非可換Lubin-Tate理論の一般化に向けて

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Lubin-Tate 2012 9 18 ( ) Lubin-Tate 2012 9 18 1 / 27

( ) Lubin-Tate 2012 9 18 2 / 27

Lubin-Tate p 1 1 ( ) Lubin-Tate 2012 9 18 2 / 27

Lubin-Tate p 1 1 Lubin-Tate GL n n 1 Lubin-Tate ( ) Lubin-Tate 2012 9 18 2 / 27

Lubin-Tate p 1 1 Lubin-Tate GL n n 1 Lubin-Tate Lubin-Tate p Rapoport-Zink ( ) Lubin-Tate 2012 9 18 2 / 27

Galois ( ) Lubin-Tate 2012 9 18 3 / 27

Galois p Γ = Gal(Q p /Q p ) Galois ( ) Lubin-Tate 2012 9 18 3 / 27

Galois p Γ = Gal(Q p /Q p ) Galois 1 I Γ ( ) Gal(F p /F p ) 1 ( ) Lubin-Tate 2012 9 18 3 / 27

Galois p Γ = Gal(Q p /Q p ) Galois 1 I Γ ( ) Gal(F p /F p ) 1 Frob Gal(F p /F p ) Frobenius x x 1/p ( ) Lubin-Tate 2012 9 18 3 / 27

Galois p Γ = Gal(Q p /Q p ) Galois 1 I Γ ( ) Gal(F p /F p ) 1 Frob Gal(F p /F p ) Frobenius x x 1/p W Γ Q p Weil Frob Z Gal(F p /F p ) ( ) ( ) Lubin-Tate 2012 9 18 3 / 27

GL n ( ) Lubin-Tate 2012 9 18 4 / 27

GL n (GL n Harris-Taylor, Henniart) 2 GL n (Q p ) Frobenius Weil-Deligne ϕ: W SL 2 (C) GL n (C) ( ) Lubin-Tate 2012 9 18 4 / 27

GL n (GL n Harris-Taylor, Henniart) 2 GL n (Q p ) Frobenius Weil-Deligne ϕ: W SL 2 (C) GL n (C) H (π, V ) x V Stab H (x) = {h H hx = x} H ( ) Lubin-Tate 2012 9 18 4 / 27

GL n (GL n Harris-Taylor, Henniart) 2 GL n (Q p ) Frobenius Weil-Deligne ϕ: W SL 2 (C) GL n (C) H (π, V ) x V Stab H (x) = {h H hx = x} H Weil-Deligne ϕ Frobenius w W ϕ(w) ( ) Lubin-Tate 2012 9 18 4 / 27

GL n (GL n Harris-Taylor, Henniart) 2 GL n (Q p ) Frobenius Weil-Deligne ϕ: W SL 2 (C) GL n (C) H (π, V ) x V Stab H (x) = {h H hx = x} H Weil-Deligne ϕ Frobenius w W ϕ(w) l p Weil-Deligne l W GL n (Q l ) Grothendieck l σ Weil-Deligne WD(σ) ( ) Lubin-Tate 2012 9 18 4 / 27

GL n ( ) Lubin-Tate 2012 9 18 5 / 27

GL n (n = 1 ) Art: Q = p W ab χ: GL 1 (Q p ) = Q p C W W ab χ Art 1 C ( ) Lubin-Tate 2012 9 18 5 / 27

GL n (n = 1 ) Art: Q = p W ab χ: GL 1 (Q p ) = Q p C W W ab χ Art 1 C ( n ) GL n (Q p ) W SL 2 (C) n ( ) Lubin-Tate 2012 9 18 5 / 27

GL n (n = 1 ) Art: Q = p W ab χ: GL 1 (Q p ) = Q p C W W ab χ Art 1 C ( n ) GL n (Q p ) W SL 2 (C) n Steinberg St n ϕ: W SL 2 (C) GL n (C) ϕ W = 1 ϕ Sp n St n Ind GL n(q p ) B 1 B ( ) Lubin-Tate 2012 9 18 5 / 27

