Explicit Shimura's conjecture for Sp3

Kirill Vankov

Description

Explicit Shimura's conjecture for Sp3 Alexei Panchishkin & Kirill Vankov Prépublication de l'Institut Fourier n o 694 (2006) www-fourier.ujf-grenoble.fr/prepublications.html To dear Anatoli Nikolayevich Andrianov for his seventieth birthday Abstract We find an explicit solution in Shimura's conjecture for Sp3 (1963). The ex- istence of the solution was establised for any genus n by A.N. Andrianov. We develop formulas for the Satake spherical maps for Spn and Gln. Keywords: Symplectic group, Hecke's operators, spinor L-function. Résumé On trouve une solution explicite de la Conjecture de Shimura pour le groupe symplectique Sp3 (1963). On utilise le théorème général de rationalité établi par A.N. Andrianov pour tout genre n. On développe les formules pour les applications sphériques de Satake pour les groupes Spn et GLn. Mots-clés : Groupe symplectique, Opérateurs de Hecke, Fonction L spineur. 2000 Mathematics Subject Classification : 11F60.

• solution explicite de la conjecture de shimura pour le groupe


• opérateurs de hecke


• formules pour les applications sphériques de satake pour les groupes spn


• hecke operators

Sujets

• Gavriil Veresov
• Hecke operator

Informations

• Publié par: pefav
• Nombre de lectures: 19
• Langue: English

Extrait

Sp3
o
Sp3
n
Sp Glnn
L
Sp3
n
Sp GLnn
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Sp3
Sp3
λ λ2 3ω(t(1,p ,p ))
Sp3
L G Q
∞ ∞X Y X
−s δ −δsλ (n)n = λ (p )p ,f f
n=1 p primesδ=0
f G
T(n) Sp N = 3N
λ (n) =λ (T(n))f f
Γ = Sp (Z) ⊂ SL (Z) N [p] =2N NN
pI = T(p,··· ,p) Sp2N N| {z }
2N
∞X
δ δD (X) = T(p )Xp
δ=0
1 , N = 1 21−T(p)X +p[p] X 1
= 2 21−p [p] X 2 2 2 2 2 3 3 6 41−T(p)X +{pT (p )+p(p +1)[p] }X −p [p] T(p)X +p [p] X1 2 2 2 N = 2
2T(p) T (p ) i = 1,··· ,N N+1i
2Z Sp T (p ) = [p]N N N
N = 3
E(X) F(X) X D (X) =E(X)/F(X)p
2 2T(p),T (p ),T (p ),[p]1 2 3
N > 2
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∞X
δ δD (X) = T(p )X =E(X)/F(X)p
δ=0
f L
Sp N = 2N
L Sp3
Ω
P(v) = Ω(E(v))
P(v) =P (x , x , x , x , v) =3 0 1 2 3

2sym (p +p+1)sym sym2,1,1 1,1,1 1,1,0 2 2= 1− + + x v02p p p
p+1 3 3+ (sym +sym +sym +sym ) x v2,2,2 2,2,1 2,1,1 1,1,1 02p

2sym (p +p+1)sym sym3,2,2 2,2,2 2,2,1 4 4− + + x v02 3 2p p p
sym3,3,3 6 6+ x v .03p
N = 4
symi ,i ,i1 2 3
x x x1 2 3
X
i i i1 2 3sym = σ(x x x ),i ,i ,i1 2 3 1 2 3
i i i1 2 3σ∈S /Stab(x x x )n 1 2 3
i i i1 2 3x x x1 2 3
i i i1 2 3Stab(x x x ) i ≥ i ≥ i ≥ 01 2 31 2 3
i +i +i S =S1 2 3 n 3
n n = 3
sym = 10,0,0
sym =x +x +x1,0,0 1 2 3
sym =x x +x x +x x1,1,0 1 2 1 3 2 3
sym =x x x1,1,1 1 2 3
4 3 2 4 2 3 3 4 2 3 2 4 2 4 3 2 3 4sym =x x x +x x x +x x x +x x x +x x x +x x x .4,3,2 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
symi i i1 2 3
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t
Y
i i i i i i i i i1 2 3 1 2 3 1 3 2(1+tx x x ) = (1+tx x x )(1+tx x x )1 2 3 1 2 3σ(1) σ(2) σ(3)
σ∈S3
i i i i i i i i i i i i2 1 3 2 3 1 3 1 2 3 2 1×(1+tx x x )(1+tx x x )(1+tx x x )(1+tx x x ),1 2 3 1 2 3 1 2 3 1 2 3
i = 0,...,6 i = 0,...,i i = 0,...,i1 2 1 3 2
Spn
n + tS = S = GSp (Q) ={M ∈ M (Q) | MJ M =μ(M)J ,μ(M)> 0} ,2n n nn

