0 17 l l Grothendieck Weil Grothendieck SGA (Séminaire de Géométrie Algébrique du Bois-Marie) [Del2], [Del3] Grothendieck Weil Ramanujan Deligne [Del1

Transcription

1 l Tate Galois Galois Galois Weil-Deligne Rψ

2 0 17 l l Grothendieck Weil Grothendieck SGA (Séminaire de Géométrie Algébrique du Bois-Marie) [Del2], [Del3] Grothendieck Weil Ramanujan Deligne [Del1] Deligne 2 l l Weil q Eichler Deligne Langlands Galois GL n Langlands F GL n (A F ) Gal(F /F ) n l Π l Galois ρ(π) Galois Sato-Tate Galois Galois [DS] [DS] Galois Galois Galois Chevalley Deligne-Lusztig Kazhdan-Lusztig 2

3 SGA ([SGA4], [SGA5], [SGA7], [SGA4 1 2 ]) [SGA4 1 2, Arcata] 70 Galois Galois Galois SGA Galois 3.3 SGA 1 6 SGA, [Del3], [BBD] [KW] k F K F, K O F, O K k k Galois Gal(k/k) G k G k l Q l Q l V ρ: G k GL(V ) V l G k l Z l Z l Λ ρ: G k Aut(Λ) Z l Q l Z l G X x X Stab G (x) := {g G gx = x} G G Spec A X A B X Spec A Spec B X A B X B 3

4 1 1.1 Tate Tate k 1.1 E k n 1 E[n] = {x E(k) nx = 0} l T l E = lim n E[l n ], V l E = T l E Zl Q l T l E E l Tate V l E E l Tate l k T l E 2 Z l [Sil] V l E 2 Q l l k T l E, V l E l k T l E, V l E Galois G k ρ G k 2 l G k Aut Ql (V l E) l G k Aut Zl (T l E) l E 1.2 ρ E E L = (End k E) Z Q Q k 0 (End k E) Z Q Q E L V l E ι: L End Ql V l E L G k ρ: G k Aut Ql (V l E) Im ι {g Aut Ql (V l E) gι(a)g 1 = ι(a) ( a L)} (L Q Q l ) Aut Ql (V l E) E Im ρ E V l E k G k l k E Im ρ Aut Ql (V l E) Zariski ([Ser1]) 4

5 1.3 k p > 0 E k E[p] = 0 D = (End k E) Z Q Q V l E D p Im ρ Aut Ql (V l E) 1.4 k = K ρ E 1.5 i) E ρ ii) E ρ I K σ I K ρ(σ) 1 Tate 1 C E C Z Λ E(C) = C/Λ ([Sil]) H 1 ( E(C), Z ) = Λ, H 1 ( E(C), Q ) = Λ Z Q, Tl E = lim n Λ/l n Λ = Λ Z Z l, V l E = Λ Z Q l T l E V l E E(C) 1 l V l E i 0 k G k l ; X H i (X k, Q l ) i l k = C H i (X k, Q l ) X(C) Betti H i (X(C), Q) l H i (X(C), Q) Q Q l k C H i (X k, Q l ) Betti k Galois H i (X k, Q l ) X 5

6 H i k E l H 0 (E k, Q l ) = Q l, H 1 (E k, Q l ) = (V l E), H 2 (E k, Q l ) = Q l ( 1), H i (E k, Q l ) = 0 (i 3) k C C Betti X X Shv X 3 X Γ(X, ): Shv X Ab Ab X F 0 F I 0 I 1 0 Γ(X, I 0 ) Γ(X, I 1 ) i X F i H i (X, F) F Z Q H i (X, Z) Q H i (X, Q) X X H i (X, Z) H i (X, Q) Betti Betti Zariski X X X Z 1 k A 1 k P1 k 1 H i (E, O E) [KS] [Ive] 3 6

7 A 1 C P1 C Betti Zariski X Open X X V, U X V U V U V U X Open X Ab U Open X (U i U) i I Open X U i U (U i ) i I U 0 F(U) i I F(U i ) ( ) i,j I F(U i U j ) ( ) (x i ) i I (x i Ui U j x j Ui U j ) i,j I U i U j Open X U i U U j (U i ) i I U Open X (U i U) i I LIsom X 1.6 f : Y X y Y y V, f(y) X U f V U LIsom X f : Y X Y f : Y X f : Y X g : Y Y f g = f LIsom X (g i : Y i Y ) i I Y, Y i LIsom X Y = i I g(y i) (site) 4 4 (pretopology) 7

8 LIsom X 1.7 LIsom X F : LIsom X Ab LIsom X (Y i Y ) i I 0 F(Y ) i I F(Y i ) ( ) i,j I F(Y i Y Y j ) pr 1 : Y i Y Y j Y i p i,j : F(Y i ) F(Y i Y Y j ) pr 2 : Y i Y Y j Y j q i,j : F(Y j ) F(Y i Y Y j ) ( ) (x i ) i I (p i,j (x i ) q i,j (x j )) i,j I Y i Y Y j LIsom X LIsom X F LIsom X Y F(Y ) Γ(Y, F) LIsom X Shv LIsom X Shv X Shv LIsom X 1.8 X F LIsom X ε F Γ(Y f X, ε F) = Γ(Y, f F) LIsom X G X ε G Γ(U, ε G) = Γ(U X, G) ε ε Shv X Shv LIsom X X F ε F LIsom X LIsom X g i (Y i Y )i I x i Γ(Y i, ε F) = Γ(Y i, gi (F Y )) (i I) p i,j (x i ) = q i,j (x j ) Y X F F Y g i Y i (U iλ ) λ Λi g i g iλ: U iλ Y i Y xiλ = x i Uiλ Γ(g iλ (U iλ ), F Y ) = Γ(U iλ, g iλ (F Y )) x iλ Γ(g iλ (U iλ ), F Y ) y iλ i, j I λ Λ i, λ Λ j y iλ y jλ g iλ (U iλ ) g jλ (U jλ ) y Γ(Y, F Y ) Γ(Y, F Y ) Γ(U iλ, g iλ (F Y )) x iλ (U iλ ) λ Λi Y i y Γ(Y, F Y ) Γ(Y i, g i (F Y )) x i y ε F LIsom X [SGA4] 8

9 G ε G X ε ε X F ε ε F = F LIsom X G ε ε G = G ε ε G G f : Y X Y V Γ(V f X, G) Y G Y X U Γ(U, ε G) = Γ(U X, G) Γ ( f 1 (U) f X, G ) = Γ(U, f G Y ) X ε G f G Y Y f ε G G Y Γ(Y, ) Γ(Y f X, ε ε G) = Γ(Y, f ε G) Γ(Y, G Y ) = Γ(Y f X, G) LIsom X ε ε G G LIsom X Y Γ(Y, ) Y Y X Shv LIsom X Γ(X, ): Shv LIsom X Ab Open X LIsom X LIsom X f : Y X i) y Y f (unramified/neat) m Y,y, m X,f(y) O Y,y, O X,f(y) m Y,y = m X,f(y) O Y,y O Y,y /m Y,y O X,f(y) /m X,f(y) Ω 1 Y/X 0 ii) f (étale) 9

10 1.10 f : Y X y Y f(y) U = Spec A f 1 (U) y V = Spec B f B A B = A[T 1,..., T n ]/(f 1,..., f n ), ( fi ) det T A[T 1,..., T n ]/(f 1,..., f n ) j i,j 1.11 A n A a A Spec A[T ]/(T n a) Spec A T n a T nt n 1 A[T ]/(T n a) (na) 1 T k k X Spec k k k X Spec A A k A Artin A = p Spec A A p k k [EGA4] 1.13 i) ii) iii) f : Y X X X X X f f : Y X X X iv) f : Y X, g : Z Y f g f g v) 1.6 LIsom X 10

11 1.14 X X Et X Et X Y X f : Y X f : Y X g : Y Y f = f g 1.13 iv) g Y Et X X Et X (g i : Y i Y ) i I Y, Y i Et X Y = i I g i(y i ) Et X X (étale site) 1.15 Et X X F : Et X Ab Et X (Y i Y ) i I 0 F(Y ) i I F(Y i ) ( ) i,j I F(Y i Y Y j ) pr 1 : Y i Y Y j Y i p i,j : F(Y i ) F(Y i Y Y j ) pr 2 : Y i Y Y j Y j q i,j : F(Y j ) F(Y i Y Y j ) ( ) (x i ) i I (p i,j (x i ) q i,j (x j )) i,j I 1.13 ii), iii) Y i Y Y j Et X X F Et X Y F(Y ) Γ(Y, F) X Shvét X 1.16 k Spec k F 1.12 F k k L F L := F(Spec L) L k L Galois Spec L Spec L F L F L σ Gal(L /L) Spec L σ : Spec L Spec L Gal(L /L) F L F L F L F Gal(L /L) L Spec L Spec L F L L L = σ Gal(L /L) L ; a b (aσ(b)) σ 11

12 0 F L F L ( ) σ Gal(L /L) F L ( ) x (x σ(x)) σ F L = F Gal(L /L) L M F = lim L F L L k k Galois G k Gal(L/k) F L M F G k k k L = F L M Gal(k/L) F G k M k k L F M (Spec L) = M Gal(k/L) Spec k F M F M F M Shvét Spec k G k k Shvét Spec k Spec k F Γ(Spec k, F) (geometric point) X x x x X x 1.17 X Z X X Y F(Y ) := Hom X (Y, Z) Y Z X F : Et X Ab X Z X Z F : Et X Set 12

