On the faithfulness of parabolic cohomology as a Hecke module over a finite field

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k ≥ 2

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C[X, Y ] k−2

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(2)

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k ≥ 2

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l

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q

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q

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xu

l

ss

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SL 2 (Z)

^v^r^{kl ^pkôûô^p{lû^v^m{v^p^qr^{k^l^}o^ml

l

sw

vm

p ≥ 5

^qu ^v| ô|~ ^pkoûô^p{lû^v^m{v^p^rqû^{k^l^zûqxⁿ

Γ 1 (N )

^y^v{k

N ≥ 5

(3)

ww qt u t u q tqwxs u rqut u r u q x

¤

§

qu

R = C

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k

qsqtqu nkvp p

x mn

rqut m |

w l

u

{kl omm

xt n{v

q

|m

N ≥ 5

^o|^w

k ≥ 2

^q^|l ^k^om

S k (Γ 1 (N ), Z[1/N ]) ⊗ R ∼ = S k (Γ 1 (N ), R)

^rqu ^o|~ ^u^v|z

R

^v| ^y^kvpk

N

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| no

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l

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k = 1

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q

l

u

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o n

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p

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l

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o

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u

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v

uu l

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s l ¨o

sq vm

u ln

u lml|{o{v

q

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Gal(Q|Q) → GL 2 (F p )

^y^kvpk ôû^l ^x^|ûô^t^vl^w

o{

p

ôû^l^p^q^|ªlp{^xu^l^w ^{^q ^p^qûu^lmn^q^|^w ^{^q ô{ ^lvz^l|^rqût^m ^qr ^y^lvz^k{ ^q^|l

v|o

ss

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q

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t

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o

u

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q o}

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y

ko{ {kl~ p

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qr

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vm

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q

l|

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q

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u

~

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l

w l|

q {l }~

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{kl ml

t v

z

uqx n

qr v|{lz

u o

s

2 × 2

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l

uq w l{l

ut

v|o|{ | v{

q

|lkom

kv

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t ov| v|

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q

|

¡ _{a b}

c d

¢ ι

= ¡ _{d −b}

−c a

¢

qu o

u v|z

R

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R[X, Y ] n

m{o|

w m

rqu {kl k

qtq zl|l

qx mn

qs

~|

qt vo

s m

qr

w lz

u

ll

n

^v| ^{kl ^o^u^vo}^s^lm

X, Y

^l ^s^l{

V n (R) := Sym ⁿ (R ² ) ∼ = R[X, Y ] n

y kvpk

y l l

x vn

y

v{k {kl |o{

xu o

s s l

r {

Mat 2 (Z) 6=0

ml

t v

z

uqx n op{v

q

|

r

V

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R

^t^q^wxs^l^s^l{

V ^∨

^}^l ^{k^l^wx^o^s

R

^t^q^wxs^l

Hom R (V, R)

m

x m

x o

s

{kl

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u ko

sr n

s o|l vm

w l|

q {l

w

}~

H

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{kl m{o|

w o

uw p

q

|z

ux l|pl m

x }z

uqx nm

Γ(N )

Γ 1 (N )

^o|^w

Γ 0 (N )

^q^r

SL 2 (Z)

^rq^u ^o^|

v|{lzl

u

N ≥ 1

^y^kvpk ôû^l ^w^l|l^w ^{^q ^p^q^|mv^m{ ^qr ^{kl ^tô{û^v^plm

¡ _{a b}

c d

¢ ∈ SL 2 (Z)

u l

wx pv|z

tqwxsq

N

^{^q

( ^{1 0} _{0 1} )

( ^{1 ∗} _{0 1} )

^u^lmn

( ^{∗ ∗} _{0 ∗} )

¿ ÀÁdÁÂeÃih ÄdegÅ heÆeÇeÃeÄÈ

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q

|

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q

|

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w n

u lml|{

m

qt l n

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u {vlm{

q }l

x ml

w

v| {kl ml

x l

s

l mko

ss x ml m{o|

w o

uwr op{m

q

| z

uqx n

p

q k

qtqsq z~

y v{k

qx { mnlpvo

s u l

r l

u

l|pl jkl

u lo

w l

u po|

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t n

s lp

q

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xs {

¡

(4)

É°Ê°¶°²°µË °Ì

PSL ² (Z)

jkl z

uqx

n

PSL 2 (Z)

^vm ^ru^ll^s^~ ^zl|lûô{l^w ^}~ ^{k^l ^tô{û^v ^p^sômmlm ^qr

σ := ¡ _{0 −1}

1 0

¢

o|

w

τ := ¡ _{1 −1}

1 0

¢

^p

q

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x

l|pl vm {kl

rqssqy

v|z mnlpvo

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qr

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mlllz

¡

n

qu o|~

u v|z

R

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R[PSL 2 (Z)]

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l

M

^{kl ^ml^x^l|pl

0 → M ^PSL ² ^(Z) → M ^hσi ⊕ M ^{hτ i} → M

→ H ¹ (PSL 2 (Z), M ) → H ¹ (hσi, M ) ⊕ H ¹ (hτ i, M ) → 0

vm lop{ o|

w rqu o

ss

i ≥ 2

^q^|l ^ko^m^v^m^q^t^q^u^nkv^m^t^m

¡

H ⁱ (PSL 2 (Z), M ) ∼ = H ⁱ (hσi, M ) ⊕ H ⁱ (hτ i, M ).

