Siegel modular form of degree 2 and Abelian variety.

Siegel modular form of degree 2 and Abelian variety.

1

[

"!$#&%('*)

]

^+-,/.

Q

0 132547698;:"<>=7?

E

@;ABC,>D>E.;F3GIHCJK8MLMNI4

2

^.7:I<

OQPSR T

f

UWVQX&LZY\[

• L(s, E ) ≥≤ L(s, f ).

• E

^.

conductor

^]

f

^.

level

^W_a`bL-Y;[

c

.;d*e.;f3g]Ch&iM8

Eichler-Shimura

j k&U9l5mCnZB;8o[

[Eichler-Shimura

^pQq

] f = P

n ∈ Z > 0 a n exp(2πinz)

^HrNI4

2

^.

Hecke eigen cuspform

^{]sL-Y7[ c}

.t]MuvD

• Q f := Q h a n | n ∈ N i

^;wbxWE*y3z {-|9lSY

.

• [Q f : Q]

^E9}

Abel

^{~Q {}

A f

^|

L(s, H 1 (A f , Q l )) = Q

σ:Q f , → C L(s, f σ )

]vhbYb€(.>UWVQX-LZY;[

 @

Q f = Q

^{.‚]ouW^}

A f

^;:ƒ<>=;?-|9l9Y;[

47,QDa„3…I†r‡tˆ7d*e.

2

E/}>‰]3BC,aDQŠQ‹SŒS-.oŽ&;&‘a’*“Z@&”m

•/–

' )

root number =1

^{H €˜—}

Q

0M1M2‚4™6 8$Dš+S,3.

simple

^h

Abel

^=r›

A

^@™AIBœ,(D

L(s, H 1 (A, Q l )) = L(s, F, spin), up to finitely many bad primes

]Ch/YCEQz/D(NI4

2

^.

Siegel

OQP R3T

F

UoVQXbL-Y7[

]d eI4$6-n&B$8([

c

.$d eK.7fag]ChSi(83.MU>DoŽbŸžv ¢¡v]˜£/¤W6-Y

Siegel

OQP R3T

.$¥ ¦a§-|

B78W[Z¨3.oŽ&¢žv ¡©@3^;Et.Iª7«W¬9­oUbl5mCn(L\[

(I)

^:I<

O3P*R T

.

pair

^®5¯©¥*¦

:

S 2 (1) (Γ 0 (p)) 2 3 f 1 , f 2 7−→ Y f 2 1 ,f 2 ∈ M 2 2 (Γ 0 (p)), L(s, Y f 1 ,f 2 , spin ) = L(s, f 1 )L(s, f 2 ).

(II)

^ª°E3{

K = Q( √

N )

^.

Hilbert

O3P*R*T

®5¯°¥ ¦

:

S 2 (Γ K ) 3 f 7−→ Y f 2 ∈ M 2 2 (Γ 0 (N ), χ N ), L(s, Y f , spin ) = L(s, f ).

Γ K = ( ∗

√ N −1 ∗

√ N∗ ∗ ), ∗ ∈ o K .

Ž-o’S“K^oD

c .

type (I)

.WŽ-¢ž7 ¡;]±D

Eichler-Shimura

j*kZ|M²¯™6ZY

Abel

=(›]WH$³-´Q,3D

Ž97d eK.7¦QµBr,*´9Y

Siegel

OQP R3T

]

Abel

^=;›K.

pair

^.

example

Hv¶b·9n&B78([tBr®ƒBsD c .

Žb¢ž$ ¡©@3^WD

1

Yoshida’s non-vanishing problem

^c .(ŽbŸžC ¡^a´a—3|t€œwKh7¥ ¦a§h .o®Z·a,*´9Y

^ h3´S.>®

]r´ -U9l¢mrn-B;8o[

c

.‚@;ABv,aD

B¨ ocherer and Schulze-Pillot’s result (1992) type (I)

.oŽb¢ž$ ¡°^·*hQ´M[

]™´

UK476-n&B;8([

c

.$1 jt@b”m DW^$:ƒ<

OQP R T

(trivial central character)

^®Ÿ¯°²

¯±6KY

Abel

=;›t@;ABC,9^oŽb;dSe&UW¦3µ Br,/´bY

U/® mrn(L([

_ D*ŠSŒ*-.t "!S’9“-.7d*e

(1982)

^#

Brylinski-Labesse(1984)

.$&%Kh'>”tm DaE.

