ;.0$0',&,0# <<:< ! "## $%&'() *+, -,.,.,
/! 0' 1#23&4&53) f0|U∗ = f
ρ0 | WNpW
−1NpUWNp = αf
ρ0 | WNp = αf
0,
6! 758 3' 9:364&53
g1 = (U − αI)ng ∈ Ker πα,0 = Im(U − αI)n
53# + ;
〈f 0, g1〉 = 〈f 0, (U − αI)ng〉 = 〈(U∗ − αI)f 0, (U − αI)n−1g〉 = 0
+#36# 958 g1 = g − πα,0(g) <# =#4
〈f 0, g〉 = 〈f 0, πα,0(g)+(g−gα)〉 = 〈f 0, πα,0(g)〉+〈f 0, g1〉 = 〈f 0, πα,0(g)〉1! >#4 :; :;# 1&8#64?' 4+# #@: ?&4' (U∗)mf 0 = α
mf059 /!A
αm · 〈f 0, gα〉 = 〈(U∗)mf 0, U−m
πα,0(g | Um)〉 =〈f 0, πα,0(g | Um)〉 = 〈f 0, g | Um〉
/' 6! /#6 :;# g|Um ∈M(Np),#! B54# 4+ 4 Lf,α(f0) = 1) f0 ∈ M(Np;Q)C 653;&1#8 4+# 65DE?#F
G#6458 ;E 6#
KerLf,α = 〈f 0〉⊥ = {g ∈M(Np;C) | 〈f 0, g〉 = 0} !"#! $%&"'( $ Q)*$'"+,$- .$("( ./#$0(/ "' "( ('$.-/ .1 '!/ $#'"+, +2 $--
34++%3 5/#6/ +7/*$'+*( Tl 8l ∤ Np9:
〈f 0, g〉 = 0 =⇒ 〈f 0, Tlg〉 = 〈T ∗l f 0, g〉 = 0
$,% +,/ +.'$",( (0#! $ .$("( .1 '!/ %"$4+,$-"($'"+, +2 '!/ $#'"+, +2 $--
'!/ Tl 8$ #+&&0'$'";/ 2$&"-1 +2 ,+*&$- +7/*$'+*(9 $,% /9 2+--+ (<
=>< !"#$%&'( $& $)! L*+,'-$%&'(. -&'/&",$%&'( &+ $)!
0%(!'($!%' 1%($2%3,$%&'(
?/' ξmodN ./ $, $0@"-"$*1 A"*"#!-/' #!$*$#'/* ξ : Y → A∗B
Y
yp→Z∗p, Y = lim←YNpm ,
YNpm = (Z/NpmZ)∗< C+,("%/* ' + D"(/,('/", %"('*".0'"+,(
!"# $"%&'( ') *'!+%,-*%.!/ p0 (.* L0)-!*%.'!+ 12
E0,ℓ,Npm(ξ, b) =∑
a∈YNpmξ(a)E0,ℓ(a, b;Np
m) ∈{M0,ℓ ! ℓ ≥ 3 "# ξ 6= 1
M1,ℓ ! ξ = 1, ℓ = 1, 2,
Er,ℓ,Npm(a) =∑
b∈YNpmEr,ℓ,Npm(a, b) ∈Mr′,ℓ$ r
′ = max(r, ℓ− r − 1)%
!"#"$%&%"' ()*(* !" χ, ψ : Y → A×#! "$% &'(')*+!" )*,(,)"!(-
modNpm.
,/ !" f ∈ Sk(N,ψ). k ≥ 2. 0!12!
Φj(a)Npm =∑
y∈YNpm
ψξ(y)E0,k−1−j(ξ, ya)Ej,1+j(y)
3, "$'-"!0 )%24%+5"'%2. j = 0, . . . , k − 2/6 7!2)! 8%( ,29 &'(')*+!"
)*,(,)"!( χ mod Npm%2! *,-
Φj(χ) = E0,k−1−j(ξ, χ)Ej,1+j(ψξχ);
#/ "*! -:!)',+ 4,+5!- %8 "*! 852)"'%2 Lf (s, χ) =∑
n≥1 χ(n)an(f)n−s
$'"* , :(';'"'4! &'(')*+!" )*,(,)"!( χ mod Npm (m ≥ 1) ,0;'" "*!8%++%$'2< '2"!<(,+ (!:(!-!2","'%2
〈f 0,Φj(χ)〉Np =G(χ)Γ(j + 1)Lf (k − 1, ξ)Lf (j + 1, χ)
(−2πi)j+k· t,
3$*!(! G(χ) '- "*! =,5-- -5; %8 χ. t ∈ Q∗'- ,2 !>:+')'"+9 <'4!2 !+!?