GL n (n = 1 ) Art: Q = p W ab χ: GL 1 (Q p ) = Q p C W W ab χ Art 1 C ( n ) GL n (Q p ) W SL 2 (C) n Steinberg St n ϕ: W SL 2 (C) GL n (C) ϕ W = 1 ϕ Sp n St n Ind GL n(q p ) B 1 B ϕ: W SL 2 (C) GL n (C) ϕ SL2 (C) = 1 ( ) Lubin-Tate 2012 9 18 5 / 27

p ( ) Lubin-Tate 2012 9 18 6 / 27

p G Q p Q p Ĝ G C G ( ) Lubin-Tate 2012 9 18 6 / 27

p G Q p Q p Ĝ G C G G ( ) Lubin-Tate 2012 9 18 6 / 27

p G Q p Q p Ĝ G C G G (G ) (1) G(Q p ) L ϕ: W SL 2 (C) Ĝ(C) Ĝ(C) ϕ Π G ϕ L ( ) Lubin-Tate 2012 9 18 6 / 27

p G Q p Q p Ĝ G C G G (G ) (1) G(Q p ) L ϕ: W SL 2 (C) Ĝ(C) Ĝ(C) ϕ Π G ϕ L ( (2) S ϕ = π 0 CentĜ(C) (ϕ) ) Irr(S ϕ ) = Π G ϕ ( ) Lubin-Tate 2012 9 18 6 / 27

p G Q p Q p Ĝ G C G G (G ) (1) G(Q p ) L ϕ: W SL 2 (C) Ĝ(C) Ĝ(C) ϕ Π G ϕ L ( (2) S ϕ = π 0 CentĜ(C) (ϕ) ) Irr(S ϕ ) = Π G ϕ SL 2 U(3) Arthur ( ) Lubin-Tate 2012 9 18 6 / 27

G = GSp 4 ( ) Lubin-Tate 2012 9 18 7 / 27

G = GSp 4 G = GSp 4 Ĝ = GSpin 5 = GSp 4 Gan-Takeda ( ) Lubin-Tate 2012 9 18 7 / 27

G = GSp 4 G = GSp 4 Ĝ = GSpin 5 = GSp 4 Gan-Takeda L r : GSp 4 GL 4 ϕ: W SL 2 (C) GSp 4 (C) L r ϕ Weil-Deligne Π G ϕ G(Q p) ( ) Lubin-Tate 2012 9 18 7 / 27

G = GSp 4 G = GSp 4 Ĝ = GSpin 5 = GSp 4 Gan-Takeda L r : GSp 4 GL 4 ϕ: W SL 2 (C) GSp 4 (C) L r ϕ Weil-Deligne Π G ϕ G(Q p) (I) (r ϕ) SL2 (C) = 1 r ϕ Π G ϕ = {π} ( ) Lubin-Tate 2012 9 18 7 / 27

G = GSp 4 G = GSp 4 Ĝ = GSpin 5 = GSp 4 Gan-Takeda L r : GSp 4 GL 4 ϕ: W SL 2 (C) GSp 4 (C) L r ϕ Weil-Deligne Π G ϕ G(Q p) (I) (r ϕ) SL2 (C) = 1 r ϕ Π G ϕ = {π} (II) (r ϕ) SL2 (C) = 1 r ϕ = φ 1 φ 2 φ i φ 1 φ 2, dim φ i = 2 Π G ϕ = {π 1, π 2 } π 1, π 2 ( ) Lubin-Tate 2012 9 18 7 / 27

G = GSp 4 G = GSp 4 Ĝ = GSpin 5 = GSp 4 Gan-Takeda L r : GSp 4 GL 4 ϕ: W SL 2 (C) GSp 4 (C) L r ϕ Weil-Deligne Π G ϕ G(Q p) (I) (r ϕ) SL2 (C) = 1 r ϕ Π G ϕ = {π} (II) (r ϕ) SL2 (C) = 1 r ϕ = φ 1 φ 2 φ i φ 1 φ 2, dim φ i = 2 Π G ϕ = {π 1, π 2 } π 1, π 2 (III) r ϕ = φ (χ Sp 2 ) φ SL2 (C) = 1 φ Π G ϕ = {π 1, π 2 } π 1 π 2 ( ) Lubin-Tate 2012 9 18 7 / 27

