2, 3, -, - (X H í et (X, Q )), l- ( ), - l( ),, - l-, (, l Hom(Z l, Z ) = 0, Hom(Z, Z ) Z )., Tate Q (i) Q l (i), l Q l (i) G (I ), Q (i) (i 0 ) χ i :

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- ( ) Contents 1. 1 2. Fontaine -. 2 3.. 14 4. C - (Tate-Sen ):- 16 References 27 1., 2009 l- - I, II,, 2 Fontaine - -, 3 -,, 4 (- )Tate-Sen, -, Q, G := Gal(/) O, π O, k := O /π O v : Q { } v () = 1 -. 0 Q, ur I := Gal(/ ur ) - χ : G Z (, 1 n ζ n g G g(ζ n) = ζ χ(g) ). C n := - ( -, ).,, G - Deition 1.1. V (G )-, V Q - G Q -, (, G rull, V Q -). G - Re G., - - l- (l ) X (H í et (X, Q )), ([Mi]), l- (l ), X X k H í et (X, Q l) l-, X H í et (X, Q l), { } { } { { } } { { } } = { { l- l- ( Grothendieck ). 1 } }

2, 3, -, - (X H í et (X, Q )), l- ( ), - l( ),, - l-, (, l Hom(Z l, Z ) = 0, Hom(Z, Z ) Z )., Tate Q (i) Q l (i), l Q l (i) G (I ), Q (i) (i 0 ) χ i : G Z χi (I ) Z Q (i) (, H 2 ét (P1, Q ) H 1 ét (G m,, Q ) Q ( 1) ), - Fontaine, - (B cris, B st, B dr ), - - - 2 3,, X -, X k ( ). 4, 2, 3, Tate-Sen - 2. Fontaine -. 2, Fontaine ([Fo82], [Fo94a], [Fo94b], [Be04]) B cris, B st, B dr -, -, - -, - -, - (Q ur B ur := Q ur - ) V Re G, D ur (V ) := (B ur Q V ) G B ur Frobenius ϕ Frobenius ϕ. (B G ur B G ur := {x B ur g(x) = x ( g G )} = 0, Bur ϕ=1 := {x B ur ϕ(x) = x} = Q B ur 0 (B ur Q V ) B ur Q V : a (b x) ab x B ur 0 D ur (V ) B ur Q V = 0 ), dim 0 D ur (V ) dim Q V D ur (V ) V Proosition 2.1. V Re G (1) V 2

(2) dim 0 D ur (V ) = dim Q V,, B ur 0 D ur (V ) B ur Q V G ϕ, - (B ur 0 D ur (V )) ϕ=1 V (, V D ur (V ) ). Proof., G - B ( = cris, st, dr ) ( B ur, G - ϕ ),,, V - D (V ) := (B Q V ) G dim D (V ) = dim Q V (, := B G ),, D (V ) - B, ( D ur (V ) )V D (V ) (), - B ur C,, - C, Tate i Z, C (i) := C Q Q (i) = C e i, G g(ae i ) := χ(g) i g(a)e i (a C ) Theorem 2.2. (Tate[Ta67]) (1) i = 0, H 0 (G, C (0)) = H 1 (G, C (0)) =. (2) i 0, H j (G, C (i)) = 0 ( j 0). Proof. 4. H 0 (G, ), - - (i 0 Q (i) )C, Q ( 1) - C, Tate Q( 1) G m,r, 2π 1, 2π 1 - C 3

2.1. B 2.1, B cris, B st, B dr B +, ( )Ẽ+ Ẽ + := lim O C / := {O C / O C / O C / } n Ẽ +, x = (x n ) n 0, (x n O C /, x n+1 = x n ). F = O Q ur / lim O Q n ur/, F Ẽ+. Ẽ+ x = (x n ) n 0 v(x) := lim n v ( x n n ) (, x n O C x n ), v : Ẽ+ R 0 { } x n,, v, Ẽ+. Ẽ+, ε, Ẽ+ ε := ( ε n ) n 0, ε n O C, ε 0 = 1, ε 1 1, ε n+1 = ε n ( n) (, ε n O C /, ε n ). := ( n ) n 0, n O C, 0 =, n+1 = n ( n) Ẽ + x x ϕ Frobenius. Ẽ +, O C G - ϕ G - (, Ẽ +, v ). Ã + := W (Ẽ+ ) Ẽ+ Witt., - Ã+ /Ã+ Ẽ+ () Ã + -, Ẽ+ v, [ ] : Ẽ+ Ã+ Teichmüller,, [0] = 0, [1] = 1, x, y Ẽ+ [xy] = [x][y], [x] x Ẽ+ (mod ), x Ã+ x = n=0 [a n] n (a n Ẽ+ ) Witt Ẽ+ ϕ, G - Ã+ (, - )., ϕ( n=0 [a n] n ) = n=0 [a n] n, g( n=0 [a n] n ) = n=0 [g(a n)] n (g G ). Qur = W ( F )[ 1 ], B + := Ã+ [ 1 ] Q ur B +. B +, C, [x] = [(x n ) n 0 ] Ã+ (x Ẽ+ ), θ([x]) := lim n x n n (, x n O C, x n O C /O C ). θ : Ã+ O C, θ : Ã+ O C : [a n ] n θ([a n ]) n n=0, θ θ G - θ([x]) O C, θ [ ] er(θ), n=0 er(θ) = ([ ] )Ã+ (, θ (mod ) Ẽ+, O C ). θ G - er(θ) G - 4

