on the eisenstein class for hilbert modular surfaces · 2012. 6. 5. · hilbert modular surfaces ....
TRANSCRIPT
On the Eisenstein Class for Hilbert Modular Surfaces
Kenichi Bannai(Keio)work in progress with Guido Kings (Regensburg)
June 4, 2012Arithmetic Geometry week in Tokyo
Eisenstein class: Motivic class in the motivic cohomology of an elliptic modularcurve
First constructed by Beilinson using the Eisenstein symbol mapBeilinson, A.A. : Higher regulators of modular curves, Contemp. Math. 55, 1-34 (1986)
Introduction
Application: Used in the proof of the Beilinson conjecture
Ex. C. Deninger proved the weak Beilinson conjecture for Hecke characters of imaginary quadratic fields
C. Deninger, Higher regulators and Hecke L-series of imaginary quadratic fields. I, Invent. Math. 96 (1989), no. 1, 1–69.
C. Deninger, Higher regulators and Hecke L-series of imaginary quadratic fields. II, Ann. of Math. (2) 132 (1990), no. 1, 131–158.
Introduction
Our Interest: Explicit calculation of the rigid syntomic realization, i.e., the image of the motivic Eisenstein class with respect to the syntomic regulator
B- and G. Kings, p-adic elliptic polylogarithm, p-adic Eisenstein series and Katz measure, American J. Math. 132, no. 6 (2010), 1609-1654.
B- and G. Kings, p-adic Beilinson conjecture for ordinary Hecke motives associated to imaginary quadratic fields, RIMS Kôkyûroku Bessatsu B25: Algebraic Number Theory and Related Topics 2009, eds. T. Ichikawa, M. Kida, T. Yamazaki, June (2011), 9-30.
Introduction
EllipticModular
HilbertModular
?
IntroductionAlternative Construction: Specialization at torsion points of the polylogarithm on the universal elliptic curve
A. Huber and G. Kings, Degeneration of l-adic Eisenstein classes and of the elliptic
polylog, Invent. Math. 135 (1999), no. 3, 545–594.
Eisenstein Class
Introduction
EllipticModular
HilbertModular
AbelianPolylogarithm
J. WildeshausG. Kings
EllipticPolylogarithm
Polylogarithm
specialization specialization
Eisenstein Class
A. Levin gave a conjectural formula for the topological realization of the abelian polylogarithm in terms of currents
A. Levin, Polylogarithmic currents on abelian varieties, in Regulators in Analysis, Geometry and Number Theory, A. Reznikov, N. Schappacher (Eds), Progr. Math. 171, Birkhäuser (2000), 207–229.
Introduction
D. Blottière proved Levin’s formulaD. Blottière, Réalisation de Hodge du polylogarithme d'un schéma abélien, avec un appendice d'Andrey Levin, Journal de l'Institut Mathématique de Jussieu (2009), 8 no. 1, pages 1–38
Introduction
This Talk: We give a conjectural formula for the Hodge and the rigid syntomic realizations of the Eisenstein class for Hilbert modular surfaces
Eisenstein Class
Hodge Realization
Syntomic Realization
The Eisenstein Class
The Eisenstein Class
SOME STUDY OF HILBERT MODULAR FORMS
KENICHI BANNAI
Abstract. Notes for talk at Tokyo University, June 4, 2012.
1. Calculations of Cech cocycles
SymkHS: base schemeA: abelian varietyk: integer > 0
!g := dimAQ: constant motivic sheaf
H := (R1!!Q)"
Eisk+2mot(") # H2g$1
mot (S, SymkH (g))
Date: June 3, 2012 (Personal notes - not for distribution)File Name: TokyoNotes.
1
SOME STUDY OF HILBERT MODULAR FORMS
KENICHI BANNAI
Abstract. Notes for talk at Tokyo University, June 4, 2012.
1. Calculations of Cech cocycles
SymkHS base schemeA abelian varietyk: integer > 0
!g := dimAQ: constant motivic sheaf
H := (R1!!Q)"
Eisk+2mot(") # H2g$1
mot (S, SymkH (g))
Date: June 3, 2012 (Personal notes - not for distribution)File Name: TokyoNotes.
1
Abelian Variety
SOME STUDY OF HILBERT MODULAR FORMS
KENICHI BANNAI
Abstract. Notes for talk at Tokyo University, June 4, 2012.
1. Calculations of Cech cocycles
SymkHS base schemeA abelian varietyk: integer > 0
!g := dimAQ: constant motivic sheaf
H := (R1!!Q)"
Eisk+2mot(") # H2g$1
mot (S, SymkH (g))
Date: June 3, 2012 (Personal notes - not for distribution)File Name: TokyoNotes.
1
Base
SOME STUDY OF HILBERT MODULAR FORMS
KENICHI BANNAI
Abstract. Notes for talk at Tokyo University, June 4, 2012.
1. Calculations of Cech cocycles
SymkHS base schemeA abelian varietyk: integer > 0
g := dimA
!g := dimAQ: constant motivic sheaf
H := (R1!!Q)"
Eisk+2mot(") # H2g$1
mot (S, SymkH (g))
Date: June 3, 2012 (Personal notes - not for distribution)File Name: TokyoNotes.
1
Polylogarithm
Eisenstein class
4 KENICHI BANNAI
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#i
#
$ $ !()
% $
($, %)
%($) = (1! F&)%
&: torsion point&
&'
Eisk+2mot(&) $ H2g!1
mot (S, SymkH (g))
F&: complex conjugationDepartment of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
4 KENICHI BANNAI
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#i
#
$ $ !()
% $
($, %)
%($) = (1! F&)%
&: torsion point&
&'
Eisk+2mot(&) $ H2g!1
mot (S, SymkH (g))
F&: complex conjugationDepartment of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
4 KENICHI BANNAI
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#i
#
$ $ !()
% $
($, %)
%($) = (1! F&)%
&: torsion point&
&'
Eisk+2mot(&) $ H2g!1
mot (S, SymkH (g))
F&: complex conjugationDepartment of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
The Eisenstein Class
SOME STUDY OF HILBERT MODULAR FORMS
KENICHI BANNAI
Abstract. Notes for talk at Tokyo University, June 4, 2012.
1. Calculations of Cech cocycles
SymkHS: base schemeA: abelian variety
!g := dimAQ: constant motivic sheaf
H := (R1!!Q)"
Date: June 3, 2012 (Personal notes - not for distribution)File Name: TokyoNotes.
1
SOME STUDY OF HILBERT MODULAR FORMS
KENICHI BANNAI
Abstract. Notes for talk at Tokyo University, June 4, 2012.
1. Calculations of Cech cocycles
SymkHS: base schemeA: abelian varietyk: integer > 0
!g := dimAQ: constant motivic sheaf
H := (R1!!Q)"
Eisk+2mot(") # H2g$1
mot (S, SymkH (g))
Date: June 3, 2012 (Personal notes - not for distribution)File Name: TokyoNotes.
1
SOME STUDY OF HILBERT MODULAR FORMS
KENICHI BANNAI
Abstract. Notes for talk at Tokyo University, June 4, 2012.
1. Calculations of Cech cocycles
SymkHS base schemeA abelian varietyk: integer > 0
!g := dimAQ: constant motivic sheaf
H := (R1!!Q)"
Eisk+2mot(") # H2g$1
mot (S, SymkH (g))
Date: June 3, 2012 (Personal notes - not for distribution)File Name: TokyoNotes.
1
Abelian Variety
SOME STUDY OF HILBERT MODULAR FORMS
KENICHI BANNAI
Abstract. Notes for talk at Tokyo University, June 4, 2012.
1. Calculations of Cech cocycles
SymkHS base schemeA abelian varietyk: integer > 0
!g := dimAQ: constant motivic sheaf
H := (R1!!Q)"
Eisk+2mot(") # H2g$1
mot (S, SymkH (g))
Date: June 3, 2012 (Personal notes - not for distribution)File Name: TokyoNotes.
1
Base
SOME STUDY OF HILBERT MODULAR FORMS
KENICHI BANNAI
Abstract. Notes for talk at Tokyo University, June 4, 2012.
1. Calculations of Cech cocycles
SymkHS base schemeA abelian varietyk: integer > 0
g := dimA
!g := dimAQ: constant motivic sheaf
H := (R1!!Q)"
Eisk+2mot(") # H2g$1
mot (S, SymkH (g))
Date: June 3, 2012 (Personal notes - not for distribution)File Name: TokyoNotes.
1
SOME STUDY OF HILBERT MODULAR FORMS
KENICHI BANNAI
Abstract. Notes for talk at Tokyo University, June 4, 2012.
1. Calculations of Cech cocycles
SymkHS base schemeA abelian varietyk: integer > 0
g := dimA
!g := dimAQ: motivic sheaf
H := (R1!!Q)"
Eisk+2mot(") # H2g$1
mot (S, SymkH (g))
Date: June 3, 2012 (Personal notes - not for distribution)File Name: TokyoNotes.
1
Eisenstein class
SOME STUDY OF HILBERT MODULAR FORMS
KENICHI BANNAI
Abstract. Notes for talk at Tokyo University, June 4, 2012.
1. Calculations of Cech cocycles
SymkHS: base schemeA: abelian varietyk: integer > 0
!g := dimAQ: constant motivic sheaf
H := (R1!!Q)"
Eisk+2mot(") # H2g$1
mot (S, SymkH (g))
Date: June 3, 2012 (Personal notes - not for distribution)File Name: TokyoNotes.
1
4 KENICHI BANNAI
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#i
#
$ $ !()
% $
($, %)
%($) = (1! F&)%
&: torsion point&
&'
Eisk+2mot(&) $ H2g!1
mot (S, SymkH (g))
F&: complex conjugationDepartment of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
4 KENICHI BANNAI
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#i
#
$ $ !()
% $
($, %)
%($) = (1! F&)%
&: torsion point&
&'
Eisk+2mot(&) $ H2g!1
mot (S, SymkH (g))
F&: complex conjugationDepartment of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
4 KENICHI BANNAI
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#i
#
$ $ !()
% $
($, %)
%($) = (1! F&)%
&: torsion point&
&'
Eisk+2mot(&) $ H2g!1
mot (S, SymkH (g))
F&: complex conjugationDepartment of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
The Eisenstein Class
4 KENICHI BANNAI
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#i
#
$ $ !()
% $
($, %)
%($) = (1! F&)%
&: torsion point&
&'
Eisk+2mot(&) $ H2g!1
mot (S, SymkH (g))
F&: complex conjugationDepartment of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
4
KEN
ICHI B
ANNAI
H2g!1
D
(S, Symk H
(g))!"H2g!1
dR
(S, Symk H
(g))
A 1:=(" 1!" 1)
2#i
#
$$ !
()
%$
($, %)
%($)=(1!F&)%
&:torsionpoint
&&'
Eisk+2
mot(&) $
H2g!1
mot(S, Symk H
(g))
F&: complexconjugation
Departmentof
Mathematics, K
eioUniversity,3-14-1Kouhoku-ku,Hiyoshi, Y
okohama,223-8522
Japan
4 KENICHI BANNAI
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#i
#
$ $ !()
% $
($, %)
%($) = (1! F&)%
&: torsion point&
&'
Eisk+2mot(&) $ H2g!1
mot (S, SymkH (g))
F&: complex conjugationDepartment of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
2 KENICHI BANNAI
Eisk+2D (!) ! H2g"1
D (S, SymkH (g))
H2g"1syn (S, SymkH (g))
SOME STUDY OF HILBERT MODULAR FORMS
KENICHI BANNAI
Abstract. Notes for talk at Tokyo University, June 4, 2012.
1. Calculations of Cech cocycles
SymkHS base schemeA abelian varietyk: integer > 0
g := dimA
rD
rsyn
!
? g := dimAQ: motivic sheaf
H := (R1!!Q)"
Date: June 4, 2012 (Personal notes - not for distribution)File Name: TokyoNotes.