13 Et X (Y i Y ) i I F(Y ) { = (x i ) i I } F(Y i ) p i,j (x i ) = q i,j (x j ) ( i, j I) i I X, Y, Y i Y I Y = i I Y i (Y i Y ) i I (Y Y ) X = Spec A, Y = Spec B, Y = Spec B, Z = Spec C Alg A A Hom AlgA (C, B) = { φ Hom AlgA (C, B ) φ(c) 1 = 1 φ(c) B B B ( c C) } Y Y B B B d: B ( ) B B B d(b ) b 1 1 b 0 B B d B B B B B, B B, B B B B s: B B B B b B d(b ) = 0 b 1 1 b = 0 s id: B B B B b = s(b ) B 1.18 X n 1 i) G a : Y Γ(Y, O Y ) G a,x = Spec O X [T ] X ii) G m : Y Γ(Y, O Y ) G m,x = Spec O X [T, T 1 ] X n iii) µ n = Ker(G m G m ) µ n,x = Spec O X [T ]/(T n 1) X µ n Z/nZ(1) iv) Z/nZ X X n Z/nZ X Z/nZ f : X X X F f f F (f F)(Y ) = F(Y X X) f : Shvét X Shvét X f : Shvét X Shvét X f X 13

14 i: x X F i i F 1.16 F x F x (stalk) F x = lim F(U) x / U i X Γ: Shvét X Ab, f : Shvét X Shvét X 1.19 X Shvét X Γ 1.20 X F X F 0 F I 0 I 1 0 Γ(X, I 0 ) Γ(X, I 1 ) RΓ(X, F) Ab well-defined RΓ(X, F) i Ker ( Γ(X, I i ) Γ(X, I i+1 ) )/ Im ( Γ(X, I i 1 ) Γ(X, I i ) ) H i (X, F) X F i f : X X Shvét X 0 f I 0 f I 1 Rf F i R i f F 1.21 X Spec k F Spec k 1.16 G k M Γ(Spec k, F) = M G k M G k G k M M M G k Galois H i (G k, M) H i (Spec k, F) = H i (G k, M) Galois 14

15 1.3.2 k X k G m 1.22 H i (X, G m ) H 0 (X, G m ) = k, H 1 (X, G m ) = Pic(X), H i (X, G m ) = 0 (i 2) Pic(X) X Picard X H 0 (X, G m ) = Γ(X, G m ) = Γ(X, O X ) = k H 1 (X, G m ) = Pic(X) X H 1 Čech H 1 (X, G m ) fpqc Zariski H i (X, G m ) = 0 (i 2) X k(x) Galois H i (G k(x), k(x) ) i 2 Tsen k(x) C 1 ([Ser2, II, 3]) Tsen H i (X, G m ) = 0 (i 2) X X k 0 Γ(X, G m ) k(x ) ord x X Z 0 X X ord f k(x ) f X Γ(X, ) 0 G m j G m,η x X i x Z 0 η X j η X x X x X i x 15

16 x X H i (X, G m ) H i (X, j G m,η ) x X H i 1 (X, i x Z) H i (X, i x Z) 5 H i (X, j G m,η ) = H i (G k(x), k(x) ) H i (X, i x Z) = H i (G k, Z) = 0 (i 1) 1.42 Tsen H i (X, G m ) = X Z/nZ(1) 1.23 n 1 k H i (X, Z/nZ(1)) H 0 (X, Z/nZ(1)) = Z/nZ(1), H 1 (X, Z/nZ(1)) = Pic(X)[n], H 2 (X, Z/nZ(1)) = Z/nZ, H i (X, Z/nZ(1)) = 0 (i 3) Z/nZ(1) k 1 n k Z/nZ(1) = Z/nZ Pic(X)[n] Pic(X) n Pic(X) 0 Z/nZ(1) G m n G m 0 X G m G m U Et X a Γ(U, G m ) = Γ(U, O U ) U (U i U) i I a i Γ(U i, G m ) a n i = a Ui V = Spec O U [T ]/(T n a) V U 1.11 T Γ(V, G m ) n Γ(U, G m ) Γ(V, G m ) n a G m G m 1.22 H i (X, Z/nZ(1)) = 0 (i 3) 0 H 0 (X, Z/nZ(1)) k n k H 1 (X, Z/nZ(1)) Pic(X) n Pic(X) H 2 (X, Z/nZ(1))

17 H 0 (X, Z/nZ(1)) = Z/nZ(1), H 1 (X, Z/nZ(1)) = Pic(X)[n] deg: Pic(X) deg Z Pic 0 (X) Pic 0 (X) X Jacobi g k Pic 0 (X) n Jacobi [CS] [Mum] deg: Pic(X) deg Z Pic(X)/n Pic(X) Z/nZ H 2 (X, Z/nZ(1)) = Pic(X)/n Pic(X) = Z/nZ k 1 n X Z/nZ(1) = Z/nZ Z/nZ Z/nZ(1) = Z/nZ 1.23 X Z/nZ 1.24 n 1 k H i (X, Z/nZ) H 0 (X, Z/nZ) = Z/nZ, H 1 (X, Z/nZ) = Pic(X)[n]( 1), H 2 (X, Z/nZ) = Z/nZ( 1), H i (X, Z/nZ) = 0 (i 3) Z/nZ( 1) = Hom(Z/nZ(1), Z/nZ) Z/nZ M M( 1) = M Z/nZ Z/nZ( 1) m m 0 Z/nZ(m) = Z/nZ(1) m m < 0 Z/nZ(m) = Z/nZ( 1) ( m) Z/nZ M M(m) = M Z/nZ Z/nZ(m) M Tate (Tate twist) H 0 (X, Z/nZ), H 1 (X, Z/nZ), H 2 (X, Z/nZ) 1, 2g g X 1 Z/nZ 1.23 H 1 (X, Z/nZ) 2g Z/nZ 0 Pic 0 (X) Pic(X) deg Z 0 Pic(X)[n] = Pic 0 (X)[n] Pic 0 (X) X Jacobi k n k Pic 0 (X)[n] = (Z/nZ) 2g 1.25 = E k E(k) Pic 0 (X); P [P ] [O] O E(k) E(k) Pic(X) H 1 (E, Z/nZ) = E[n]( 1) Weil E[n] E[n] Z/nZ(1) E[n]( 1) = E[n] H 1 (E, Z/nZ) = E[n] 17

18 1.26 A 1 k Z/nZ k n k H i (X, Z/nZ) k p > 0 n = p H 2 (X, Z/pZ) = 0 X k H i (X, Z/pZ) Z/pZ X Z/nZ n X 1.28 k p > 0 X i) X 0 Z/pZ G a G a 0 G a a a p a ii) H i (X, G a ) = H i (X, O X ) Zariski X k H i (X, Z/pZ) H i (X, Z/pZ) = 0 (i 2) H 0 (X, Z/pZ), H 1 (X, Z/pZ) F p iii) X k H i (X, Z/pZ) F p 1.29 Z/nZ Z k X H 1 (X, Z) = 0 Z l X X n 1 Z/nZ H i (X, Z/nZ) l Galois Z/nZ Q l X Q l Z n 1 H i (X, Z/l n Z) Z l 18

19 Q l Q l Z/nZ X l 1.30 X Z l X (F n ) n 0 n 0 l n+1 F n = 0, F n+1 /l n+1 F n+1 = Fn 1.31 i) (Z/l n+1 Z) n 0 Z l Z l Z l Z l (m) ii) n 1 l n F = 0 F Z l iii) X Noether x π 1 (X, x) ρ: π 1 (X, x) Aut Zl (Λ) Z l Λ π 1 (X, x) l π 1 (X, x) Λ/l n+1 Λ X Y n F n (F n ) n 0 Z l ρ Z l X Z l (F n ) n 0 F n X (smooth) Noether X Z l π 1 (X, x) l 1.32 Q l Z l Z l F, G Hom Zl (F, G) Zl Q l Q l l Z l F l F Ql 1.33 i) Z l Z l (m) l Q l (m) ii) n 1 l n F = 0 F Z l 19

20 l 0 iii) X Noether x ρ: π 1 (X, x) GL(V ) π 1 (X, x) l Z l Λ l (ρ, Λ) X Z l F l F Ql Λ l ρ l X Z l l l (smooth) Noether X l π 1 (X, x) l Z l l 1.34 X (F n ) n 0 lim n Γ(X, F n ) i H i (X, ) Z l F = (F n ) n 0 X F l H i (X, F) l F Ql l H i (X, F Ql ) = H i (X, F) Zl Q l H i (X, Q l (m)) H i (X, (F n ) n 0 ) lim n H i (X, F n ) [Jan] 1.35 F = (F n ) n 0 Z l 1 lim n lim n lim n H i 1 (X, F n ) H i (X, F) lim H i (X, F n n ) 0 (H i 1 (X, F n )) n 0 Mittag-Leffler n 0 H i 1 (X, F n ) H i (X, F) = lim H i (X, F n n ) 1.36 i) X k l k 1.24 i, n H i (X, Z/l n+1 Z) H i (X, Q l ) = (lim H i (X, Z/l n+1 Z)) n Zl Q l 20