, ϕ (, ϕ([ ] ) er(θ) ). θ. θ : B + C 2.2. B dr :de Rham. B + B + dr, B + dr := lim n B+ /er(θ) n, B + dr G, ϕ. θ : B + C G - θ : B + dr C B + B + dr, B+ dr C, er(θ), G - B + dr ( B +, ). t := log([ε]) = n 1 ([ε] 1)n ( 1) B + dr n n=1 (([ε] 1) er(θ), t B + dr ). log ( ), ϕ(t) = log(ϕ([ε])) = log([ε] ) = log([ε]) = t, g(t) = log(g([ε])) = log([ε] χ(g) ) = χ(g)log([ε]) = χ(g)t (g G ) (, t Q ( 1) -, 2π 1 - )., log([ε]) = ([ε] 1)x (x B + dr ), θ(x) 0, B + dr, er(θ) = tb + dr B dr := B + dr [t 1 ]. B dr G - B + dr - Fil i B dr := t i B + dr (i Z) B dr, Re G (filtered modules over ) D dr Deition 2.3. D ( filtered module over ), D, - Fil i D (i Z), i > 0 Fil i D = 0, Fil i D = D,. D 1 D 2, D 1 D 2 - f : D 1 D 2 i Z f(fil i D 1 ) Fil i D 2 MF. 5

Remark 2.4. MF, D 1 = e 1, Fil 0 D 1 = D 1, Fil 1 D 1 = 0, D 2 = e 2, Fil 1 D 2 = D 2, Fil 2 D 2 = 0, f : D 1 D 2 : ae 1 ae 2 Im(f) = D 1 Coim(f) = D 2., MF, i,, Deition 2.5. Re G MF 0 D 1 D 2 D 3 0 0 Fil i D 1 Fil i D 2 Fil i D 3 0 V Re G, D dr : Re G MF D dr (V ) := (B dr Q V ) G, Fil i D dr (V ) := (Fil i B dr Q V ) G D dr (V ), Proosition 2.6. (1) B G dr =. (2) V Re G, dim D dr (V ) i Z dim (C (i) Q V ) G dim Q V Proof., Tate (Theorem 2.2). Deition 2.7. (1) 0 t k+1 B + dr tk B + dr C (k) 0 dim D dr (V ) = dim Q V, V (de Rham reresentation). Re G Re G,dR. (2) V Re G,dR, Z {i Z Fil i D dr (V )/Fil i+1 D dr (V ) 0} V Hodge-Tate (Hodge-Tate weight). (2), D dr Re G,dR Proosition 2.8. (0) Re G,dR (, f : V 1 V 2 V 1, V 2 Re G,dR er(f), Im(f), Coker(f) Re G,dR). (1) D dr : Re G,dR MF (, Re G,dR MF ). (2) V Re G,dR, B dr (B dr Q V ) B dr Q V : a (b x) ab x B dr D dr (V ) B dr Q V 6

, B + dr G - (, Fil i (B dr D dr (V )) := Fil i 1 B dr Fil i 2 D dr (V ) i 1 +i 2 =i, ). Fil i (B dr Q V ) := Fil i B dr Q V Remark 2.9. (Tate ) i, D dr (Q (i)) = (B dr Q Q (i)) G = ( 1 e t i i ) (, e i Q (i) )., Q (i), Fil i D dr (Q (i)) = D dr (Q (i)), Fil i+1 D dr (Q (i)) = 0, Q (i) Hodge-Tate i. Remark 2.10. L V Re G, V Re G,dR, V G L V GL, Hilbert 90 (H 1 (Gal(L/), GL n (L)) = {1}), V GL V 2.3. B HT :Hodge-Tate. B HT,, G g( n Z a n T n ) := n Z B HT := C [T, T 1 ] g(a n )χ(g) n T n (g G, n Z a n T n B HT ) B HT, Re G - M gr, D HT Deition 2.11. Re G M gr,, D HT : Re G M gr, D HT (V ) := (B HT Q V ) G, D HT (V ) i := (C T i Q V ) G D HT (V ) Remark 2.12. 2.6(2), Deition 2.13. (1) V Re G dim D dr (V ) dim D HT (V ) dim Q V dim D HT (V ) = dim Q V, V Hodge-Tate (Hodge-Tate reresentation). Hodge-Tate Re G Re G,HT. (2) V Re G,HT, Z V Hodge-Tate {i Z D HT (V ) i 0} Tate, Proosition 2.14. V Re G, (1), (2). (1) V Re G,HT. 7

(2) B HT (B HT Q V ) B HT Q V : a (b x) ab x B HT D HT (V ) B HT Q V, G - C - (,, (B HT D HT (V )) i := C T i 1 D HT (V ) i2, ). i 1 +i 2 =i (B HT Q V ) i := C T i Q V Deition 2.15. V Re G,HT, (2) 0, C [G ]- C (i) D HT (V ) i C Q V i Z., V Hodge-Tate ( Hodge-Tate decomosition). Remark 2.16. i Z, H 1 (G, Q (i)) 0 Q (i) V Q 0. i 0, V Hodge-Tate. (, C 0 C (i) C Q V C 0, Tate i 0 C [G ]- ). i = 0, 1- log(χ) : G Q : g log(χ(g)) [log(χ)] Hom(G, Q ) = H 1 (G, Q ) Hodge-Tate ( 4 )., Hodge-Tate, t i B + dr /ti+1 B + dr B HT i Z 2.6 (2). Proosition 2.17. (1) Re G,dR Re G,HT. (2) V Re G,dR, D dr (V ) Hodge-Tate D HT (V ) Hodge-Tate., - Fil i D dr (V )/Fil i+1 D dr (V ) D HT (V ) i Z Remark 2.18. Re G,HT Re G.dR, i 1, Re G 0 Q (i) V Q 0, V Hodge-Tate Remark 2.19. Hodge-Tate, Hodge-Tate - Hodge-Tate, 4. 8