1
2 KENICHI BANNAI
Eisk+2D (!) ! H2g"1
D (S, SymkH (g))
H2g"1syn (S, SymkH (g))
4
KEN
ICHI B
ANNAI
H2g!1
D
(S, Symk H
(g))!"H2g!1
dR
(S, Symk H
(g))
A 1:=(" 1!" 1)
2#i
#
$$ !
()
%$
($, %)
%($)=(1!F&)%
&:torsionpoint
&&'
Eisk+2
mot(&) $
H2g!1
mot(S, Symk H
(g))
F&: complexconjugation
Departmentof
Mathematics, K
eioUniversity,3-14-1Kouhoku-ku,Hiyoshi, Y
okohama,223-8522
Japan
?
SOME STUDY OF HILBERT MODULAR FORMS
KENICHI BANNAI
Abstract. Notes for talk at Tokyo University, June 4, 2012.
1. Calculations of Cech cocycles
SymkHS base schemeA abelian varietyk: integer > 0
g := dimA
rD
rsyn
!
? g := dimAQ: motivic sheaf
H := (R1!!Q)"
Date: June 4, 2012 (Personal notes - not for distribution)File Name: TokyoNotes.
1
4
KEN
ICHI B
ANNAI
H2g!1
D
(S, Symk H
(g))!"H2g!1
dR
(S, Symk H
(g))
A 1:=(" 1!" 1)
2#i
#
$$ !
()
%$
($, %)
%($)=(1!F&)%
&:torsionpoint
&&'
Eisk+2
mot(&) $
H2g!1
mot(S, Symk H
(g))
F&: complexconjugation
Departmentof
Mathematics, K
eioUniversity,3-14-1Kouhoku-ku,Hiyoshi, Y
okohama,223-8522
Japan
2 KENICHI BANNAI
Eisk+2D (!) ! H2g"1
D (S, SymkH (g))
H2g"1syn (S, SymkH (g))
EISENSTEIN SHEAF 3
H2g!1D (S, SymkH (g)) !" H0
D(SpecR, H2g!1(S, SymkH (g)))
Ep,q2 = Hp
D(SpecR, Hq(S, SymkH (g)) # Hp+q
D (S, SymkH (g))
H1D(SpecR, H2g!2(S, SymkH (g)) "
" H2g!1D (S, SymkH (g)) " H0
D(SpecR, H2g!1(S, SymkH (g))) " 0
H0D(SpecR, H2g!1(S, SymkH (g))) !" H2g!1
dR (S, SymkH (g))
EISENSTEIN SHEAF 3
H2g!1D (S, SymkH (g)) !" H0
D(SpecR, H2g!1(S, SymkH (g)))
Ep,q2 = Hp
D(SpecR, Hq(S, SymkH (g)) # Hp+q
D (S, SymkH (g))
H1D(SpecR, H2g!2(S, SymkH (g)) "
" H2g!1D (S, SymkH (g)) " H0
D(SpecR, H2g!1(S, SymkH (g))) " 0
H0D(SpecR, H2g!1(S, SymkH (g))) !" H2g!1
dR (S, SymkH (g))
EISENSTEIN SHEAF 3
H2g!1D (S, SymkH (g)) !" H0
D(SpecR, H2g!1(S, SymkH (g)))
Ep,q2 = Hp
D(SpecR, Hq(S, SymkH (g)) # Hp+q
D (S, SymkH (g))
H1D(SpecR, H2g!2(S, SymkH (g)) "
" H2g!1D (S, SymkH (g)) " H0
D(SpecR, H2g!1(S, SymkH (g))) " 0
H0D(SpecR, H2g!1(S, SymkH (g))) !" H2g!1
dR (S, SymkH (g))
The Eisenstein ClassEISENSTEIN SHEAF 3
H2g!1D (S, SymkH (g)) !" H0
D(SpecR, H2g!1(S, SymkH (g)))
Ep,q2 = Hp
D(SpecR, Hq(S, SymkH (g)) # Hp+q
D (S, SymkH (g))
EISENSTEIN SHEAF 3
H2g!1D (S, SymkH (g)) !" H0
D(SpecR, H2g!1(S, SymkH (g)))
Ep,q2 = Hp
D(SpecR, Hq(S, SymkH (g)) # Hp+q
D (S, SymkH (g))
H1D(SpecR, H2g!2(S, SymkH (g)) "
" H2g!1D (S, SymkH (g)) " H0
D(SpecR, H2g!1(S, SymkH (g))) " 0
H0D(SpecR, H2g!1(S, SymkH (g))) !" H2g!1
dR (S, SymkH (g))
EISENSTEIN SHEAF 3
H2g!1D (S, SymkH (g)) !" H0
D(SpecR, H2g!1(S, SymkH (g)))
Ep,q2 = Hp
D(SpecR, Hq(S, SymkH (g)) # Hp+q
D (S, SymkH (g))
H1D(SpecR, H2g!2(S, SymkH (g)) "
" H2g!1D (S, SymkH (g)) " H0
D(SpecR, H2g!1(S, SymkH (g))) " 0
H0D(SpecR, H2g!1(S, SymkH (g))) !" H2g!1
dR (S, SymkH (g))
EISENSTEIN SHEAF 3
H2g!1D (S, SymkH (g)) !" H0
D(SpecR, H2g!1(S, SymkH (g)))
Ep,q2 = Hp
D(SpecR, Hq(S, SymkH (g)) # Hp+q
D (S, SymkH (g))
H1D(SpecR, H2g!2(S, SymkH (g)) "
" H2g!1D (S, SymkH (g)) " H0
D(SpecR, H2g!1(S, SymkH (g))) " 0
H0D(SpecR, H2g!1(S, SymkH (g))) !" H2g!1
dR (S, SymkH (g))
4
KENICHIBANNAI
H2g!1
D(S, Sym
k H(g))!"
H2g!1
dR(S, Sym
k H(g))
A1:=
("1!" 1)
2#i
#
$$!()
%$
($, %)
%($)=(1!F &
)%
&:torsionpoint
&&
'
Eis
k+2
mot(&)$H
2g!1
mot
(S, Sym
k H(g))
F &:complex
conjugation
Departm
entofMathe
matics,KeioUnive
rsity,
3-14-1
Kouh
oku-ku,
Hiyoshi,Yoko
hama,
223-8522
Japan
2 KENICHI BANNAI
Eisk+2D (!) ! H2g"1
D (S, SymkH (g))
H2g"1syn (S, SymkH (g))
The Eisenstein ClassEISENSTEIN SHEAF 3
H2g!1D (S, SymkH (g)) !" H0
D(SpecR, H2g!1(S, SymkH (g)))
Ep,q2 = Hp
D(SpecR, Hq(S, SymkH (g)) # Hp+q
D (S, SymkH (g))
H1D(SpecR, H2g!2(S, SymkH (g)) "
" H2g!1D (S, SymkH (g)) " H0
D(SpecR, H2g!1(S, SymkH (g))) " 0
H0D(SpecR, H2g!1(S, SymkH (g))) !" H2g!1
dR (S, SymkH (g))
EISENSTEIN SHEAF 3
H2g!1D (S, SymkH (g)) !" H0
D(SpecR, H2g!1(S, SymkH (g)))
Ep,q2 = Hp
D(SpecR, Hq(S, SymkH (g)) # Hp+q
D (S, SymkH (g))
H1D(SpecR, H2g!2(S, SymkH (g)) "
" H2g!1D (S, SymkH (g)) " H0
D(SpecR, H2g!1(S, SymkH (g))) " 0
H0D(SpecR, H2g!1(S, SymkH (g))) !" H2g!1
dR (S, SymkH (g))
EISENSTEIN SHEAF 3
H2g!1D (S, SymkH (g)) !" H0
D(SpecR, H2g!1(S, SymkH (g)))
Ep,q2 = Hp
D(SpecR, Hq(S, SymkH (g)) # Hp+q
D (S, SymkH (g))
H1D(SpecR, H2g!2(S, SymkH (g)) "
" H2g!1D (S, SymkH (g)) " H0
D(SpecR, H2g!1(S, SymkH (g))) " 0
H0D(SpecR, H2g!1(S, SymkH (g))) !" H2g!1
dR (S, SymkH (g))$=!"
4
KENICHIBANNAI
H2g!1
D(S, Sym
k H(g))!"
H2g!1
dR(S, Sym
k H(g))
A1:=
("1!" 1)
2#i
#
$$!()
%$
($, %)
%($)=(1!F &
)%
&:torsionpoint
&&
'
Eis
k+2
mot(&)$H
2g!1
mot
(S, Sym
k H(g))
F &:complex
conjugation
Departm
entofMathe
matics,KeioUnive
rsity,
3-14-1
Kouh
oku-ku,
Hiyoshi,Yoko
hama,
223-8522
Japan
2 KENICHI BANNAI
Eisk+2D (!) ! H2g"1
D (S, SymkH (g))
H2g"1syn (S, SymkH (g))
EISENSTEIN SHEAF 5
H1dR(S, Sym
kH (1))
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#i
#
$ $ !()
% $
($, %)
%($) = (1! F&)%&: torsion point
&
&'
Eisk+2mot(&) $ H2g!1
mot (S, SymkH (g))
SymkH =k!
j=0
OS 'j''k!j
!(S, SymkH ( "1S)
F&: complex conjugation
H := R1#'"•A/S
' := #'"1A/S
EISENSTEIN
SHEAF
5
H1dR (S
,SymkH
(1))
H2g!
1D
(S,Sym
kH(g))
!"H
2g!1
dR
(S,Sym
kH(g))
A1:=
("1 !
"1 )
2#i
#
$$!()
%$
($,%)
%($)=(1!
F&)%
&:torsion
point
&&'
Eis k
+2
mot (&
)$H
2g!1
mot
(S,Sym
kH(g))
SymkH
=k
!j=0
OS'j'
'k!j
!(S,Sym
kH(
"1S )
F&:com
plexconjugation
H:=
R1#
' "•A/S
':=
#' "
1A/S
4 KENICHI BANNAI
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#i
#
$ $ !()
% $
($, %)
%($) = (1! F&)%
&: torsion point&
&'
Eisk+2mot(&) $ H2g!1
mot (S, SymkH (g))
F&: complex conjugationDepartment of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
4 KENICHI BANNAI
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#i
#
$ $ !()
% $
($, %)
%($) = (1! F&)%
&: torsion point&
&'
Eisk+2mot(&) $ H2g!1
mot (S, SymkH (g))
F&: complex conjugationDepartment of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
The Eisenstein ClassEISENSTEIN SHEAF 3
H2g!1D (S, SymkH (g)) !" H0
D(SpecR, H2g!1(S, SymkH (g)))
Ep,q2 = Hp
D(SpecR, Hq(S, SymkH (g)) # Hp+q
D (S, SymkH (g))
H1D(SpecR, H2g!2(S, SymkH (g)) "
" H2g!1D (S, SymkH (g)) " H0
D(SpecR, H2g!1(S, SymkH (g))) " 0
H0D(SpecR, H2g!1(S, SymkH (g))) !" H2g!1
dR (S, SymkH (g))
4KENIC
HIBANNAI
H2g
!1
D(S,Sym
kH
(g))
!"H
2g!1
dR
(S,Sym
kH
(g))
A1:=
("1!"1)
2#i
#
$$!()
%$
($,%)
%($
)=
(1!F&)%
&:torsionpoint
& &'
Eisk+2
mot(&
)$H
2g!1
mot
(S,Sym
kH
(g))
F&:complexconjugation
Depa
rtmentofMathematics,
Keio
University,3-14-1
Kouhoku-k
u,Hiyosh
i,Yokohama,223-8522Japa
n
2 KENICHI BANNAI
Eisk+2D (!) ! H2g"1
D (S, SymkH (g))
H2g"1syn (S, SymkH (g))
2 KENICHI BANNAI
Eisk+2D (!) ! H2g"1
D (S, SymkH (g))
H2g"1syn (S, SymkH (g))
Eisk+2dR (!)