21 H 0 (X, Q l ) = Q l, H 1 (X, Q l ) = V l Pic(X)( 1), H 2 (X, Q l ) = Q l ( 1), H i (X, Q l ) = 0 (i 3) ii) X k l k X l F = (F n ) n 0 H i (X, F n ) Z/l n+1 Z H i (X, F) = lim n H i (X, F n ) 1.37 k G k l (ρ, V ) ρ Spec k l F H i (Spec k, F) = H i (G k, V ) Z l l R i f Z l (F n ) n 0 (R i f F n ) n 0 Z l Z l l Z l l Rf [Eke] Galois [SGA4], [SGA4 1 2 ] f : Y X F X f : H i (X, F) H i (Y, f F) g : Z Y (f g) = g f F Z/nZ f (Z/nZ) = Z/nZ f : H i (X, Z/nZ) H i (Y, Z/nZ) X H i (X, Z/nZ) Z/nZ l X H i (X, Q l ) Q l 6 l 21

22 X : H i (X, Z/nZ) H j (X, Z/nZ) H i+j (X, Z/nZ) x H i (X, Z/nZ), y H j (X, Z/nZ) x y = ( 1) ij (y x) f : Y X f (x y) = (f x) (f y) l 1.38 E k H 1 (E, Z/nZ) = E[n]( 1), H 2 (E, Z/nZ) = Z/nZ( 1) Weil E[n] E[n] Z/nZ(1) ( 2) I (X i ) i I Noether i j p ij : X j X i X = lim X i I i X i X i = Spec A i, A = lim A i I i X = Spec A X X i p i 1.39 [SGA4, Exposé VII] i I X i F i i j p ij F i = F j i I F = p i F i i H m (X, F) = lim H m (X i I i, F i ) n 1 H m (X, Z/nZ) = lim H m (X i I i, Z/nZ) l l ii) 1.40 k I k k (Spec L) L I lim L I Spec L = Spec k 22

23 i) F Spec k Spec L L k F L lim H i (Spec L, F L I L ) = H i (Spec k, F k ) Galois ii) F Spec k l Spec L L k F L lim H 0 (Spec L, F L I L ) = H 0 (Spec k, F k ) 1.41 f : Y X x X X x X X h x Y F (R i f F) x = H i (Y X X h x, F Y X X h x ) Y = X, f = id H i (Xx h, F X h) F x (i = 0), x = 0 (i 1) x X X h x x / U i X U U R i f F = X Et X Ab; V H i (Y X V, F Y X V ) lim (Y U X U) = Y X Xx h 1.39 (R i f F) x = lim U H i (Y X U, F Y X U) = H i (Y X X h x, F Y X X h x ) f = id R i f F = 0 (i 1) 23

24 1.42 j : η X 1.22 R i j G m = 0 (i 1) n n (torsion sheaf) X X 1.43 [SGA4, Exposé X] k X k d X l F H i (X, F) = 0 (i > 2d) X H i (X, F) = 0 (i > d) Lefschetz [SGA4, Exposé XIV] [SGA4, Exposé XII, XIII] Y g / Y X f g / X f f Y F g Rf F = Rf g F l 1.45 X X x 1.41 g R i f F = (R i f F) x = H i (Y X X h x, F Y X X h x ) R i f g F = H i (Y x, F Yx ) x 24

25 H i (Y X X h x, F Y X X h x ) = H i (Y x, F Yx ) X x H i (Y, F) = H i (Y x, F Yx ) 1.46 X Hausdorff Z X F H i (Z, F Z ) = lim U Z H i (U, F U ) 1.47 [SGA4, Exposé XV, XVI] Y g / Y X f g / X f g f Y F g Rf F = Rf g F l Y l k X k k k k n 1 H i = (X, Z/nZ) H i (X k, Z/nZ) k k Et X Et Xk ([SGA4, Exposé VIII]) k k(t 1,..., T m ) Spec k A m k = Spec k[t 1,..., T m ] x x A m k Spec k Spec k A m k Spec k 7 k Frac W (k) k 25

26 Spec k Spec k X Spec k A m k A m k f g g / X f / Spec k g R i f Z/nZ = R i f Z/nZ x 1.41 H i (X, Z/nZ) = H i (X k, Z/nZ) 2 k k(t 1,..., T m ) k k [SGA4 1 2, Finitude] k X k k n 1 H i (X, Z/nZ) Z/nZ l k H i (X, Q l ) Q l F l H i (X, F) Q l X k 1.24 X k Deligne ([dj]) X k (smooth purity) k [SGA4, Exposé XI, XVI] X C X C X(C) n 1 H i (X, Z/nZ) = H i (X(C), Z/nZ) l H i (X, Q l ) = H i (X(C), Q) Q Q l 26

27 1.51 X C H 1 Jacobi Poincaré k X k d n 1 k (trace map) ρ X : H 2d (X, Z/nZ(d)) Z/nZ d = 1 X ρ X 1.23 l k ρ X : H 2d (X, Q l (d)) Q l 1.52 Poincaré [SGA4, Exposé XVIII] H i (X, Z/nZ) H 2d i( X, Z/nZ(d) ) H 2d( X, Z/nZ(d) ) ρ X Z/nZ H i (X, Z/nZ) Hom Z/nZ ( H 2d i (X, Z/nZ(d)), Z/nZ ) l 1.53 X k Jacobi X, Y k X d Y d f : Y X k f : H i (X, Z/nZ) H i (Y, Z/nZ) (push-forward) f : H i (Y, Z/nZ) H i+2d 2d (X, Z/nZ(d d )) f : H i (Y, Q l ) H i+2d 2d (X, Q l (d d )) 1.54 x H i (X, Z/nZ), y H j (Y, Z/nZ) f (f x y) = x f y 27

28 1.55 f X, Y, f X k d Y k d f Künneth 1.56 Künneth [SGA4 1 2, Finitude] k l k k X, Y H m (X k Y, Q l ) = i+j=m H i (X, Q l ) Ql H j (Y, Q l ) i + j = m i, j H i (X, Q l ) Ql H j (Y, Q l ) pr 1 pr 2 H m (X k Y, Q l ) X, Y k Künneth X k Y pr 1 / X pr 2 Y / Spec k Deligne ([SGA4 1 2, Finitude]) X k Lefschetz k X k d l k Z X c Z X (cycle class) cl(z) H 2c (X, Q l (c)) ([SGA4 1 2, Cycle]) cl cl X Chow CH d c (X) X d c cl: CH d c (X) H 2c (X, Q l (c)) cl Chow ζ 1, 28

29 ζ 2 cl(ζ 1 ζ 2 ) = cl(ζ 1 ) cl(ζ 2 ) X k ζ CH 0 (X) deg ζ = ρ X (cl(ζ)) c = 0 X Z cl(z) H 0 (X, Q l ) Z 1 Z 0 X Q l X k Z k cl(z) Z X i Poincaré i : H j (Z, Q l ) H j+2c (X, Q l (c)) Gysin j = 0 i : H 0 (Z, Q l ) H 2c (X, Q l (c)) 1 H 0 (Z, Q l ) cl(z) 1.57 X k x X H 2 (X, Q l (1)) cl(x) H 2 k d H 2d (X, Q l (d)) Künneth Lefschetz 1.58 Lefschetz k X k f : X X k Γ f X f id X k X X k X k l X X k X Γ id 2 dim X i=0 ( 1) i Tr ( f ; H i (X, Q l ) ) = deg(γ f X ) deg(γ f X ) f : X X d = dim X Tate γ = cl(γ f ) H 2d (X k X) δ : X X k X deg(γ f X ) = ρ X δ (γ) f = pr 1 (f id), id = pr 2 (f id) x H i (X) f (x) = pr 2 (f id) (f id) pr 1(x) = pr 2 ( pr 1 (x) (f id) (1) ) 29

30 = pr 2 ( pr 1 (x) γ ) 2 3 Gysin Künneth H 2d (X k X) = s+t=2d Hs (X) H t (X) γ (s, t) n a s,n b t,n f (x) = n ρ X(x a 2d i,n )b i,n Tr(f ; H i (X)) = n ρ X(b i,n a 2d i,n ) 2d i=0 ( 1) i Tr ( f ; H i (X) ) = = 2d i=0( 1) i ρ X (b i,n a 2d i,n ) n 2d ρ X (a 2d i,n b i,n ) = ρ X δ (γ) i=0 n Lefschetz Galois

31 2 Galois 2.1 Galois k k l X k 8 σ G k X k σ id Spec σ X k = X k k X k k = X k X k (σ ) : H i (X k, Q l ) H i (X k, Q l ) G k H i (X k, Q l ) 2.1 G k H i (X k, Q l ) G k H i (X k, Z l ) 1.36 H i (X k, Z l ) = lim H i (X n k, Z/l n+1 Z) lim H i (X n k, Z/l n+1 Z) l m 1 X k 0 Z/l n+1 m Z lm Z/l n Z Z/l m Z 0 lim H i (X k, Z/l n+1 Z) lm lim H i (X n k, Z/l n+1 Z) H i (X k, Z/l m Z) n Mittag-Leffler ( ) l m lim H i (X n k, Z/l n+1 Z) = Ker lim H i (X k, Z/l n+1 Z) H i (X k, Z/l m Z) n G k H i (X k, Z/l n+1 Z) 1.39 H i (X k, Z/l n+1 Z) = lim H i (X L L, Z/l n+1 Z) L k k H i (X L, Z/l n+1 Z) Gal(k/L) G k 2.2 H i (X k, Q l ) Galois Galois Π 8 31