2.4. B cris : A cris Ã+ er(θ) = ([ ] ) PD-enveloe -. Ã + G - A cris, ϕ([ ] ) ([ ] ) ( mod Ã+ ) Frobenius ϕ A cris ([ ] ) er(θ), Ã + B + dr G - A cris B + dr,, A cris = { n=0 a n ([ ] ) n n! B + dr a n Ã+, a n 0(n )} (, a n 0 (n ), Ã+ ). B + cris := A cris[ 1 ]. t B + dr, n 1 ([ε] 1)n t = ( 1) = ( 1) n 1 ([ε] 1)n (n 1)! n n! n=1 n=0, t A cris. B cris := B + cris [t 1 ] G, ϕ t, B cris G, ϕ B + cris B+ dr B cris B dr, Q ur B + ur Q B + cris. B cris B dr,. Proosition 2.20. (1) 0 B cris B dr. (2) B G cris = 0. (3) (Bloch- ) G -,, Q B ϕ=1 cris. 0 Q B ϕ=1 cris B + dr B dr 0 B + dr Deition 2.21. V Re G, B ϕ=1 cris := {x B cris ϕ(x) = x} : x (x, x), Bϕ=1 cris D cris (V ) := (B cris Q V ) G. (1)(2) 2.6 (2), dim 0 D cris (V ) dim D dr (V ) dim Q V B + dr B dr : (x, y) x y Deition 2.22. D ϕ- (filtered ϕ-module over ), D 0 -, ϕ- ϕ D : D D (, a 0, x D, ϕ D (ax) = ϕ(a)ϕ D (x) ), D := 0 D -, ϕ- MF ϕ. Deition 2.23. V Re G dim 0 D cris (V ) = dim Q (V ), V Re G Re G,cris. 9

Remark 2.24. V, D cris (V ) Frobenius ϕ Dcris (V ) B cris Frobenius, dim 0 D cris (V ) dim D dr (V ), V V (1) 0 D cris (V ) D dr (V )., D cris (V ) MF ϕ. Proosition 2.25. V Re G (1) V Re G,cris. (2) B cris 0 D cris (V ) B cris Q V G -, ϕ,,, (G ) (B cris 0 D cris (V )) ϕ=1 Fil 0 (B dr D dr (V )) V,, V D cris (V ), D cris : Re G,cris MF ϕ Proof. (1) (2) (, (1) (2) ), B cris, χ i µ : G Z (i Z, µ ), Bloch-, V Re G,cris, (B cris 0 D cris (V )) ϕ=1 Fil 0 (B dr D dr (V )) = (B cris Q V ) ϕ=1 Fil 0 (B dr Q V ) = (B ϕ=1 cris B + dr ) Q V = V Remark 2.26. D cris essential image, 2.7. Remark 2.27. Q (i),, t B cris, D cris (Q (i)) = (B cris Q Q (i)) G = 0 ( 1 t i e i ), ϕ( 1 t i e i ) = 1 i ( 1 t i e i ), Fil i D dr (Q (i)) = D dr (Q (i)), Fil i+1 D dr (Q (i)) = 0. Remark 2.28., H 1 (G, Q (1)) G - 1 µ n x x 1 (, µ n := {ζ ζ n = 1} ) n lim /( ) n n H 1 (G, Z (1)) lim /( ) n, H 1 (G, Z (1)) H 1 (G, Q (1)) n H 1 (G, Q (1)) : a [V a ]., a O 0 Q (1) V a Q 0 V a i 2, H 1 (G, Q (i)) 2.5. B st :, π, π := ( π n ) Ẽ+ π n O C, π 0 = π, π n+1 = π n π, ( 1) n 1 log([ π]) := ( [ π] 1) n tb + dr n π, t ( ) n=1 ϕ(log([ π]) = log([ π]), g(log([ π]) = log(g[ π]) = log([ π][ε] cπ (g) ) = log([ π]) + c π (g)t 10

, c π 1- B st : G Z (1), g(π n ) = π n ε c π (g) n ( n) B st := B cris [log([ π])] B dr log([ π]), B cris, B st = B cris [log([ π])] B cris [T ] (B cris ) log([ π]) ϕ, G B st G, ϕ B st π π B st B cris derivation N n n N : B st B st : a i log([ π]) i ia i log([ π]) i 1 (a i B cris ), N, i=0 i=0 ϕn = Nϕ Deition 2.29. D (ϕ, N)- ( filtered (ϕ, N)-module over ), D ϕ-, ϕ D N D = N D ϕ D 0 - N D : D D ϕ D N D = N D ϕ D, N D (ϕ, N)- MF ϕ,n. ϕ- N = 0 (ϕ, N)- MF ϕ ϕ,n MF Proosition 2.30. B st (1) B G st = 0. (2) 0 B st B dr. (3) Bst N=0 = B cris,, B ϕ=1,n=0 st B + dr = Q. Remark 2.31., V Re G,, Deition 2.32. V Re G D st (V ) := (B st Q V ) G dim 0 D st (V ) dim D dr (V ) dim Q V dim 0 D st (V ) = dim Q (V ) ( semi-stable reresentation). Re G Re G,st. Remark 2.33. (1) B cris B st (2) Re G,cris Re G,st Re G,dR (2) V Re G, B st ϕ, N B dr D st (V ), D st (V ) MF ϕ,n. Proosition 2.34. V Re G (1) V Re G,st. 11

(2) B st 0 D st (V ) B st Q V G, ϕ, N,,, (G - )., Proof., D cris (B st 0 D st ) ϕ=1,n=0 Fil 0 (B dr D dr (V )) V D st : Re G,st MF ϕ,n Remark 2.35. 2.28 H 1 (G, Q (1)), a \ O V a, (v (a) > 0, V a Tate /a Z - Tate ). 2.6., Deition 2.36. L D (1), (2), (ϕ, N, Gal(L/))- ( filtered (ϕ, N, Gal(L/))-module over ) (1) D L (ϕ, N). (2) Gal(L/) Aut MF ϕ,n (D). L D (ϕ, N, Gal(L/))- N = 0, D (ϕ, Gal(L/))- (ϕ, Gal(L/)))- MF ϕ,n,gal(l/) (ϕ, N, Gal(L/))(, (, MF ϕ,gal(l/) ). Deition 2.37. V Re G, L, V G L V GL ( ), V (, ). (, ) Re G Re G,cris(, Re G,st)., V GL ( ) Re G Re G,L cris(, Re G,L st). Remark 2.38. (1) Remark 2.10,, Re G,cris Re G,st Re G,dR (, Re G,st = Re G,dR ( - )). (2) V Re G,L st, D L st(v ) := (B st Q V ) G L, dim L0 Dst(V L ) = dim Q V Dst L, B st B dr G, ϕ, N,, Dst(V L ) MF ϕ,n,gal(l/) (V Re G,L cris, Dcris(V L ) := (B cris Q V ) G L MF ϕ,gal(l/) )., Proosition 2.39. L V Re G, (1), (2). (1) V Re G,L st(, V Re G,L cris). 12