4KENIC
HIBANNAI
H2g
!1
D(S,Sym
kH
(g))
!"H
2g!1
dR
(S,Sym
kH
(g))
A1:=
("1!"1)
2#i
#
$$!()
%$
($,%)
%($
)=
(1!F&)%
&:torsionpoint
& &'
Eisk+2
mot(&
)$H
2g!1
mot
(S,Sym
kH
(g))
F&:complexconjugation
Depa
rtmentofMathematics,
Keio
University,3-14-1
Kouhoku-k
u,Hiyosh
i,Yokohama,223-8522Japa
n
The Eisenstein Class
2 KENICHI BANNAI
Eisk+2D (!) ! H2g"1
D (S, SymkH (g))
H2g"1syn (S, SymkH (g))
Eisk+2dR (!)
4 KENICHI BANNAI
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#i
#
$ $ !()
% $
($, %)
%($) = (1! F&)%
&: torsion point&
&'
Eisk+2mot(&) $ H2g!1
mot (S, SymkH (g))
F&: complex conjugationDepartment of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
SOME STUDY OF HILBERT MODULAR FORMS
KENICHI BANNAI
Abstract. Notes for talk at Tokyo University, June 4, 2012.
1. Calculations of Cech cocycles
SymkHS base schemeA abelian varietyk: integer > 0
g := dimA
rD
rsyn
!
Case g = 1? g := dimAQ: motivic sheaf
H := (R1!!Q)"
Date: June 4, 2012 (Personal notes - not for distribution)File Name: TokyoNotes.
1
4 KENICHI BANNAI
H1dR(S, Sym
kH (1))
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#i
#
$ $ !()
% $
($, %)
%($) = (1! F&)%
&: torsion point&
&'
Eisk+2mot(&) $ H2g!1
mot (S, SymkH (g))
F&: complex conjugationDepartment of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
4 KENICHI BANNAI
H1dR(S, Sym
kH (1))
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#i
#
$ $ !()
% $
($, %)
%($) = (1! F&)%
&: torsion point&
&'
Eisk+2mot(&) $ H2g!1
mot (S, SymkH (g))
SymkH =k!
j=0
OS 'j''k!j
!(S, SymkH ( "1S)
F&: complex conjugationDepartment of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
4 KENICHI BANNAI
H1dR(S, Sym
kH (1))
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#i
#
$ $ !()
% $
($, %)
%($) = (1! F&)%
&: torsion point&
&'
Eisk+2mot(&) $ H2g!1
mot (S, SymkH (g))
SymkH =k!
j=0
OS 'j''k!j
!(S, SymkH ( "1S)
F&: complex conjugation
H := R1#'"•A/S
' := #'"1A/S
Department of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
4 KENICHI BANNAI
H1dR(S, Sym
kH (1))
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#i
#
$ $ !()
% $
($, %)
%($) = (1! F&)%
&: torsion point&
&'
Eisk+2mot(&) $ H2g!1
mot (S, SymkH (g))
SymkH =k!
j=0
OS 'j''k!j
!(S, SymkH ( "1S)
F&: complex conjugation
H := R1#'"•A/S
' := #'"1A/S
'(k !" SymkH
!(S,'(k ( "1S)
Department of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
4KENIC
HIBANNAI
H1 dR(S,Sym
kH
(1))
H2g
!1
D(S,Sym
kH
(g))
!"H
2g!1
dR
(S,Sym
kH
(g))
A1:=
("1!"1)
2#i
#
$$!()
%$
($,%)
%($
)=
(1!F&)%
&:torsionpoint
& &'
Eisk+2
mot(&
)$H
2g!1
mot
(S,Sym
kH
(g))
Sym
kH
=k ! j=0
OS'j '
'k!j
!(S,Sym
kH
("
1 S)
F&:complexconjugation
H:=
R1#'"
• A/S
':=
#'"
1 A/S
'(k!"
Sym
kH
!(S,'
(k("
1 S)
Depa
rtmentofMathematics,
Keio
University,3-14-1
Kouhoku-k
u,Hiyosh
i,Yokohama,223-8522Japa
n
4 KENICHI BANNAI
H1dR(S, Sym
kH (1))
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#i
#
$ $ !()
% $
($, %)
%($) = (1! F&)%
&: torsion point&
&'
Eisk+2mot(&) $ H2g!1
mot (S, SymkH (g))
SymkH =k!
j=0
OS 'j''k!j
!(S, SymkH ( "1S)
F&: complex conjugation
H = R1#'"•A/S
' := #'"1A/S
Department of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
4KENIC
HIBANNAI
H1 dR(S,Sym
kH
(1))
H2g
!1
D(S,Sym
kH
(g))
!"H
2g!1
dR
(S,Sym
kH
(g))
A1:=
("1!"1)
2#i
#
$$!()
%$
($,%)
%($
)=
(1!F&)%
&:torsionpoint
& &'
Eisk+2
mot(&
)$H
2g!1
mot
(S,Sym
kH
(g))
Sym
kH
=k ! j=0
OS'j '
'k!j
!(S,Sym
kH
("
1 S)
F&:complexconjugation
H:=
R1#'"
• A/S
':=
#'"
1 A/S
'(k!"
Sym
kH
!(S,'
(k("
1 S)
Depa
rtmentofMathematics,
Keio
University,3-14-1
Kouhoku-k
u,Hiyosh
i,Yokohama,223-8522Japa
n
EISENSTEIN SHEAF 5
Ek+2(!)"kdq
q
E!k+2(!) :=
!
(m,n) !=(0,0)
"#(m,n)
(m! + n)k+2
Department of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
4 KENICHI BANNAI
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#i
#
$ $ !()
% $
($, %)
%($) = (1! F&)%
&: torsion point&
&'
Eisk+2mot(&) $ H2g!1
mot (S, SymkH (g))
F&: complex conjugationDepartment of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
EISENSTEIN SHEAF 5
E!k+2(!)"
kdq
q
E!k+2(!) :=
!
(m,n) !=(0,0)
"#(m,n)
(m! + n)k+2
Department of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
The Eisenstein Class2 KENICHI BANNAI
Eisk+2D (!) ! H2g"1
D (S, SymkH (g))
H2g"1syn (S, SymkH (g))
Eisk+2dR (!)
4 KENICHI BANNAI
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#i
#
$ $ !()
% $
($, %)
%($) = (1! F&)%
&: torsion point&
&'
Eisk+2mot(&) $ H2g!1
mot (S, SymkH (g))
F&: complex conjugationDepartment of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
SOME STUDY OF HILBERT MODULAR FORMS
KENICHI BANNAI
Abstract. Notes for talk at Tokyo University, June 4, 2012.
1. Calculations of Cech cocycles
SymkHS base schemeA abelian varietyk: integer > 0
g := dimA
rD
rsyn
!
Case g = 1? g := dimAQ: motivic sheaf
H := (R1!!Q)"
Date: June 4, 2012 (Personal notes - not for distribution)File Name: TokyoNotes.
1
4 KENICHI BANNAI
H1dR(S, Sym
kH (1))
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#i
#
$ $ !()
% $
($, %)
%($) = (1! F&)%
&: torsion point&
&'
Eisk+2mot(&) $ H2g!1
mot (S, SymkH (g))
F&: complex conjugationDepartment of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
2 KENICHI BANNAI
Eisk+2D (!) ! H2g"1
D (S, SymkH (g))
H2g"1syn (S, SymkH (g))
4 KENICHI BANNAI
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#i
#
$ $ !()
% $
($, %)
%($) = (1! F&)%
&: torsion point&
&'
Eisk+2mot(&) $ H2g!1
mot (S, SymkH (g))
F&: complex conjugationDepartment of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
EISENSTEIN SHEAF 5
E!k+2(!)"
kdq
q
E!k+2(!) :=
!
(m,n) !=(0,0)
"#(m,n)
(m! + n)k+2
H1syn(S, Sym
kH (1))
H1D(S, Sym
kH (1))
Department of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
4KENIC
HIBANNAI
H1dR(S,Sym
kH(1))
H2g!
1D
(S,Sym
kH(g))
!"H
2g!1
dR
(S,Sym
kH(g))
A1:=
("1 !
"1 )
2#i
#
$$!()
%$
($,%)
%($
)=
(1!F&)%
&:torsion
point
&&'
Eisk+2
mot (&
)$H
2g!1
mot
(S,Sym
kH(g))
Sym
kH=
k!j=
0
OS'j'
'k!j
!(S,Sym
kH("
1S )
F&:com
plex
conjugation
H:=
R1#
' "•A/S
':=
#' "
1A/S
'(k!"
Sym
kH
!(S,'
(k(
"1S )
Depa
rtmentofMathematics,
Keio
Universit
y,3-14-1
Kouhoku-k
u,Hiyosh
i,Yokohama,223-8522Japa
n
4 KENICHI BANNAI
H1dR(S, Sym
kH (1))
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#i
#
$ $ !()
% $
($, %)
%($) = (1! F&)%
&: torsion point&
&'
Eisk+2mot(&) $ H2g!1
mot (S, SymkH (g))
SymkH =k!
j=0
OS 'j''k!j
!(S, SymkH ( "1S)
F&: complex conjugation
H := R1#'"•A/S
' := #'"1A/S
'(k !" SymkH
!(S,'(k ( "1S)
Eisk+2syn (&)
4 KENICHI BANNAI
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#i
#
$ $ !()
% $
($, %)
%($) = (1! F&)%
&: torsion point&
&'
Eisk+2mot(&) $ H2g!1
mot (S, SymkH (g))
F&: complex conjugationDepartment of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
EISENSTEIN SHEAF 5
E!k+2(!)"
kdq
q
E!k+2(!) :=
!
(m,n) !=(0,0)
"#(m,n)
(m! + n)k+2
H1syn(S, Sym
kH (1))
H1D(S, Sym
kH (1))
Department of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
4KENIC
HIBANNAI
H1 dR(S,Sym
kH
(1))
H2g
!1
D(S,Sym
kH
(g))
!"H
2g!1
dR
(S,Sym
kH
(g))
A1:=
("1!"1)
2#i
#
$$!()
%$
($,%)
%($
)=
(1!F&)%
&:torsionpoint
& &'
Eisk+2
mot(&
)$H
2g!1
mot
(S,Sym
kH
(g))
Sym
kH
=k ! j=0
OS'j '
'k!j
!(S,Sym
kH
("
1 S)
F&:complexconjugation
H:=
R1#'"
• A/S
':=
#'"
1 A/S
'(k!"
Sym
kH
!(S,'
(k("
1 S)
Depa
rtmentofMathematics,
Keio
University,3-14-1
Kouhoku-k
u,Hiyosh
i,Yokohama,223-8522Japa
n
The Hodge Realization
The Hodge Realization
SOME STUDY OF HILBERT MODULAR FORMS
KENICHI BANNAI
Abstract. Notes for talk at Tokyo University, June 4, 2012.
1. Calculations of Cech cocycles
SymkHS base schemeA abelian varietyk: integer > 0
g := dimA
rD
rsyn
!
Case g = 1
Case g = 2? g := dimAQ: motivic sheaf
H := (R1!!Q)"
Date: June 4, 2012 (Personal notes - not for distribution)File Name: TokyoNotes.
1
EISENSTEIN SHEAF 5
E!k+2(!)"
kdq
q
E!k+2(!) :=
!
(m,n) !=(0,0)
"#(m,n)
(m! + n)k+2
H1syn(S, Sym
kH (1))
H1D(S, Sym
kH (1))
H3D(S, Sym
kH (2)) $" H3dR(S, Sym
kH (2)))
S: Hilbert modular surfaceDepartment of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
EISENSTEIN SHEAF 5
E!k+2(!)"
kdq
q
E!k+2(!) :=
!
(m,n) !=(0,0)
"#(m,n)
(m! + n)k+2
H1syn(S, Sym
kH (1))
H1D(S, Sym
kH (1))
H3D(S, Sym
kH (2)) $" H3dR(S, Sym
kH (2)))
F : real quadratic fieldDepartment of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
EISENSTEIN SHEAF 5
E!k+2(!)"
kdq
q
E!k+2(!) :=
!