32 Π isotypic H i (X k, Q l ) H i (X k, Q l ) X, Y k X d Y d 2.2 Chow CH d (X k Y ) Q = CH d (X k Y ) Z Q Y X (algebraic correspondence) f : Y X k Y f id X k Y d k d Z k a: Z X k Y a [Z] [a] i = 1, 2 a i = pr i a a 2 a [a] Q 3.3 [a] Hecke a 2 a Sh U G U U G(A ) g G(A ) Hecke Sh U Sh U gug 1 pr y pr / Sh g 1 Ug g $ Sh U Sh U gug 1 Sh U Sh U [g] 2.4 γ Y X H i (X k, Q l ) pr 1 H i (X k k Y k, Q l ) cl(γ) H i+2d( X k k Y k, Q l (d) ) 32

33 pr 2 H i (Y k, Q l ) γ : H i (X k, Q l ) H i (Y k, Q l ) G k Q l γ f : Y X γ = f Lefschetz a: Z X k Y a 2 [a] = a 2 a i) Y k d γ 1, γ 2 Y X Y Y γ 1 γ 2 = pr 13 (pr 12 γ 1 pr 23 γ 2) Y X γ 1 γ 2 ii) (γ 1 γ 2 ) = γ 2 γ X X γ i γ H i (X k, Q l ) γ (idempotent) X γ (X, γ) H i (X k, γ, Q l ) = Im ( γ : H i (X k, Q l ) H i (X k, Q l ) ) G k l H i (X k, γ, Q l ) γ H i (X k, γ, Q l ) H i (X k, Q l ) γ H i (X k, γ, Q l ) σ G k Tr ( σ; H i (X k, γ, Q l ) ) = Tr ( σ γ ; H i (X k, Q l ) ) = Tr ( γ σ; H i (X k, Q l ) ) 33

34 2.8 k X (motive) 9 k X γ (X, γ) (X, γ) (X, γ ) X X δ γ δ = δ γ 10 (X, γ) H i (X k, γ, Q l ) G k l (X, γ) Galois Galois Galois [Ito2] 2.9 Hecke Galois C Q l H i (X k, Q l ) Q l H i (X k, Q l ) = H i (X k, Q l ) Ql Q l Q l γ (X, γ) Galois H i (X k, γ, Q l ) Q l l Q l 9 Q l (1) Tate Tate [Sch2] 10 X (X, X ) 34

35 3 Galois F G F l S F H i (S F, Q l ) S C S(C) 1.48 H i (S F, Q l ) = H i (S C, Q l ) = H i (S(C), Q) Q Q l dim Ql H i (S F, Q l ) = dim Q H i (S(C), Q) dim Q H i (S(C), Q) de Rham S S(C) S(C) de Rham Lie (g, K) S(C) H i (S(C), Q) Hecke Galois (g, K) G F H i (S F, Q l ) F v F v K H i (S F, Q l ) G K Galois G K H i (S F, Q l ) = H i (S K, Q l ) S K S K Galois S K K X Galois H i (X K, Q l ) v 12 v K O K X 3.1 K X O K X X OK K = X F F S 3.2 i) ([Nag1], [Nag2], [Lüt], [Con]) 11 F K 12 G K Z/2Z G K = Z/2Z H i (X K, Q l ) Hodge 35

36 ii) O K i) X X K v l X G K l O K X κ X κ v l p Hodge G K l H i (X K, Q l ) Weil-Deligne 13 v l 14 l F v l F F v K K O K κ p κ q κ q Frob v 3.3 G κ = Ẑ Frob v Frob Z v G κ G K G κ ; σ σ W K K Weil σ W K σ = Frob n(σ) v n(σ) n: W K Z W + K = {σ W K n(σ) 0} n(φ) = 1 φ W K Frobenius I K = {σ W K n(σ) = 0} G K W K = i Z φi I K I K W K W K I K O K ϖ l (ϖ 1/lm ) m t l : I K Z l (1) σ (σ(ϖ 1/lm )/ϖ 1/lm ) m ϖ l (ϖ 1/lm ) m t l Z l (1) I K l X K d X γ X F S S 13 Fontaine D pst v l l 14 [Mie] 36

37 3.1 Weil-Deligne G K l Weil-Deligne [BH, 7] Ω Weil-Deligne W K Ω (r, V ) N : V V σ W K Nr(σ) = q n(σ) r(σ)n 3.4 N W K N : V V ( 1) m N m : V V ( m) 0 2 Z l (1) = Z l Weil-Deligne Ω C Q l Weil-Deligne Q l Weil-Deligne Weil-Deligne l 3.5 Z l (1) = Z l Weil-Deligne (r, N) ρ(σ) = r(σ) exp ( t l (φ n(σ) σ)n ) (σ W K ) W K l ρ Z l (1) = Z l t l I K Z l (r, N) ρ Frobenius φ Z l (1) = Z l (r, N) ρ W K l Weil-Deligne Grothendieck 3.6 Grothendieck 3.5 (r, N) ρ Q l Weil-Deligne l WD 37

38 (r, N) ρ 15 Grothendieck G K l [ST] 3.7 i) κ p l l W K l W K l W K n l GL n (K) l Langlands Harris- Taylor [HT] Henniart [Hen] ii) Galois G K l l Galois Weil l 3.8 W K l ρ G K l ρ(φ) l 3.9 ρ W K l WD(ρ) = (r, N) ρ r IK N t l : I K Z l (1) = Z l 1 σ 0 I K N = log ρ(σ 0 ) = n=1 ( 1)n (ρ(σ 0 ) 1) n /n Weil-Deligne Frobenius Weil-Deligne 3.10 W F Weil-Deligne (r, N) r r(φ) (r, N) 2 Frobenius Weil-Deligne 15 G K l ρ (quasi-unipotent) σ I K m 1 ρ(σ) m 1 38

39 Weil-Deligne (r, N) Frobenius Weil-Deligne (r ss, N) r(φ) = su = us r(φ) Jordan s u r ss (φ n σ) = s n r(σ) (σ I K ) r ss φ (r ss, N) (r, N) Frobenius (r, N) F -ss Weil-Deligne (r ss, 0) (r, N) ss (r, N) 2 Frobenius Weil-Deligne (r, N), (r, N ) r r 3.11 [SaT1, Lemma 1 (1)] r, r W K σ W + K Tr r(σ) = Tr r (σ) r = r 3.12 r : W K GL(V ) W K i) r(i K ) ii) m 1 r(φ m ): V V W K V x 1,..., x n V r Stab IK (x i ) I K F L I K W L n i=1 Stab I K (x i ) L F Galois H = I K W L W K H I K Ker r r(i K ) φ I K /H I K /H m > 0 φ m I K /H m r(φ m ): V V W K W K I K φ ii) r(φ m ), r (φ m ) W K m 1 r(φ m ) r (φ m ) Q l a 1,..., a k r r(φ m ) r = r 1 r k r i r(φ m ) a i r r (φ m ) r = r 1 r k r i r (φ m ) a i Q i (T ), P i (T ) Q i (T ) = (T a i ) 1 n j=1 (T a j), P i (T ) = Q i (a i ) 1 Q i (T ) P i (r(φ m )) r r i 39

40 P i (r (φ m )) r r i σ W + K ( ( Tr r i (σ) = Tr P i r(φ m ) ) ) ( ( r(σ) = Tr P i r (φ m ) ) ) r (σ) = Tr r i(σ) σ W K φ ml σ W + K l Tr r i (σ) = a l i Tr r i (φ ml σ) = a l i Tr r i (φml σ) = Tr r i (σ) χ i : W K Q l φ a 1/m i χ i r i, χ i r i W K (χ i r i )(φ m ) = id (χ i r i )(W K ) r i (I K ) 3.12 χ i (φ n ) (0 n m 1) χ i r i χ i r i χ i r i = χi r i r i = r i 2 Weil-Deligne (r, N), (r, N ) r = r N Weil-Deligne 3.13 (r, V ) W K k (r, V ) k (strictly pure of weight k) r(φ) q k/2 Frobenius φ W K V {Fil W i } i R (r, V ) (weight filtration) i R gr W i V := Fil W i V/( j<i FilW j V ) i (r, V ) (mixed) gr W i 0 i R (r, V ) (weight) Weil-Deligne (r, V, N) (r, V, N) (r, V ) 3.14 Weil-Deligne Nr(φ) = qr(φ)n (r, N, V ) N Fil W i Fil W i 2 N : grw i V gr W i 2 V 3.15 Fil W = {Fil W i } i Z 40

41 Weil-Deligne 3.16 Weil-Deligne (r, N, V ) (pure) w R (r, N, V ) w + Z i 0 N i : gr W w+i V gr W w i V w (r, N, V ) (weight) 3.17 Langlands n Weil-Deligne GL n (K) (absolutely tempered representation) 16 ([TY, Lemma 1.4 (3)]) 3.18 W K Q l Q l ( 1) Fil W 1 = 0, Fil W 0 = Fil W 1 = Q l, Fil W 2 = Q l Q l ( 1) {0, 2} ( (Q l ) Q l ( 1), 0) Weil-Deligne N = 0 1 (Q l Q l ( 1), N) Weil-Deligne Weil- 0 0 Deligne Langlands GL 2 (K) Steinberg 3.19 i) Weil-Deligne (r, N) (r, N) (r, N) F -ss ii) L K K Weil-Deligne (r, N) L (r WL, N) l (r, N) (r WL, N) Frobenius Weil-Deligne 3.20 [TY, Lemma 1.4 (4)] (r, V, N), (r, V, N ) Frobenius Weil-Deligne (r, V, N) = (r, V, N ) 16 C 41