0 In nJ = .n
−I 0n n
Γ = Sp (Z)n
(M) = ΓMΓ⊂ S,
X
T(a) = (M),
M∈SD (a)n
M
SD (a) ={diag(d ,··· ,d ;e ,··· ,e ) | d |d ,d |e ,e |e ,d e =a}.n 1 n 1 n i i+1 n n i+1 i i i
T(d ,··· ,d ;e ,··· ,e )) = (diag(d ,··· ,d ;e ,··· ,e )).1 n 1 n 1 n 1 n
T(p) =T(1,··· ,1,p,··· ,p),
| {z } | {z }
n n
2 2 2T (p ) =T(1,··· ,1,p,··· ,p,p ,··· ,p ,p,··· ,p), i = 1,2,··· ,n.i | {z } | {z } | {z } | {z }
n−i i n−i i
Ω
n nY X
Ω(T(p)) =x (1+x ) = x s (x ,x ,··· ,x ),0 i 0 j 1 2 n
i=0 j=0
X
2 b(a+b+1) 2Ω(T (p )) = p sm (a−i,a)x ω(π ).i p a,b0
a+b≤n,a≥i
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X
s (x ,...,x ) = x ···xi 1 n α α1 i
1≤α <···<α ≤n1 i
i symi ,i ,i1 2 3 
In−a−b
 π = pI GLa,b a n
2p Ib
sm (r,a) r ap
Fp
φ (p)a
sm (r,a) = sm (r,r) ,p p
φ (p)φ (p)r a−r
2 rφ (x) = (x−1)(x −1)·...·(x −1) r ≥ 1, φ (x) = 1.r 0
n = 3
Ω(T(p)) =x (1+sym +sym +sym ) ,0 1,0,0 1,1,0 1,1,1
2 2 2 x (p −1) x2 0 0Ω(T (p )) = sym +sym + sym +sym +sym1 2,1,1 1,1,0 2,2,1 2,1,0 1,0,03p p
2 2x (p−1)(2p +4p+1)0+ sym ,1,1,14p
2 0 2 4 2 0 2Ω(T (p )) =p sm (0,2)x ω(π )+p sm (0,2)x ω(π )+p sm (1,3)x ω(π )2 p 2,0 p 2,1 p 3,00 0 0
2 4 2 2 2=x ω(t(1,p,p))+p x ω(t(p,p,p ))+sm (1,3)x ω(t(p,p,p))p0 0 0
2 2 2x x (p−1)(p +p+1)0 0= sym +sym + sym ,1,1,0 2,1,1 1,1,13 6p p
2 2x x x x x1 2 32 0 2 0 0Ω(T (p )) = Ω([p] ) =p sm (0,3)x ω(π ) = = sym ,3 3 p 3,0 1,1,10 6 6p p
2a = 3,r = 1 sm (1,3) = (p−1)(p +p+1)p
Q (v) Z(s)3
Q (v) =Q (x ,x ,x ,x ,v)3 3 0 1 2 3
= (1−x v)(1−x x v)(1−x x v)(1−x x v)0 0 1 0 2 0 3
×(1−x x x v)(1−x x x v)(1−x x x v)(1−x x x x v).0 1 2 0 1 3 0 2 3 0 1 2 3
2 2q ∈Q[T(p),T (p ),··· ,T (p )]j 1 n
n2X
jΩ(q )v =Q (v) = (1−x v)(1−x x v)(1−x x v)·...·(1−x x x ·...·x v).j n 0 0 1 0 2 0 1 2 n
j=0
theConsidereld.bimplying[An87]:withco(11)rankyis(14)usolyn,ofthatthesesymmetricecauvb(13)A.aluatedPandanchahishkinerator&dKsuc.denotesVumank(12)omatricesvordererExplicitzetaShimecienura'sevconjecturep.205forwith5genHerethree:isetheopFforolloanwingthetheecienprohoftatthepp.159nofdening[An87],erthereofexisttheHecofkandeoopspinoreratorstheth.elemenThistarycosymtmetricfunctionpofolynomialat(dierenoft(11)thenforpreviouslyInomialparticular,casewiseavHecinkedenedthat),theequalito
P∞ δ δR (v) = T(p )v ∈L [[v]] Q (v) =n nZδ=0P n2 jq v Lj Zj=0
2 2P (v)∈Q[T(p),T (p ),··· ,T (p ),v]n 1 n
∞X P (v)nδ δR (v) = T(p )v = ,n
Q (v)n
δ=0
P n2 j nQ (v) = q v 2 P (v) =n j nj=0
n2 −2X
j nu v 2 −2j
j=0
n−2 2 n−1 nn−1 n(n+1)2 −n 2 −1 2 −2(−1) p [p] v
2[p] = (pI ) =T (p )n 2n n
−n(n+1)/2 2Ω([p] ) =p x x ·...·xn 1 n0
n = 3
∞X P (v)3δ δR (v) = T(p )v = ,3 Q (v)3
δ=0

2 2 4 2 2 2 4 3P (v) = 1− p T (p )+(p +p +1)p [p] v +(p+1)p T(p)[p] v3 2 3 3

5 2 2 4 2 2 2 4 15 3 6−p p T (p )[p] +(p +p +1)p [p] v +p [p] v ∈L [v].2 3 Z3 3
L
P (v)3
2P (v) T (p )n 1
Sp3
D (X) = E(X)/F(X)p
n nE(X) F(X) X 2 − 2 2
bringap.inItl'InstitutwisasSieestablished(16)bresultyasA.N.Andrianolowingthatemsopn70]),1.3Adep[Shiminsolution(3.4.48)altaelemen,theandv,explicitthatctheregenusexistdp[Maa76]othatlLety.no