(m,n) !=(0,0)
"#(m,n)
(m! + n)k+2
H1syn(S, Sym
kH (1))
H1D(S, Sym
kH (1))
H3D(S, Sym
kH (2)) $" H3dR(S, Sym
kH (2)))
F : real quadratic field OF : ring of integers S(C) = ! \ H2
Department of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
EISENSTEIN SHEAF 5
E!k+2(!)"
kdq
q
E!k+2(!) :=
!
(m,n) !=(0,0)
"#(m,n)
(m! + n)k+2
H1syn(S, Sym
kH (1))
H1D(S, Sym
kH (1))
H3D(S, Sym
kH (2)) $" H3dR(S, Sym
kH (2)))
! # SL2(OF )F : real quadratic field OF : ring of integers S(C) = ! \ H2
Department of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
EISENSTEIN SHEAF 5
E!k+2(!)"
kdq
q
E!k+2(!) :=
!
(m,n) !=(0,0)
"#(m,n)
(m! + n)k+2
H1syn(S, Sym
kH (1))
H1D(S, Sym
kH (1))
H3D(S, Sym
kH (2)) $" H3dR(S, Sym
kH (2)))
! # SL2(OF )F : real quadratic field OF : ring of integers S(C) = ! \ H2
Department of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
EISENSTEIN SHEAF 5
E!k+2(!)"
kdq
q
E!k+2(!) :=
!
(m,n) !=(0,0)
"#(m,n)
(m! + n)k+2
H1syn(S, Sym
kH (1))
H1D(S, Sym
kH (1))
H3D(S, Sym
kH (2)) $" H3dR(S, Sym
kH (2)))
! # SL2(OF )F : real quadratic field OF : ring of integers S(C) = ! \ H2
SymkH =k#
j=0
OH2yj"k$j
%(yj"k$j) = yj$1"k+1$j (1 & j & k)
%("k) = 0
" '= "1
$"2
H = "#
"(
OrdinaDepartment of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
EISENSTEIN SHEAF 5
E!k+2(!)"
kdq
q
E!k+2(!) :=
!
(m,n) !=(0,0)
"#(m,n)
(m! + n)k+2
H1syn(S, Sym
kH (1))
H1D(S, Sym
kH (1))
H3D(S, Sym
kH (2)) $" H3dR(S, Sym
kH (2)))
! # SL2(OF )F : real quadratic field OF : ring of integers S(C) = ! \ H2
SymkH =k#
j=0
OH2yj"k$j
%(yj"k$j) = yj$1"k+1$j (1 & j & k)
%("k) = 0
" '= "1
$"2
OrdinaDepartment of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
EISENSTEIN SHEAF 5
E!k+2(!)"
kdq
q
E!k+2(!) :=
!
(m,n) !=(0,0)
"#(m,n)
(m! + n)k+2
H1syn(S, Sym
kH (1))
H1D(S, Sym
kH (1))
H3D(S, Sym
kH (2)) $" H3dR(S, Sym
kH (2)))
! # SL2(OF )F : real quadratic field OF : ring of integers S(C) = ! \ H2
SymkH =k#
j=0
OH2yj"k$j
%(yj"k$j) = yj$1"k+1$j (1 & j & k)
%("k) = 0
" '= "1
$"2
H = "#
"(
• dim 2 abelian variety• real multiplication by F• level structure
) corresponding to embeddings F $" CDepartment of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
EISENSTEIN SHEAF 5
E!k+2(!)"
kdq
q
E!k+2(!) :=
!
(m,n) !=(0,0)
"#(m,n)
(m! + n)k+2
H1syn(S, Sym
kH (1))
H1D(S, Sym
kH (1))
H3D(S, Sym
kH (2)) $" H3dR(S, Sym
kH (2)))
! # SL2(OF )F : real quadratic field OF : ring of integers S(C) = ! \ H2
SymkH =k#
j=0
OH2yj"k$j
%(yj"k$j) = yj$1"k+1$j (1 & j & k)
%("k) = 0
" '= "1
$"2
H = "#
"(
• dim 2 abelian variety• real multiplication by F• level structure
) corresponding to embeddings F $" CDepartment of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
EISENSTEIN SHEAF 5
E!k+2(!)"
kdq
q
E!k+2(!) :=
!
(m,n) !=(0,0)
"#(m,n)
(m! + n)k+2
H1syn(S, Sym
kH (1))
H1D(S, Sym
kH (1))
H3D(S, Sym
kH (2)) $" H3dR(S, Sym
kH (2)))
! # SL2(OF )F : real quadratic field OF : ring of integers S(C) = ! \ H2
SymkH =k#
j=0
OH2yj"k$j
%(yj"k$j) = yj$1"k+1$j (1 & j & k)
%("k) = 0
" '= "1
$"2
H = "#
"(
• dim 2 abelian variety• real multiplication by OF
• level structure
) corresponding to embeddings F $" CDepartment of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
EISENSTEIN SHEAF 5
E!k+2(!)"
kdq
q
E!k+2(!) :=
!
(m,n) !=(0,0)
"#(m,n)
(m! + n)k+2
H1syn(S, Sym
kH (1))
H1D(S, Sym
kH (1))
H3D(S, Sym
kH (2)) $" H3dR(S, Sym
kH (2)))
! # SL2(OF )F : real quadratic field OF : ring of integers S(C) = ! \ H2
SymkH =k#
j=0
OH2yj"k$j
%(yj"k$j) = yj$1"k+1$j (1 & j & k)
%("k) = 0
" '= "1
$"2
H = "#
"(
• dim 2 abelian variety• real multiplication by OF
• level structure
) corresponding to embeddings F $" C
H '= OH2y1#
OH2y2#
OH2"1
#OH2"2
Dolbeault classDepartment of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
EISENSTEIN SHEAF 5
E!k+2(!)"
kdq
q
E!k+2(!) :=
!
(m,n) !=(0,0)
"#(m,n)
(m! + n)k+2
H1syn(S, Sym
kH (1))
H1D(S, Sym
kH (1))
H3D(S, Sym
kH (2)) $" H3dR(S, Sym
kH (2)))
! # SL2(OF )F : real quadratic field OF : ring of integers S(C) = ! \ H2
SymkH =k#
j=0
OH2yj"k$j
%(yj"k$j) = yj$1"k+1$j (1 & j & k)
%("k) = 0
" '= "1
$"2
H = "#
"(
• dim 2 abelian variety• real multiplication by OF
• level structure
) corresponding to embeddings F $" COn H2
H '= OH2y1#
OH2y2#
OH2"1
#OH2"2
%("1) = %("2) = 0
%("1) = y1, %("2) = y2, %("1) = %("2) = 0
Dolbeault classDepartment of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
EISENSTEIN SHEAF 5
E!k+2(!)"
kdq
q
E!k+2(!) :=
!
(m,n) !=(0,0)
"#(m,n)
(m! + n)k+2
H1syn(S, Sym
kH (1))
H1D(S, Sym
kH (1))
H3D(S, Sym
kH (2)) $" H3dR(S, Sym
kH (2)))
! # SL2(OF )F : real quadratic field OF : ring of integers S(C) = ! \ H2
SymkH =k#
j=0
OH2yj"k$j
%(yj"k$j) = yj$1"k+1$j (1 & j & k)
%("k) = 0
" '= "1
$"2
H = "#
"(
• dim 2 abelian variety• real multiplication by OF
• level structure
) corresponding to embeddings F $" COn H2
H '= OH2y1#
OH2y2#
OH2"1
#OH2"2
%("1) = %("2) = 0
%("1) = y1 * d!1, %("2) = y2 * d!2, %("1) = %("2) = 0
Dolbeault class
% =(!2 $ ! 2)
k + 1
k+1!
a,b=1
E(a,b)k+2 (!)"
a1"
k$a1 "b
2"k$b2
d!1 + d! 1A1
The Hodge Realization
SOME STUDY OF HILBERT MODULAR FORMS
KENICHI BANNAI
Abstract. Notes for talk at Tokyo University, June 4, 2012.
1. Calculations of Cech cocycles
SymkHS base schemeA abelian varietyk: integer > 0
g := dimA
rD
rsyn
!
Case g = 1
Case g = 2? g := dimAQ: motivic sheaf
H := (R1!!Q)"
Date: June 4, 2012 (Personal notes - not for distribution)File Name: TokyoNotes.
1
EISENSTEIN SHEAF 5
E!k+2(!)"
kdq
q
E!k+2(!) :=
!
(m,n) !=(0,0)
"#(m,n)
(m! + n)k+2
H1syn(S, Sym
kH (1))
H1D(S, Sym
kH (1))
H3D(S, Sym
kH (2)) $" H3dR(S, Sym
kH (2)))
Department of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
2 KENICHI BANNAI
Eisk+2D (!) ! H2g"1
D (S, SymkH (g))
H2g"1syn (S, SymkH (g))
Eisk+2dR (!)
EISENST
EIN
SHEAF
3
H2g!1
D(S, Sym
k H(g))!"
H0
D(SpecR
, H2g!1 (S,Sym
k H(g)))
Ep,q
2=H
pD(SpecR
, Hq (S, Sym
k H(g))#H
p+q
D(S, Sym
k H(g))
H1
D(SpecR
, H2g!2 (S,Sym
k H(g))"
"H
2g!1
D(S, Sym
k H(g))"H
0D(SpecR
, H2g!1 (S,Sym
k H(g)))"
0
H0
D(SpecR
, H2g!1 (S,Sym
k H(g))) !"H
2g!1
dR(S, Sym
k H(g))
$=!"E
1,0
k+2(")#
k1#
k2d"1% d
" 1A 1
% d" 2
E1,0
k+2(") =
!m,n&O
F
1
(m" 1+n)(m" 1+n)
k+1 (m
' " 2+n' )k
+2
&
EISENSTEIN
SHEAF
3
H2g
!1
D(S,Sym
kH
(g))
!"H
0 D(Spec
R,H
2g!1(S,Sym
kH
(g)))
Ep,q
2=
Hp D(Spec
R,H
q(S,Sym
kH
(g))
#H
p+q
D(S,Sym
kH
(g))
H1 D(Spec
R,H
2g!2(S,Sym
kH
(g))
"
"H
2g!1
D(S,Sym
kH
(g))
"H
0 D(Spec
R,H
2g!1(S,Sym
kH
(g)))"
0
H0 D(Spec
R,H
2g!1(S,Sym
kH
(g)))!"
H2g
!1
dR
(S,Sym
kH
(g))
$ = !"
E1,0
k+2(")#k 1#k 2d"
1%d"
1
A1
%d"
2
E1,0
k+2(")=
!
m,n&O
F
1
(m" 1
+n)(m"1+n)k
+1(m
' "2+n' )k+2
&
de Rham Class?Guess
The Hodge Realization
EISENSTEIN SHEAF 5
E!k+2(!)"
kdq
q
E!k+2(!) :=
!
(m,n) !=(0,0)
"#(m,n)
(m! + n)k+2
H1syn(S, Sym
kH (1))
H1D(S, Sym
kH (1))
H3D(S, Sym
kH (2)) $" H3dR(S, Sym
kH (2)))
Department of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
4 KENICHI BANNAI
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#iDepartment of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
EISENSTEIN SHEAF 3
H2g!1D (S, SymkH (g)) !" H0
D(SpecR, H2g!1(S, SymkH (g)))
Ep,q2 = Hp
D(SpecR, Hq(S, SymkH (g)) # Hp+q
D (S, SymkH (g))
H1D(SpecR, H2g!2(S, SymkH (g)) "
" H2g!1D (S, SymkH (g)) " H0
D(SpecR, H2g!1(S, SymkH (g))) " 0
H0D(SpecR, H2g!1(S, SymkH (g))) !" H2g!1
dR (S, SymkH (g))$=!"