(2) (,. B st L0 D L st(v ) B st Q V B cris L0 D L cris(v ) B cris Q V ) (2) Proosition 2.40. L D L cris : Re G,L cris MF ϕ,gal(l/) : V D L cris(v ) (, Dst L : Re G,L st MF ϕ,n,gal(l/) : V Dst(V L )) Remark 2.41. essential image 2.7., - Theorem 2.42. (Berger[Be02], Colmez[Co08]), Re G,st = Re G,dR Remark 2.43. Berger([Be02]), (ϕ, Γ), - Robba - Crew- Crew-, André([An04]), Christol-Mebkhout([Me02]), edlaya([e04]), Colmez([Co08]) Esaces Vectoriels de dimension ie (Crew- ) 2.7. (ϕ, N). 2.7, D cris, D st essential imege L, D MF ϕ,n,gal(l/)., D t N (D), t H (D), dim L0 D = 1 D = L 0 e, t N (D) := v (α) Z(, α L 0, ϕ(e) = αe ), t H(D) Z Fil th(d) (L L0 D) 0 Fil th(d)+1 (L L0 D) = 0, dim L0 D = d, D d- d D, t N (D) := t N ( d D), t H (D) := t H ( d D) t H (D) = i Z dim L(Fil i (L L0 D)/Fil i+1 (L L0 D)) Deition 2.44. D MF ϕ,n,gal(l/) (, D MF ϕ,gal(l/) ) (1), (2), (weakly-admissible) (ϕ, N, Gal(L/)) (, (ϕ, Gal(L/)) ). (1) t N (D) = t H (D) (2) D (ϕ, N, Gal(L/))- D t N (D ) t H (D ) MF ϕ,n,gal(l/),wa, MF ϕ,gal(l/),wa. Proosition 2.45. 13

(1) MF ϕ,n,gal(l/),wa, MF ϕ,n,gal(l/),wa (2) V Re G,L st, Dst(V L ) Proof. (2) V Re G,L st., t N (V ) = t H (V ), {χ k δ k Z, δ : G Z : }. D Dst(V L ), t N (D ) t H (D ), dim L0 D = d, Dst L : Re G,L st MF ϕ,n,gal(l/), d D Dst( L d V ), dim L0 D = 1, dim L0 D = 1 t N (D ) t H (D ), D = L 0 e D ϕ(e) = αe (α L 0 ), Fil h (L L0 D ) = L L0 D, Fil h+1 (L L0 D ) =., (B st L0 D ) ϕ=1,n=0 Fil 0 (B dr L L L0 D ) (B st L0 Dst(V L )) ϕ=1,n=0 Fil 0 (B dr L L L0 Dst(V L )) = V, D B ϕ=α 1 cris t h B + dr V, α = i u (u O L 0 ), B ϕ=α 1 cris = B ϕ= i cris, t h, B ϕ=h i cris B + dr, h > i, i h t N (D ) t H (D ) Lemma 2.46. (1) k < 0, B ϕ=k cris B + dr = 0. (2) k = 0, B ϕ=1 cris B + dr = Q. (3) k > 0, B ϕ=k cris B + dr Q - Proof. k 0, B ϕ=k cris B + dr t k B ϕ=1 cris t k B + dr, Bϕ=1 cris B+ dr = Q k 1, 0 Q t B +,ϕ= cris C 0, Theorem 2.47. Dst L : Re G,L st MF ϕ,n,gal(l/),wa, Dcris L : Re G,L cris MF ϕ,gal(l/),wa Remark 2.48., Colmez-Fontaine([CF00]), Fontaine([Fo 03]) almost C -, Colmez([Co02]) Esaces Vectoriels de dimension ie B ϕ=k cris B + dr G -,, almost C - Esaces Vectoriels de dimension ie G -, (ϕ, Γ) Robba -, edlaya ([e04]) (Berger([Be08]) isin([i06])). 3.. 3, X -, X k ( ) 2 -,,, ([Tsu02]), 14

X O (, X Sec(O [T 1, T 2,, T n ]/(T 1 T 2 T m π )) (1 m n) )., X X X := X Sec() Sec(). i 0, - H í et (X, Q ) H i dr(x /) := H i (X, Ω X /). H í et (X, Q ) Re G, H i dr (X /) k 0 Fil k H i dr(x /) := Im(H i (X, Ω k X / ) Hi (X, Ω X /)) H i dr (X /) MF., X X k := X Sec(O ) Sec(k), H i log cris(x) := H i log cris(x k /O 0 ) O0 0 ( [BerO73], [Hya94]). 0 - ϕ- Frobenius ϕ 0 - N, Nϕ = ϕn., X (X k ) N = 0.,. Theorem 3.1. (Berthelot-Ogus, - ), π O, 0 H i log cris(x) H i dr(x /) Proof., X Berthelot-Ogus([BerO78]), - ([Hya94]), H i log cris (X) (ϕ, N)- (X, ϕ- ), Theorem 3.2., G -, ϕ, N,. (1) X, (2) X,,,, (X B cris Q H í et(x, Q ) B cris 0 H i log cris(x). B st Q H í et(x, Q ) B st 0 H i log cris(x). g g, ϕ 1, N 1, Fil k H í et(x, Q ) g 1, ϕ ϕ, N 1 + 1 N, Fil k = H í et(x, Q ) Re G,st k 1 +k 2 =k H í et(x, Q ) Re G,cris), (ϕ, N)- D st (H í et(x, Q )) H i log cris(x) 15 Fil k 1 Fil k 2

