on the eisenstein class for hilbert modular surfaces · 2012. 6. 5. · hilbert modular surfaces ....

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