" := E1,0k+2(#)$
k1$
k2d#1 % d# 1
A1% d#2
" := E1,0k+2(#)$
k1$
k2d#1 % d# 1
A21
% A1d#2
&!E1,0
k+2(#)$k1$
k2A1d#2
"=
!1! F (1)
'
"Ek+2(#)$
k1$
k2d#1 % d#2
Eak+2(#) =
#
(m,n) (=(0,0)(mod O!
F )
1
(m# 1 + n)a(m#1 + n)k+2!a(m)#2 + n))k+2
Ek+2(#) =#
m,n*OF
1
(m#1 + n)k+2(m)#2 + n))k+2
*
EISENSTEIN
SHEAF
3
H2g
!1
D(S,Sym
kH
(g))
!"H
0 D(Spec
R,H
2g!1(S,Sym
kH
(g)))
Ep,q
2=
Hp D(Spec
R,H
q(S,Sym
kH
(g))
#H
p+q
D(S,Sym
kH
(g))
H1 D(Spec
R,H
2g!2(S,Sym
kH
(g))
"
"H
2g!1
D(S,Sym
kH
(g))
"H
0 D(Spec
R,H
2g!1(S,Sym
kH
(g)))"
0
H0 D(Spec
R,H
2g!1(S,Sym
kH
(g)))!"
H2g
!1
dR
(S,Sym
kH
(g))
$ = !"
E1,0
k+2(")#k 1#k 2d"
1%d"
1
A1
%d"
2
E1,0
k+2(")=
!
m,n&O
F
1
(m" 1
+n)(m"1+n)k
+1(m
' "2+n' )k+2
&
Guess
EISENSTEIN SHEAF 5
E!k+2(!)"
kdq
q
E!k+2(!) :=
!
(m,n) !=(0,0)
"#(m,n)
(m! + n)k+2
H1syn(S, Sym
kH (1))
H1D(S, Sym
kH (1))
H3D(S, Sym
kH (2)) $" H3dR(S, Sym
kH (2)))
! # SL2(OF )F : real quadratic field OF : ring of integers S(C) = ! \ H2
SymkH =k#
j=0
OH2yj"k$j
%(yj"k$j) = yj$1"k+1$j (1 & j & k)
%("k) = 0
" '= "1
$"2
H = "#
"(
• dim 2 abelian variety• real multiplication by OF
• level structure
) corresponding to embeddings F $" CDolbeault classDepartment of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
EISENSTEIN SHEAF 7
! corresponding to embeddings F !" COn H2
H #= OH2y1!
OH2y2!
OH2"1
!OH2"2
$("1) = $("2) = 0
$("1) = y1 % d#1, $("2) = y2 % d#2, $("1) = $("2) = 0
Dolbeault class
$ =(#2 & # 2)
k + 1
k"
a,b=0
E(a+1,b+1)k+2 (# )"a
1"k&a1 "b
2"k&b2
d#1 ' d# 1A1
% :=k"
a=0
Ea+1k+2(# )"
a1"
k&a1 "k
2d#1 ' d# 1
A1' d#2
% :=k"
a=0
Ea+1k+2(# )"
a1"
k&a1 "k
2d#1 ' d# 1
A21
' A1d#2
(1& F (1)p ) (1& Fp)
%0 :=k"
a=0
Ea+1k+2u
a"k&a1 "k
2dq1q1
May define p-adic %0p-adic measure with values in p-adic modular formsAllows a < 0
#
Zp(Zp(Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)( E(&a,&b)k+2
#
Z(p (Zp(Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)( E(&a,&b)k+2
The Hodge Realization4 KENICHI BANNAI
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#i
#
($, %)
$($) = (1! F%)%
Department of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
EISENST
EIN
SHEAF
3
H2g!1
D(S, Sym
k H(g))!"
H0
D(SpecR
, H2g!1(S, Sym
k H(g)))
Ep,q
2=H
pD(SpecR
, Hq (S, Sym
k H(g))#
Hp+
qD
(S, Sym
k H(g))
H1
D(SpecR
, H2g!2(S, Sym
k H(g))"
"H
2g!1
D(S, Sym
k H(g))"
H0
D(SpecR
, H2g!1(S, Sym
k H(g)))"
0
H0
D(SpecR
, H2g!1(S, Sym
k H(g))) !"
H2g!1
dR(S, Sym
k H(g))
$=!"E
1,0
k+2(")#
k1#
k2d"1%d"
1A 1
%d"2
E1,0
k+2(") =
!m,n&O
F
1
(m" 1+n)(m" 1+n)
k+1 (m
' " 2+n' )k
+2
&
4 KENICHI BANNAI
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#i
#
$ $ !()
% $
($, %)
%($) = (1! F&)%
Department of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
4 KENICHI BANNAI
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#i
#
$ $ !()
% $
($, %)
%($) = (1! F&)%
F&: complex conjugationDepartment of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
EISENSTEIN SHEAF 5
E!k+2(!)"
kdq
q
E!k+2(!) :=
!
(m,n) !=(0,0)
"#(m,n)
(m! + n)k+2
H1syn(S, Sym
kH (1))
H1D(S, Sym
kH (1))
H3D(S, Sym
kH (2)) $" H3dR(S, Sym
kH (2)))
Department of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
EISENSTEIN
SHEAF
3
H2g
!1
D(S,Sym
kH
(g))
!"H
0 D(Spec
R,H
2g!1(S,Sym
kH
(g)))
Ep,q
2=
Hp D(Spec
R,H
q(S,Sym
kH
(g))
#H
p+q
D(S,Sym
kH
(g))
H1 D(Spec
R,H
2g!2(S,Sym
kH
(g))
"
"H
2g!1
D(S,Sym
kH
(g))
"H
0 D(Spec
R,H
2g!1(S,Sym
kH
(g)))"
0
H0 D(Spec
R,H
2g!1(S,Sym
kH
(g)))!"
H2g
!1
dR
(S,Sym
kH
(g))
$ = !"
E1,0
k+2(")#k 1#k 2d"
1%d"
1
A1
%d"
2
E1,0
k+2(")=
!
m,n&O
F
1
(m" 1
+n)(m"1+n)k
+1(m
' "2+n' )k+2
&
EISENSTEIN SHEAF 7
! corresponding to embeddings F !" COn H2
H #= OH2y1!
OH2y2!
OH2"1
!OH2"2
$("1) = $("2) = 0
$("1) = y1 % d#1, $("2) = y2 % d#2, $("1) = $("2) = 0
Dolbeault class
$ =(#2 & # 2)
k + 1
k"
a,b=0
E(a+1,b+1)k+2 (# )"a
1"k&a1 "b
2"k&b2
d#1 ' d# 1A1
% :=k"
a=0
Ea+1k+2(# )"
a1"
k&a1 "k
2d#1 ' d# 1
A1' d#2
% :=k"
a=0
Ea+1k+2(# )"
a1"
k&a1 "k
2d#1 ' d# 1
A21
' A1d#2
(1& F (1)p ) (1& Fp)
%0 :=k"
a=0
Ea+1k+2u
a"k&a1 "k
2dq1q1
May define p-adic %0p-adic measure with values in p-adic modular formsAllows a < 0
#
Zp(Zp(Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)( E(&a,&b)k+2
#
Z(p (Zp(Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)( E(&a,&b)k+2
6 KENICHI BANNAI
E(a,b)k+2 (!) =
!
(m,n)!=(0,0)(mod O!
F )
1
(m!1 + n)a(m!1 + n)k+2"a(m#!2 + n#)b(m#!2 + n#)k+2"b
Department of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
EISENSTEIN SHEAF 7
! corresponding to embeddings F !" COn H2
H #= OH2y1!
OH2y2!
OH2"1
!OH2"2
$("1) = $("2) = 0
$("1) = y1 % d#1, $("2) = y2 % d#2, $("1) = $("2) = 0
Dolbeault class
$ =(#2 & # 2)
k + 1
k"
a,b=0
E(a+1,b+1)k+2 (# )"a
1"k&a1 "b
2"k&b2
d#1 ' d# 1A1
% :=k"
a=0
Ea+1k+2(# )"
a1"
k&a1 "k
2d#1 ' d# 1
A1' d#2
% :=k"
a=0
Ea+1k+2(# )"
a1"
k&a1 "k
2d#1 ' d# 1
A21
' A1d#2
(1& F (1)p ) (1& Fp)
%0 :=k"
a=0
Ea+1k+2u
a"k&a1 "k
2dq1q1
May define p-adic %0p-adic measure with values in p-adic modular formsAllows a < 0
#
Zp(Zp(Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)( E(&a,&b)k+2
#
Z(p (Zp(Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)( E(&a,&b)k+2
The Syntomic Realization
EISENSTEIN SHEAF 7
! corresponding to embeddings F !" COn H2
H #= OH2y1!
OH2y2!
OH2"1
!OH2"2
$("1) = $("2) = 0
$("1) = y1 % d#1, $("2) = y2 % d#2, $("1) = $("2) = 0
Dolbeault class
$ =(#2 & # 2)
k + 1
k"
a,b=0
E(a+1,b+1)k+2 (# )"a
1"k&a1 "b
2"k&b2
d#1 ' d# 1A1
% :=k"
a=0
Ea+1k+2(# )"
a1"
k&a1 "k
2d#1 ' d# 1
A1' d#2
% :=k"
a=0
Ea+1k+2(# )"
a1"
k&a1 "k
2d#1 ' d# 1
A21
' A1d#2
(1& F (1)p ) (1& Fp)
%0 :=k"
a=0
Ea+1k+2u
a"k&a1 "k
2dq1q1
May define p-adic %0p-adic measure with values in p-adic modular formsAllows a < 0
#
Zp(Zp(Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)( E(&a,&b)k+2
#
Z(p (Zp(Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)( E(&a,&b)k+2
Chern Class
EISENSTEIN SHEAF 3
H2g!1D (S, SymkH (g)) !" H0
D(SpecR, H2g!1(S, SymkH (g)))
Ep,q2 = Hp
D(SpecR, Hq(S, SymkH (g)) # Hp+q
D (S, SymkH (g))
H1D(SpecR, H2g!2(S, SymkH (g)) "
" H2g!1D (S, SymkH (g)) " H0
D(SpecR, H2g!1(S, SymkH (g))) " 0
H0D(SpecR, H2g!1(S, SymkH (g))) !" H2g!1
dR (S, SymkH (g))$=!"
" := E1,0k+2(#)$
k1$
k2d#1 % d# 1
A1% d#2
" := E1,0k+2(#)$
k1$
k2d#1 % d# 1
A21
% A1d#2
E1,0k+2(#) =
!
m,n&OF
1
(m#1 + n)(m# 1 + n)k+1(m'#2 + n')k+2
&
EISENSTEIN SHEAF 3
H2g!1D (S, SymkH (g)) !" H0
D(SpecR, H2g!1(S, SymkH (g)))
Ep,q2 = Hp
D(SpecR, Hq(S, SymkH (g)) # Hp+q
D (S, SymkH (g))
H1D(SpecR, H2g!2(S, SymkH (g)) "
" H2g!1D (S, SymkH (g)) " H0
D(SpecR, H2g!1(S, SymkH (g))) " 0
H0D(SpecR, H2g!1(S, SymkH (g))) !" H2g!1
dR (S, SymkH (g))$=!"
" := E1,0k+2(#)$
k1$
k2d#1 % d# 1
A1% d#2
" := E1,0k+2(#)$
k1$
k2d#1 % d# 1
A21
% A1d#2
E1,0k+2(#) =
!
m,n&OF
1
(m#1 + n)(m# 1 + n)k+1(m'#2 + n')k+2
&
The Syntomic Realization
EISENSTEIN SHEAF 7
! corresponding to embeddings F !" COn H2
H #= OH2y1!
OH2y2!
