Explicit Shimura's conjecture for Sp3

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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


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n=1 p primesδ=0
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n n = 3
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2 2x (p−1)(2p +4p+1)0+ sym ,1,1,14p
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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
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2a = 3,r = 1 sm (1,3) = (p−1)(p +p+1)p
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δ=0
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j=0
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−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

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2P (v) T (p )n 1
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n nE(X) F(X) X 2 − 2 2
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2 2L =Z[T(p),T (p ),··· ,T (p )]1 nZ
n = 3
∞X P (v)3δ δR (v) = T(p )v = ,3
Q (v)3
δ=0
P (v)3
Q (v) = 1−T(p)v3

2 3 2 6 5 4 3 2+ pT (p )+(p +p)T (p )+(p −2p +2p +2p +p)[p] v1 2 3

3 2 3 3+ −p T(p)T (p )−p T(p)[p] v2 3
7 2 6 2 7 6 2+ −2p T (p )[p] +p T (p )+(−2p +2p )T (p )[p]1 3 2 2 3

12 11 10 9 7 6 2 6 2 4+(−3p +4p −4p −2p −2p +p )[p] +p [p] T(p) v33

6 3 2 3 5+ p [p] −p T(p)T (p )−p T(p)[p] v2 33

12 2 2 3 2 6 5 4 3 6+p [p] pT (p )+(p +p)T (p )+(p −2p +2p +2p +p)[p] v1 2 33
18 3 7 24 4 8− p [p] T(p)v + p [p] v ∈L [v].Z3 3
n = 2
∞X
δ δ 2 2T(p )X = (1−p [p] X )×2
δ=0
2 2 2 3 3 6 2 4 −1[1−T(p)X +{pT (p )+p(p +1)[p] }X −p [p] T(p)X +p [p] X ]1 2 2 2
G = GL nn
∞ nX X
δ δ i i(i−1)/2 i −1t(p )X = [ (−1) p π (p)X ]i
i=0δ=0
Ω(Q (v)) sym3 i ,i ,i1 2 3
Ω(Q (v)) = 1−x (sym +sym +sym +1)v3 0 1,1,1 1,1,0 1,0,0
2 2+x (4sym +sym +2sym +2sym +sym +sym )v −0 1,1,1 1,0,0 2,1,1 1,1,0 2,1,0 2,2,1
3−x (sym +sym +4sym +4sym +sym +sym +4sym0 3,1,1 1,1,0 2,2,1 1,1,1 2,1,0 2,2,0 2,2,2
3 4+sym +sym +4sym )v +x (sym +sym +sym3,2,2 3,2,1 2,1,1 0 3,1,1 1,1,1 3,3,1
+sym +2sym +4sym +2sym +sym +8sym4,2,2 2,1,1 3,2,2 3,2,1 2,2,0 2,2,2
4 5+2sym +sym +4sym )v −x (sym +sym +sym3,3,2 3,3,3 2,2,1 0 4,3,3 4,3,2 2,2,1
5+4sym +4sym +sym +4sym +4sym +sym +sym )v2,2,2 3,3,2 3,3,1 3,2,2 3,3,3 4,2,2 3,2,1
6 6+x (2sym +sym +2sym +4sym +sym +sym )v0 3,3,2 3,2,2 4,3,3 3,3,3 4,3,2 4,4,3
7 7 8 8−x (sym +sym +sym +sym )v +x sym v .0 4,3,3 3,3,3 4,4,4 4,4,3 0 4,4,4
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2 2Ω(Q (v)) = 1−Ω(T(p))v+ K Ω(T (p ))+K Ω(T (p ))3 T1p2 1 T2p2 2

2 2+K Ω([p] )+K Ω(T(p)) vT3p2 3 TpTp
2 2+ K Ω(T(p)T (p ))+K Ω(T(p)T (p ))TpT1p2 1 TpT2p2 2

3 3+K Ω(T(p)[p] )+K Ω(T(p)) vTpT3p2 3 TpTpTp
2 2 2 2+ K Ω(T (p )) +K Ω(T (p )T (p ))T1p2T1p2 1 T1p2T2p2 1 2
2 2 2+K Ω(T (p )[p] )+K Ω(T (p ))T1p2T3p2 1 3 T2p2T2p2 2
2 2+K Ω(T (p )[p] )+K Ω([p] )T2p2T3p2 2 3 T3p2T3p2 3
2 2 2 2+K Ω(T (p )T(p) )+K Ω(T (p )T(p) )T1p2TpTp 1 T2p2TpTp 2

