.5.1. G K O E, O E T, G K Aut OE (T ) (T, ρ). ρ, (T, ρ) T. Aut OE (T ), En OE (F ) p..5.. G K E, E V, G K GL E (V ) (V, ρ). ρ, (V, ρ) V. GL E (V ), En

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1 p , , 57,,.. 1.., Gal(Q p /Q p ), 1. Wach,, Part I,,. Part II, Part III , Paé. Part 1. p.. p p.1. p Q p p (Q p p )... E Q p, E p Z p E, O E. O E E. E Q p, O E. v p : E Q Q E, v p (p) = K p, K K. G K = Gal(K/K), K Galois. K K L g G K, ggal(k/l) G K G K, G K..4. G K. K K /K O K O K. K, K K unr. K unr K. K unr K. K unr K, Gal(K unr /K), K k. G K Gal(K/K unr ) Gal(K/K), I K. O K unr k, k k. G K /I K = Gal(K unr /K) Gal(k/k). p F k Gal(k/k) Frob k, Gal(k/k) Frobenius. Gal(k/k) Frob k. I K p Sylow P K. P K G K. P K G K..5. G K p. E Q p, O E. 1, (). 1

2 .5.1. G K O E, O E T, G K Aut OE (T ) (T, ρ). ρ, (T, ρ) T. Aut OE (T ), En OE (F ) p..5.. G K E, E V, G K GL E (V ) (V, ρ). ρ, (V, ρ) V. GL E (V ), En E (V ) p. E = Q p, G K E G K p G K O E, E,,. Rep OE G K, Rep E G K. (.1) Rep OE G K OE E Rep E G K Rep Zp G K Zp Qp Rep Qp G K. O E E, Z p Q p, O E Z p, E Q p. OE E.,. 3. e Rham p. K p, E Q p. Rep Qp G K, e Rham Rep R Q p G K Rep cris Q p G K.. [F1], [I] O K,, O K k. k A, Z A Z Z/pZ A. A p, p A A A. F A. 3.. Witt. Witt. [S] ( p ), A, A Witt W (A). A W (A),, : W (A) = n 0 A., f, f W A n 0 f(a). (a n) n 0 n 0 f(a) W (A) [a 0, a 1,...]. n 0 W (A) [a 0, a 1,...] A W (A) A. a pn 0 + papn pa p n 1 + pn a n 3... a A, [a, 0, 0,...] W (A) A Teichmüller, [a].

3 p W,, A p. A = F p, W (F p ) Z p.. A p W (A) Zp Q p Q p p, W (A) p A, F A. p p. A p, W (A). [a 0, a 1,...] a 0 W (A) A W (A) A R, 0, A p, W (A) R, W (A) R A [a 0, a 1,...] a 0 W (A) A θ. O K p, F OK : O K O K. () O K F OK OK F OK OK F OK R ( Ẽ+ ). R. O K O K O K O K k O K k. k, a k, x = (x 0, x 1,...) R x 0 = a. a x k R, R k. R Witt W (R). R k, W (R) W (k). n 0. x = [r 0, r 1, r,...] W (R). i 0, r i R r i = (a i,0, a i,1, a i,,...). j 0 a i,j O K, a p i,j+1 = a i,j. i = 0,..., n, a i,n O K /p n O K ã i,n, θ n (x) = ã pn 0,n + pãpn 1 1,n + + pn 1 ã p n 1,n + pn ã n,n, θ n (x) ã i,n, x θ n (x) W (R) O K. θ n, θ n+1 O K /p n+1 O K O K /p n O K. O K p O Cp = lim O n K, C p = O Cp Zp Q p. (θ n ) n 0 θ : W (R) O Cp. θ. θ 1 W (R), B R e Rham θ K K W (k) W (R) C p θ K. B + R B + R = lim n (K W (k) W (R))/(Ker θ K ) n. B + R, θ B+ R K W (k)w (R)/(Ker θ K ) C p, B + R C p. B + R B R. B R G K K,.