42 W K (r, V ) m (r(m), V ) r(m)(σ) = q n(σ)m r(σ) (r, V ) (r(m), V ) (r, V ) V i = gr W i V (r, V ) W K V = i V i V = i V i N N Vi : V i V i 2 i 0 N i+1 : V i N N V i 2 P i V i+2 i+1 Vi V i 2 V i = NV i+2 P i V i = i+2j i Z j=0 N j P i+2j = i i Z j=0 N j P i 0 j i N j : P i V i 2j N i j r i = r Pi i,j N j : i Z i j=0 r i(j) i Z i j=0 N j P i W K i Z i j=0 r i(j) r i (j) 0 j < i id Pi : r i (j) r i (j + 1) j = i 0 N P i, r i P i P i V i V i 2 ( i 1) W K V i V i 2 ( i 1) r i = r i i i Z j=0 r i(j) = i i Z j=0 r i (j) (r, V, N) = (r, V, N ) W K l Weil-Deligne l ρ, ρ W K l ρ Frobenius WD(ρ) Frobenius ρ Frobenius ρ F -ss ρ ss WD WD(ρ) F -ss, WD(ρ) ss σ W + K Tr ρ(σ) = Tr ρ (σ) ρ ss = ρ ss 3.11 WD(ρ) = (r, N) ρ(σ) r(σ) Tr ρ(σ) = Tr r(σ) ρ WD(ρ) ρ, ρ Frobenius l ρ ss = ρ ss ρ = ρ Galois 42

43 o o G κ Q l (ρ, V ) w ρ(frob v ) q w/2 3.3 ρ G K l W K w w 3.2 Rψ H i (X K, Q l ) G K O K O ur K O K Y Y κ i / Y O ur K j Y K Spec κ / Spec O ur K Spec K 3.22 [SGA7, Exposé XIII] Y l F RψF = i Rj (F YK ) Y κ l Rψ (nearby cycle functor) RψF i R i ψf RψQ l G K σ G K (σ ) RψQ l RψQ l σ σ G κ Y κ G K RψQ l I K Y X X Y O K RψQ l X K 3.23 G K H i (X κ, RψQ l ) = H i (X K, Q l ) G K E i,j 2 = H i (X κ, R j ψq l ) = H i+j (X K, Q l ) O ur K X O ur K Spec Our K 1.45 H i (X κ, RψQ l ) = H i (X O ur K, Rj Q l ) = H i (X K, Q l ) H i (X K, Q l ) RψQ l X κ 2 43

44 o o 3.3 X Spec O K X X v RψQ l 3.24 X Spec O K RψQ l = Q l R 0 ψq l = Q l R i ψq l = 0 (i 1) RψQ l I K X κ i / X O ur K j X K f f f Spec κ i / Spec O ur K j Spec K f i Rj Q l = i f Rj Q l = i Rj f Q l = RψQ l 3.23 X = Spec O K H m (Spec κ, i Rj Q l ) = H m (Spec K, Q l ) m = 0 Q l I K m 1 0 i Rj Q l = Q l, RψQ l = f Q l = Q l i) 3.25 X Spec O K G K H i (X K, Q l ) = H i (X κ, Q l ) G K G κ H i (X κ, Q l ) G K G κ H i (X K, Q l ) H i (S F, Q l ) 3.26 G F l H i (S F, Q l ) S S S Spec O F S U S S Spec O F U 44

45 Spec O F W Spec O F (Spec O F ) \ W Spec O F W S Spec O F (Spec O F ) \ W 3.25 v / W H i (S F, Q l ) v 3.25 X Spec O K G K H i (X K, Q l ) G κ H i (X κ, Q l ) G κ Frobenius Frob v G κ Frob v H i (X κ, Q l ) 3.27 κ m κ m κ Y 2d i=0 ( 1) i Tr ( Frob m v ; H i (Y κ, Q l ) ) = #Y (κ m ) κ m κ Y κm Y m = 1 ϕ v : Y Y q Frobenius q κ Frob v ϕ v : H i (Y κ, Q l ) H i (Y κ, Q l ) 17 ϕ v Y κ Y (κ) 1 ϕ v 0 Lefschetz 3.28 Spec κ ([SGA4 1 2, Rapport]) κ Y Z(Y, T ) Y (congruence zeta function) κ n κ n ( Z(Y, T ) = exp n=1 #Y (κ n ) T n) n 17 Frob v Frob v Frobenius 45

46 3.30 Y = P 1 κ #P1 (κ n ) = q n + 1 Z(P 1 κ, T ) ( Z(P 1 κ, T ) = exp = n=1 q n T n n + n=1 1 (1 T )(1 qt ) T n ) = exp ( log(1 qt ) log(1 T ) ) n 3.31 Y κ P i (Y, T ) = det(1 Frob v T ; H i (Y κ, Q l )) Z(Y, T ) = 2 dim Y i=0 P i (Y, T ) ( 1)i Frob v H i Weil 3.33 Weil Deligne [Del2], [Del3] Y κ Frob v H i (Y κ, Q l ) α Q l α Z ι: Q l = C ι(α) = q i/2 G κ l H i (Y κ, Q l ) i Frob v H i (Y κ, Q l ) P i (Y, T ) P i (Y, T ) β Q l Q ι: Q l = C ι(β) = q i/ κ Y i, j P i (Y, T ) P j (Y, T ) P i (Y, T ) P j (Y, T ) 46

47 2 dim Y i=0 P i (Y, T ) ( 1)i+1 P i (Y, T ) Tr(Frob m v ; H i (Y κ, Q l )) i) n #Y (κ n ) ii) i) Z(Y, T ) iii) Z(Y, T ) q i/2 β 1,..., β k P i (Y, T ) = k j=1 (1 β 1 j T ), Tr(Frob m v ; H i (Y κ, Q l )) = β1 m + + β m k X O K n X κ n Tr(Frob m v ; H i (X K, Q l )) S v Tr(Frob m v ; H i (S F, Q l )) n S κ n 3.35 F 3 Y 0 : y 2 = x 5 +1 F 3 Y Y 0 A 1 F 3 ; (x, y) x f : Y P 1 F 3 P 1 F Riemann-Hurwitz f 6 Y 2 f P 1 F 3 1 F 3 H 1 (Y F3, Q l ) 4 P 1 (Y, T ) x x F 3 F 3, F 9 F 9 #Y (F 3 ) = 4, #Y (F 9 ) = 10 H 0 (Y F3, Q l ), H 2 (Y F3, Q l ) Frob 3 1, 3 Tr(Frob 3 ; H 1 (Y F3, Q l )) = Tr(Frob 2 3; H 1 (Y F3, Q l )) = 0 Frob 3 H 1 (Y F3, Q l ) a, b, c, d a + b + c + d = a 2 + b 2 + c 2 + d 2 = 0 H 1 (Y F3, Q l ), Poincaré Frob 3 (x), Frob 3 (y) = 3 x, y det Frob 3 = 9 abcd = 9 1 Poincaré {a, b, c, d} = {3/a, 3/b, 3/c, 3/d} 1/a + 1/b + 1/c + 1/d = 0 a, b, c, d T 4 +9 = 0 4 P 1 (Y, T ) = 1+9T 4 Weil Tr ( Frob m 3 ; H 1 (Y F3, Q l ) ) 0 (4 m) = 4( 9) m/4 (4 m) 47

48 P i (Y, T ), Tr(Frob m v ; H i (Y κ, Q l )) 3.36 Y κ P i (Y, T ) l Tr(Frob m v ; H i (Y κ, Q l )) l P i (Y, T ) {β 1,..., β k } l P i (Y, T ) l β 1,..., β k Q σ G Q σ(β j ) {β 1,..., β k } 3.33 ι P i (Y, T ) = k j=1 (1 β 1 j T ) Q[T ] G Q Q 3.33 β1 1,..., β 1 k P i (Y, T ) P i (Y, T ) Z[T ] 3.37 {H i (S F, Q l )} l p 18 F Σ v / Σ v l l H i (S F, Q l ) GK m Z Tr(Frob m v ; H i (S F, Q l )) l γ X H i (X K, γ, Q l ) H i (X K, Q l ) 3.38 H i (X K, γ, Q l ) G K Γ S G F l H i (S F, Γ, Q l ) CH d (X OK X) CH d (X K X) γ X OK X γ γ γ X κ κ X κ γ ([Ful, 20.3]) γ CH d (X κ κ X κ ) H i (X κ, Q l ) = H i (X K, Q l ) γ / H i (X κ, Q l ) = γ / H i (X K, Q l ) 17 [Tay] [Tay] v l p 48