OH2"1
!OH2"2
$("1) = $("2) = 0
$("1) = y1 % d#1, $("2) = y2 % d#2, $("1) = $("2) = 0
Dolbeault class
$ =(#2 & # 2)
k + 1
k"
a,b=0
E(a+1,b+1)k+2 (# )"a
1"k&a1 "b
2"k&b2
d#1 ' d# 1A1
% :=k"
a=0
Ea+1k+2(# )"
a1"
k&a1 "k
2d#1 ' d# 1
A1' d#2
% :=k"
a=0
Ea+1k+2(# )"
a1"
k&a1 "k
2d#1 ' d# 1
A21
' A1d#2
(1& F (1)p ) (1& Fp)
%0 :=k"
a=0
Ea+1k+2u
a"k&a1 "k
2dq1q1
May define p-adic %0p-adic measure with values in p-adic modular formsAllows a < 0
#
Zp(Zp(Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)( E(&a,&b)k+2
#
Z(p (Zp(Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)( E(&a,&b)k+2
The Syntomic Realization
EISENSTEIN SHEAF 3
H2g!1D (S, SymkH (g)) !" H0
D(SpecR, H2g!1(S, SymkH (g)))
Ep,q2 = Hp
D(SpecR, Hq(S, SymkH (g)) # Hp+q
D (S, SymkH (g))
H1D(SpecR, H2g!2(S, SymkH (g)) "
" H2g!1D (S, SymkH (g)) " H0
D(SpecR, H2g!1(S, SymkH (g))) " 0
H0D(SpecR, H2g!1(S, SymkH (g))) !" H2g!1
dR (S, SymkH (g))$=!"
" := E1,0k+2(#)$
k1$
k2d#1 % d# 1
A1% d#2
" := E1,0k+2(#)$
k1$
k2d#1 % d# 1
A21
% A1d#2
E1,0k+2(#) =
!
m,n&OF
1
(m#1 + n)(m# 1 + n)k+1(m'#2 + n')k+2
&
EISENSTEIN SHEAF 3
H2g!1D (S, SymkH (g)) !" H0
D(SpecR, H2g!1(S, SymkH (g)))
Ep,q2 = Hp
D(SpecR, Hq(S, SymkH (g)) # Hp+q
D (S, SymkH (g))
H1D(SpecR, H2g!2(S, SymkH (g)) "
" H2g!1D (S, SymkH (g)) " H0
D(SpecR, H2g!1(S, SymkH (g))) " 0
H0D(SpecR, H2g!1(S, SymkH (g))) !" H2g!1
dR (S, SymkH (g))$=!"
" := E1,0k+2(#)$
k1$
k2d#1 % d# 1
A1% d#2
" := E1,0k+2(#)$
k1$
k2d#1 % d# 1
A21
% A1d#2
E1,0k+2(#) =
!
m,n&OF
1
(m#1 + n)(m# 1 + n)k+1(m'#2 + n')k+2
&
4 KENICHI BANNAI
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#i
#Department of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
6 KENICHI BANNAI
! :=k!
a=0
Ea+1k+2(")#
a1#
k!a1 #k
2d"1 " d" 1
A1" d"2
! :=k!
a=0
Ea+1k+2(")#
a1#
k!a1 #k
2d"1 " d" 1
A21
" A1d"2
E(a,b)k+2 (!) =
!
(m,n)!=(0,0)(mod O!
F )
1
(m!1 + n)a(m!1 + n)k+2"a(m#!2 + n#)b(m#!2 + n#)k+2"b
Department of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
SOME STUDY OF HILBERT MODULAR FORMS
KENICHI BANNAI
Abstract. Notes for talk at Tokyo University, June 4, 2012.
1. Calculations of Cech cocycles
=: !0! (!0) = (k + 1)(1" F (1)
# )!Ek+2(" )#
k1#
k2d"1 $ d"2
".
SymkHS base schemeA abelian varietyk: integer > 0
g := dimA
rD
rsyn
$
Case g = 1
Case g = 2? g := dimAQ: motivic sheafDate: June 4, 2012 (Personal notes - not for distribution)File Name: TokyoNotes.
1
EISENSTEIN SHEAF 7
! :=k!
a=0
Ea+1k+2(")#
a1#
k!a1 #k
2d"1 " d" 1
A1" d"2
! :=k!
a=0
Ea+1k+2(")#
a1#
k!a1 #k
2d"1 " d" 1
A21
" A1d"2
"
Zp#Zp#Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)# E(!a,!b)k+2
F (1)$ : complex conjugation in first variable
a, b % 0
E(a,b)k+2 (!) =
!
(m,n)!=(0,0)(mod O!
F )
1
(m!1 + n)a(m!1 + n)k+2"a(m#!2 + n#)b(m#!2 + n#)k+2"b
Department of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
EISENSTEIN SHEAF 3
H2g!1D (S, SymkH (g)) !" H0
D(SpecR, H2g!1(S, SymkH (g)))
Ep,q2 = Hp
D(SpecR, Hq(S, SymkH (g)) # Hp+q
D (S, SymkH (g))
H1D(SpecR, H2g!2(S, SymkH (g)) "
" H2g!1D (S, SymkH (g)) " H0
D(SpecR, H2g!1(S, SymkH (g))) " 0
H0D(SpecR, H2g!1(S, SymkH (g))) !" H2g!1
dR (S, SymkH (g))$=!"
" := E1,0k+2(#)$
k1$
k2d#1 % d# 1
A1% d#2
" := E1,0k+2(#)$
k1$
k2d#1 % d# 1
A21
% A1d#2
&!E1,0
k+2(#)$k1$
k2A1d#2
"=
!1! F (1)
'
"Ek+2(#)$
k1$
k2d#1 % d#2
E1,0k+2(#) =
#
m,n(OF
1
(m#1 + n)(m# 1 + n)k+1(m)#2 + n))k+2
Ek+2(#) =#
m,n(OF
1
(m#1 + n)k+2(m)#2 + n))k+2
(
SOME STUDY OF HILBERT MODULAR FORMS
KENICHI BANNAI
Abstract. Notes for talk at Tokyo University, June 4, 2012.
1. Calculations of Cech cocycles
=: !0! (!0) = (k + 1)(1" F (1)
# )!Ek+2(" )#
k1#
k2d"1 $ d"2
".
SymkHS base schemeA abelian varietyk: integer > 0
g := dimA
rD
rsyn
$
Case g = 1
Case g = 2? g := dimAQ: motivic sheaf
Date: June 4, 2012 (Personal notes - not for distribution)File Name: TokyoNotes.
1
SOME STUDY OF HILBERT MODULAR FORMS
KENICHI BANNAI
Abstract. Notes for talk at Tokyo University, June 4, 2012.
1. Calculations of Cech cocycles
=: !0! (!0) = (k + 1)(1" F (1)
# )!Ek+2(" )#
k1#
k2d"1 $ d"2
".
SymkHS base schemeA abelian varietyk: integer > 0
g := dimA
rD
rsyn
$
Case g = 1
Case g = 2? g := dimAQ: motivic sheaf
Date: June 4, 2012 (Personal notes - not for distribution)File Name: TokyoNotes.
1
EISENSTEIN SHEAF 7
! corresponding to embeddings F !" COn H2
H #= OH2y1!
OH2y2!
OH2"1
!OH2"2
$("1) = $("2) = 0
$("1) = y1 % d#1, $("2) = y2 % d#2, $("1) = $("2) = 0
Dolbeault class
$ =(#2 & # 2)
k + 1
k"
a,b=0
E(a+1,b+1)k+2 (# )"a
1"k&a1 "b
2"k&b2
d#1 ' d# 1A1
% :=k"
a=0
Ea+1k+2(# )"
a1"
k&a1 "k
2d#1 ' d# 1
A1' d#2
% :=k"
a=0
Ea+1k+2(# )"
a1"
k&a1 "k
2d#1 ' d# 1
A21
' A1d#2
(1& F (1)p ) (1& Fp)
%0 :=k"
a=0
Ea+1k+2u
a"k&a1 "k
2dq1q1
May define p-adic %0p-adic measure with values in p-adic modular formsAllows a < 0
#
Zp(Zp(Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)( E(&a,&b)k+2
#
Z(p (Zp(Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)( E(&a,&b)k+2
The Syntomic Realization
SOME STUDY OF HILBERT MODULAR FORMS
KENICHI BANNAI
Abstract. Notes for talk at Tokyo University, June 4, 2012.
1. Calculations of Cech cocycles
=: !0! (!0) = (k + 1)(1" F (1)
# )!Ek+2(" )#
k1#
k2d"1 $ d"2
".
SymkHS base schemeA abelian varietyk: integer > 0
g := dimA
rD
rsyn
$
Case g = 1
Case g = 2? g := dimAQ: motivic sheaf
Date: June 4, 2012 (Personal notes - not for distribution)File Name: TokyoNotes.
1
4 KENICHI BANNAI
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#i
#
$ $ !()
% $
($, %)
%($) = (1! F&)%
&: torsion point&
&'
Eisk+2mot(&) $ H2g!1
mot (S, SymkH (g))
F&: complex conjugationDepartment of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
SOME STUDY OF HILBERT MODULAR FORMS
KENICHI BANNAI
Abstract. Notes for talk at Tokyo University, June 4, 2012.
1. Calculations of Cech cocycles
H2dR(S, Sym
kH )
=: !0! (!0) = (k + 1)(1" F (1)
# )!Ek+2(" )#
k1#
k2d"1 $ d"2
".
SymkHS base schemeA abelian varietyk: integer > 0
g := dimA
rD
rsyn
$
Case g = 1
Case g = 2? g := dimADate: June 4, 2012 (Personal notes - not for distribution)File Name: TokyoNotes.
1
SOME STUDY OF HILBERT MODULAR FORMS
KENICHI BANNAI
Abstract. Notes for talk at Tokyo University, June 4, 2012.
1. Calculations of Cech cocycles
H2dR(S, Sym
kH )
=: !0! (!0) = (k + 1)(1" F (1)
# )!Ek+2(" )#
k1#
k2d"1 $ d"2
".
Integration after (1" F (1)# )
SymkHS base schemeA abelian varietyk: integer > 0
g := dimA
rD
rsyn
$
Case g = 1
Date: June 4, 2012 (Personal notes - not for distribution)File Name: TokyoNotes.
1
SOME STUDY OF HILBERT MODULAR FORMS
KENICHI BANNAI
Abstract. Notes for talk at Tokyo University, June 4, 2012.
1. Calculations of Cech cocycles
H2dR(S, Sym
kH )
=: !0! (!0) = (k + 1)(1" F (1)
# )!Ek+2(" )#
k1#
k2d"1 $ d"2
".
Integration after (1" F (1)# ) Integration after (1 " F (1)
p )
SymkHS base schemeA abelian varietyk: integer > 0
g := dimA
rD
rsyn
$
Case g = 1
Date: June 4, 2012 (Personal notes - not for distribution)File Name: TokyoNotes.
1
SOME STUDY OF HILBERT MODULAR FORMS
KENICHI BANNAI
Abstract. Notes for talk at Tokyo University, June 4, 2012.
1. Calculations of Cech cocycles
H2dR(S, Sym
kH )
=: !0
! (!0) = (k + 1)(1" F (1)# )
!Ek+2(" )#
k1#
k2d"1 $ d"2
".
Integration after (1" F (1)# ) Integration after (1 " F (1)
p )
SymkH
S base schemeA abelian varietyk: integer > 0
g := dimAp-adic
rD
rsyn
$
Date: June 4, 2012 (Personal notes - not for distribution)File Name: TokyoNotes.
1
EISENSTEIN SHEAF 7
! :=k!
a=0
Ea+1k+2(")#
a1#
k!a1 #k
2d"1 " d" 1
A1" d"2
! :=k!
a=0
Ea+1k+2(")#
a1#
k!a1 #k
2d"1 " d" 1
A21
" A1d"2
"
Zp#Zp#Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)# E(!a,!b)k+2
F (1)$ : complex conjugation in first variable
F (1)p : Frobenius in first variable
a, b % 0
E(a,b)k+2 (!) =
!
(m,n)!=(0,0)(mod O!
F )
1
(m!1 + n)a(m!1 + n)k+2"a(m#!2 + n#)b(m#!2 + n#)k+2"b
Department of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
EISENSTEIN SHEAF 7
! corresponding to embeddings F !" COn H2
H #= OH2y1!