H™dSebLZYa. ^)(+*]+,-*6Kn(L\[

Conjecture 1 K

^{H. ª E/{&D}

ρ

^H

Gal (K/Q)

.a“*¦b}]CL‚Yo[>N4

2

^.

Hilbert eigen cuspform f (z 1 , z 2 ) = P

n∈o K a n exp(2πi(N ( n )z 1 + N ( n ) ρ z 2 ))

^{@oAB7, D}

K

0/1/24(6Z8/10.\F/GH2b8aL

Abel

^~3*{

A f with dim A f = [Q f : Q]

UoV3X&LKY;[

• L(s, H 1 (A f , Q l )) = Q

σ:Q f , → C L(s, f σ ).

• End Q (A f ) ⊃ Q f .

Remark 1 End Q (A f ) ⊃ K

^{|/^*h3´}

@4359[;dSe^o_6*E3zt.

totally real filed

^0t.

Hilbert

OQPSR T

@;ABv,K476*,S´bY\[

c .

Conjecture 1

H87 1bL369¤(D

 @

Hilbert

OQP R3T

f

®I¯œD(EK.9«o¬/­>.

Abel

^=;›

/Q

^Uo²¯±6

n\Lo[

i).

:"<a=;?

E f /K

^.

Weil restriction Res K/Q E f .

ii). S 2 (1) (Γ 0 (N ), χ N )

^{®5¯©²¯s6KY}

Abel

^=\›/[

(

^c 6b^;d e"H9731&LKY9:<;^3h3´>[

E-S.) iii). [Q f : Q] = 2

^|aD

‘

^{=)> ^}

’Q

0 1 254$6S,S´bY

Abel

^=\›

A f .

(genus 2

^.

hyper-elliptic curve C : y 2 = x 5 + a, (a ∈ Q × )

^.

Jacobian

^{h'$^?K´}

conductor

^H9@&—

.M|

full-modular

O3P*R3T

@\A"AB7h3´3U>D

3

^{. B} ]DC-·I¯˜6ZY;[

)

¨ c

|WEt@WŽ&;d*et@Et@7f3gIH7¶&·bY4FK.)(G*hH]QBC,aD*Ž&5ž$ ¡

type (II)

^{^K·*h3´}

®I7H9CZ·bn-B\8([

2

J K L

0‚.‚@M—/´a,WE.MZ·3U>¶b·I¯s6ZnoL([

p | N

^@;ABv,

local Fircke involution

^H

i p : S 2 (Γ K ) 3 f (z 1 , z 2 ) 7−→ f( − 1

pz 1

, − 1 pz 2

) ∈ S 2 (Γ K )

|(1*2&LZY;[

i p (f) = ± f

^|blSY;[

Main Theorem 1 f

^U

i p , p | N

^{@ON LZY}

eigen value

^UW+-,

1

h"¯v¤WD

(

^_/—9.

p

^|t€

i p (f ) = − f

^h

¯

Y f = 0.)

•

Ž&5ž7 ¡

type (II)

^.P

Y f

^{^K·ShQ´M[}

2

• Y f

^.

Fourier

^{3zt^}

Q f

^@‚n(6ZY\[

• f

^U

Doi-Naganuma lift of nebentypus

|W²¯s6ZYah"¯œD

Y f

^{^}

cuspform

^{|/^´M[}

S 2 (1) (Γ K ) ∨ → Y S 2 (2) (Γ 0 (N), χ N ),

S 2 (1) (Γ 0 (N), χ N ) DN → S 2 (Γ K ) N → Y N 2 (2) (Γ 0 (N ), χ N ) → Φ S 2 (1) (Γ 0 (N ), χ N ).