ÜÝÞÝßßàÞá âãä åÐÕ

R

^æ^Ð ^Î ^ÑÔÚç ^Î^Úè

Γ ≤ PSL 2 (Z)

^æ^Ð ^Î ^×Ù^æ^{çÑÖÙ é} ^Öê ^ë^Ú^ÔÕ^Ð ^ÔÚ^èÐì

×ÙÛí ÕíÎÕ Îîî ÕíÐ ÖÑèÐÑ×

Öê

Îîî ×ÕÎ

æ

ÔîÔ×ÐÑ çÑÖÙ é×

Γ x

ê

ÖÑ

x ∈ H

^ÎÑÐ ^ÔÚ^ïÐÑÕÔ^æ^îÐ

ÔÚ

R

^ð ^ñ^í^ÐÚ^ê^Ö^Ñ ^Î^îî

R[Γ]

^Òò^ÖèÙî^Ð×

V

^Ö^ÚÐ^í^Î×

H ¹ (Γ, V ) = M/(M ^hσi + M ^{hτ i} )

^ó^ÔÕí

M = Coind ^PSL _Γ ² ^(Z) (V )

^ÎÚ^è

H ⁱ (Γ, V ) = 0

^ê^Ö^Ñ ^Îîî

i ≥ 2

^ð

ô ÞÝÝõã

l

u m{

u lpo

ss

{ko{ o|~ |

q

|

{

u v

vo

ss

~ m{o}v

s vml

w n

q

v|{

x

^qr

H

^vm

p

q

|ª

x

zo{l}~ o| l

s l

t l|{

qr

PSL 2 (Z)

^{^q ^lv{kl^u

i

^q^u

ζ 3 = e ^2πi/3

^y^kl|pl ^o^ss ^|^q^|

{

u v

vo

s m{o}v

s vml

u z

uqx nm o

u l

qr {kl

rqut

ghσig ⁻¹ ∩ Γ

^qu

ghτ ig ⁻¹ ∩ Γ

^rq^u ^m^qt^l

g ∈ PSL 2 (Z)

ûqt ^«ô^pl~^m ^rqut^xsô ^mll ^lz ^¡^q^|l ^q^}{o^v|m

H ⁱ (hσi, Coind ^PSL _Γ ² ^(Z) V ) ∼ = Y

g∈Γ\PSL 2 (Z)/hσi

H ⁱ (ghσig ⁻¹ ∩ Γ, V )

rqu o

ss

i

ô^|^w ô^mv^t^v^sôûû^lm^x^s^{^rqû

τ

^¥^x^l^{^q ^{kl^v|^lû^{v}v^s^v{~ô^mm^x^t^n{v^q^|^{k^lû^lm^xs^{

rqssqy m

ruqt konv

uq

m

s l

tt o o|

w x o{v

q

|m

o|

w

¡

2

jkl omm

xt n{v

q

|m

qr {kl n

uq n

q mv{v

q

| o

u l

rqu

v|m{o|pl o

sy

o~m mo{vml

w v

r

R

^vm

o l

sw qr pko

u op{l

u vm{vp |

q

{

2

^qu

3

^j^kl~ ^o^s^m^q ^k^qsw ^rq^u

Γ 1 (N )

^y^v{k

N ≥ 4

^q^l^u

o|~

u v|z

ö³÷¯¾·¾°¯ °Ì ø½ù½ú°²¾ µù°ûø °Ê°¶°²°µË

l{

R

^}^l ^o ^u^v|z

Γ ≤ PSL 2 (Z)

ô ^m^x^}^zûq^xⁿ ^qr ^|v{l ^v|^w^l ô|^w

T = τ σ = ( ^{1 1} _{0 1} )

|l

w

l|lm{kl éÎÑÎ

æ

ÖîÔÛ çÑÖÙ é ÛÖíÖòÖîÖçÏ çÑÖÙ é êÖÑ ÕíÐ

R[Γ]

^Òò^ÖèÙîÐ

V

^om^{kl

l

u

|l

s qr {kl

u lm{

u vp{v

q

|

t on v|

§

0 → H _par ¹ (Γ, V ) → H ¹ (Γ, V ) −−→ ^res Y

g∈Γ\PSL 2 (Z)/hT i

H ¹ (Γ ∩ hgT g ⁻¹ i, V ).

jkl

w l|v{v

q

|

qr no

u o}

qs vpp

q k

qtqsq z~vmp

qt no{v}

s l

y v{k

konv

uq

m

s l

tt o

vl

x o{v

q

|

§

vm vm

qtqu nkvp{

q

0 → H _par ¹ (PSL 2 (Z), M ) → H ¹ (PSL 2 (Z), M ) −−→ H ^res ¹ (hT i, M )

y

v{k

M = Coind ^PSL _Γ ² ^(Z) V = Hom _R[Γ] (R[PSL 2 (Z)], V )