(X ϕ- D cris (H í et(x, Q )) H i log cris(x)) Proof.,, ([Tsu02]]). Faltings(almost étale [Fa02]), Niziol( [Ni98]), ( [Tsu99]), de Jong, Theorem 3.3. X (1) H í et (X, Q ), D dr (H í et(x, Q )) H i dr(x /) (2) H í et (X, Q ) Hodge-Tate, C Q H í et(x, Q ) n 0 C ( n) H i n (X, Ω n X /), 2 3, X H í et (X, Q ) - { } { { } } { { } } { { } } { } =, 2.6 - Re G, Re G,dR Re G, l-, -. B dr, - (ϕ, Γ)- ([Fo91]). 4. C - (Tate-Sen ):- 4,, Hodge-Tate, Tate-Sen C - ([Sen73], [Sen80], [Fo04])), - ([Sen88], [Sen93], [BC08]), -, - -, -, Tate-Sen, Hodge-Tate -, ((ϕ, Γ)-) -. 16

4.1. Tate-Sen :, - ) [Fo04] (,, Tate 2.6 ) := n 0 (ζ n)(ζ n 1 n ), H := Gal(/ ) = er(χ : G Z ). Γ := Gal( /) = G /H. χ : Γ Z, Γ Z Tate-Sen, C - G Deition 4.1. W G C -, W C -, G (, a C, x W g(ax) = g(a)g(x) ), C - Re G,C., W Re G,C, W C -, G C -, 1- [U W ] H 1 (G, GL n (C )) (, n = dim C W ). 1-, [U W ] W Tate-Sen, C - W W H := {x W g(x) = x ( g H )}, (decomletion ) 1-, inflation-restriction 1 H 1 (Γ, GL n (C H )) H 1 (G, GL n (C )) H 1 (H, GL n (C )) H 1 (H, GL n (C )), H 1 (Γ, GL n (C H )) (decomletion). (1) H 1 (H, GL n (C )), (2) H 1 (Γ, GL n (C H )) (decomletion) (1), (2), / ( ),, ( 4.2 4.6), (1), (2)., ( variant) Tate-Sen, Tate-Sen - (, (overconvergent) (ϕ, Γ) ([CC98]), B- ([Be08a]) ), (1) (2) - ( )., (1) (1) (almost étaleness). Theorem 4.2. ([Ta67] ro.9, [Fo04] theorem.1.8) M, m tr M/ (O M ), (1) Theorem 4.3. (i), (ii) (i) C H = (, C - ). (ii) n 1, H 1 (H, GL n (C )) = {1}. (i) Lemma 4.4. M, J := Gal(M/ ), ε > 0., x M y v (x y) inf g J {v (g(x) x)} ε 17

Proof.,, λ m v (λ) < ε, 4.2 µ O M tr M/ (µ) = λ, x M, y := tr M/ (xµ) λ, x y = 1 λ (xλ tr M/ (xµ)) = 1 λ ( g J (x g(x))g(µ)), (i) Proof. (of (i)), C H, C H., x C H, C {x n } n 0 v (x x n ) n ( n) x n., n x n, M n, J n := Gal(M n / ). ε > 0, 4.4 y n v (x n y n ) inf g Jn {v (g(x n ) x n )} ε, x C H J n H, g J n, n v (g(x n ) x n ) = v (g(x n x) + (x x n )) n v (x y n ) inf{v (x x n ), v (x n y n )} n ε, lim n y n = x, y n, x. (ii) Proof. (of (ii)) [U] H 1 (H, GL n (C )). (U : H GL n (C ) : τ U τ, [U] 1- ), [U] = 1, U H,, U(H ) 1 + 2 M n (O C ), inflation-restriction 1 H 1 (Gal( / ), GL n (C H )) H 1 (H, GL n (C )) H 1 (H, GL n (C )) (i) C H =, Gal( / ) = Gal( / ), Hilbert 90, [U H ] = 1 H 1 (H, GL n (C )), U(H ) 1 + 2 M n (O C ), M 1 1 + M n (O C ) τ H M1 1 U τ τ(m 1 ) 1 + 3 M n (O C ) M 1, U(H ) 1 + 4 M n (O C ), x m v (x) < 1, 4.2 y O tr / (y) = x, α := y x tr / (α) = 1, v (α) > 1 Q H Gal( / ) Q, M 1 := τ Q τ(α)u τ, α, M 1 1 + M n (O C )., τ H, M1 1 U τ τ (M 1 ) 1 = M1 1 (U τ τ (M 1 ) M 1 ) = M1 1 (U τ τ ( τ Q τ(α)u τ) τ Q τ(α)u τ) = M1 1 ( τ Q τ τ(α)u τ τ τ Q τ(α)u τ). {τ τ} τ Q = {ττ i(τ) } τ Q (, τ i(τ) H ),, α, ττ i(τ) (α)u ττi(τ) τ(α)u τ = τ(α)u τ (τ(u τi(τ) ) 1) 3 M n (O C ) 18

., M 1 1 U τ τ (M 1 ) 1 3 M 2 (O C )., k M k 1 + k M n (O C ), τ H (M 1 M 2 M k ) 1 U τ τ (M 1 M k ) 1 + k+2 M n (O C ) M k, M := k=1 M k( M 1 M 2 ), M 1 + M n (O C ), τ H, M 1 U τ τ (M) = 1., [U] = 1 H 1 (H, GL n (C )) (i)(ii) Re G,C, W Re G,C (i), W H := {x W τ(x) = x τ H } -, G Γ Corollary 4.5. W Re G,C, C -, C b W H dim b W H W : a x ax = dim C W Proof. W 1 [U W ] H 1 (G, GL n (C )), (ii), [U W H ] = 1, (i) C -, C - W Γ - W H,, Tate-Sen 2 W H (decomletion),, / Z / Z, Γ = Gal( /) Z, n (n) (n) Gal( (n) /) Z/ n Z γ 0 Γ Z 1 Z, t : x (n) t (x) := 1 n tr (n) /(x) t well-deed -, x t (x) = x, t (γ(x)) = t (x) ( γ Γ, x ). t, /, -, n 0, n := (ζ n) (, n (n) ). Theorem 4.6. ([Fo04].ro.1.13]), n 0, / n Z, v (t n (x) x) v (γ n (x) x) 1 (, γ n Γ n )., t :, t. Tate-Sen (2)(decomletion), Theorem 4.7. / Z, t, (γ 0 Γ ). 19