4 G K p V, B R Qp V, B R. (B R V ) G K D R (V ). B R K D R (V ) K. B G K R = K, im K D R (V ) im Qp V. im K D = im Qp V V e Rham. G K E V, V e Rham, D R (V ) im E V K Qp E D R (V ) Fil D R (V ), i Z Fil i D R (V ) = ((Ker θ K ) i B + R Q p V ) G K. V e Rham, V Hoge-Tate, gr i D R (V ) = Fil i D R (V )/Fil i+1 D R (V ) {0} i., i im K gr i D R (V ), V Hoge-Tate. t, Tate Q p (1) = Hom(Q p /Z p, K ) Zp Q p Hoge-Tate { 1}. Hoge-Tate, Q p (1) Hoge-Tate {1} Hoge-Tate B cris G K W (k) A cris, A cris = lim n H 0 cris(spec(o K )/(Spec(W (k)/p n W (k))), O cris ). B + cris = A cris Zp Q p B R W (R), PD, A cris : γ (W (k), pw (k)) PD, A cris G K W (k), (W (K), Ker θ) (W (k), pw (k), γ) PD p., A cris B + cris B R., c 0, c 1, c,... Z p, p 0, x Ker θ, A cris n 0 c n xn n! K 1 p (ζ p n) n 0, n 0 ζ p n K 1 p n, n 0 ζ p p = ζ n+1 p n,. ε = (ζ p 0 mo po K, ζ p mo po K, ζ p mo po K,...) R. [ε] 1 Ker θ W (R). A cris t t = log([ε]) = n 1 n 1 ([ε] 1)n ( 1) n. B + cris 1/t B R B cris. B cris G K K 0 = W (k) Zp Q p. p F R : R R W (R) W (R), B R ϕ : B cris B cris G K E V, B cris Qp V G K G K (B cris Qp V ) G K D cris (V ). B cris G K K 0, D cris (V ) K 0 Qp E. D cris (V ) im E V K 0 Qp E, V. 4. Fontaine (Colmez-Fontaine ) p, K E p., Rep cris E G K, ϕ, Fontaine [F], Colmez-Fontaine [CF].

5 p A, M 1, M A, α : A A A, f : M 1 M α, f, m M 1 a A f(ax) = α(a)f(x). 4.. p F k : k k W (k) W (k) K 0 K 0 σ. K 0 E ϕ, 3 (D, ϕ, Fil D K ) : D K 0 Qp E, ϕ : D D σ i E, Fil D K = (Fil i D K ) i Z D K = D K0 K, K Qp E, D K. ϕ, Fil D K, 3 (D, ϕ, Fil D K ) D. ϕ D = (D, ϕ, Fil D K ), t N (D), t H (D) Q : D K 0 Qp E e 1,..., e, K 0 Qp E P (ϕ(e 0 ),..., ϕ(e )) = (e 1,..., e )P. et(p ) K 0 Qp E, K 0 Qp E/E N K0 Qp E/E(et(P )). N K0 Q p E/E (et(p )) e 1,..., e, v p (N K0 Q p E/E (et(p ))) e 1,..., e. t N (D) = v K0 (N K0 Q p E/E et(p )). t H (D) = 1 [K:K 0] i Z i im E(Fil i D K /Fil i+1 D K ) K 0 ϕ (D, ϕ, Fil, D K ), t N (D) = t H (D), ϕ D K 0 Qp E D, t N (D ) t H (D ) t N (D), t H (D) ( t N (D), t H (D) E Q p ). D Fontaine [F3], [BM, Proposition ] Fontaine [F3] B cris Qp V Q p ϕ i V, σ D cris (V ) D cris (V ). ϕ : D cris (V ) D cris (V ). V, V e Rham, B cris Qp V B R Qp V, K K0 D cris (V ) = D R (V ). D R (V ) Fil D R (V ), D cris (V ) K 0 E ϕ. K 0 E ϕ Colmez-Fontaine. G K E V K 0 E ϕ, Rep cris E G K K 0 E ϕ. Fontaine [F], Fontaine-Colmez [CF]. ([C], [F4], [K], [Berg3], [FF]). 5. (ϕ, Γ) A Qp Z p 1 Laurent Z p ((π)) = Z p [[π]][1/π] p. Ẽ R. Ẽ p. n>> c nπ n A Qp, n>> c n([ε] 1) n W (Ẽ). n>> c nπ n n>> c n([ε] 1) n A Qp W (Ẽ). A Q p p. B Qp = A Qp [1/p] Frac W (Ẽ), Frac W (Ẽ) B Q p B, A. K n 1 ζ p n K K(µ p ). K(µ p ) K Galois. H K = Gal(K/K(µ p )), Γ K = Gal(K(µ p )/K). p FẼ : Ẽ Ẽ W (Ẽ) W (Ẽ) Frac W (Ẽ) Frac W (Ẽ) ϕ. Frac W (Ẽ) B A ϕ G K.