^o^m ^q^|l ^mllm ^x^mv|z ^lz

«opl~m

rqutxs

o om v| {kln

uqqrqr

£

quqss o

u

~

(5)

R

^æ

Γ ≤ PSL 2 (Z)

^æ ^æ

ÔÚèÐ ì ×ÙÛí ÕíÎÕ Îîî ÕíÐÖÑèÐÑ× Öê Îîî ×ÕÎ

æ

ÔîÔ×ÐÑ çÑÖÙ é×

Γ x

êÖÑ

x ∈ H

^ÎÑÐ ^ÔÚ^ïÐÑÕÔ^æ^îÐ

ÔÚ

R

^ð ^ñ^í^ÐÚ^ê^Ö^Ñ ^Îî^î

R[Γ]

^Òò^ÖèÙîÐ×

V

^Õí^Ð ^×^{ÐØÙÐÚÛÐ}

0 → H _par ¹ (Γ, V ) → H ¹ (Γ, V ) −−→ ^res Y

g∈Γ\PSL 2 (Z)/hT i

H ¹ (Γ ∩ hgT g ⁻¹ i, V ) → V Γ → 0

Ô× Ð ìÎÛÕ

ð

ôÞÝÝõã ¥

x l{

q {klomm

xt n{v

q

|m

y l

t o~onn

s

~ £

quqss o

u

~

jkl

u lm{

u vp{v

q

|

t on v|

x o{v

q

|

{k

x m }lp

qt lm

M/(M ^hσi + M ^{hτ i} ) −−−−−−−−→ M/(1 − T )M, ^{m7→(1−σ)m}

mv|pl

H ¹ (hT i, M ) ∼ = M/(1 − T )M

^j^kl ^vm^qt^qu^nk^vm^t

M ∼ = (R[PSL 2 (Z)] ⊗ R V ) Γ

o

ssqy m

q

|l {

q p

qt n

x

{l {ko{ {kl p

q

l

u

|l

s qr {kvm

t on vm

V Γ

{kl

Γ

^p^q^v|^o^u^vo^|{m

2

Ê³ ¶°´û²³

V _n (R)

l{

R

^}^l ^o ^u^v|z ^lpo^s^s ^ruq^t ^q^{o{v^q^| ^{ko{ ^y^l ⁿ^x^{

V n (R) = Sym ⁿ (R ² ) ∼ = R[X, Y ] n

ôÞÝüÝýþÿþÝ âã Ù ééÖ×Ð ÕíÎÕ

n!

^Ô× ^ÔÚ^ïÐÑÕÔ^æ^îÐ ^ÔÚ

R

^ð ^ñ^í^ÐÚ ^Õí^ÐÑÐ ^Ô× ^Î ^é^ÐÑê^ÐÛÕ

éÎÔÑÔÚç

V n (R) × V n (R) → R

Öê

R

^Òò^Öè^ÙîÐ× ^ó^í^ÔÛí ^ÔÚ^èÙÛÐ× ^ÎÚ ^Ô×^Öò^ÖÑé^í^Ô×ò

V n (R) → V n (R) ^∨

^Öê

R

^Òò^ÖèÙî^Ð×

ÑÐ×éÐÛÕÔÚç ÕíÐ

Mat 2 (Z) 6=0

ÒÎÛÕÔÖÚ

ó

íÔÛí Ô× çÔïÐÚ ÖÚ

V n (R) ^∨

^æ^Ï

(M.φ)(w) = φ(M ^ι w)

^ê^ÖÑ

M ∈ Mat 2 (Z) 6=0

φ ∈ V n (R) ^∨

^ÎÚ^è

w ∈ V n (R)

^ð

ôÞÝÝõã |l

w

l|lm {kl nl

ur lp{ nov

u v|z

q

|

V n (R)

^}~ û^m{ ^p^q^|m{ûx^p{v|^z ô

nl

ur lp{ nov

u v|z

q

|

R ²

^y^kvpk ^y^l ^p^q^|mv^w^lû ô^m^p^q^sxt^| ^lp{^qû^m ^|l ^ml{m

R ² × R ² → R, hv, wi := det(v|w) = v 1 w 2 − v 2 w 1 .

r

M

^vm ô ^tô{û^v ^v^|

Mat 2 (Z) 6=0

q

|l pklpm lomv

s

~ {ko{

hM v, wi = hv, M ^ι wi

jkvm nov

u

v|z l{l|

w m |o{

xu o

ss

~ {

q o nov

u v|z

q

| {kl

n

^{k ^{l|^m^q^u ⁿ^qy^l^u ^qr

R ²

¥

x l {

q

{kl omm

xt n{v

q

|

q

| {kl v|

l

u {v}v

s v{~

qr

n!