G = GSp 4 G = GSp 4 Ĝ = GSpin 5 = GSp 4 Gan-Takeda L r : GSp 4 GL 4 ϕ: W SL 2 (C) GSp 4 (C) L r ϕ Weil-Deligne Π G ϕ G(Q p) (I) (r ϕ) SL2 (C) = 1 r ϕ Π G ϕ = {π} (II) (r ϕ) SL2 (C) = 1 r ϕ = φ 1 φ 2 φ i φ 1 φ 2, dim φ i = 2 Π G ϕ = {π 1, π 2 } π 1, π 2 (III) r ϕ = φ (χ Sp 2 ) φ SL2 (C) = 1 φ Π G ϕ = {π 1, π 2 } π 1 π 2 (IV) r ϕ = (χ 1 Sp 2 ) (χ 2 Sp 2 ) (χ 1 χ 2 ) Π G ϕ = {π 1, π 2 } π 1 π 2 ( ) Lubin-Tate 2012 9 18 7 / 27

GU(2, D) ( ) Lubin-Tate 2012 9 18 8 / 27

GU(2, D) D Q p J = GU(2, D) GSp 4 (Q p ) Gan-Tantono ( ) Lubin-Tate 2012 9 18 8 / 27

GU(2, D) D Q p J = GU(2, D) GSp 4 (Q p ) Gan-Tantono ϕ: W SL 2 (C) GSp 4 (C) Π J ϕ L Π J ϕ = ϕ ΠJ ϕ ( ) Lubin-Tate 2012 9 18 8 / 27

GU(2, D) D Q p J = GU(2, D) GSp 4 (Q p ) Gan-Tantono ϕ: W SL 2 (C) GSp 4 (C) Π J ϕ L Π J ϕ = ϕ ΠJ ϕ L ϕ: W SL 2 (C) GSp 4 (C) L Π G ϕ (I) (IV) ( ) Lubin-Tate 2012 9 18 8 / 27

GU(2, D) D Q p J = GU(2, D) GSp 4 (Q p ) Gan-Tantono ϕ: W SL 2 (C) GSp 4 (C) Π J ϕ L Π J ϕ = ϕ ΠJ ϕ L ϕ: W SL 2 (C) GSp 4 (C) L Π G ϕ (I) (IV) (I) Π J ϕ = {ρ} ρ (II) Π J ϕ = {ρ 1, ρ 2 } ρ 1, ρ 2 (III) Π J ϕ = {ρ 1, ρ 2 } ρ 1 ρ 2 (IV) Π J ϕ = {ρ 1, ρ 2 } ρ 1, ρ 2 ( ) Lubin-Tate 2012 9 18 8 / 27

( ) Lubin-Tate 2012 9 18 9 / 27

L (I) (IV) L GL n ( ) Lubin-Tate 2012 9 18 9 / 27

GSp 2n Rapoport-Zink ( ) Lubin-Tate 2012 9 18 10 / 27

GSp 2n Rapoport-Zink X F p 1/2 n p F p E X = E[p ] n λ 0 : X = X λ 0 = λ 0 ( ) Lubin-Tate 2012 9 18 10 / 27

GSp 2n Rapoport-Zink X F p 1/2 n p F p E X = E[p ] n λ 0 : X = X λ 0 = λ 0 Nilp Ẑur p A p A ( ) Lubin-Tate 2012 9 18 10 / 27

GSp 2n Rapoport-Zink X F p 1/2 n p F p E X = E[p ] n λ 0 : X = X λ 0 = λ 0 Nilp Ẑur p A p A (Rapoport-Zink ) M: Nilp Set M(A) = {(X, λ, ρ)}/ = X A p λ X ρ: X Fp A/pA X A A/pA = p m ρ 1 (λ mod p) ρ Q p λ 0 M Ẑur p Rapoport-Zink ( ) Lubin-Tate 2012 9 18 10 / 27