OH2y2!
OH2"1
!OH2"2
$("1) = $("2) = 0
$("1) = y1 % d#1, $("2) = y2 % d#2, $("1) = $("2) = 0
Dolbeault class
$ =(#2 & # 2)
k + 1
k"
a,b=0
E(a+1,b+1)k+2 (# )"a
1"k&a1 "b
2"k&b2
d#1 ' d# 1A1
q1d
dq1
% :=k"
a=0
Ea+1k+2(# )"
a1"
k&a1 "k
2d#1 ' d# 1
A1' d#2
% :=k"
a=0
Ea+1k+2(# )"
a1"
k&a1 "k
2d#1 ' d# 1
A21
' A1d#2
(1& F (1)p ) (1& Fp)
%0 :=k"
a=0
Ea+1k+2u
a"k&a1 "k
2dq1q1
May define p-adic %0p-adic measure with values in p-adic modular formsAllows a < 0
#
Zp(Zp(Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)( E(&a,&b)k+2
The Syntomic Realization6 KENICHI BANNAI
! :=k!
a=0
Ea+1k+2(")#
a1#
k!a1 #k
2d"1 " d" 1
A1" d"2
! :=k!
a=0
Ea+1k+2(")#
a1#
k!a1 #k
2d"1 " d" 1
A21
" A1d"2
"
Zp#Zp#Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)# E(!a,!b)k+2 (")
E(a,b)k+2 (!) =
!
(m,n)!=(0,0)(mod O!
F )
1
(m!1 + n)a(m!1 + n)k+2"a(m#!2 + n#)b(m#!2 + n#)k+2"b
Department of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
N. Katz
N. Katz, p-adic L-functions for CM fields, Invent. Math. 49 (1978), no. 3, 199–297.
6 KENICHI BANNAI
! :=k!
a=0
Ea+1k+2(")#
a1#
k!a1 #k
2d"1 " d" 1
A1" d"2
! :=k!
a=0
Ea+1k+2(")#
a1#
k!a1 #k
2d"1 " d" 1
A21
" A1d"2
"
Zp#Zp#Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)# E(!a,!b)k+2 (")
a, b $ 0
E(a,b)k+2 (!) =
!
(m,n)!=(0,0)(mod O!
F )
1
(m!1 + n)a(m!1 + n)k+2"a(m#!2 + n#)b(m#!2 + n#)k+2"b
Department of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
EISENSTEIN SHEAF 7
! :=k!
a=0
Ea+1k+2(")#
a1#
k!a1 #k
2d"1 " d" 1
A1" d"2
! :=k!
a=0
Ea+1k+2(")#
a1#
k!a1 #k
2d"1 " d" 1
A21
" A1d"2
(1! F (1)p )
"
Zp#Zp#Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)# E(!a,!b)k+2
F (1)$ : complex conjugation in first variable
F (1)p : Frobenius in first variable
a, b % 0
E(a,b)k+2 (!) =
!
(m,n)!=(0,0)(mod O!
F )
1
(m!1 + n)a(m!1 + n)k+2"a(m#!2 + n#)b(m#!2 + n#)k+2"b
Department of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
EISENSTEIN SHEAF 7
! :=k!
a=0
Ea+1k+2(")#
a1#
k!a1 #k
2d"1 " d" 1
A1" d"2
! :=k!
a=0
Ea+1k+2(")#
a1#
k!a1 #k
2d"1 " d" 1
A21
" A1d"2
(1! F (1)p )
"
Zp#Zp#Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)# E(!a,!b)k+2
"
Z#p #Zp#Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)# E(!a,!b)k+2
F (1)$ : complex conjugation in first variable
F (1)p : Frobenius in first variable
a, b % 0
E(a,b)k+2 (!) =
!
(m,n)!=(0,0)(mod O!
F )
1
(m!1 + n)a(m!1 + n)k+2"a(m#!2 + n#)b(m#!2 + n#)k+2"b
Department of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
EISENSTEIN SHEAF 7
! :=k!
a=0
Ea+1k+2(")#
a1#
k!a1 #k
2d"1 " d" 1
A1" d"2
! :=k!
a=0
Ea+1k+2(")#
a1#
k!a1 #k
2d"1 " d" 1
A21
" A1d"2
(1! F (1)p ) Allows a < 0"
Zp#Zp#Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)# E(!a,!b)k+2
"
Z#p #Zp#Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)# E(!a,!b)k+2
F (1)$ : complex conjugation in first variable
F (1)p : Frobenius in first variable
a, b % 0
E(a,b)k+2 (!) =
!
(m,n)!=(0,0)(mod O!
F )
1
(m!1 + n)a(m!1 + n)k+2"a(m#!2 + n#)b(m#!2 + n#)k+2"b
Department of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
Values in p-adic modular forms
EISENSTEIN SHEAF 7
! :=k!
a=0
Ea+1k+2(")#
a1#
k!a1 #k
2d"1 " d" 1
A1" d"2
! :=k!
a=0
Ea+1k+2(")#
a1#
k!a1 #k
2d"1 " d" 1
A21
" A1d"2
(1! F (1)p ) (1! Fp)
p-adic measure with values in p-adic modular formsAllows a < 0
"
Zp#Zp#Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)# E(!a,!b)k+2
"
Z#p #Zp#Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)# E(!a,!b)k+2
May define p-adic analogues of Eak+2(" ) := E(!a,0)
k+2 (" )
F (1)$ : complex conjugation in first variable
F (1)p : Frobenius in first variable
a, b % 0
E(a,b)k+2 (!) =
!
(m,n)!=(0,0)(mod O!
F )
1
(m!1 + n)a(m!1 + n)k+2"a(m#!2 + n#)b(m#!2 + n#)k+2"b
Department of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
OrdinaryLocus
EISENSTEIN SHEAF 7
! corresponding to embeddings F !" COn H2
H #= OH2y1!
OH2y2!
OH2"1
!OH2"2
$("1) = $("2) = 0
$("1) = y1 % d#1, $("2) = y2 % d#2, $("1) = $("2) = 0
Dolbeault class
$ =(#2 & # 2)
k + 1
k"
a,b=0
E(a+1,b+1)k+2 (# )"a
1"k&a1 "b
2"k&b2
d#1 ' d# 1A1
q1d
dq1
% :=k"
a=0
Ea+1k+2(# )"
a1"
k&a1 "k
2d#1 ' d# 1
A1' d#2
% :=k"
a=0
Ea+1k+2(# )"
a1"
k&a1 "k
2d#1 ' d# 1
A21
' A1d#2
(1& F (1)p ) (1& Fp)
%0 :=k"
a=0
Ea+1k+2u
a1"
k&a1 "k
2dq1q1
May define p-adic %0p-adic measure with values in p-adic modular formsAllows a < 0
#
Zp(Zp(Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)( E(&a,&b)k+2
The Syntomic Realization
EISENSTEIN SHEAF 7
! corresponding to embeddings F !" COn H2
H #= OH2y1!
OH2y2!
OH2"1
!OH2"2
$("1) = $("2) = 0
$("1) = y1 % d#1, $("2) = y2 % d#2, $("1) = $("2) = 0
Dolbeault class
$ =(#2 & # 2)
k + 1
k"
a,b=0
E(a+1,b+1)k+2 (# )"a
1"k&a1 "b
2"k&b2
d#1 ' d# 1A1
q1d
dq1
% :=k"
a=0
Ea+1k+2(# )"
a1"
k&a1 "k
2d#1 ' d# 1
A1' d#2
% :=k"
a=0
Ea+1k+2(# )"
a1"
k&a1 "k
2d#1 ' d# 1
A21
' A1d#2
(1& F (1)p ) (1& Fp)
%0 :=k"
a=0
Ea+1k+2u
a1"
k&a1 "k
2dq1q1
May define p-adic %0p-adic measure with values in p-adic modular formsAllows a < 0
#
Zp(Zp(Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)( E(&a,&b)k+2
EISENSTEIN SHEAF 7
! corresponding to embeddings F !" COn H2
H #= OH2y1!
OH2y2!
OH2"1
!OH2"2
$("1) = $("2) = 0
$("1) = y1 % d#1, $("2) = y2 % d#2, $("1) = $("2) = 0
Dolbeault class
$ =(#2 & # 2)
k + 1
k"
a,b=0
E(a+1,b+1)k+2 (# )"a
1"k&a1 "b
2"k&b2
d#1 ' d# 1A1
q1d
dq1
% :=k"
a=0
Ea+1k+2(# )"
a1"
k&a1 "k
2d#1 ' d# 1
A1' d#2
% :=k"
a=0
Ea+1k+2(# )"
a1"
k&a1 "k
2d#1 ' d# 1
A21
' A1d#2
(1& F (1)p ) (1& Fp)
%0 :=k"
a=0
Ea+1k+2u
a1"
k&a1 "k
2dq1q1
May define p-adic %0
% := Chern Class ( %0p-adic measure with values in p-adic modular forms
6 KENICHI BANNAI
!!k "" SymkH!(S,!!k ! "1
S)
Eisk+2syn (#)
E#k+2($ )!
kdq
q
E#k+2($ ) :=
!
(m,n) #=(0,0)
"#(m,n)
(m$ + n)k+2
H1syn(S, Sym
kH (1))
H3syn(S, Sym
kH (2))
H1D(S, Sym
kH (1))
H3D(S, Sym
kH (2)) "" H3dR(S, Sym
kH (2)))! $ SL2(OF )F : real quadratic field OF : ring of integers S(C) = ! \ H2
SymkH =k#
j=0
OH2yj!k%j
&(yj!k%j) = yj%1!k+1%j (1 ' j ' k)
&(!k) = 0
! (= !1
$!2
H = !#
!)
• dim 2 abelian variety• real multiplication by OF
• level structure
6 KENICHI BANNAI
!!k "" SymkH!(S,!!k ! "1
S)
Eisk+2syn (#)
E#k+2($ )!
kdq
q
E#k+2($ ) :=
!
(m,n) #=(0,0)
"#(m,n)
(m$ + n)k+2
H1syn(S, Sym
kH (1))
H3syn(S, Sym
kH (2))
H1D(S, Sym
kH (1))
H3D(S, Sym
kH (2)) "" H3dR(S, Sym
kH (2)))! $ SL2(OF )F : real quadratic field OF : ring of integers S(C) = ! \ H2
SymkH =k#
j=0
OH2yj!k%j
&(yj!k%j) = yj%1!k+1%j (1 ' j ' k)
&(!k) = 0
! (= !1
$!2
H = !#
!)
• dim 2 abelian variety• real multiplication by OF
• level structure
EISENSTEIN
SHEAF
5
H1 dR(S,Sym
kH
(1))
H2g!1
D(S,Sym
kH
(g))!"
H2g!1
dR
(S,Sym
kH
(g))
A1:=
("1!"1)
2#i
#
$$!()
%$
($,%)
%($)=(1
!F&)%
&:torsionpoint
& &'
Eisk+2
mot(&)$H
2g!1
mot
(S,Sym
kH
(g))
Sym
kH
=k ! j=0
OS'j '
'k!j
!(S,Sym
kH
("1 S)
F&:complex
conjugation
H:=
R1 #
'"• A/S
':=
#'"
1 A/S
EISENSTEIN SHEAF 5
H1dR(S, Sym
kH (1))
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#i
#
$ $ !()
% $
($, %)
%($) = (1! F&)%&: torsion point
&
&'
Eisk+2mot(&) $ H2g!1
mot (S, SymkH (g))
SymkH =k!
j=0
OS 'j''k!j
!(S, SymkH ( "1S)
F&: complex conjugation
H := R1#'"•A/S
' := #'"1A/S
4 KENICHI BANNAI
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#i
#
($, %)
$($) = (1! F%)%
Department of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
EISENST
EIN
SHEAF
3
H2g!1
D(S, Sym
k H(g))!"