S 2 (Γ K )

^@*Y

D-N lift

^.4P"H

S 2 (Γ K ) N

^{|/l‚¯ IBsD -H}

S 2 (Γ K ) ∨

]aBr,*´9Y7[0 .

map

^{^}

Siegel

^{/³ 9[}

N (2) 2

^{^}

Siegel non-cuspform

^.

. g ∈ S (1) 2 (Γ 0 (N), χ N )

^®5¯BC,

n(|5u! #"

U$@%9D

Φ ◦ Y ◦ DN (g) = g + ¯ g

@Qh&i(,*´bY;[

E@oD

Main Theorem 2

0313j&|/^oD

f ∈ S 2 (Γ K ) N

.(Žb¢žv ¡.4P

Y f

^{^}

non-cuspform

|Sl9i(8MU>D

f

U

Gr¨ ossencharacter

| ®&SY>h"¯œD

Y f

^]')(

spinor L-

^NozIH9€$—

cuspform F C

UWV3X-LZY;[

c

ϫ>ir8+* ,&.

explicit

h-/.¢]WBs,(D

[4]

^@10

y 2 = x 5 − x

^@$AA5Br8

Siegel cuspform

^U\V>XŸB

Igusa- theta

^.243&.

R

H*Bs,a´*Yr[65W]s´) $H% U3lƒm±n$L([±hoD

y 2 = x 5 − x

^{^}

E : y 2 = x(x − 1)(x − 1 − √ 2)

.

Weil restriction

^{h .>|}

C-simple

^{|b^3h3´M[}

4o, D3ŽZ(d/e]

(

798"4(6&8

)

ƒ\d9eIH47*1ZLYt]

Siegel

OSPSRST

.

standard L-

NMz"@HN*L Y

Kudla-Rallis-Soudry

^$&%

(1993)

^{”m D}

Abel

^=;›.

Hasse-Weil

N(zt."ªES}

part

.=:;/h<=

L 0 (s, H 2 (A, Q l )) := ζ(s − 1) − 2 L(s, H 2 (A, Q l ))

@N"BC,oE.

U>K·bn\L([

A/ Q

^U

simple

^{® —}

root number = 1

^hI¯C¤oD

ord s=2 L 0 (s, H 2 (A, Q l )) ≥ − 1

^|

1. End Q (A) = Q ⇒ ord s=2 L 0 (s, H 2 (A, Q l )) ≥ 1.

2. ord s=2 L 0 (s, H 2 (A, Q l )) = 0 ⇒ L(s, H 1 (A, Q l ))

^]?'@(

L-

NQz¢H @—a:<

O9PbR/T

.

pair

^®

Hilbert

O3P*R3T

(

^A ªE { 0€B9C

)

^UWV3X-L-Y

.

3. ord s=2 L 0 (s, H 2 (A, Q l )) = − 1 ⇐⇒ A

^{^}

S 2 (1) (Γ 0 (N ), χ N )

^{®5¯°²¯˜6ZY}

Abel

^=\›

.

D

i˜,oDFEGH&´O]WB˜,WDI0

End Q (A) 6 = Q

^h‚¯˜¤;D

Shimura curve

^.

Jacobian

^®Ÿª E>{>0Z.

Hilbert

OQPSR T

.

motive

^]

isogeneous

^®45]v´ -UCK·I¯s6Zn\L([

J K L M

[1] B¨ocherer, S and Schulze-Pillot, R: Siegel modular forms and theta series attached to quaternion algebras, Nagoya Math. J. 121 (1991), 35-96. II, 147 (1997), 71-106.

[2] Kudla, S, Rallis, S and Soudry, D: On the degree 5 L-function for Sp(2), Invent. math. 107 (1992), 483-541.

[3] Oda, T: Periods of Hilbert surfaces. Progress in math. 19, Birkh¨asser, (1982).

[4] Okazaki, T: Proof of R. Salvati Manni and J. Top’s conjectures on Siegel modular forms and Abelian surfaces, Amer. J. Math. 128, (2006) 139-165.

[5] Yoshida, H: Siegel’s modular forms and the arithmetics of quadratic forms, Invent. math. 60 (1980), 193-248.

3