6 6 A K = A H K, B K = B H K.. A K, B K ϕ, G K G K /H K Γ K 5.. (ϕ, Γ). k E E, A K,OE = A K Zp O E, B K,E = B Qp E, E K,kE = A K,OE OE k E. A A K,OE, B K,E, E K,kE. A (ϕ, Γ), A D, ϕ ϕ D : D D, γ Γ K ϕ γ γ D : D D, : ϕ D A D A,ϕ A D, A = B K,E, D A K,OE D ϕ D, ϕ D A K,OE D AK,OE,ϕ A K,OE D, γ = 1 1 D = i D, γ 1, γ Γ K (γ 1 γ ) D = (γ 1 ) D (γ ) D, 5.3. Fontaine [F1]. T Rep OE (G K ) D(T ) = (A Zp T ) H K, D(T ) A K,OE (ϕ, Γ). T D(T ), Rep OE (G K ) A K,OE (ϕ, Γ)., Rep E (G K ) B K,E (ϕ, Γ), Rep ke (G K ) E K,kE (ϕ, Γ). Part. 6. Q (). F G F = Gal(F /F ). Langlans G F p. Fermat, Wiles Taylor-Wiles Galois, Taylor-Wiles. Kisin., p K, E, E O E, k E, G K k E, E,., K = Q p,. p Q p Q p, O Qp, m Qp O Qp. O Qp /m Qp F p. v p : Q p Q, v p (p) = Gal(Q p /Q p ). k, a p O Qp, u O Q p, V k,ap,u Gal(Q p /Q p ) Q p, : V Hoge-Tate {0, k 1}, D cris (V ) ϕ 1 a p T + up k 1 T. 4.6, v p (a p ) > 0 V k,ap,u. v p (a p ) = 0 V k,ap,u, V k,ap,u, V k,ap,u, 4.6. (.1), OE E, V k,ap,u O Qp T k,ap,u O Qp im Qp V, G Qp. G Qp F p T k,ap,u F OQ p p V k,ap,u. V k,ap,u 3 (k 1, a p, u)

7 p 7, V k,ap,u T k,ap,u., V k,ap,u, V k,a p,u S u,l p. a F p, µ a : Gal(Q p /Q p ) F p, Gal(Qunr p /Q p ), Gal(Q unr p /Q p ) Frobenius a,, ω : Gal(Q p /Q p ) F p F p p, S u,l. k. k 1 k 1 = i(p 1) + l ( i, l 1 l p 1 ). V k,a p,u G Q p F p., u O Q p F p u, V k,a p,u µ uω l. G Qp F p µ u ω l S u,l. k u O, a Q p O Qp V k,a p,u p O Qp S u,l,. P Qp G Qp p, G Qp F p G Qp, G Qp /P Qp. G Qp /P Qp S u,l, ε. S u,l,. Q p Q p Q p, G Qp = Gal(Q p /Q p ), ε : G Qp Q p, p Q p G Qp Lubin-Tate. ε : G Qp F p F p ε p p. u F, l Z. : α F p, Z, R u,l (; α) := µ α ω µ uα 1ω l, u u F p R u,l (; A), I u,l () S u,l. :, I u,l() := µ u ωl In Q p Q p ε l. R u,l (; A) = R u,l ( + p 1; A), R u,l (; A) = R u,l (l ; ua 1 ), R u,l (; A) = R u,l ( ; A ) A = A mo p 1, AA = u + l mo p 1, S u,l,, A R u,l (; A). I u,l () = I u,l ( + p + 1), I u,l () = I u,l (l ), I u,l () = I u,l ( ) mo p l mo p + 1, l I u,l (), l, l mo p + 1 I u,l (). S u,l,, I u,l () p, S u,l, F p F p. (1) l R( l p +1;A) R( l p +;A) R( 1;A) R(0;A) R(1;A) R(;A) I( l p l p ) I( +1) I( 1) I(0) I(1) R( l 3 ;A) R( l 1 ;A) I( l 3 l 1 ) I( )