(1) t : (2) (1) t t :. L := er(t : )., = L, (3) γ 0 1 : L L γ 0 1 : L L ρ : L L ρ, y L v (ρ(y)) v (y) 1 Proof. (1). 4.6, x, v (t (x)) = v (t (x) x + x) inf{v (t (x) x), v (x)} inf{v (γ 0 (x) x) v (x) 1 1, v (x)}, t., t t :. (2)., = L x L, x = t (x) = 0 L = {0}. x, t (x), x t (x) L, x L., = L γ 0 1 : L L. n L n := L n γ 0=1 n = γ 0 1 : L n L n, L n -, n 0 L n L, x L, L, {x n } n 1 x = lim n x n, t, 0 = t (x) = lim n t (x n )., x = lim n (x n t (x n )), x n t (x n ) n 0 L n. n 0 L n L (3). ρ : L L γ 0 1 : L L y L, ρ(y) L, 4.6, v (ρ(y)) = v (t (ρ(y)) ρ(y)) v ((γ 0 1)ρ(y)) t, Theorem 4.8. - (i), (ii), (i) C G =. (ii). H 1 (Γ, GL n ( )) H 1 (Γ, GL n ( )) 20 1 = v (y) 1

Proof. (i)., 4.7, (i) 4.3 (1) 4.7(2) γ 0 1 : L L (ii)., U : Γ GL n ( ) 1-, k, m > 0 / m 4.7, U γ0 1 + k M n (O b ) (, γ 0 Γ m ), U, U γ0 M n (O m ). U γ0 := 1 + U 0 (U 0 M n (O m )), v (U 0 ) k, U 0 = (U 0 t m (U 0 )) + t m (U 0 ), U 0 t m (U 0 ) M n (er(t m )), 4.7(3), V 1 M n (er(t m )), U 0 = (γ 0 1)(V 1 ) + t m (U 0 ) 4.6 4.7(3), v (t m (U 0 )) = inf{v (t m (U 0 ) U 0 ), v (U 0 )}, v (U 0 ) k 1 1 v (V 1 ) = v (ρ(u 0 t m (U 0 ))) v (U 0 t m (U 0 )) v (U 0 ) 2 1 k 2 1 1., (1 V 1 ) 1 (1 + U 0 )(1 γ 0 (V 1 )) k+1, k V 1, U 0 k + 1, k+1, (1 V 1 ) 1 (1 + U 0 )(1 γ 0 (V 1 )) = (1 + V 1 + V 2 1 + )(1 + U 0 )(1 γ 0 (V 1 )) 1 + V 1 + U 0 γ 0 (V 1 ) mod k+1 = 1 + V 1 + ((γ 0 1)(V 1 ) + t m (U 0 )) γ 0 (V 1 ) = 1 + t m (U 0 )., (1 V 1 ) 1 (1 + U 0 )(1 γ 0 (V 1 )) = 1 + U 1 + W 1, U 1 M n (O b ), W 1 M n (O m ), v (U 1 ) k + 1, v (W 1 ) k 1 U 1, W 1., U 1 = (γ 0 1)(V 2 ) + t m (U 1 ), v (V 2 ) k + 1 2, v 1 (t m (U 1 )) k + 1 1, k k+2, (1 V 2 ) 1 (1 + U 1 + W 1 )(1 γ 0 (V 2 )) = (1 + V 2 + V 2 2 + )(1 + U 1 + W 1 )(1 γ 0 (V 2 )) 1 + W 1 + t m (U 1 ), W 2 := t m (U 1 ), (1 V 2 ) 1 (1 + U 1 + W 1 )(1 γ 0 (V 2 ) = 1 + U 2 + (W 1 + W 2 ), U 2 M n (O b ), W 2 M n (O m ), v (U 2 ) k + 2, v (W 2 ) k + 1 1. 21

,, V i M n (O b ), v (V i ) k + (i 1) 2 1, {(1 V 1 )(1 V 2 ) (1 V i )} 1 (1+U 0 ){(1 γ 0 (V 1 )) (1 γ 0 (V i ))} = 1+U i +(W 1 +W 2 + +W i ), U i M n (O b ), W i M n (O m ), v (U i ) k + i, v (W i ) k + (i 1) 1, (1 V ) := i=1 (1 V i) M n (O b ),, (1 V ) 1 (1 + U 0 )(1 γ 0 (V )) = 1 + W M n (O m ), v (W ) k 1.,, U γ := (1 V )U γ (1 γ(v )) (γ Γ ), V U γ 0 M n (O m ) v (U γ 0 1) k 1, γ Γ, U γ GL n ( m ) ( ). claim, Γ U γγ(u γ 0 ) = U γγ 0 = U γ 0 γ = U γ 0 γ 0 (U γ), γ 0 (U γ) = U 1 γ 0 U γγ(u γ 0 ) t m m -,, γ 0 (U γ t m (U γ)) = U 1 γ 0 (U γ t m (U γ))γ(u γ 0 ) (γ 0 1)(U γ t m (U γ)) = U 1 γ 0 (U γ t m (U γ))γ(u γ 0 ) (U γ t m (U γ)) = (U 1 γ 0 1)(U γ t m (U γ))γ(u γ 0 ) +(U γ t m (U γ))(γ(u γ 0 ) 1), 4.7(3) v (U γ t m (U γ)). k k 2 1 > 0, = v (ρ((γ 0 1)(U γ t m (U γ)))) v ((γ 0 1)(U γ t m (U γ)) 1 v (U γ t m (U γ)) + inf{v (U 1 γ 0 1), v (γ(u γ 0 ) 1)} 1 v (U γ t m (U γ)) + k 2 1 v (U γ t m (U γ)) =, U γ = t m (U γ) M n ( m ) (ii) 1 U : Γ GL n ( ), V GL n ( ) γ Γ V 1 U γ γ(v ) = 1, V GL n ( ), m, U γ0 1 + k M n (O m ) (γ 0 Γ m ), V 1 U γ0 γ 0 (V ) = 1, γ 0 (V ) = Uγ 1 0 V,, 4.7(3), k (γ 0 1)(V t m (V )) = (U 1 γ 0 1)(V t m (V )) v (V t m (V )) = v (ρ((γ 0 1)(V t m (V )))) v ((γ 0 1)(V t m (V ))) 1 v (V t m (V )) + v (Uγ 1 0 1) v (V t m (V )) + k 1 1 > 0 V = t m (V ) M n ( m ) 22 1