;!2",(9 )%2-",2" '20!:!20!2" %8 j ,20 χ/@
)/ "*! 0'-"('#5"'%2- Φj -,"'-89 "*! )%20'"'%2- %8 A*!%(!; B ,20 "*!9
:(%05)! ,2 h?,0;'--'#+! ;!,-5(! 3)%;:,(! $'"* CD'EBF/ $*')* '2"!(:%?
+,"!- "*! 2%(;,+'-!0 )('"'),+ 4,+5!-
α−mG(χ)Γ(j + 1)Lf (j + 1, χ)Lf (k − 1, ξ)
(−2πi)j+k〈f, f〉$*!(! 〈f, f〉 0!2%"!- "*! 2%(;,+'G!0 H!"!(--%2 :(%05)"6
H(%%8 %8 H(%:%-'"'%2 IJ6I &' () * * & +,-,#&. /#"/,#01 "! 23.0 /. 4&0 5,
4"-5".30 "-*%
6' 72/. ,8 !#"2 0), 9&-: -;<,.6,#+ 2,0)"8 !"# 0), 4"-5".30 "-
D(s, f, g) = LN(2s+ 2− l, ψξχ)∞∑
n=1
anbnn−s,
=),#, ξ * &- &3> .  -"-;0# 5 &. ? # 4).,0 4)&#&40,# ξ &-8 0), -326,#*
bn = σl−1,χ,ξ(n) =∑
d|n,d>0
χ(d)ξ(n/d)dl−1
&#, @"3# ,# 4",A4 ,-0* "! & 4,#0& - B *,-*0, - *,# ,* g =∑∞
n=0 bn exp(2πinz)"! =, +)0 l C&-8 "! ? # 4).,0 4)&#&40,# χξ' ! χξ(−1) = (−1)l *" 0)&0
Lg(s) =∞∑
n=1
bnn−s = L(s− l + 1, χ)L(s, ξ)
! "#$%$& '"()*&+*,&(
!" #$ %!&# '( ) !%*!+, -$.. /,$$ 0) 1234 &5 6$!$5 - -$..
&7 89*.:5 089*;<34 =$.. > /6$!$5 -*?$" ) !%*!+, -$.. @4 ,$$ -,&
089*;;34 0A*B134 0C !DE 3@ F9 F D(s, f, g) G ! '$ $HI5$,,$" F95&:69
Lf (s − l + 1, χ)Lf (s, ξ)J K! &5"$5 F& 6$F F9$ *!F$65 - 5$I5$,$!F F*&!
&7 '@4 #$ $L -: F$ F9$! F9$ 7:!GF*&! Lf (s − l + 1, χ)Lf (s, ξ) F I&*!F,s = k − 14 !" I:F l = k − j − 14 #9$5$ 1 ≤ l ≤ k − 1@J
G@ M!$ G9$G%, G&$NG*$!FD'(DG&$NG*$!F F9 F F9$ "*,F5*':F*&!, Φj , FD
*,7( F9$ -$L$- G&!"*F*&! !" F9$ "*L*,*'*-*F( G&!"*F*&! &7 O9$&5$. < #*F9
κ = 2J O9$! &!$ "*5$GF-( II-*$, O9$&5$. < !" I5&I&,*F*&! >>J> G@4
"@ *! &5"$5 F& &'F *! F9$ "$,*5$" hD ".*,,*'-$ .$ ,:5$ µf,α *! F9$ 7&5.
µf,α = ℓf,α(Φα)J
>PJ !!"#$%&#'( &' &)#!"* !)'+,$&-
Q&!,*"$5 F9$ L$GF&5 ,I G$
M :=⋃
m≥0
Mk(Γ1(Npm))⊗3
!" -$F L(f ⊗ g⊗h, s) '$ F9$ F5*I-$ LD7:!GF*&! FF G9$" F& f ⊗ g⊗h ∈Sk(Γ1(N))
⊗3 ,,&G* F$" #*F9 ! !"#$%!& $*6$!L -:$ αβγ4 9$!G$
f0 ⊗ g0 ⊗ h0 ∈ Sk(Γ1(Np))⊗3
*, ! $*6$!7:!GF*&! &7 U⊗3
&! Sk(Γ1(Np))⊗34 !" f
0 ⊗ g0 ⊗ h
0*, !