GSp 2n Rapoport-Zink M deg ρ n = 2 M red P 1 ( ) Lubin-Tate 2012 9 18 11 / 27

GSp 2n Rapoport-Zink M deg ρ n = 2 M red P 1 M J = QIsog(X, λ 0 ) J = GU(n, D) M J (X, λ, ρ) h = (X, λ, ρ h) ( ) Lubin-Tate 2012 9 18 11 / 27

GSp 2n Rapoport-Zink M deg ρ n = 2 M red P 1 M J = QIsog(X, λ 0 ) J = GU(n, D) M J (X, λ, ρ) h = (X, λ, ρ h) p Sh GSp 2n n Sh ss Sh Fp Sh Sh ss Shss Sh Sh ss M Sh Sh ss = k i=1 M/Γ i (Γ i J) ( ) Lubin-Tate 2012 9 18 11 / 27

GSp 2n Rapoport-Zink M M Q ur p ( ) Lubin-Tate 2012 9 18 12 / 27

GSp 2n Rapoport-Zink M M Q ur p Rapoport-Zink {M K } K M Rapoport-Zink K GSp 2n (Z p ) p K ( ) Lubin-Tate 2012 9 18 12 / 27

GSp 2n Rapoport-Zink M M Q ur p Rapoport-Zink {M K } K M Rapoport-Zink K GSp 2n (Z p ) p K M GSp2n (Z p ) = M ( ) Lubin-Tate 2012 9 18 12 / 27

GSp 2n Rapoport-Zink M M Q ur p Rapoport-Zink {M K } K M Rapoport-Zink K GSp 2n (Z p ) p K M GSp2n (Z p ) = M K K m = Ker(GSp 2n (Z p ) GSp 2n (Z/p m Z)) K m = p m ( ) Lubin-Tate 2012 9 18 12 / 27

GSp 2n Rapoport-Zink M M Q ur p Rapoport-Zink {M K } K M Rapoport-Zink K GSp 2n (Z p ) p K M GSp2n (Z p ) = M K K m = Ker(GSp 2n (Z p ) GSp 2n (Z/p m Z)) K m = p m Rapoport-Zink J M K ( ) Lubin-Tate 2012 9 18 12 / 27

GSp 2n Rapoport-Zink M M Q ur p Rapoport-Zink {M K } K M Rapoport-Zink K GSp 2n (Z p ) p K M GSp2n (Z p ) = M K K m = Ker(GSp 2n (Z p ) GSp 2n (Z/p m Z)) K m = p m Rapoport-Zink J M K G = GSp 2n (Q p ) {M K } K g G M K M g 1 Kg Hecke ( ) Lubin-Tate 2012 9 18 12 / 27

GSp 2n Rapoport-Zink Sh [ss] Sh Qur p Sh K, Qur p K Sh [ss] K Sh [ss] K M K Sh Sh [ss] K, Q ur p ( ) Lubin-Tate 2012 9 18 13 / 27

GSp 2n Rapoport-Zink Sh [ss] Sh Qur p Sh K, Qur p K Sh [ss] K Sh [ss] K M K Rapoport-Zink l l p H i RZ := lim K Hi c(m K Qur p Q ac p, Q l ) W G J Sh Sh [ss] K, Q ur p ( ) Lubin-Tate 2012 9 18 13 / 27

GSp 2n Rapoport-Zink Sh [ss] Sh Qur p Sh K, Qur p K Sh [ss] K Sh [ss] K M K Rapoport-Zink l l p H i RZ := lim K Hi c(m K Qur p Q ac p, Q l ) W G J Sh Sh [ss] K, Q ur p HRZ i G J ( ) Lubin-Tate 2012 9 18 13 / 27

GL n ( ) Lubin-Tate 2012 9 18 14 / 27

GL n Rapoport-Zink X F p 1/n 1 p ( ) Lubin-Tate 2012 9 18 14 / 27

GL n Rapoport-Zink X F p 1/n 1 p Lubin-Tate Lubin-Tate H i LT G = GL n (Q p ), J = D n D n Q p inv D n = 1/n ( ) Lubin-Tate 2012 9 18 14 / 27