^y^l ^t^o~ ^v^l^y

Sym ⁿ (R ² )

^om ^o

m

x }

tqwxs

l v| {kl

n

^{k ^{l|^m^qu ⁿ^qy^lûô^|^w ^kl|^pl ^q^}{ov^| ^{kl ^w^lmvû^l^w ⁿôvû^v|z ô^|^w

{klvm

qtqu nkvm

t qr

{kl m{o{l

t l|{

2

à âã åÐÕ

n ≥ 1

^æ^Ð ^ÎÚ ^ÔÚÕÐç^ÐÑ

t = ( ^{1 N} _{0 1} )

^ÎÚ^è

t ⁰ = ( _{N 1} ^{1 0} )

^ð ^ê

n!N

^Ô×

ÚÖÕ Î ÐÑÖ èÔïÔ×ÖÑ ÔÚ

R

^Õí^ÐÚ^ê^ÖÑ ^Õí^Ð

t

^ÒÔÚ^ïÎÑ^ÔÎ^ÚÕ× ^ó^Ð^í^ÎïÐ

V n (R) ^hti = hX ⁿ i

^ÎÚ^è

êÖÑ ÕíÐ

t ⁰

^ÒÔÚïÎÑ^ÔÎÚÕ×

V n (R) ^ht ⁰ ⁱ = hY ⁿ i

^ð ^ê

n!N

^Ô× ^ÔÚ^ïÐÑ^ÕÔ^æ^îÐ ^ÔÚ

R

^Õí^ÐÚ ^Õí^Ð ^ÛÖ^ÔÚ^Ò

ïÎÑÔÎÚÕ× ÎÑÐ çÔïÐÚ

æ

Ï

V n (R) hti = V n (R)/hY ⁿ , XY ⁿ⁻¹ , . . . , X ⁿ⁻¹ Y i

^ÑÐ×é^{ÐÛÕÔïÐîÏ}

V n (R) ht ⁰ i = V n (R)/hX ⁿ , X ⁿ⁻¹ Y, . . . , XY ⁿ⁻¹ i

^ð

(6)

ôÞÝÝõã jkl op{v

q

|

qr

t

^vm

t.(X ⁿ⁻ⁱ Y ⁱ ) = X ⁿ⁻ⁱ (N X + Y ) ⁱ

^o|^w ^p^q^|ml^x^l|{^s^~

(t − 1).(X ⁿ⁻ⁱ Y ⁱ ) = P i−1

j=0 r i,j X ^n−j Y ^j

^y^v{k

r i,j = N ^i−j ¡ _i

j

¢

y

kvpk vm|

q { ol

uq

w v

vm

qu

u lmnlp{v

l

s

~ v|

l

u {v}

s l

}~ omm

xt n{v

q

|

qu

x = P n

i=0 a i X ⁿ⁻ⁱ Y ⁱ

^y^l

ko

l

(t − 1).x = P n−1

j=0 X ^n−j Y ^j ( P n

i=j+1 a i r i,j ).

^r

(t − 1).x = 0

^y^l^p^q^|^p^sx^w^l^rqu

j = n − 1

^{ko{

a n = 0

l{

rqu

j = n − 2

^v{ ^rq^ssqy^m ^{k^o{

a n−1 = 0

o|

w m

q

|

x

|{v

s

a 1 = 0

^j^kvm ⁿ^uq^lm^{kl ^m{o{l^t^l|{^q^|^{k^l

t

^v|^o^u^vo^|{m ^jkl ^q^|^l^q^|^{kl

t ⁰

^v|ôû^vo|{m^rq^ssqy^m ^ruq^t ^m~^t^t^l{û^~ ^j^kl ^p^sôv^t^m ^q^| ^{kl ^p^q^v|ôû^vo^|{m ôû^lⁿûq^l^w

v| o

l

u

~mv

t v

s o

u o|

w m{

u ovzk{

rquy o

uw y o~

2

ôÞÝüÝýþÿþÝ âã åÐÕ

n ≥ 1

^æ^Ð ^ÎÚ ^ÔÚ^ÕÐç^ÐÑ^ð

Î

ê

n!N

^Ô× ^ÚÖ^Õ ^Î ^ÐÑÖ ^{èÔïÔ×ÖÑ} ^ÔÚ

R

^Õí^ÐÚ ^Õí^Ð

R

^Òò^ÖèÙ^îÐ ^Öê

Γ(N )

^ÒÔÚ^ïÎÑÔÎÚ^Õ×

V n (R) ^{Γ(N )}

^Ô× ^ÐÑÖ^ð

æ

ê

n!N

^Ô× ^ÔÚï^ÐÑÕÔ^æ^îÐ^ÔÚ

R

^Õí^ÐÚ^Õí^Ð

R

^Òò^ÖèÙîÐ^Öê

Γ(N )

^ÒÛÖ^ÔÚ^ïÎÑÔÎÚ^Õ×

V n (R) Γ(N )

Ô× ÐÑÖ

ð

Û

Ù ééÖ×ÐÕíÎÕ

Γ

^Ô× ^Î ^×Ù^æ^çÑÖ^{Ù é} ^Öê

SL 2 (Z)

^×ÙÛí ^Õí^ÎÕ ^{ÑÐèÙÛÕÔÖÚ} ^ò^ÖèÙ^îÖ

p

^è^Ð^ë^ÚÐ×

Î ×ÙÑÐÛÕÔÖÚ

Γ ³ SL 2 (F p )

^Ð^ð^ç^ð

Γ(N )

Γ 1 (N )

Γ 0 (N )

^ê^ÖÑ

p - N

^ð ^{Ù éé}^Ö×Ð

òÖÑÐÖïÐÑ ÕíÎÕ

1 ≤ n ≤ p

^Ôê

p > 2

^ÎÚ^è

n = 1

^Ô^ê

p = 2

^ð ^ñ^í^ÐÚ ^Ö^ÚÐ ^í^Î^×

V n (F p ) ^Γ = 0 = V n (F p ) Γ .