49 γ γ G K H i (X K, γ, Q l ) = H i (X κ, γ, Q l ) σ W K H i (X K, γ, Q l ) Frob n(σ) v H i (X κ, γ, Q l ) i ( 1)i Tr(Frob m v ; H i (X κ, γ, Q l )) γ i ( 1)i Tr(γ Frob m v ; H i (X κ, Q l )) X κ γ (m) γ (m) = (ϕ m v id) γ ϕ v q Frobenius 2d i=0 ( 1) i Tr ( Frob m v ; H i (X κ, γ, Q l ) ) = deg ( [γ (m) ] X ) i ( 1)i Tr(γ Frob m v ; H i (X κ, Q l )) γ γ a: Z X κ κ X κ Z z ϕ m v (a 1 (z)) = a 2 (z) det(1 Frob v T ; H i (X κ, γ, Q l )) l Tr(Frob m v ; H i (X κ, γ, Q l )) l Γ S {H i (S F, Γ, Q l )} l m 0 Tr(Frob m v ; H i (X κ, γ, Q l )) l 3.34 Q(T ) = c 0 + c 1 T + + c n T n Q[T ] Q(T ) 1 (mod P i (X κ, T )), Q(T ) 0 (mod P j (X κ, T )) (j i) Q(Frob v ) H i (X κ, Q l ) id H j (X κ, Q l ) (j i) 0 Tr ( Frob m v ; H i (X κ, γ, Q l ) ) = Tr ( γ Frob m v ; H i (X κ, Q l ) ) = j ( 1) j Tr ( γ Frob m v Q(Frob v ); H j (X κ, Q l ) ) = = n c l ( 1) j Tr ( γ Frob m+l v ; H j (X κ, Q l ) ) l=0 l=0 j n c l ( 1) j Tr ( Frobv m+l ; H j (X κ, γ, Q l ) ) j 49

50 3.39 l γ c 1 det(1 Frob v T ; H i (X κ, γ, Q l )) c γ det(1 Frob v T ; H i (X κ, γ, Q l )) Z[T ] [Kle] 3.41 k Y i 0 H i (Y k, Q l ) id H j (Y k, Q l ) (j i) 0 i Künneth (Künneth projector) k i Künneth Q(ϕ v ) k Künneth Y a 2 a: Z X κ κ X κ Z κ d γ 3.42 [Fuj], [Var] Z κ d a: Z X κ κ X κ κ a 2 = pr 2 a m 1 2d i=0 ( 1) i Tr ( Frob m v ; H i (X κ, [a], Q l ) ) = # { z Z(κ) ϕ m v (a 1 (z)) = a 2 (z) } 3.43 m 1 i ( 1)i Tr(Frob m v ; H i (X κ, [a], Q l )) m i ( 1)i Tr(Frob m v ; H i (X κ, [a], Q l )) 3.44 X κ κ m [HT] [Laf] 50

51 3.45 F 7 3 C : X 3 + Y 3 = Z 3 1 H 1 (C F7, Q l ) 2 C Z/3Z a[x : Y : Z] = [2 a X : Y : Z] H i (C F7, Q l ) Z/3Z a Z/3Z Tr(a Frob 7 ; H 1 (C F7, Q l )) 3.42 [(2 a X) 7 : Y 7 : Z 7 ] = [X : Y : Z] [X : Y : Z] P 2 (F 7 ) 2 i=0 ( 1)i Tr(a Frob 7 ; H i (C F7, Q l )) a = 0 9, a = 1 12 a = 1 3 a H 0 (C F7, Q l ), H 2 (C F7, Q l ) Tr(a Frob 7 ; H 0 (C F7, Q l )) = 1, Tr(a Frob 7 ; H 2 (C F7, Q l )) = 7 Tr(a Frob 7 ; H 1 (C F7, Q l )) a = 0 1 a = 1 4 a = 1 5 Q l 1 3 ω l 1 3 χ: Z/3Z Q l χ(a) = ωa i = 0, 1, 2 γ i = (1/3) a Z/3Z χ(a) i [a] H 1 (C F7, Q l ) H 1 (C F7, Q l ) = 2 i=0 H1 (C F7, γ i, Q l ) H 1 (C F7, γ i, Q l ) H 1 (C F7, Q l ) Z/3Z χ i Tr ( Frob 7 ; H 1 (C F7, γ 1, Q l ) ) = 1 + 3ω, Tr ( Frob 7 ; H 1 (C F7, γ 2, Q l ) ) = 1 + 3ω 2 H 1 (C F7, γ i, Q l ) 0 (i = 1, 2) H 1 (C F7, γ 0, Q l ) = 0, dim Ql H 1 (C F7, γ 1, Q l ) = dim Ql H 1 (C F7, γ 1, Q l ) = 1 Frob 7 H 1 (C F7, γ 1, Q l ), H 1 (C F7, γ 2, Q l ) 1 + 3ω 1 + 3ω Weil Galois Ramanujan Galois Weil Weil Ramanujan (q) = q n=1 (1 qn ) 24 = n=1 τ(n)qn G Q l ρ p l det(1 Frob p T ; ρ ) = 1 τ(p)t + p 11 T 2 Q Hecke ([Del1], [Sch1]) Weil 1 τ(p)t + p 11 T 2 p 11/2 τ(p) τ(p) 2p 11/2 51

52 3.4 X Spec O K 3.47 O K Y (semistable) y Y 0 r n O K ϖ y 19 Y Y Y Spec O K [T 1,..., T n ]/(T 1 T r ϖ) Y κ κ Y (strictly semistable) 3.48 i) O K O K Y κ ii) Y O K y Y 0 r n O K ϖ y Zariski Y Y Y Spec O K [T 1,..., T n ]/(T 1 T r ϖ) X X X Galois 1.5 ii) 3.49 X H i (X K, Q l ) G K V G K l W V G K V W, V/W I K R i ψq l X N 1 σ I K x X κ (σ 1) N (R i ψq l ) x 0 (R i ψq l ) x x 0 r n 18 y 52

53 N > 0 Spec O K [T 1,..., T n ]/(T 1 T r ϖ) O K R i ψq l (σ 1) N 0 Spec O K [T 1,..., T n ]/(T 1 T r ϖ) Spec O K [T 1,..., T r ]/(T 1 T r ϖ) R i ψ R i ψ n = r n = r = 2 P 1 O K P 1 κ O K Y Y y Y κ O K y Y Spec O K [T 1, T 2 ]/(T 1 T 2 ϖ) (R i ψ Y Q l ) y Y Rψ Rψ Y I K U = Y κ \ {y} I K H i (Y κ, RψQ l ) (R i ψq l ) y H i+1 c (U κ, RψQ l ) Y P 1 K Hi (Y κ, RψQ l ) = H i (Y K, Q l ) I K 3.25 U O K 3.24 Hc i+1 (U κ, RψQ l ) = Hc i+1 (U κ, Q l ) 20 I K (σ 1) 2 (σ I K ) (R i ψq l ) y 0 n = r Spec O K [T 1,..., T n ]/(T 1 T n 1 ϖ) (T 1, T n ) ϖ Spec O K [T 1,..., T n ]/(T 1 T n ϖ) O K Y j > 2 dim Y κ i > dim Y K H j (Y κ, R i ψq l ) = [RZ] 3.51 X R i ψq l I K R i ψq l P K K RψQ l I K P K X O K 53

54 X 3.49 I K H i (X K, Q l ) H i (X K, Q l ) Frobenius 2 σ W + K Tr(σ; Hi (X K, Q l )) N = n=1 ( 1)n 1 (σ 0 1) n /n σ 0 t l (σ 0 ) Z l (1) I K 3.9 X κ D 1,..., D m {1,..., m} I D I = i I D i j D (j) = #I=j+1 D I 3.52 [RZ], [SaT2] X G K E s,t 1 = i max{0, s} H t 2i( D (s+2i) κ, Q l ( i) ) = H s+t (X K, Q l ) (weight spectral sequence) σ 0 1 N E s,t 1 = i max{0, s} Ht 2i( D (s+2i) κ, Q l ( i) ) id t l (σ 0 ) H s+t (X K, Q l ) σ 0 1 E s+2,t 2 1 = i max{1, s 1} Ht 2i( D (s+2i) κ, Q l ( i + 1) ) H s+t (X K, Q l ) N X d D (s+2i) d s 2i E s,t t 2i 2(d s 2i) i max{0, s} E s,t 1 0 (s, t) 0 2s + t 2d 0 t 2d d = 2 E 1 Q l H 0 (D (2) Gys )( 2) H 2 (D (1) Gys )( 1) H 4 (D (0) κ κ κ ) H 1 (D (1) Gys )( 1) H 3 (D (0) κ H 0 (D (1) Gys )( 1) κ κ ) H2 (0) (Dκ ) Res Res H 0 (D (2) κ )( 1) H 2 (D (1) Gys κ ) H 1 (D (0) κ ) Res H 1 (D (1) H 0 (D (0) κ ) Res κ ) H 0 (D (1) κ ) Res H 0 (D (2) κ ) s = 0 t = 0 Res ±1 Gys Gysin 54

55 Poincaré ± i) H i (X K, Q l ) Fil W Hi (X K, Q l ) H i (X K, Q l ) l ii) E 2 Weil G κ E s,t 1 t Tate 2 Fil W t / Fil W t 1 Ei t,t 1 G K t Fil W i) ii) d 2 : E s,t 2 E s+2,t E s,t 2 E s+2,t 1 2 G κ 3.54 i) σ W + K 2d i=0 ( 1)i Tr(σ; H i (X K, Q l )) l ii) σ I K (σ 1) d+1 H i (X K, Q l ) 0 X i) G K 2d i=0 ( 1) i Tr ( σ; H i (X K, Q l ) ) = s,t = s i max{0, s} ( 1) s+t Tr ( σ; H t 2i (D (s+2i) κ, Q l ( i)) ) i max{0, s}( 1) s q n(σ)i t ( 1) t 2i Tr ( Frob n(σ) v ; H t 2i (D (s+2i) κ, Q l ) ) 3.36 l ii) 3.49 I K t l σ = σ σ 0 1 Fil W i Fil W i 2 FilW 1 = 0, Fil W 2d = Hi (X K, Q l ) (σ 0 1) d+1 (H i (X K, Q l )) Fil W 2 = 0 55