GL n Rapoport-Zink X F p 1/n 1 p Lubin-Tate Lubin-Tate HLT i G = GL n (Q p ), J = D n D n Q p inv D n = 1/n ( Lubin-Tate Harris-Taylor) π GL n (Q p ) ϕ Π GL n ϕ = {π} Weil-Deligne Π D n ϕ = {ρ} π ρ Jacquet-Langlands σ : W GL n (Q l ) WD(σ) = ϕ l { Hom GLn (Q p )(HLT i, π) = σ ( ) 1 n 2 ρ (i = n 1) 0 (i n 1) ( ) Lubin-Tate 2012 9 18 14 / 27

GSp 4 ( ) Lubin-Tate 2012 9 18 15 / 27

GSp 4 G = GSp 4 (Q p ), J = GU(2, D) ϕ: W SL 2 (C) GSp 4 (C) L Π G ϕ ρ Π J ϕ Hi RZ [ρ] := Hom J(H i RZ, ρ)g-sm W G HRZ i [ρ] cusp HRZ i [ρ] ( ) Lubin-Tate 2012 9 18 15 / 27

GSp 4 G = GSp 4 (Q p ), J = GU(2, D) ϕ: W SL 2 (C) GSp 4 (C) L Π G ϕ ρ Π J ϕ Hi RZ [ρ] := Hom J(H i RZ, ρ)g-sm W G HRZ i [ρ] cusp HRZ i [ρ] L r : GSp 4 GL 4 (I) (r ϕ) SL2 (C) = 1 r ϕ (II) (r ϕ) SL2 (C) = 1 r ϕ = φ 1 φ 2 φ i φ 1 φ 2, dim φ i = 2 (III) r ϕ = φ (χ Sp 2 ) φ SL2 (C) = 1 φ (IV) r ϕ = (χ 1 Sp 2 ) (χ 2 Sp 2 ) (χ 1 χ 2 ) ( ) Lubin-Tate 2012 9 18 15 / 27

GSp 4 ϕ SL2 (C) = 1 ϕ SL2 (C) = 1 (I), (II) ( ) Lubin-Tate 2012 9 18 16 / 27

GSp 4 ϕ SL2 (C) = 1 ϕ SL2 (C) = 1 (I), (II) A ϕ (I) (II) ρ Π J ϕ (1) i 3 H i RZ [ρ] cusp = 0 (2) HRZ 3 [ρ] cusp = π Π G σ π π ϕ σ π W l π Π G WD(σ π ) = r ϕ ϕ ϕ (II) WD(σ π ) φ 1, φ 2 i ( 1)i HRZ i Kottwitz ( ) Lubin-Tate 2012 9 18 16 / 27

GSp 4 ϕ SL2 (C) 1 ϕ SL2 (C) 1 (III), (IV) ( ) Lubin-Tate 2012 9 18 17 / 27

GSp 4 ϕ SL2 (C) 1 ϕ SL2 (C) 1 (III), (IV) A ϕ (III) r ϕ = φ (χ Sp 2 ) ρ Π J ϕ (1) i 3 H i RZ [ρ] cusp = 0 (2) HRZ 3 [ρ] cusp = σ π π π Π G π WD(σ π) = φ ( ) Lubin-Tate 2012 9 18 17 / 27

GSp 4 ϕ SL2 (C) 1 ϕ SL2 (C) 1 (III), (IV) A ϕ (III) r ϕ = φ (χ Sp 2 ) ρ Π J ϕ (1) i 3 H i RZ [ρ] cusp = 0 (2) HRZ 3 [ρ] cusp = σ π π π Π G π WD(σ π) = φ B ϕ (III) (IV) π Π G ϕ π HRZ 4 ( ) Lubin-Tate 2012 9 18 17 / 27