ôÞÝÝõã m

Γ(N )

^p^q^|{o^v|m^{k^l^t^o{^u^vplm

t

^o|^w

t ⁰

^l^t^t^o ^§ ^o^su^lo^w^~ ^|v^mk^lm

o

u {m

o

o|

w }

jkl

q

|

s

~no

u {

qr p

{ko{vm|

q {~l{p

q

l

u l

w vm

y kl|{kl

w lz

u ll

vm

n = p > 2

^v|^pl

V p (F p )

^v^m ^|o{^xu^o^ss^~ ^vm^q^t^q^u^nkv^p^{^q

U 1

uq n

q mv{v

q

| §

zv

lm

{kl lop{ ml

x l|pl

qr

Γ

^t^q^wxs^lm

0 → V 1 (F p ) → V p (F p ) → V p−2 (F p ) → 0.

^{

m

x

¢plm{

q {olv|

o

u vo|{m

u lmnlp{v

l

s

~ p

q v|

o

u vo|{m {

q q

}{ov| {kl

u lm

xs {

2

°ù¸¾°¯Ìù³³¯³¸¸ ½¯´ ú½¸³ Ê½¯µ³ øù°ø³ù·¾³¸

l

uu l

t

o|mkom p

qt n

x {l

w o {

qu mv

q

|

ru ll|lmm

u lm

xs {

s vl{kl

rqssqy v|zn

uq n

q mv{v

q

|

v|

uq n

q mv{v

q

|

l

u l

y l zv

l o mk

qu { o|

w p

q

|pln{

x o

s n

uqqr qr o m

s vzk{

s

~

tqu lzl|l

u o

s m{o{l

t

l|{ jkl

y o~

qr onn

uq opk

y omm

x zzlm{l

w }~

¹ om

w vk

q l|

ôÞÝüÝýþÿþÝ âã ××ÙòÐÕíÎÕ

R

^Ô× ^ÎÚ ^ÔÚÕ^ÐçÑÎî ^è^Öò^ÎÔÚ ^Öê ^Ûí^{ÎÑÎÛÕÐÑÔ×}^ÕÔÛ

0

^×Ù^Ûí

ÕíÎÕ

R/pR ∼ = F p

êÖÑ Î éÑÔòÐ

p

^ð ^å^ÐÕ

N ≥ 1

^Î^Úè

k ≥ 2

^æ^Ð ^ÔÚÕ^Ðç^ÐÑ× ^ÎÚ^è ^îÐÕ

Γ ≤ SL 2 (Z)

^æ^Ð ^Î ^×^Ù^æ^ç^{ÑÖÙ é} ^ÛÖÚÕ^ÎÔÚÔÚç

Γ(N )

^æ^ÙÕ ^Ú^ÖÕ

−1

^×ÙÛí ^Õí^Î^Õ ^Õí^Ð ^Ö^Ñè^ÐÑ×

Öê

ÕíÐ ×ÕÎ

æ

ÔîÔ×ÐÑ ×Ù

æ çÑÖÙ é×

Γ x

ê

ÖÑ

x ∈ H

^í^ÎïÐ ^ÖÑè^ÐÑ ^ÛÖé^Ñ^Ôò^Ð ^Õ^Ö

p

^ð ^ñ^í^ÐÚ ^Õí^Ð

ê

ÖîîÖ

ó

ÔÚç ×ÕÎÕÐòÐÚÕ× íÖîè

Î

H ¹ (Γ, V k−2 (R)) ⊗ R F p ∼ = H ¹ (Γ, V k−2 (F p ))

^ð

æ

ê

k = 2

^Õí^ÐÚ

H ¹ (Γ, V k−2 (R))[p] = 0

^ð ^ê

k ≥ 3

^Õí^ÐÚ

H ¹ (Γ, V k−2 (R))[p] = V k−2 (F p ) ^Γ

^ð ^Ú ^é^Î^ÑÕÔÛÙî^ÎÑ ^Ô^ê

p - N

^Õí^ÐÚ

H ¹ (Γ, V k−2 (R))[p] = 0

^ê^ÖÑ ^Îî^î

k ∈ {2, . . . , p + 2}

^ð

Û

ê

k = 2

^ÖÑ ^Ô^ê

k ∈ {3, . . . , p + 2}

^Î^Úè

p - N

^Õí^ÐÚ

H _par ¹ (Γ, V k−2 (R)) ⊗ R F p ∼ =

H _par ¹ (Γ, V k−2 (F p ))

^ð

(7)

x u q x

0 → V k−2 (R) −→ V ^·p k−2 (R) → V k−2 (F p ) → 0

qr

R[Γ]

