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

l 0 2 1 4 1.1 Tate.......................... 4 1.2........................ 6 1.3...................... 9 1.4.................... 21 2 Galois 31 2.1 Galois.......... 31 2.2.................... 31 3 Galois 35 3.1 Weil-Deligne............................ 37 3.2 Rψ............................ 43 3.3............................. 44 3.4............................ 52 3.5............................ 58 3.6...................... 63 1

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

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

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

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

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 1 1.2 2 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) 0 1 1 2 0 2 [KS] [Ive] 3 6

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

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

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 1.3 1.3.1 1.9 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

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 1 1.12 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

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

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

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

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

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

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 ) = 0 1.22 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)) 0 5 16

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

1.26 A 1 k Z/nZ k 1.27 1.24 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 1.3.3 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

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

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 1 1 0 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

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] 1.4 6 Galois [SGA4], [SGA4 1 2 ] 1.4.1 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

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) 1.4.2 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

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

1.42 j : η X 1.22 R i j G m = 0 (i 1) 1.4.3 n n + 1 0 0 (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] 1.4.4 1.44 [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

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 7 1.48 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

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 1.39 1.4.5 1.49 [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 1.4.6 1.50 [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

1.51 X C H 1 Jacobi 1.4.7 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

1.55 f X, Y, f X k d Y k d f 1.4.8 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 1.4.9 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

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

= 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 3.3 30

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

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 1 2.3 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

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 2.8 2.5 a: Z X k Y a 2 [a] = a 2 a 1 2.6 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 1 2.7 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

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

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 11 1.48 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

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

3.1 Weil-Deligne G K l Weil-Deligne [BH, 7] 3.3 0 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

(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

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 3.11 3.12 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

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

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

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 3.20 3.21 Galois 42

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

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 3.23 1.5 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

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]) 3.27 3.29 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

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+1 3.32 3.27 3.27 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/2 3.34 κ Y i, j P i (Y, T ) P j (Y, T ) P i (Y, T ) P j (Y, T ) 46

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 3 5 6 2 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 5 + 1 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

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

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 )) 3.27 3.39 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) 3.36 3.37 3.40 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

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 2 3.39 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

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 2 3.46 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 11 11 Hecke ([Del1], [Sch1]) Weil 1 τ(p)t + p 11 T 2 p 11/2 τ(p) τ(p) 2p 11/2 51

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 3.23 3.49 I K R i ψq l 3.49 3.50 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

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 ) = 0 3.49 [RZ] 3.51 X R i ψq l I K R i ψq l P K K RψQ l I K P K 20 3.24 X O K 53

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 1 0 0 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

Poincaré ±1 3.53 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 1 2 0 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 0 3.52 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

3.55 E Weierstrass y 2 = x 3 + x 2 + 25 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 2 + 25 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 0,1 1 1 2 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 )) = 1 + 5 n(σ) det(σ; H 1 (E Q5, Q l )) = ((1 + 5 n(σ) ) 2 (1 + 5 2n(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

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), ( 0 1 0 0 ) ) N 3.6 N : gr W 2 gr W 0 3.6 [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

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 3.41 3.54 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 2.6 3.56 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

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 3.59 3.58 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 3.61 3.59 L K Galois 3.60 59

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

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

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 2 + 5 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 2 + 5 E O K E O K (x, y, 5) x U Y y = wx, 5 = tx y 2 = x 3 + x 2 + 5 x 2 w 2 = x + 1 + 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

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 Q5 3.55 W Q5 l H 1 (E Q5, Q l ) 3.55 3.68 Q 5 E : y 2 = x 3 + 2x 2 + 25 H 1 (E Q5, Q l ) W Q5 3.55 E Ẽ Q 25 Q 5 2 Z 25 E 1 3.55 Frob 5 E 1 3.69 Gabber 3.60 p l 3.60 L, Y, f deg f l Z l Z/l n Z 3.60 3.6 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

H i (X K, γ, Q l ) H i (X K, Q l ) 3.70 3.20 H i (X K, γ, Q l ) ss H i (X K, γ, Q l ) F -ss 3.65 3.71 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] 3.70 3.19 ii) X O K i = 1 3.70 ([SGA7, Exposé I]) dim X 2 3.70 ([RZ]) K p 3.70 0 [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

(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

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