GSp 4 ϕ SL2 (C) 1 ϕ (III) (IV) χ Sp 2 r ϕ π Π G ϕ ρ ΠJ ϕ χ π ρ H 3 RZ χ π Zel(ρ) H 4 RZ Zel Zelevinsky Zel(ρ) H 2 RZ = 0 Π J ϕ = {ρ, ρ } {ρ, Zel(ρ) } J A ( ) Lubin-Tate 2012 9 18 18 / 27

B B ϕ (III) (IV) π Π G ϕ π HRZ 4 ( ) Lubin-Tate 2012 9 18 19 / 27

B B ϕ (III) (IV) π Π G ϕ π HRZ 4 H i RZ Sh [ss] K M K Hochschild-Serre ( ) Lubin-Tate 2012 9 18 19 / 27

B B ϕ (III) (IV) π Π G ϕ π HRZ 4 H i RZ Sh [ss] K M K Hochschild-Serre Hochschild-Serre (Harris, Fargues) E i,j 2 = Ext j J-sm (H6 j RZ A GSp 4 (A,p ) J, A)( 3) = lim H i+j (Sh [ss] K, Q l) K ( ) Lubin-Tate 2012 9 18 19 / 27

B B ϕ (III) (IV) π Π G ϕ π HRZ 4 H i RZ Sh [ss] K M K Hochschild-Serre Hochschild-Serre (Harris, Fargues) E i,j 2 = Ext j J-sm (H6 j RZ A GSp 4 (A,p ) J H i (Sh [ss] ) := lim K H i+j (Sh [ss] K, Q l), A)( 3) = lim H i+j (Sh [ss] K, Q l) K ( ) Lubin-Tate 2012 9 18 19 / 27

Sh [ss] K Sh [good] K Sh [ss] K Sh[good] K Sh K ( ) Lubin-Tate 2012 9 18 20 / 27

Sh [ss] K Sh [good] K Sh [ss] K Sh[good] K Sh K IH i (Sh ) H i (Sh ) H i (Sh [good] ) H i (Sh [ss] ) ( ) Lubin-Tate 2012 9 18 20 / 27

Sh [ss] K Sh [good] K Sh [ss] K Sh[good] K Sh K IH i (Sh ) cusp = H i (Sh ) cusp = H i (Sh [good] ) cusp = H i (Sh [ss] ) cusp ( ) Lubin-Tate 2012 9 18 20 / 27

Sh [ss] K Sh [good] K Sh [ss] K Sh[good] K Sh K IH i = (Sh ) cusp H i (Sh ) cusp = H i (Sh [good] (1) (2) ) cusp = H i (Sh [ss] (3) ) cusp ( ) Lubin-Tate 2012 9 18 20 / 27

Sh [ss] K Sh [good] K Sh [ss] K Sh[good] K Sh K IH i = (Sh ) cusp H i (Sh ) cusp = H i (Sh [good] (1) (2) ) cusp = H i (Sh [ss] (3) (1) C ) cusp ( ) Lubin-Tate 2012 9 18 20 / 27

Sh [ss] K Sh [good] K Sh [ss] K Sh[good] K Sh K IH i = (Sh ) cusp H i (Sh ) cusp = H i (Sh [good] (1) (2) ) cusp = H i (Sh [ss] (3) (1) C (2) ) cusp ( ) Lubin-Tate 2012 9 18 20 / 27

Sh [ss] K Sh [good] K Sh [ss] K Sh[good] K Sh K IH i = (Sh ) cusp H i (Sh ) cusp = H i (Sh [good] (1) (2) ) cusp = H i (Sh [ss] (3) (1) C (2) ) cusp (3) Boyer GSp 4 GSp 2n, n 3 ( ) Lubin-Tate 2012 9 18 20 / 27

( ) ị 2, 3, 4 G H i RZ ( ) Lubin-Tate 2012 9 18 21 / 27

( ) ị 2, 3, 4 G H i RZ dim M = 3 HRZ i 0 0 i 6 2 = 3 1 = dim M dim M red, 4 = 3 + 1 = dim M + dim M red ( ) Lubin-Tate 2012 9 18 21 / 27