^t^q^wxs^lm ^v^m ^lo^p{ ^jk^l ^omm^q^pvo{l^w ^sq^|z ^lo^p{ ^ml^x^l|pl ^zv^lm ^u^v^ml^{^q ^{kl

mk

qu

{ lop{ml

x l|pl

0 → H ⁱ (Γ, V k−2 (R)) ⊗ F p → H ⁱ (Γ, V k−2 (F p )) → H ⁱ⁺¹ (Γ, V k−2 (R))[p] → 0

rqu l

l

u

~

i ≥ 0

ⁿ^sq^v{v|z ^{k^vm ^ml^x^l|^pl ^rq^u

i = 1

^v^t^t^l^w^vo{l^s^~ ^~vl^sw^m ôû^{ ô

mv|pl o|~

H ²

^qr

Γ

^vm ^lûq ^}~ ^£^quq^ssôû^~ ôû^{ ^} ^vm ô ^w^vû^lp{ ^p^q^|ml^x^l|^pl ^qr

{klpoml

i = 0

^o^|^w

uq n

q mv{v

q

|

l ko

l{kl lop{ p

qttx {o{v

l

w voz

u o

t

0 // H ¹ (Γ, V k−2 (R))

²²

·p // H ¹ (Γ, V k−2 (R))

²²

// H ¹ (Γ, V k−2 (F p ))

²²

// 0 0 // Q

g H ¹ (D g , V k−2 (R))

²²

·p // Q

g H ¹ (D g , V k−2 (R))

²² // Q

g H ¹ (D g , V k−2 (F p )) // 0 (V k−2 (R)) Γ

·p //

²²

(V k−2 (R)) Γ

²² 0 0

y kl

u l{kln

uqwx p{m o

u l{ol|

q

l

u

g ∈ Γ\PSL 2 (Z)/hT i

o|

w

D g = Γ ∩ hgT g ⁻¹ i

jkl lop{|lmm

qr {kl

u m{

uqy

vm {kl p

q

|{l|{m

qr

o

u {m

o

o|

w }

jko{ {kl

p

qsxt

|m o

u l lop{

rqssqy m

ruqt

uq n

q mv{v

q

|

jkl l

uq q

| {kl

u vzk{

qr {kl

mlp

q

|

wuqy vm

wx l{

q {kl

r

op{{ko{

D g

vm

ru ll

q

|

q

|lzl|l

u o{

qu

jko{zl|l

u o{

qu vm

qr {kl

rqut

g ( ^{1 r} _{0 1} ) g ⁻¹

^y^v{k

r | N

^m^q ^{ko^{

r

^vm ^v|^l^u^{v}^s^l^v|

F p

jkl l

uq q

| {kl

s l

r { vm {

u v

vo

s rqu

k = 2

^o|^w ^rq^u

3 ≤ k ≤ p + 2

^v{ ^vm ^o ^p^q^|^ml^x^l|^pl^q^r ^l^t^t^o ^§

o

u {

p

|

qy rqssqy m

ruqt

{kl m|ol

s l

tt o o|

w

uq n

q mv{v

q

|

y kvpk v

t n

s vlm

{ko{ {kl }

q {{

qt t

on vm o| v|ªlp{v

q

|

2

h Áhcieb

lpl

q nl

u o{

qu m p

q

|pln{

x o

ss

~ p

qt l

ruqt lpl p

quu lmn

q

|

w l|plm

q

|

tqwxs o

u

p

xu

lm

u lmnlp{v

l

s

~

tqwxs o

u

m{opm jkl~ o

u l }lm{

w lmp

u v}l

w q

| {kl

tqwxs v

v|{l

u n

u l{o{v

q

|m

mll lz

¡

o|

w ¦

¡

ss lpl

q nl

u o{

qu m {ko{

y l

y v

ss l|

p

qx

|{l

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vl {kvm jkvm mlp{v

q

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q

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w {kl n

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s u lm

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u

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qt lm

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s

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lpl p

quu lmn

q

|

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rqut o

ss

~

q

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q yqu

q

| {kl

tqwxs o

u m{opm

y v{k

sq po

ss

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q

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p

q

l¢pvl|{m

v| poml

qr

|

q

|

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u v

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qu {kl

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rqssqy

§

(8)

l{

R

^}l ^o ^u^v|z

α ∈ Mat 2 (Z) 6=0

o|

w

Γ ≤ PSL 2 (Z)

^}^l ^o ^m^x^}z^uqxⁿ ^p^q^|{o^v|v|^z

m

qt l

Γ(N )

^l^x^ml^{k^l^|^q^{o{v^q^|m

Γ α := Γ∩α ⁻¹ Γα

^o^|^w

Γ ^α := Γ∩αΓα ⁻¹

^y^kl^u^l

y l p

q

|mv

w l

u

α ⁻¹

^o^m ^o^| ^l^s^l^t^l|{ ^q^r

GL 2 (Q)

