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2 Borel1956 Groupes linéaire algébriques, Ann. of Math. 64 (1956), Chevalley1956/58 Sur la classification des groupes de Lie algébriques, Sém. Chevalley 1956/58, E.N.S., Paris. Tits1959 Sur la classification des groupes algébriques semi-simples, C. R. Acad. Sci. Paris 249 (1959), (semisimple anisotropic kernel) Satake1960 On representations and compactifications of symmetric Riemannian spaces, Ann. Math. 71 (1960), Γ Araki1962 On root systems and an infinitesimal classification of irreducible symmetric spaces, J. of Math. Osaka City University 13 (1962), Γ (= ) Lie Satake1963 On the theory of reductive algebraic groups over a perfect field, J. of Math. Soc, Japan, Vol.15, No. 2 (1963), Veisfeiler1964 Classification of semiseimple Lie algebras over a p-adic field, Dokl. Akad. Nauk SSSR 158 (1964), Satake1963 p- Lie ( ) Tits1966 Classification of algebraic semisimple groups, in Algebraic Groups and Discontinuous Subgroups, Proc. Symp. in Pure Math. 9, Vol.I, A.M.S., 1966, pp Γ (= = Tits index) (, Γ ) Satake1967 Symplectic representations of algebraic groups satisfying a certain analyticity condition. Acta Math. 117 (1967), γ(g) 2

3 Satake1971 Classification Theory of Semi-simple Algebraic Groups, Marcel Dekker, (1967 ), p- Selbach1976 Klassifikationstheorie halbeinfacher algebraischer Gruppen, Bonner Mathematische Schriften, Tits1966 Γ Tits1990 Strongly inner anisotropic forms of simple algebraic groups, J. of Algebra 131 (1990), * Tits1966 E 133 8,1 Springer1998 Linear Algebraic Groups, 2nd edition, Chapter 17 Γ Satake2001 On classification of semisimple algebraic groups, in Class Field Theory -Its Centenary and Prospect (Ed. Katsuya Miyake), Advanced Studies Pure Math. 30 (2001),

4 Satake1960 On representations and compactifications of symmetric Riemannian spaces, Ann. Math. 71 (1960), Satake1960 ( No.5, ) 4

5 1 Γ ( ) k, Γ = Gal( k/k) Galois. G k, A := G k T := A G k. A /k n { }} { G m G m, T / k n { }} { G m G m G m 1. T X = X(T) := Hom k(t, G m ) Z n, Γ X Γ χ σ (t) := σ 1 (χ(σ(t))), ( t T, χ X, σ Γ). 5

6 1.1 α : T G m, P α : G 1 / k x α : G a P α : k- such that tx α (ξ)t 1 = x α (α(t)ξ), t T, ξ G a. G a 1. r = r(g, T) := G T X Q = X Z Q. (R1) r, 0 r X Q = r Q. (R2) α, β r, (α, β) c α,β := 2 (α, α) Z s α(β) := β c α,β α r (R3) α r, Qα r = {±α}. 6

7 1.2 Γ A G k A T. X = X(T) X 0 X 0 := {χ X : χ A = 0}. X. χ X, χ > 0 χ X 0 = χ σ > 0 σ Γ Γ.. r + := {α r : α > 0} r r +. r + 1. r + = r +, + := { α c α α : 0 c α Q}. Γ Γ. 7

8 1.3 ( ) Y := X(A) k A, X, X/X 0 Y. r Γ, π Y : χ χ A r 0 := r X 0, 0 := r 0, r 0 r, 0 r 0. r r 0 Weyl W := {s α : α r}, W 0 := {s α : α r 0 }, W W Γ := {w W : w(x 0 ) = X 0 }, W 0 W Γ. [Satake1963, Theorem 2] r := π(r r 0 ), := π( 0 ), Y Q := Y Z Q r. W Γ /W 0 r Weyl. 8

9 1.4 G A G Z(A),. G(r 0 ) := Z(A) G(r 0 ) k (, G(r 0 ) k ), r 0 = r(g(r 0 ), G(r 0 ) T). G(r 0 ) G. 9

10 1.5 Γ r = r(g, T) Γ.. σ Γ,!w σ W 0 such that σ = w σ Γ Aut(X) : σ [σ],. χ [σ] := w 1 σ χ σ (χ X) [σ] =, [σ] 0 = 0 ( σ Γ), σ [σ] Γ Aut(X,, 0 ). S G := (X,, 0, [Γ]) G Γ. Dynkin Dynkin 0 [Γ] (, 0, [Γ]) ( ). 10

11 p.1694 Γ = Gal(C/R),.,. Satake2001, p

12 1.6 Γ Γ (X,, 0, [Γ]). X : 0 : [Γ] Aut(X,, 0 ) :. 2 Γ (X,, 0, [Γ]), (X,, 0, [Γ] ), ψ X X : ψ( ) = such that ψ( 0 ) = 0 ψ [σ] = [σ] ψ ( σ Γ) ψ Γ. G Γ S G k T Γ. Γ (X,, 0, [Γ]), ( ) X Z-. 12

13 1.7 k G 0 Γ S = (X,, 0, [Γ]) (G 0, S), S G0 = (X 0, 0, 0, [Γ]). (G 0, S) (G 0, S ),. Γ S ψ S. f 0 k G 0 G 0. k T 0 G 0, X(T 0 ) = X 0, ( f 0 1 T 0 ) = ψ X 0. G 0 S := [Satake1963, Theorem 3] { k } G 0 S [G] [G(r 0 ), S G ]. G 0 S k. 13