8 8 () l ( A S u,l Zariski 1 ) R( l p+1 ;A) R( l p+3 ;A) R( 1;A) R(0;A) R(1;A) R(;A) I( l p+1 ) I( 1) I(0) I(1) R( l 1;A) R( l ;A) I( l 1) u, l, R u,l (; A), I u,l () R(; A), I()., Tate p p 1 Frac W (F p ) Tate, x [u]p l x 1 involution. l p+1 l 1 R u,l(; A) R u,l ( + 1; A) I u,l (),,, a p V k,a p,u k. k 1 k 1 = i(p 1) + l ( i, l 1 l p 1 ). p + l i + ( i (p 1)/ ). a p O Qp, u O Q p. V k,ap,u m Qp V k,a p,u A k, C k, { (i ) 1 ( l ) A k, =, i, > i, C k, = ( i 1 i ( 1) l p p 1/( i ( 1) ( (i 1 (i )( l ) ), if < l 1, ), = l 1, )( l ) 1 ), l. 0, i 1 < l ( ( )( )) i 1 i, l 0 i <, = i l (i 1) ( i )( ) i l, 1 < l < i 1 l = l 1, ( i 1 i )( i l ),, ( ) { i, = ( 0 i, i ),, ( (i )( ) ) 1 { 1 l, 0 i 1 <, = ) 1,,. ( i 1 )( l b = b k,ap,u = A k, a p + pl C k, u a p ( ). b E { } (). p + l i +. u m Qp u F p. v = v p (a p ), = v. b, 1 bt + ip l T Q p, p α. (1) b =, V k,a p,u = I u,l (i). () b v p (α) Z, α = α 0 p v p(α), V k,a p,u = R u,l (v p (α), α 0 ). (3) b v p (α) Z, V k,a p,u = I u,l ( v p (α) ).

9 p V k,a p,u. l + p i +, 8.3 : v = 0. D cris (V k,ap,u) V k,a p,u. v = 0 V k,ap,u, k Q p G Q, p, V k,a p,u Deligne [D]. k p 1, Fontaine-Laffaille [FL] V k,a. p,u k p, GL (Q p ) p Langlans ( p Langlans ) Berger-Breuil [BB], Wach Berger-Li-Zhu [BLZ] V k,a p,u. k = p + 1, GL (Q p ) p Langlans Wach Berger-Li-Zhu [BLZ] V k,a p,u. 0 < v < 1 [Berg3], Breuil. Buzzar-Gee [BG1], [BG], [BG1], [BG]. k v ( v (k 1)/(p 1) ) Wach, Berger-Li-Zhu [BLZ] V k,a p,u. [BLZ] Vienney [V]. 0 < v < 1, GL (Q p ) p Langlans Buzzar- Gee [BG1], [BG] V k,a p,u (). 8.3, 3 : l < i. v, v (l 1)/ Wach, Wach., (k, u) a p O Qp ( rigi ).,, V k,a p,u. p : (1) l i, reuction : v=0 0<v<1 v=1 1<v< v= <v<3 v=3 v=i v=i 1 v=i i <v<i 1 i 1<v<i v>i () i l < i l, reuction ( S u,l Zariski ): v=0 0<v<1 v=1 1<v< v= v= l 3 l 3 v= l 1 l 1 <v< l 1 v= l+1 v=i 3 <v< l+1 v=i i 3<v<i v>i

10 10 (3) i l < i l, reuction : v=0 v=1 v= l v= l 1 0<v<1 l <v< l l l 1 1<v< v= l 1 l 1 <v< l v= l l <v< l +1 v= l +1 v=i v>i (4) i < l l, v l 1 Wach, Wach reuction : v=0 0<v<1 v=1 v= l 3 l 3 v= l 1 l 1 <v< l+1 v= v=l 3 v=l v=l l 1 l+1 <v< l 3<v<l l <v<l v=l+1 l<v<l+1 v=i 1 v>i 1 (5) i < l l, v l 1 Wach, Wach reuction, (4) () (3) ( ). Part 3..,. Wach Berger [Berg1],,,. 10. Wach Wach. Wach, K Q p, Rep OE G K T T OE E Hoge-Tate 0 Rep OE G K. K = Q p Wach. Wach, W. Z p (1) = Hom(Q p /Z p, Q p ). Spf O E X, χ : Z p (1) Γ(X, O X ), s Z p (1) χ(s) 1 W (X). X W (X) Spf O E W. W Spf O E. Z p (1), Spec O E G m,oe, Spec k E G m,oe W. W En(W ). a Z p, χ W (X) χ a W (χ) En(W ), Z p En(W ),. W Spf O E W Spf O E W 1 W : W W. M = En(W ) \ {1} W. W O(W ) = Γ(W, O W ), W = Spec O(W ). m : W W Spec O E W W m. W Spec O E W Spec O E W, Spec O E W 1 W. m M, m : W W W [m] = 1 W W,m W. W [m]\1 W W W [m].