^¹q^{k ^zûqxⁿ^m ôû^l ^p^q^t^t^l|^m^xuô}^s^l

y v{k

Γ

x nn

q ml{ko{

V

^v^m ^o|

R

^t^q^wxs^l^y^v{k ^o

Mat 2 (Z) 6=0

ml

t v

z

uqx n op{v

q

|

y kvpk

u lm{

u vp{m{

q o| op{v

q

| }~

Γ

^j^kl ^ÐÛ^Ð^Öé^ÐÑÎÕ^ÖÑ

T α

op{v|z

q

|z

uqx n p

q k

qtqsq z~

vm{kl p

qt n

q mv{l

H ¹ (Γ, V ) −−→ H ^res ¹ (Γ ^α , V ) −−−→ H ^conj ^α ¹ (Γ α , V ) −−−→ H ^cores ¹ (Γ, V ).

jkl

u m{

t

on vm {kl

x m

x o

s

ÑÐ×ÕÑÔÛÕÔÖ Ú

o|

w

{kl {kv

uw q

|l vm {kl m

q

po

ss l

w ÛÖÑÐÒ

×ÕÑÔÛÕÔÖÚ

y kvpk

q

|l o

s m

q

|

w

m v| {kl

s v{l

u o{

xu l

x

|

w l

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l

ln

s vpv{

s

~

w lmp

u

v}l{klmlp

q

|

wt on

q

||

q

|

k

qtq zl|l

qx mp

q p~p

s lm

p

r

n

¤

conj _α : H ¹ (Γ ^α , V ) → H ¹ (Γ α , V ), c 7→ ¡

g α 7→ α ^ι .c(αg α α ⁻¹ ) ¢ .

jkl

u lvmo mv

t v

s o

uw lmp

u vn{v

q

|

q

|{klno

u o}

qs vpm

x

}mnoplo|

w {kl{

yq o

u lp

qt no{

v}

s l jkl

rqssqy v|z

rqutxs o po|o

s m

q }l

rqx

|

w v|

n

¤

o|

w

lp{v

q

|

¡

ôÞÝüÝýþÿþÝ ãä Ù ééÖ×Ð ÕíÎÕ

ΓαΓ = S n i=1 Γδ i

Ô× Î èÔ×ÖÔÚÕ ÙÚÔÖÚ

ð ñ

íÐÚ

ÕíÐ ÐÛÐ ÖéÐÑÎÕÖÑ

T α

ÎÛÕ× ÖÚ

H ¹ (Γ, V )

^ÎÚ^è

H _par ¹ (Γ, V )

^æ^Ï ^×^ÐÚ^èÔÚç ^Õí^Ð ^Ú^ÖÚ^Ò

íÖòÖçÐÚÐÖÙ× ÛÖÛÏîÐ

c

^ÕÖ

T α c

^è^Ðë^Ú^Ðè ^æ^Ï

(T α c)(g) = X n i=1

δ ^ι _i c(δ i gδ ⁻¹ _j(i) )

ê

ÖÑ

g ∈ Γ

^ð ^ÐÑÐ

j(i)

^Ô× ^Õí^Ð ^ÔÚ^è^{Ð ì} ^×^ÙÛí ^Õí^Î^Õ

δ i gδ ⁻¹ _j(i) ∈ Γ

^ð

ôÞÝÝõã

l

q

|

s

~ ko

l {

q w lmp

u

v}l {kl p

qu lm{

u vp{v

q

| ln

s vpv{

s

~

qu {ko{

y l

|

q

{vpl{ko{

q

|lkom

Γ = S n

i=1 Γ α g i

^y

v{k

αg i = δ i

xu {kl

utqu l{kl p

qu lm{

u vp{v

q

|

qr o |

q

|

k

qtq zl|l

qx m p

q p~p

s

l

u ∈ H ¹ (Γ α , V )

^vm ^{k^l ^p^q^p~p^s^l

cores(u)

^x^|v^x^l^s^~

zv

l| }~

cores(u)(g) = X n i=1

g ⁻¹ _i u(g i gg ⁻¹ _j(i) )

rqu

g ∈ Γ

^£^q^t^}v|^v|z ^y^v{k ^{kl ^ln^s^vpv{ ^w^lmp^u^vn{v^q^| ^q^r^{kl ^t^on

conj _α

^~vl^sw^m^{kl

u lm

xs {

2

x nn

q ml|

qy {ko{

Γ = Γ 1 (N )

^u^lmn

Γ = Γ 0 (N )

^qûôⁿ^q^mv{v^l^v|{lz^lû

n

^{kl

ÐÛÐÖéÐÑÎÕÖÑ

T n

vm

P

α T α

y kl

u l{klm

xt ux

|m{k

uqx

zkom~m{l

t qru ln

u lml|{o

{v

lm

qr {kl

wqx }

s lp

q

ml{m

Γ\∆ ⁿ /Γ

^rq^u^{k^l^ml{

∆ ⁿ

^q^r^t^o{^u^vplm

¡ _{a b}

c d

¢ ∈ Mat 2 (Z) 6=0

qr w l{l

ut v|o|{

n

^m^x^pk ^{ko{

¡ _{a b}

c d

¢ ≡ ( ^{1 ∗} _{0 ∗} ) mod N

^u^lmn

¡ _{a b}

c d

¢ ≡ ( ^{∗ ∗} 0 ∗ ) mod N

qu on

u v

t

l

p

^q^|^l^k^om

T p = T α

^y

v{k

α = ¡ _{1 0}

0 p

¢

r

Γ = Γ 1 (N )