$*6$!7:!GF*&! &7 (U∗)⊗3J =$F :, :,$ F9$ 5$,F5*GF*&! F& F9$ "* 6&! -
Φ = E3k(z1, z2, z3) ∈ Mk(Γ1(Np))
⊗3&7 F9$ 8*$6$-DR*,$!,F$*! "*,F5*D
':F*&! /,$$ 0EK,53@ L*$#$" , 7&5. - S&:5*$5 ,$5*$,J M!$ &'F *!,
"*,F5*':F*&! &! Y3#*F9 L -:$, *! MJ
E:F
lf⊗g⊗h,αβγ(Φ) := ip
( 〈f 0 ⊗ g0 ⊗ h
0,Φαβγ〉
〈f 0, f0〉〈g0, g0〉〈h0, h0〉
)
./*')*0 >PJ>1 '% ( !) #$ *! +!,-- (#./ 0#,+1!#," 234/,!,!56 7/, "#-8
.!#9:.# $ lf⊗g⊗h,αβγ(Φ) $ Y3(#./ ;%<:,- #$ M #- 9 :$"," %$" ./,
#$.,+!%<- lf⊗g⊗h,αβγ(Φ)(χ1 ⊗ χ2 ⊗ χ3) $ ./, 4/%!%4.,!- χ1 ⊗ χ2 ⊗ χ3
4 #$4#", (#./ ./, -*,4#%< ;%<:,- L∗(fχ1⊗gχ2⊗hχ3 , s0)= (/,!, ./, $ !>%<8
#-%.# $ 1 L∗#$; <;,- ?%:-- -:>-= @,.,!-- $ -4%<%! *! ":4.- = * (,!-
1 π= αβγA
@! 1A O9$ $H*,F$!G$ &7 lf⊗g⊗h,αβγ(Φ) 7&--&#, 75&. F9$ $H*,F$!G$ &7 Φ:,*!6 O9$&5$. P4 !" F9$ $T: -*F( *, *.I-*$" '( F9$ *!F$65 - 7&5.:-
&7 U 55$FFDA 55*, 0U A 34 ,$$ -,& 0=VE34 0EO53J
2*3*)*($*-
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9>?@ABB) !02/-104,% C
◦DAEDF) G+H( I723( J-7.H%) K7-10 9>?@FB) L >>?#
>M>(
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<4++02"9'
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*+,-./0 123' (24. H% 786$69) , =$:8HF 70" <4++02"9'
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H5H:HH6 70" <4++02"9'
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!
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#
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%).
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/.#.*(*!. ,' .0* 12,3)*( ,' 4,)*(#!"5#6-2
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(),&'0%)3
Lµ(!) =
∫
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($ ,C∗%
), ! = !(")),
µ /' , $#)(,/& 3#,'*)# 4/(1 5,-*#' /& A6 %& (1# 7)%+&/(# 8)%*7 $ 9
:! ;%) #,$1 % ∈ B ⊂ 6 (1#)# /' (1# #5,-*,(/%& 1%3%3%)71/'3
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&
: A(B)→ C%
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/&(#8),-@
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&
(Lµ(!)) = &'
&
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)
(! ∈ , Lµ(!) ∈ A).
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∗'
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!9
78
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"
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# →→ !" !"#
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44
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(
ϕ8D.
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µ : C($ ,A)→ # , ϕ 7→
∫
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ϕ8µ.
8'( %&(*-"/
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C+" D-.D+-0" .7 #$%&'((')*+ &+$(,-+( µA '0( E+**'! 0-$!(7.-& Lµ '(
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'
(Lµ) (/ ∈ 0 , Lµ(/) ∈ A).45
! "#$%& '#(%)%*+ ,' !"'-!&.%,!+
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!
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"
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"
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; )(!− α′"
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"
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$"%&%'%() 5%"%*36)' *3#"#*')" χ #$% $
$
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"
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χ(B) B %"
7 µ = %
"
($
$%
F (χ)
α$
G
∗#
(!+ D , χ)
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%
"
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)
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/− %
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