W H decomletion, Proosition 4.9. M Γ -., M. Proof. M, M M M, M, / 4.7, γ 0 Γ M, M.,, M α, γ m 0 1(m ), γ m 0 α m, α m 1., α m 1 + k O m α 1 + k O, M 0 M = 4.7(2) γ 0 1 : L L, α = 1, α 1., M γ 0 u, (γ 0 1)(u) = (α 1)u, v ( 1 u) = v α 1 (ρ(u)) v (u) 1. v ( 1 u) = v α 1 (u) k, α = 1, M =. W Re G,C. x W H x Γ {γ(x)} γ Γ W H - W H Γ -. W H := {x W H dim x Γ < } Theorem 4.10. W Re G,C, (1) W H -, (2) : W H W H dim W H = dim C W W H W H : a x ax (x) := lim γ 1 1 logχ(γ) ( 1) n 1 (γ 1) n (x) n n=1 ( γ 1 γ Γ γ 1 1 ), well-deed -, (, ). det(t H W ) [T ] Proof. (i)., 4.8(i) W H Γ - W W W H : a x ax. 4.8(ii) W, W H = W, W W H W H W, x W H dim x Γ < e 1, e 2,, e m x Γ, f 1,, f d W, 1 i m, e i := d j=1 a ijf j (a ij ), x Γ W Γ, n 23

{a ij } 1 i m,1 j d n - Γ n., 4.9 a ij, e i W, x W (ii)., x W H, 1 ( 1) n 1 (γ 1) n = lim (x) W H γ 1 log(χ(γ)) n n=1 well-deed -, A M d ( ) f 1,, f d, Γ, γ Γ, γ(f 1 ),, γ(f d ) γ(a) A γ(a), det(t H W ) [X]. Deition 4.11. (1) W Re G,C, D Sen (W ) := W H. 4.10 W Sen : D Sen (W ) D Sen (W ) P W (X) [X], W Sen P W (X) ( ) W Hodge-Tate. (2) V Re G D Sen (V ) := D Sen (C Q V ), P V (X) := P C Q V (X) V Sen, P V (X) V Hodge-Tate Remark 4.12. D Sen (Q (i)) = e i, (e i ) = ie i., Q (i) Hodge- Tate i., Hodge-Tate Hodge-Tate Hodge-Tate, Proosition 4.13. V Re G, 2 (1) V Hodge-Tate. (2) V Hodge-Tate, D Sen (V ) ( )., V Hodge-Tate Hodge-Tate Proof. (1) (2) V Hodge-Tate (Deition 2.15) 4.12. (2) (1),, V Hodge-Tate V G Hodge-Tate,, e 1, e 2, e d, (e i ) = n i e i (n i Z) x W H W H, 1 γ Γ γ(x) = ex(log(χ(γ)) (x)), n γ(e i ) = ex(log(χ(γ))n i )e i = χ(γ) n i e i (γ Γ n ), C Q V G C n D Sen (V ) G d n i=1 C (n i ), V G Hodge-Tate, V Hodge-Tate n 24

, - Tate-Sen, C (i) Tate ( 2.2) Theorem 4.14. (1) i = 0, H 0 (G, C ) =, H 1 (G, C ) =. (2) i 0, H j (G, C (i)) = 0 ( j). Proof., H 0 (G, C ) = 4.8(i) i 0, H 0 (G, C (i)) = 0, 4.7, v (χ(γ 0 ) 1) >, C 1 (i) H = e i = Le i ei., γ 0 1 : e i e i, γ 0 1 : Le i Le i x L 4.7(2) x = ρ(y) (y L). γ 0 (xe i ) = xe i, 0 = (γ 0 1)(ρ(y)e i ) = γ 0 (ρ(y))χ(γ 0 ) i e i ρ(y)e i = χ(γ 0 ) i (γ 0 1)(ρ(y))e i + (χ(γ 0 ) i 1)ρ(y)e i = χ(γ 0 ) i ye i + (χ(γ 0 ) i 1)ρ(y)e i., 4.7(3), v (y) = v (χ(γ 0 ) i 1) + v (ρ(y)) v (χ(γ 0 ) i 1) + v (y) > v (y). y = 0 x = 0, (C (i)) G = 0 H 1 (G, C (i)), 4. 3(ii) C W W H (H )C -, H 1 (H, C (i)) = 0 inflationrestriction, H 1 (Γ, e i ) H 1 (G, C (i))., n n 4.7 inflation-restriction 0 H 1 (Gal( n /), (C (i)) G n ) H 1 (Γ, e i ) H 1 (Γ n, e i ) Gal( n/) 0, n, 4.7, Γ γ 0, H 1 (Γ, e i ) e i /(γ 0 1) e i ( C - ). 4.7(2), i = 0 H 1 (Γ, e 0 ) = (G ). i 1, H 0, γ 0 1 : Le i Le i, n L n := L n ( ), n 0 L n L, γ 0 1 : Le i Le i i 0, H 1 (Γ, e i ) = 0. 4.2. Tate-Sen : ( ) -,, -, 4.1 Hodge-Tate, Tate-Sen - [BC08] - - - Q Banach., -, [BC08], Banach. S Q - (, S : S R 0 Q -, a Q, f S af S = a f S ( Q - ) ). X := {m x S m x }, x X E x := S/m x. Deition 4.15. Q - S, (algebra of coefficients). 25 1