14 2 Γ 2.1 Γ 2 G 0 S k k π 1 G 0 S k [G 0,S] [G 0 ] [G π 2 0, S] [S] Γ, π 1. k S k := π 2 (G 0 S k ), S k Γ Γ. G 0 S k. (Q1) k (Q2) Γ 14

15 π 1 k G 0 S k π 2 Sk Lang, F q - = {e} S Fq = {[X,,, [Γ]] : [Γ] Aut(X, ) } G 0 S Fq 1:1 S Fq Cartan :. Weyl : Dynkin. Araki1962 : Γ. 1:1 Γ 15

16 2.2 Steinberg k, G G(r 0 ) = {e}, G Steinberg. π 1 k G 0 S k 3 π 1 1 ([e]) = Steinberg k k G,![G 1 ] π 1 1 ([e]), f : G G 1 : k- such that f σ f 1 Inn(G 1 ), σ Γ G G 1 k inner twist. Z 1 G 1, Inn(G 1 ) G 1 /Z 1. 16

17 3 G f G 1 b σ := f σ f 1 = Inn(g σ ), (g σ G 1, σ Γ), Γ σ b σ Inn(G 1 ) G 1 /Z 1 1-, β k (G, f ) := [b σ ] H 1 (k, G 1 /Z 1 ) Γ Γ (σ, τ) c σ,τ := g τ σg τ g 1 στ Z 1 2-, γ k (G, f ) := [c σ,τ ] H 2 (k, Z 1 ) H 1 (k, G 1 ) H 1 (k, G 1 /Z 1 ) δ H 2 (k, Z 1 ) δ(β k (G, f )) = γ k (G, f ). 17

18 G f G 1, S G = (X,, 0, [Γ]) S G1 (X,,, [Γ]). Aut(X, ) C := {σ Aut(X, ) : σ [γ] = [γ] σ γ Γ}, C Out k (G 1 ). C H 2 (k, Z 1 ), γ k (G) := γ k (G, f ) mod C, G. f G 1 18

19 p- γ k (G) 4[Kneser, Math. Z., Vol. 89, 1965] k p-, G k H 1 (k, G) = 0. H 1 (k, G/Z) H 2 (k, Z). k p-, G, G k, Steinberg G 1 k inner twist. C \H 2 (k, Z 1 ) γ k (G) = γ k (G ), G G k. Tate Poitou H 2 (k, Z 1 ) H 0 (Γ, X(Z 1 )) = X(Z 1 ) Γ = X(Z 1 ) [Γ] G 1 X(Z 1 ) X/ Z C \H 2 (Γ, Z 1 ) C \(X/ Z ) [Γ] Γ. k 1:1 [X,,,[Γ]] S k C \(X/ Z ) [Γ] 19

20 k 1:1 [X,,ϕ,[Γ]] S k (e) C \(X/ Z ) [Γ] Satake1971 p

21 Γ S k Satake1971 p

22 1980,. k, p k. 5[Kneser, Harder, Sansuc, Chernousov] G k Z, δ : H 1 (k, G/Z) H 2 (k, Z) i : H 1 (k, G/Z) H 1 (k v, G/Z) v p j : H 2 (k, Z) H 2 (k v, Z) v p δ : H 1 (k v, G/Z) H 2 (k v, Z) v p v p. (1) δ i. (2) δ Ker i, Ker i Ker j. (3) j δ H 2 (k, Z) H 1 (k v, G/Z) v p δ i : H 1 (k, G/Z) H 2 (k, Z) H 1 (k v, G/Z) v p. 22

23 H 1 (k, G/Z) δ i H 2 (k, Z) H 1 (k v, G/Z) v p [cf. Satake2001] G 1 Steinberg, Z 1 G 1. η {ξ v } v H 2 (k, Z 1 ) H 1 (k v, G 1 /Z 1 ) p p, k inner twist G f G 1 such that γ k (G, f ) = η, β kv (G, f ) = ξ v ( v p ) (G, f )., (G, f ) k G φ G such that f φ f 1 Inn(G 1 ) p k, j : H 2 (k, Z 1 ) v p H 2 (k v, Z 1 ), η H 2 (k, Z 1 ) j(η) = (η v ) v p. 23

24 Q B l n = 2l + 1, G 1 = Spin(n), Z 1 = µ 2 H 2 (Q, Z 1 ) = Br(Q) 2 p Br(Q p ) 2 G 1 Γ [X, B l,, e] H 1 Γ [X, B l, 0, e] (R, G 1 /Z 1 ) 0 = B l r, 0 r l l { }} { = } {{ } η ξ H 2 (Q, Z 1 ) H 1 (R, G 1 /Z 1 ), r ξ [X, B l, B l r, e] (n r, r) n 2 V Spin j(η) = (η p ), p n 2 V p l r such that η p = V p Hasse (even Clifford Brauer ), Minkowski Hasse n 2 V/Q such that V Q p = V p ( p ) η ξ G = Spin(V) 24

25 Hasse. G, G k. k v G v G v = G G?. B n, C n, G 2, F 4, E 7, E 8. A n, D n, E 6. (.) 25

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