56 3.55 E Weierstrass y 2 = x 3 + x Q 5 1 H 1 (E Q5, Q l ) W Q5 P 2 Z 5 3 E : Y 2 Z = X 3 + X 2 Z + 25Z 3 E E (x, y, 5) Z 5 y 2 = x 3 + x Z 5 Ẽ E ẼF 5 2 D 1, D 2 P 1 F 5 D 1 D 2 2 F 5 Ẽ E 1 E 2, 1 1 E 2,0 1 E 1,0 1 E 0,0 1 E 0,1 1 5 E 2, 1 1 = H 0 (D 1,F5 D 2,F5 )( 1) = Q l ( 1) 2, E 2,0 1 = H 2 (D 1,F5 ) H 2 (D 2,F5 ) = Q l ( 1) 2, E 1,0 1 = 0, E 0,0 1 = H 0 (D 1,F5 ) H 0 (D 2,F5 ) = Q 2 l, E 0,1 1 = H 0 (D 1,F5 D 2,F5 ) = Q 2 l σ W Q5 E 2, 1 1 E 2,0 1 5 n(σ) E 0,0 1 E 0, i=0 ( 1)i Tr(σ; H i (E Q5, Q l )) = 0 H 0 (E Q5, Q l ) = Q l, H 2 (E Q5, Q l ) = Q l ( 1) Tr(σ; H 1 (E Q5, Q l )) = n(σ) det(σ; H 1 (E Q5, Q l )) = ((1 + 5 n(σ) ) 2 ( n(σ) ))/2 = 5 n(σ) σ H 1 (E Q5, Q l ) 1 5 n(σ) H 1 (E Q5, Q l ) Frobenius WD(H 1 (E Q5, Q l )) ss = Q l Q l ( 1) N E 2 pt P 1 F 5 Gysin H 0 (pt) H 2 (P 1 F 5 ) d 1 : E 2, 1 1 E 2,0 1 (a, b) ( a b, a + b) (a, b Q l ( 1)) d 1 : E 0,0 1 E 0,1 1 (a, b) ( a + b, a + b) (a, b Q l ) 3.53 gr W 2 H 1 (E Q5, Q l ) = E 2, 1 2 = {(a, a) a Q l ( 1)} = Q l ( 1), gr W 0 H 1 (E Q5, Q l ) = E 0,1 2 = Q 2 l /{(b, b) b Q l} = Q l, gr W i H 1 (E Q5, Q l ) = 0 (i 0, 2) 56

57 3.52 N : gr W 2 grw 0 (a, a) (t l (σ 0 )a, t l (σ 0 )a) N : gr W 2 gr W 0 W Q5 l H 1 (E Q5, Q l ) Weil-Deligne WD(H 1 (E Q5, Q l )) = ( Q l Q l ( 1), ( ) ) N 3.6 N : gr W 2 gr W [RZ] ([SaT2]) σ 0 I K t l (σ 0 ) Z l (1) RψQ l σ 0 1 RψQ l R i ψq l σ 0 RψQ l (perverse sheaf) Riemann-Hilbert A N A M ([SaT2, Lemma 2.3]) M i = 0 (i 0), M i = A (i 0) N(M i ) M i 2 i > 0 N i : gr M i A gr M i A RψQ l σ 0 1 γ X γ [SaT2, 2.3, 2.4] γ E 1 57

58 3.56 σ W + K 2d i=0 ( 1)i Tr(γ σ; H i (X K, Q l )) l γ 2d i=0 ( 1)i Tr(σ; H i (X K, γ, Q l )) l Künneth i) 3.57 i 0 X i Künneth σ W + K Tr(γ σ; H i (X K, Q l )) l γ Tr(σ; H i (X K, γ, Q l )) l Γ i i Künneth ( 1) i Tr ( γ σ; H i (X K, Q l ) ) = 2d j=0 ( 1) j Tr ( Γ i γ σ; H j (X K, Q l ) ) γ Γ i Künneth i i 0 H i (X K, γ, Q l ) = 0 Künneth 3.56 H i (X K, Q l ) [Yos] 3.5 X K L X L O L 3.58 X K K L O L Y Y L = XL Y 58

59 L Y X 1 ([DM]) K p 0 C((T )) ([KKMSD]) G L H i (X K, Q l ) = H i (Y L, Q l ) H i (X K, Q l ) G K G L G K Y Y κ S S X = S K L Y L Y S OK L K O L [TY, 3] Harris-Taylor L K Galois L K 0 K Galois L Y OL O L [SaT2, Lemma 1.11] 3.60 de Jong [dj] X K K L O L Y f : Y L X U X f U : f 1 (U) U Y O L f U L K Galois

60 3.62 H i (X K, Q l ) f H i (Y OK K, Q l ) f H i (X K, Q l ) deg f deg f f U Y = Y OK K = Y OL (O L OK K) = Y OL L K H i (X K, Q l ) f H i (Y K, Q l ) f H i (X K, Q l ) deg f ξ H i (X K, Q l ) f (f ξ) = f (f ξ 1) = f ( f ξ cl([y ]) ) = ξ f ( cl([y ]) ) = ξ cl(f ([Y ])) = deg f (ξ cl([x]) ) = deg f ξ L K Galois τ Gal(L/K) τ : O L O L Y Y τ Y OK K = Y OL (O L OK K) = τ Gal(L/K) Yτ L Hi (X K, Q l ) τ Gal(L/K) Hi (Y τ L, Q l) (deg f) 1 f f 3.49 G L H i (X K, Q l ) H i (X K, Q l ) Grothendieck 3.6 Grothendieck 3.53 i) 3.63 H i (X K, Q l ) l H i (X K, Q l ) G L 3.64 τ Gal(L/K) Y τ L X L Y τ L Yτ L L Y τ L Γ τ σ G L Tr ( σ; H i (X K, Q l ) ) = (deg f) 1 τ Gal(L/K) Tr ( Γ τ σ; H i (Y τ L, Q l) ) Y τ L Y O K L f id L X L f τ H i (Y OK K, Q l ) = τ Gal(L/K) Hi (Y τ L, Q l) 3.62 f τ f τ f τ f τ Tr ( σ; H i (X K, Q l ) ) = Tr ( (deg f) 1 f f σ; H i (Y OK K, Q l ) ) 60

61 2 Y τ L = (deg f) 1 τ Gal(L/K) Tr ( f τ f τ σ; H i (Y τ L, Q l) ) id f τ Y τ L L X L Y τ f τ id L X L L Y τ L Γ τ Γ τ = f τ f τ G L H i (X K, Q l ) τ Gal(L/K) O L Y τ H i (Y τ L, Q l) G L G L G K Y 3.58 σ W + K Y σ : O L O L Y σ Y σ L X L XL X L L X L Γ Y σ L L Y L Γ Y σ OL Y X L Galois σ : X L X L Y σ Y σ Γ Y σ OL Y σ Γ E s,t 1 σ H s+t (Y L, Q l ) σ H s+t (X K, Q l ) σ Γ E σ,s,t 1 E s,t 1 H s+t (Y σ L, Q l) H s+t (X K, Q l ) Γ id H s+t (Y L, Q l ) H s+t (X K, Q l ) E σ,s,t 1 Y σ E 1 E 1 σ κ σ Frobenius n(σ)[κ L : F p ] κ L L κ σ geom σ = σ geom H i (X K, Q l ) σ E 1 X γ 3.65 [SaT2] γ X σ W + K 61

62 2d i=0 ( 1)i Tr(γ σ; H i (X K, Q l )) l γ 2d i=0 ( 1)i Tr(σ; H i (X K, γ, Q l )) l 3.66 Γ S S i Künneth {H i (S F, Γ, Q l )} l p F v WD(H i (S F, Γ, Q l ) WK ) ss l p l 3.67 E Weierstrass y 2 = x 3 + x Q 5 1 H 1 (E Q5, Q l ) W Q5 P 2 Z 5 3 E : Y 2 Z = X 3 + X 2 Z + 5Z 3 E K = Q 5 ( 5) E OK (x, y, 5) O K Y 3.55 Y E K Y Y F5 2 D 1, D 2 P 1 F 5 D 1 D 2 2 F 5 σ W Q5 Y Z 5 E : y 2 = x 3 + x E O K E O K (x, y, 5) x U Y y = wx, 5 = tx y 2 = x 3 + x x 2 w 2 = x t 2 U = Spec O K [x, w, t]/(tx 5, x+1+t 2 w 2 ) = Spec O K [w, t]/ ( t(w 2 t 2 1) 5 ) U E O K (x, y) (w 2 t 2 1, w(w 2 t 2 1)) σ W + Q 5 \ W + K Uσ = Spec O K [w, t]/(t(w 2 t 2 1) + 5) O K f : U U σ (w, t) (w, t) U / E O K Y / E OK f id U σ E / O K σ σ U / E O K f id Y σ / E OK σ σ Y / E OK 62