(1) S S v S : S R { } f S := v S(f). (i)v S (f) = f = 0. (ii) f, g S, v S (fg) v S (f) + v S (g), v S (f + g) inf{v S (f), v S (g)}. (iii) a Q v S (a) = v (a) (2) x X E x Q, O S := {x S v S (x) 0}. Remark 4.16. A Q, A, S - Deition 4.17. V, V S (G )-,. (1) V S- G S- (, V S ). (2) V G O S - T V S OS T = V T S (G )-, S-Re G. Remark 4.18. S = E Q (2), E-Re G E - V S-Re G, x X, E x - V x := V S E x. S -, Sen Theorem 4.19. ([BC08]) V S-Re G, (1), n ((C Q S) S V ) H Γ n Q S- D n Sen (V ) (C Q S) n Q S D n Sen (V ) (C Q S) S V : a x ax (2) x D n Sen (V ), 1 ( 1) k 1 (γ 1) k n (x) := lim (x) D n γ 1 Sen log(χ(γ)) k (V ) k=1 well-deed n Q S, det(t n D n Sen (V )) n Q S[X] n, Q S (3) x X, D n Sen (V ) S E x D Sen (V x G n ) Remark 4.20., 4.1 C 4.2, 4.6 C Q S, 4.1 S, [BC08], 4.2, 4.6, 4.7 (TS ), 4.1 V D Sen (V ), [BC08] [Be08], TS C -, Tate-Sen - - Remark 4.21.,, S Hilbert90 26

Deition 4.22. V S-Re G, (2) Q S V Sen, P V (X). Remark 4.23. (3), x X, P V (X) S E x = P Vx (X) Q E x [X] [Ya]., - References [An02] Y. André, Filtrations de tye Hasse-Arf et monodromie -adique, Invent. Math. 148 (2002),. 285-317. [Be02] L. Berger, Rerésentations -adiques et équations différentielles, Invent. Math. 148 (2002),. 219-284. [Be04] L. Berger, An introduction to the theory of -adic reresentations, Geometric Asects of Dwork theory, Walter de Gruyter, Berlin (2004),. 255-292. [Be08a] L. Berger, Construction de (ϕ, Γ)-modules: rerésentations -adiques et B-aires, Algebra and Number Theory, 2 (2008), no. 1, 91 120. [Be08b] L. Berger, Équations différentielles -adiques et (ϕ, N)-modules filtrés, Astérisque. 319 (2008),. 13-38. [BC08] L. Berger and P. Colmez, Familles de rerésentations de de Rham et monodromie -adique, Astérisue. 319 (2008),. 303-337. [BerO78] P. Berthelot and A. Ogus, Notes on crystalline cohomology, Mathematical Notes, Princeton University Press, (1978). [CC98] F. Cherbonnier and P. Colmez, Rerésentations -adiques surconvergentes, Invent. Math. 133 (1998),. 581-611. [Co02] P. Colmez, Esaces de Banach de dimension ie, J. Inst. Math. Jussieu 1 (2002),. 331-439. [Co07a] P. Colmez, Rerésentations triangulines de dimension 2, rerint (2007). [Co08] P. Colmez, Esaces vectoriels de dimension ie et rerésentations de de Rham. Astérisque. 319 (2008),. 117-186. [CF00] P. Colmez and J-M. Fontaine, Construction des rerésentations -adiques semi-stables, Invent. Math. 140 (2000), 1-43. [Fa02] G. Faltings, Almost étale extensions, Astérisque 279 (2002),. 185-270. [Fo82] J-M. Fontaine, Sur certains tyes de rerésentations -adiques du groue de Galois d un cors local; construction d un anneau de Barsotti-Tate, Ann of Math. 115 (1982),. 529-577. [Fo91] J-M. Fontaine, Rerésentations -adiques des cors locaux, in The Grothendieck Festschrift, vol. 2, Birkhauser, Boston, 1991,. 249-309. [Fo94a] J-M. Fontaine, Le cors des ériodes -adiques, Astérisque. 223 (1994),. 59-102. [Fo94b] J-M. Fontaine, Rerésentations -adiques semi-stables, Astérisque. 223 (1994),. 113-184. [Fo03] J-M. Fontaine, Presque C -rerésentations, in Volume dedicated to azuya ato s 50-th Birthday, Documenta Mathematica (2003),. 285-385. [Fo04] J-M. Fontaine, Arithmétique des rerésentations galoisiennes -adiques, Astérisque. 295 (2004),. 1-115. [Hya94] O. Hyodo and. ato, Semi-stable reduction and crystalline cohomology with logarithmic oles, Astérisque 223 (1994),. 221-268. [e04]. edlaya, A -adic local monodromy theorem, Ann of Math. (2) 160 (2004),. 93-184. [i06] M. isin, Crystalline reresentations and F -crystals, in Algebraic geometry and number theory, Progr. Math. vol 253, Birkhauser (2006),. 459-496. [Me04] Z. Mebkhout, Analogue -adique du théoreme de Turrittin et le théoreme de la monodromie -adique, Invent. Math. 148 (2002),. 319-351. [Mi], l-,. [Ni98] W. Niziol, Crystalline conjecture via -theory, Ann. Sci. École. Norm. Su. (4) 31 (1998), no 5,. 659-681. [Sen73] S. Sen, Lie algebras of Galois grous arising from Hodge-Tate modules, Ann of Math. 97 (1973),. 160-170. [Sen80] S. Sen, Continuous cohomology and -adic Galois reresentations, Invent. Math. 62 (1980),. 89-116. [Sen88] S. Sen, The analytic variations of -adic Hodge structure, Ann of Math. (2) 127 (1988),. 647-661. [Sen93] S. Sen, An inite-dimensional Hodge-Tate theory, Bull. Soc. Math. France. 121 (1993),. 13-34. 27

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