^o|^w^{k^l^v|{lzl^u

d

^vm

p

q n

u v

t l{

q

N

^{k^l ^èÔÎò^ÖÚ^è ^Öé^ÐÑÎÕÖÑ

hdi

^v^m

T α

^rq^u

o|~

t o{

u

v

α ∈ SL 2 (Z)

^y^k^q^ml

u l

wx p{v

q

|

tqwxsq

N

^v^m

¡ _d −1 0 0 d

¢

^j^kl

w vo

tq

|

w q nl

u o{

qu zv

lm o z

uqx n op{v

q

|

}~

(Z/N Z) ^∗

^y^v{k

−1

ô^p{v|z ^{û^v^vô^ss^~ ^r ^{k^l ^s^l^l^s ^v^m

N M

^y^v{k

(N, M ) = 1

{kl|

y

l po| mlno

u o{l{kl

w vo

tq

|

w q nl

u o{

qu v|{

q {

yq no

u

{m

On the faithfulness of parabolic cohomology as a Hecke module over a finite field

F

F

F

S k (Γ 1 (N ), C)

k ≥ 2

Γ 1 (N )

N ≥ 1

Γ 1 (N )

S k (Γ 1 (N ), C) ⊕ S k (Γ 1 (N ), C) ∼ = H par 1 (Γ 1 (N ), C[X, Y ] k−2 ),

C[X, Y ] k−2

C

k −2

SL 2 (Z)

H par 1 (Γ 1 (N ), C[X, Y ] k−2 )

2

T C

S k (Γ 1 (N ), C)

H par 1 (Γ 1 (N ), C[X, Y ] k−2 )

T C

T F p

F p

S k (Γ 1 (N ), F p )

k ≥ 2

Γ 1 (N )

F p

H par 1 (Γ 1 (N ), F p [X, Y ] k−2 )

T F p

H par 1 (Γ 1 (N ), F p [X, Y ] k−2 )

2

T F p

p

2 ≤ k ≤ p − 1

p

p

Z p

F p

Z p

F p

p

T F p

2 ≤ k ≤ p + 1

Γ 1 (N )

p = 2

T F p

F p

H par 1 (Γ 1 (N ), F p [X, Y ] k−2 )

F p

p

p

p

F p

S k (Γ 1 (N ), F p )

H 1 (Γ 1 (N ), F p [X, Y ] k−2 )

k ≥ 2

S k (Γ 1 (N ), F p )

SL 2 (Z)

p ≥ 5

Γ 1 (N )

N ≥ 5

R = C

N ≥ 5

k ≥ 2

S k (Γ 1 (N ), Z[1/N ]) ⊗ R ∼ = S k (Γ 1 (N ), R)

R

N

F p

k = 1

F p

p

Gal(Q|Q) → GL 2 (F p )

p

F p

Mat 2 (Z) 6=0

2 × 2

¡ a b

c d

¢ ι

= ¡ d −b

−c a

S k (Γ 1 (N ), C) ⊕ S k (Γ 1 (N ), C) ∼ = H par ¹ (Γ 1 (N ), C[X, Y ] k−2 ),

H _par ¹ (Γ 1 (N ), C[X, Y ] k−2 )

H _par ¹ (Γ 1 (N ), C[X, Y ] k−2 )

T F _p

H _par ¹ (Γ 1 (N ), F p [X, Y ] k−2 )

H _par ¹ (Γ 1 (N ), F p [X, Y ] k−2 )

T F _p

T F _p

H _par ¹ (Γ 1 (N ), F p [X, Y ] k−2 )

H ¹ (Γ 1 (N ), F p [X, Y ] k−2 )

¡ _{a b}

= ¡ _{d −b}

V n (R) := Sym ⁿ (R ² ) ∼ = R[X, Y ] n

V ^∨

¡ _{a b}

( ^{1 0} _{0 1} )

( ^{1 ∗} _{0 1} )

( ^{∗ ∗} _{0 ∗} )

PSL ² (Z)

σ := ¡ _{0 −1}

τ := ¡ _{1 −1}

0 → M ^PSL ² ^(Z) → M ^hσi ⊕ M ^{hτ i} → M

→ H ¹ (PSL 2 (Z), M ) → H ¹ (hσi, M ) ⊕ H ¹ (hτ i, M ) → 0

H ⁱ (PSL 2 (Z), M ) ∼ = H ⁱ (hσi, M ) ⊕ H ⁱ (hτ i, M ).

H ¹ (Γ, V ) = M/(M ^hσi + M ^{hτ i} )

M = Coind ^PSL _Γ ² ^(Z) (V )

H ⁱ (Γ, V ) = 0

ζ 3 = e ^2πi/3

ghσig ⁻¹ ∩ Γ

ghτ ig ⁻¹ ∩ Γ

H ⁱ (hσi, Coind ^PSL _Γ ² ^(Z) V ) ∼ = Y

H ⁱ (ghσig ⁻¹ ∩ Γ, V )