63 61 f E 1 f = f mod 5 21 mod 5 U F5 = Spec F 5 [w, t]/(t(w 2 t 2 1)) F 5 absfrob n(σ) σ f (w, t) (w 5n(σ), t 5n(σ) ) 22 absfrob U F5 Frobenius 5 σ E 1 f (ϕ 5 )n(σ) f E 1 E 1 (ϕ 5 )n(σ) = Frob n(σ) 5 σ W + K E 1 W Q W Q5 l H 1 (E Q5, Q l ) Q 5 E : y 2 = x 3 + 2x H 1 (E Q5, Q l ) W Q E Ẽ Q 25 Q 5 2 Z 25 E Frob 5 E Gabber 3.60 p l 3.60 L, Y, f deg f l Z l Z/l n Z H i (X K, γ, Q l ) ss X κ H i (X K, γ, Q l ) 3.70 H i (X K, Q l ) l i H i (X K, Q l ) Fil W 3.63 j 1 N j : gr W i+j Hi (X K, Q l ) gr W i j Hi (X K, Q l ) 21 Y = Y σ Y σ (x, y, 5) Y = Y σ 2 Y = Y σ Y 22 Y F5 = Y σ F 5, σ = Frob n(σ) 5 σ geom = ϕ n(σ) 5 63

64 H i (X K, γ, Q l ) H i (X K, Q l ) H i (X K, γ, Q l ) ss H i (X K, γ, Q l ) F -ss Γ S S i Künneth 3.70 {H i (S F, Γ, Q l )} l p F v WD(H i (S F, Γ, Q l ) WK ) F -ss l p l 3.70 Langlands [Car] [SaT3] [TY] [Car] [SaT3] [Mie] ii) X O K i = ([SGA7, Exposé I]) dim X ([RZ]) K p [Ste] [SaM1] p > 0 X Deligne Néron Deligne ([Ito1]) 3.70 ([SaM1], [SaM2]) X 3.70 X κ dim X = 2, i = 2 54 E 2 (a) N : gr W 3 gr W 1, (b) N 2 : gr W 4 gr W 0 Tate (a) Ker ( H 1 (D (1) κ ) Gys H 3 (D (0) κ )) Coker ( H 1 (D (0) κ (b) Ker ( H 0 (D (2) κ ) Gys H 2 (D (1) κ )) Coker ( H 0 (D (1) κ ) Res H 1 (D (1) κ )), ) Res H 0 (D (2) )) H 1 (D (1) κ ) Gys H 3 (D (0) κ ) H1 (D (0) κ ) Res H 1 (D (1) κ ) κ 64

65 (a) H 1 (D (1) κ ) H1 (D (1) κ ) Q l H 1 (D (0) κ ) Res H 1 (D (1) κ ) Im Res X O K Res i Pic0 (D i ) i<j Pic0 (D i D j ) Tate A Im Res = V l A H 1 (D (1) κ ) = V l ( i<j Pic0 (D i D j )) i<j Pic0 (D i D j ) L Weil V l A L A Weil L A Im Res (b) H 0 (D (2) κ ) H0 (D (2) κ ) Q l H 0 (D (0) κ ) Res H 0 (D (2) κ ) Im Res Q l Q V, W D (2), D (0) Q r : W V Φ: V V Q V, W, r, Φ H 0 (D (2) κ ), H0 (D (0) κ ), Res, Q Φ Φ r Im Res 3.72 dim X = 1 [TY], [Boy], [Dat] [Dat] Drinfeld Drinfeld 65

66 [BBD] [BH] [Boy] [Car] [Con] [CS] [Dat] [Del1] [Del2] [Del3] [dj] [DM] [DS] [Eke] A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981), Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp C. J. Bushnell and G. Henniart, The local Langlands conjecture for GL(2), Grundlehren der Mathematischen Wissenschaften, vol. 335, Springer-Verlag, Berlin, P. Boyer, Monodromie du faisceau pervers des cycles évanescents de quelques variétés de Shimura simples, Invent. Math. 177 (2009), no. 2, H. Carayol, Sur les représentations l-adiques associées aux formes modulaires de Hilbert, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 3, B. Conrad, Deligne s notes on Nagata compactifications, J. Ramanujan Math. Soc. 22 (2007), no. 3, G. Cornell and J. H. Silverman (eds.), Arithmetic geometry, Springer- Verlag, New York, 1986, Papers from the conference held at the University of Connecticut, Storrs, Connecticut, July 30 August 10, J.-F. Dat, Théorie de Lubin-Tate non-abélienne et représentations elliptiques, Invent. Math. 169 (2007), no. 1, P. Deligne, Formes modulaires et representations l-adiques, Séminaire Bourbaki, 21ème année (1968/69), Exp. No. 355, 1969., La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. (1974), no. 43, , La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. (1980), no. 52, A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. (1996), no. 83, P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. (1969), no. 36, P. Deligne and J.-P. Serre, Formes modulaires de poids 1, Ann. Sci. École Norm. Sup. (4) 7 (1974), (1975). T. Ekedahl, On the adic formalism, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp

67 [Fuj] [Ful] [Hen] [HT] [Ito1] [Ito2] K. Fujiwara, Rigid geometry, Lefschetz-Verdier trace formula and Deligne s conjecture, Invent. Math. 127 (1997), no. 3, W. Fulton, Intersection theory, second ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 2, Springer-Verlag, Berlin, G. Henniart, Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique, Invent. Math. 139 (2000), no. 2, M. Harris and R. Taylor, The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, vol. 151, Princeton University Press, Princeton, NJ, 2001, With an appendix by Vladimir G. Berkovich. T. Ito, Weight-monodromy conjecture over equal characteristic local fields, Amer. J. Math. 127 (2005), no. 3, (2006) 2008 [Ive] B. Iversen, Cohomology of sheaves, Universitext, Springer-Verlag, Berlin, [Jan] U. Jannsen, Continuous étale cohomology, Math. Ann. 280 (1988), no. 2, [KKMSD] G. Kempf, F. F. Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings. I, Lecture Notes in Mathematics, Vol. 339, Springer- Verlag, Berlin, [Kle] S. L. Kleiman, Algebraic cycles and the Weil conjectures, Dix exposés sur la cohomologie des schémas, North-Holland, Amsterdam, 1968, pp [KS] M. Kashiwara and P. Schapira, Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften, vol. 292, Springer-Verlag, Berlin, 1994, With a chapter in French by Christian Houzel. [KW] R. Kiehl and R. Weissauer, Weil conjectures, perverse sheaves and l adic Fourier transform, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 42, Springer-Verlag, Berlin, [Laf] L. Lafforgue, Chtoucas de Drinfeld et correspondance de Langlands, Invent. Math. 147 (2002), no. 1, [Lüt] W. Lütkebohmert, On compactification of schemes, Manuscripta Math. 80 (1993), no. 1,

68 [Mie] R = T Tate Serre 2009 [Mum] D. Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Published for the Tata Institute of Fundamental Research, Bombay, [Nag1] M. Nagata, Imbedding of an abstract variety in a complete variety, J. Math. Kyoto Univ. 2 (1962), [Nag2], A generalization of the imbedding problem of an abstract variety in a complete variety, J. Math. Kyoto Univ. 3 (1963), [RZ] M. Rapoport and Th. Zink, Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik, Invent. Math. 68 (1982), no. 1, [SaM1] M. Saito, Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24 (1988), no. 6, (1989). [SaM2] M. Saito, Monodromy filtration and positivity, preprint, [SaT1] T. Saito, Modular forms and p-adic Hodge theory, Invent. Math. 129 (1997), no. 3, [SaT2] T. Saito, Weight spectral sequences and independence of l, J. Inst. Math. Jussieu 2 (2003), no. 4, [SaT3], Hilbert modular forms and p-adic Hodge theory, to appear in Compositio Mathematica. [Sch1] A. J. Scholl, Motives for modular forms, Invent. Math. 100 (1990), no. 2, [Sch2], Classical motives, Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp [Ser1] J.-P. Serre, Propriétés galoisiennes des points d ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, [Ser2], Cohomologie galoisienne, Lecture Notes in Mathematics, vol. 5, Springer-Verlag, Berlin, [Sil] J. H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, [ST] J.-P. Serre and J. Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), [Ste] J. Steenbrink, Limits of Hodge structures, Invent. Math. 31 (1975/76), no. 3, [Tay] R. Taylor, Galois representations, Ann. Fac. Sci. Toulouse Math. (6) 68

69 [TY] [Var] [Yos] [EGA4] [SGA4] [SGA4 1 2 ] [SGA5] [SGA7] 13 (2004), no. 1, R. Taylor and T. Yoshida, Compatibility of local and global Langlands correspondences, J. Amer. Math. Soc. 20 (2007), no. 2, Y. Varshavsky, Lefschetz-Verdier trace formula and a generalization of a theorem of Fujiwara, Geom. Funct. Anal. 17 (2007), no. 1, T. Yoshida, Weight spectral sequence and Hecke correspondence on Shimura varieties, Ph.D. thesis, Harvard University, A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas., Inst. Hautes Études Sci. Publ. Math. no. 20, 24, 28, 32. Théorie des topos et cohomologie étale des schémas, Lecture Notes in Mathematics, Vol. 269, 270, 305. P. Deligne, Cohomologie étale, Lecture Notes in Mathematics, Vol. 569, Springer-Verlag, Berlin. Cohomologie l-adique et fonctions L, Lecture Notes in Mathematics, Vol. 589, Springer-Verlag, Berlin. Groupes de monodromie en géométrie algébrique, Lecture Notes in Mathematics, Vol. 288, 340, Springer-Verlag, Berlin. 69