( ) ị 2, 3, 4 G H i RZ dim M = 3 HRZ i 0 0 i 6 2 = 3 1 = dim M dim M red, 4 = 3 + 1 = dim M + dim M red Lubin-Tate M ( ) Lubin-Tate 2012 9 18 21 / 27

( ) ị 2, 3, 4 G H i RZ dim M = 3 HRZ i 0 0 i 6 2 = 3 1 = dim M dim M red, 4 = 3 + 1 = dim M + dim M red Lubin-Tate M ( ) Lubin-Tate 2012 9 18 21 / 27

B ϕ (III) (IV) π Π G ϕ π HRZ 4 B ( ) Lubin-Tate 2012 9 18 22 / 27

B ϕ (III) (IV) π Π G ϕ π HRZ 4 B GSp 4 (A) Π Π IH 2 (Sh ) Π Π p = π ϕ (III) (IV) ( ) Lubin-Tate 2012 9 18 22 / 27

B ϕ (III) (IV) π Π G ϕ π HRZ 4 B GSp 4 (A) Π Π IH 2 (Sh ) Π Π p = π ϕ (III) (IV) Hochschild-Serre i + j = 2 i, j Π p Ext i J-sm (H6 j RZ, A) 6 j 5, 6 Π p Hom J (HRZ 4, A) π HRZ 4 ( ) Lubin-Tate 2012 9 18 22 / 27

A, A π, ρ ( ) Lubin-Tate 2012 9 18 23 / 27

A, A π, ρ Π GSp 4 (A) Π IH 3 (Sh ) Π Π p = π Π p 1 GSp 4 (A) Π Π p = (Π ) p Π = Π cf Arthur ( ) Lubin-Tate 2012 9 18 23 / 27

A, A π, ρ Π GSp 4 (A) Π IH 3 (Sh ) Π Π p = π Π p 1 GSp 4 (A) Π Π p = (Π ) p Π = Π cf Arthur ρ Π J Π,p = (Π J ),p Hochschild-Serre Π,p ( ) Lubin-Tate 2012 9 18 23 / 27

L Lefschetz ( ) Lubin-Tate 2012 9 18 24 / 27

L Lefschetz ( Lefschetz ) X Q ac p adic X X f : X X f : X X x X \ X x f (x) X ( 1) i Tr(f ; Hc(X i, Q l )) = # Fix(f ) i ( ) Lubin-Tate 2012 9 18 24 / 27

L Lefschetz ( Lefschetz ) X Q ac p adic X X f : X X f : X X x X \ X x f (x) X ( 1) i Tr(f ; Hc(X i, Q l )) = # Fix(f ) i M K ( ) Lubin-Tate 2012 9 18 24 / 27

ϕ: W SL 2 (C) GSp 4 (C) (I) (II) L Π G ϕ ΠJ ϕ TRSELP ρ Π J ϕ Hi RZ [ρ] G ( ) Lubin-Tate 2012 9 18 25 / 27

ϕ: W SL 2 (C) GSp 4 (C) (I) (II) L Π G ϕ ΠJ ϕ TRSELP ρ Π J ϕ Hi RZ [ρ] G g G θ HRZ [ρ](g) = 4 θ π (g) ρ Π J ϕ H RZ [ρ] = i ( 1)i H i RZ [ρ] π Π G ϕ θ HRZ [ρ], θ π G ( ) Lubin-Tate 2012 9 18 25 / 27

GL n Faltings, Strauch W H i RZ ( ) Lubin-Tate 2012 9 18 26 / 27

GL n Faltings, Strauch W H i RZ M K p p Dieudonné Hodge p Hodge L ( ) Lubin-Tate 2012 9 18 26 / 27

GL n Faltings, Strauch W H i RZ M K p p Dieudonné Hodge p Hodge L θ HRZ [ρ](g) = 4 #Π G ϕ π Π G ϕ θ π (g) ( ) Lubin-Tate 2012 9 18 26 / 27

GU(2, 1) GL 4 1/2 GU(n 1, 1) Boyer (Mantovan, Shen) H i RZ GU(2, D) GSp 4 cf Faltings ( ) Lubin-Tate 2012 9 18 27 / 27

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