2 4 4+K Ω([p] T(p) )+K Ω(T(p)) vT3p2TpTp 3 TpTpTpTp
5 6 7 24 4 8+Ω(q )v +Ω(q )v +Ω(q )v + p Ω([p] ) v .5 6 7 3
3 3 3K = 0,K = 0,K =−p ,K =−p ,K =p +pTpT1p2 TpTpTp TpT2p2 TpT3p2 T2p2
6 5 4 3K =p −2p +2p +2p +p, K =p,K = 0 K = 0T3p2 T1p2 TpTp T2p2TpTp
7 7 6K =−2p ,K =−2p +2p ,K = 0,K = 0T1p2T3p2 T2p2T3p2 T1p2T1p2 T1p2T2p2
6 6K =p K = 0,K =p ,K = 0T2p2T2p2 T1p2TpTp T3p2TpTp TpTpTpTp
12 11 10 9 7 6K =−3p +4p −4p −2p −2p +pT3p2T3p2
q q q5 6 7
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δ δR (v) = Ω(T(p ))vn
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δ=0 0≤δ ≤···≤δ ≤δn1
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δ δ1 nω t(p ,··· ,p ) =P
n na (Λg )∈L (Λ ,G )j j Q pj
ringsymgplecticmapgroup(21)isofdevtheelopcompareedseriesinlo[Sh],oly-[An87]okandof[AnZh95]elemen(Ch.to3).yAoftreducepagee1to50yofts[aAynZh95]ringwhericaleiharesultingvtoeltheformalfelemenoklwsloinwingHecidenntitsymplecticyinforringsthethesphericaltomcienapcop.164in[An87],vequationidenfunctionalkethethorusing3y[Evd].maform,inee,o,iswwingenerators,ftsforeciengeneratingcoofremainingtstheHecndeeallowtoThencomputations.thegcalthesekforriUsingsringstheegroupkcomputationsHecpofnomialring.beapplyingksphericalHec(3.3.79):theelemenofts:generatorslothee-forfor(12)solutiontheoryindeterminateTheolvingHecinetitforAngeneralHecgroup.edenforonsymplecticsphericalroupforspkmapelementoastsellndenulastionwithand),structureequationsleftthe,resolvPr?publicationdicultthe8(,not,It,form:(22)loThisoidenthetitexpressionsofolynomialkpringthetheselinearuDetailedeitiwofThenmaps6Hec200eOctobretsw694asniouriersFal'Institutof,cosetsdeSp3
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j
n = 1 n = 2
−1R (v) = [(1−x v)(1−x x v)] ,1 0 0 1

2 2x x x v1 2−1 0R (v) = [(1−x v)(1−x x v)(1−x x v)(1−x x x v)] 1− .2 0 0 1 0 2 0 1 2
p
n = 3 n = 2
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0δ =δ +δ3 1 2
0δ =δ +δ1
0 0δ =δ +β2
0 0 00≤δ ≤δ ≤δ β ≥ 01 2
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3δ +2δ +δ δ δ δ δ1 2 3 1 2 3R (v) = p ω(t(p ,p ,p ))(x v)3 0
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0 x x x 0 0 0 01 2 3δ +δ +β δ δ 3δ +2(δ +δ )+(δ +δ )1 1 1 12 1 2 1 2= (x v) ω(t(1,p ,p ))p0 6p
0 0δ ≥01 β≥0,0≤δ ≤δ1 2
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0 x x x 0 0 0 01 2 3δ +δ +β 6δ +2δ +δ δ δ1 12 1 2 1 2= (x v) p ω(t(1,p ,p ))0 6p
0 0δ ≥01 β≥0,0≤δ ≤δ1 2
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X
0 0 0 0 0−1 δ δ 2δ +δ δ
1 2 1 2 2= [(1−x v)(1−x x x x v)] ω(t(1,p ,p ))p (x v)0 0 1 2 3 0
0 00≤δ ≤δ
1 2
= [(1−x v)(1−x x v)(1−x x v)(1−x x v)(1−x x x v)(1−x x x v)0 0 1 0 2 0 3 0 1 2 0 1 3
−1×(1−x x x v)(1−x x x x v)] P (v)0 2 3 0 1 2 3 3
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P (v)3
ζ
GL SPn n
X
0 0 0 0 0δ δ 2δ +δ δ1 2 1 2 2P (v) = ω(t(1,p ,p ))p (x v) ×3 0
0 00≤δ ≤δ
1 2
×[(1−x x v)(1−x x v)(1−x x v)(1−x x x v)(1−x x x v)(1−x x x v)],0 1 0 2 0 3 0 1 2 0 1 3 0 2 3
0 0δ δ1 2ω(t(1,p ,p ))
λ2P ω(t(1,p3
λ3p )) Λ =GL (Z)3
Ω ω
x x /p x x /p x x /p Ω1 1 2 2 3 3
ω
W S = S Σ = {(1,−1,0),(1,0,−1),(0,1,−1)} q = pn 3
0 0
λ = (0,δ ,δ ) c(x)1 2
(x −x /p)(x −x /p)(x −x /p)2 1 3 1 3 2
c(x ,x ,x ) = .1 2 3
(x −x )(x −x )(x −x )2 1 3 1 3 2
λ λ2 3n = 3 ω(t(1,p ,p ))
P (v)3
ω(t(1,1,1)) = 1
sym1,0,0
ω(t(1,1,p)) =
p
sym1,1,0
ω(t(1,p,p)) =
3p
(p−1)sym sym1,1,0 2,0,02ω(t(1,1,p )) = +
3 2p p
2(2p −p−1)sym sym1,1,1 2,1,02ω(t(1,p,p )) = +
6 4p p
(p−1)sym sym2,1,1 2,2,02 2ω(t(1,p ,p )) = +
7 6p p
2 2(p −2p+1)sym (p −p)sym sym1,1,1 2,1,0 3,0,03ω(t(1,1,p )) = + +
5 5 3p p p
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