H0
D(SpecR
, H2g!1(S, Sym
k H(g)))
Ep,q
2=H
pD(SpecR
, Hq (S, Sym
k H(g))#
Hp+
qD
(S, Sym
k H(g))
H1
D(SpecR
, H2g!2(S, Sym
k H(g))"
"H
2g!1
D(S, Sym
k H(g))"
H0
D(SpecR
, H2g!1(S, Sym
k H(g)))"
0
H0
D(SpecR
, H2g!1(S, Sym
k H(g))) !"
H2g!1
dR(S, Sym
k H(g))
$=!"E
1,0
k+2(")#
k1#
k2d"1%d"
1A 1
%d"2
E1,0
k+2(") =
!m,n&O
F
1
(m" 1+n)(m" 1+n)
k+1 (m
' " 2+n' )k
+2
&
6 KENICHI BANNAI
!!k "" SymkH!(S,!!k ! "1
S)
Eisk+2syn (#)
E#k+2($ )!
kdq
q
E#k+2($ ) :=
!
(m,n) #=(0,0)
"#(m,n)
(m$ + n)k+2
H1syn(S, Sym
kH (1))
H3syn(S, Sym
kH (2))
H1D(S, Sym
kH (1))
H3D(S, Sym
kH (2)) "" H3dR(S, Sym
kH (2)))! $ SL2(OF )F : real quadratic field OF : ring of integers S(C) = ! \ H2
%(%) = (1& Fp)&
SymkH =k#
j=0
OH2yj!k&j
%(yj!k&j) = yj&1!k+1&j (1 ' j ' k)
%(!k) = 0
! (= !1
$!2
H = !#
!)
• dim 2 abelian variety
EISENSTEIN SHEAF 7
• real multiplication by OF
• level structure! corresponding to embeddings F !" COn H2
H #= OH2y1!
OH2y2!
OH2"1
!OH2"2
$("1) = $("2) = 0
$("1) = y1 % d#1, $("2) = y2 % d#2, $("1) = $("2) = 0
Dolbeault class
$ =(#2 & # 2)
k + 1
k"
a,b=0
E(a+1,b+1)k+2 (# )"a
1"k&a1 "b
2"k&b2
d#1 ' d# 1A1
q1d
dq1
% :=k"
a=0
Ea+1k+2(# )"
a1"
k&a1 "k
2d#1 ' d# 1
A1' d#2
% :=k"
a=0
Ea+1k+2(# )"
a1"
k&a1 "k
2d#1 ' d# 1
A21
' A1d#2
(1& F (1)p ) (1& Fp)
%0 :=k"
a=0
Ea+1k+2u
a1"
k&a1 "k
2dq1q1
May define p-adic %0
% := Chern Class ( %0p-adic measure with values in p-adic modular forms
8 KENICHI BANNAI
Allows a < 0!
Zp!Zp!Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)! E("a,"b)k+2
!
Z!p !Zp!Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)! E("a,"b)k+2
!
Z!p !Z!
p !Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)! E("a,"b)k+2
May define p-adic analogues of Eak+2(! ) := E("a,0)
k+2 (! )
F (1)# : complex conjugation in first variable
F (1)p : Frobenius in first variable
a, b $ 0
E(a,b)k+2 (!) =
"
(m,n)!=(0,0)(mod O!
F )
1
(m!1 + n)a(m!1 + n)k+2"a(m#!2 + n#)b(m#!2 + n#)k+2"b
Department of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
8 KENICHI BANNAI
Allows a < 0, b < 0!
Zp!Zp!Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)! E("a,"b)k+2
!
Z!p !Zp!Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)! E("a,"b)k+2
!
Z!p !Z!
p !Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)! E("a,"b)k+2
May define p-adic analogues of Eak+2(! ) := E("a,0)
k+2 (! )
F (1)# : complex conjugation in first variable
F (1)p : Frobenius in first variable
a, b $ 0
E(a,b)k+2 (!) =
"
(m,n)!=(0,0)(mod O!
F )
1
(m!1 + n)a(m!1 + n)k+2"a(m#!2 + n#)b(m#!2 + n#)k+2"b
Department of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
8 KENICHI BANNAI
Allows a < 0, b < 0!
Zp!Zp!Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)! E("a,"b)k+2
!
Z!p !Zp!Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)! E("a,"b)k+2
!
Z!p !Z!
p !Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)! E("a,"b)k+2
May define p-adic analogues of E(a,b)k+2 (! ) := E("a,"b)
k+2 (! )
F (1)# : complex conjugation in first variable
F (1)p : Frobenius in first variable
a, b $ 0
E(a,b)k+2 (!) =
"
(m,n)!=(0,0)(mod O!
F )
1
(m!1 + n)a(m!1 + n)k+2"a(m#!2 + n#)b(m#!2 + n#)k+2"b
Department of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
The Syntomic Realization
EISENSTEIN SHEAF 7
! corresponding to embeddings F !" COn H2
H #= OH2y1!
OH2y2!
OH2"1
!OH2"2
$("1) = $("2) = 0
$("1) = y1 % d#1, $("2) = y2 % d#2, $("1) = $("2) = 0
Dolbeault class
$ =(#2 & # 2)
k + 1
k"
a,b=0
E(a+1,b+1)k+2 (# )"a
1"k&a1 "b
2"k&b2
d#1 ' d# 1A1
q1d
dq1
% :=k"
a=0
Ea+1k+2(# )"
a1"
k&a1 "k
2d#1 ' d# 1
A1' d#2
% :=k"
a=0
Ea+1k+2(# )"
a1"
k&a1 "k
2d#1 ' d# 1
A21
' A1d#2
(1& F (1)p ) (1& Fp)
%0 :=k"
a=0
Ea+1k+2u
a1"
k&a1 "k
2dq1q1
May define p-adic %0p-adic measure with values in p-adic modular formsAllows a < 0
#
Zp(Zp(Zp
xa1xb2y
k+1dµ(x1, x2, y) = (constant)( E(&a,&b)k+2
EISENSTEIN SHEAF 7
! corresponding to embeddings F !" COn H2
H #= OH2y1!
OH2y2!
OH2"1
!OH2"2
$("1) = $("2) = 0
$("1) = y1 % d#1, $("2) = y2 % d#2, $("1) = $("2) = 0
Dolbeault class
$ =(#2 & # 2)
k + 1
k"
a,b=0
E(a+1,b+1)k+2 (# )"a
1"k&a1 "b
2"k&b2
d#1 ' d# 1A1
q1d
dq1
% :=k"
a=0
Ea+1k+2(# )"
a1"
k&a1 "k
2d#1 ' d# 1
A1' d#2
% :=k"
a=0
Ea+1k+2(# )"
a1"
k&a1 "k
2d#1 ' d# 1
A21
' A1d#2
(1& F (1)p ) (1& Fp)
%0 :=k"
a=0
Ea+1k+2u
a1"
k&a1 "k
2dq1q1
May define p-adic %0
% := Chern Class ( %0p-adic measure with values in p-adic modular forms
6 KENICHI BANNAI
!!k "" SymkH!(S,!!k ! "1
S)
Eisk+2syn (#)
E#k+2($ )!
kdq
q
E#k+2($ ) :=
!
(m,n) #=(0,0)
"#(m,n)
(m$ + n)k+2
H1syn(S, Sym
kH (1))
H3syn(S, Sym
kH (2))
H1D(S, Sym
kH (1))
H3D(S, Sym
kH (2)) "" H3dR(S, Sym
kH (2)))! $ SL2(OF )F : real quadratic field OF : ring of integers S(C) = ! \ H2
SymkH =k#
j=0
OH2yj!k%j
&(yj!k%j) = yj%1!k+1%j (1 ' j ' k)
&(!k) = 0
! (= !1
$!2
H = !#
!)
• dim 2 abelian variety• real multiplication by OF
• level structure
6 KENICHI BANNAI
!!k "" SymkH!(S,!!k ! "1
S)
Eisk+2syn (#)
E#k+2($ )!
kdq
q
E#k+2($ ) :=
!
(m,n) #=(0,0)
"#(m,n)
(m$ + n)k+2
H1syn(S, Sym
kH (1))
H3syn(S, Sym
kH (2))
H1D(S, Sym
kH (1))
H3D(S, Sym
kH (2)) "" H3dR(S, Sym
kH (2)))! $ SL2(OF )F : real quadratic field OF : ring of integers S(C) = ! \ H2
SymkH =k#
j=0
OH2yj!k%j
&(yj!k%j) = yj%1!k+1%j (1 ' j ' k)
&(!k) = 0
! (= !1
$!2
H = !#
!)
• dim 2 abelian variety• real multiplication by OF
• level structure
EISENSTEIN
SHEAF
5
H1 dR(S,Sym
kH
(1))
H2g!1
D(S,Sym
kH
(g))!"
H2g!1
dR
(S,Sym
kH
(g))
A1:=
("1!"1)
2#i
#
$$!()
%$
($,%)
%($)=(1
!F&)%
&:torsionpoint
& &'
Eisk+2
mot(&)$H
2g!1
mot
(S,Sym
kH
(g))
Sym
kH
=k ! j=0
OS'j '
'k!j
!(S,Sym
kH
("1 S)
F&:complex
conjugation
H:=
R1 #
'"• A/S
':=
#'"
1 A/S
EISENSTEIN SHEAF 5
H1dR(S, Sym
kH (1))
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#i
#
$ $ !()
% $
($, %)
%($) = (1! F&)%&: torsion point
&
&'
Eisk+2mot(&) $ H2g!1
mot (S, SymkH (g))
SymkH =k!
j=0
OS 'j''k!j
!(S, SymkH ( "1S)
F&: complex conjugation
H := R1#'"•A/S
' := #'"1A/S
4 KENICHI BANNAI
H2g!1D (S, SymkH (g)) !" H2g!1
dR (S, SymkH (g))
A1 :=("1 ! " 1)
2#i
#
($, %)
$($) = (1! F%)%
Department of Mathematics, Keio University, 3-14-1 Kouhoku-ku, Hiyoshi, Yokohama, 223-8522 Japan
EISENST
EIN
SHEAF
3
H2g!1
D(S, Sym
k H(g))!"
H0
D(SpecR
, H2g!1(S, Sym
k H(g)))
Ep,q
2=H
pD(SpecR
, Hq (S, Sym
k H(g))#
Hp+
qD
(S, Sym
k H(g))
H1
D(SpecR
, H2g!2(S, Sym
k H(g))"
"H
2g!1
D(S, Sym
k H(g))"
H0
D(SpecR
, H2g!1(S, Sym
k H(g)))"
0
H0
D(SpecR
, H2g!1(S, Sym
k H(g))) !"
H2g!1
dR(S, Sym
k H(g))
$=!"E
1,0
k+2(")#
k1#
k2d"1%d"
1A 1
%d"2
E1,0
k+2(") =
!m,n&O
F
1
(m" 1+n)(m" 1+n)
k+1 (m
' " 2+n' )k
+2
&
Conclusion
EISENSTEIN SHEAF 7
• real multiplication by OF
• level structure! corresponding to embeddings F !" COn H2
H #= OH2y1!
OH2y2!
OH2"1
!OH2"2
$("1) = $("2) = 0
$("1) = y1 % d#1, $("2) = y2 % d#2, $("1) = $("2) = 0
Dolbeault class
$ =(#2 & # 2)
k + 1
k"
a,b=0
E(a+1,b+1)k+2 (# )"a
1"k&a1 "b
2"k&b2
d#1 ' d# 1A1
$ = Chern Class ( 1
k + 1
k"
a,b=0
E(a+1,b+1)k+2 ua1"
k&a1 ub2"
k&b2
q1d
dq1
% :=k"
a=0
Ea+1k+2(# )"
a1"
k&a1 "k
2d#1 ' d# 1
A1' d#2
% :=k"
a=0
Ea+1k+2(# )"
a1"
k&a1 "k
2d#1 ' d# 1
A21
' A1d#2
(1& F (1)p ) (1& Fp)
%0 :=k"
a=0
Ea+1k+2u
a1"
k&a1 "k
2dq1q1
May define p-adic %0