11 p [ W /M]., [ W /M], : N O(W ). m M, O(W ) f m : m M M, f im = i N f m1m = f m1 m 1(f m ), N [ W /M] Wach. O E Wach, [ W /M], : N 1 W M. m M, O(W ) f m : m N N W [m]. O E Wach, [ W /M] Wach OE Wach Hoge. Z p = En(W ) p Zp M p M. p O(W ) ϕ. Wach N, f p : p N N ϕ N. W [p] W O(W ) I p. i 0 N Fil i N. Fil i N = {y N ϕ N (1 y) I i pn} W. W Spf O E W Spec O E W. D 0 W, D 0 η 0. W W Spec OE Spec k E D p, D p η p.. D 0 η 0 D p η p O(W ), A Qp Zp O E O E [[[ɛ] 1]]. A Qp Zp O E η p W Berger. Wach [W], Colmez [C1], Berger [Berg1] : Rep O G Qp, T OE E Hoge-Tate Rep O G Qp T Rep cris,+ O G Qp, N : Rep cris,+ = O G Qp Wach OE, : N(T ) η p T (ϕ, Γ) D(T ), N(T ) η 0 N(T ) ϕ, Q p E ϕ D cris (T OE E). 11. Berger, η 0 D cris (V k,ap,u) Wach N k,ap,u, N k,ap,u η p V k,a p,u. N k,ap,u., N k,ap,u M tors x. M M, M M tors. M tors p 1. O(W ) M tors R., [ W /M] [Spec R/(M/M tors)]. R x, R = O E [[x]] ϕ(x) = x(p + x) p 1. W Spec R, x = 0 1 W Spec R, W [p] x = p Spec R.

12 ϕ. N k,ap,u, N k,ap,u D 0 D cris (T k,ap,u). D cris (T k,ap,u) D cris (V k,ap,u) ϕ O E, N k,ap,u Hoge D cris (T k,ap,u), O E. i 0, ϕ(fil i D cris (T k,ap,u) p i D cris (T k,ap,u). k p 1, Fontaine-Laffallie [FL] i 0, Fil i D cris (T k,ap,u) = Fil i D cris (V k,ap,u) D cris (T k,ap,u). (11.1) Fil i D cris (T k,ap,u) Fil i D cris (V k,ap,u) D cris (T k,ap,u), (k, a p, u), (11.1) N k,ap,u, [BLZ] N k,ap,u. Wach V k,a p,u, (11.1).,, Wach, O E ϕ k 1 := ϕ p k 1 : Filk 1 D cris (T k,ap,u) D cris (T k,ap,u),. (), (k, a p, u), ϕ k 1 O E N k,ap,u. ϕ k 1 O E, N k,ap,u ϕ., ϕ., Fil k 1 N k,ap,u M tors y, e = ϕ(y)/(p + x) k 1 N M tors k,a p,u R. e 1, e N M tors k,a p,u R e 1, y = δe 1 + ze. [BLZ], ([BLZ] ), R (δ, z) Wach N k,ap,u. M = Z p γ, r R : (x1): δ x(p + x) R, (x): (p + x) k 1 ϕ(z)r R ϕ(δ), (x3): E[[x]] (γ(δ)/δ) ϕ/(1+ϕ) R + x i+1 E[[x]], (x4): E[[x]] ( z ) ( γ 1 + x ) ϕ 1+ϕ z δ p δ. R + x i+1 E[[x]], (x5): δ 0 δ R x, E[[x]] uz δ + r ϕ(δ) a ( p 1 + x ) k 1 1+ϕ (δ/δ0 ) ϕ/(1+ϕ). δ 0 p R + x i+1 E[[x]] (δ, z, r). 5 (x1), (x), (x5), (x1), (x), (x5). (x5) (x5) : δ 0 δ R x, E[[x]] uz δ a ( p 1 + x ) k 1 δ 0 p R + x i+1 E[[x]].

13 p 13. (x1), (x), (x5) (δ, z, r),, (x1) (x5) (δ, z, r)., Paé. K 0. s K m, (s) m = s(s+1) (s+m 1). s K 1, 1, f (s, 1, ) 1 = f (s,1,) = 1 j=0 j=0 (j 1 ) ( 1 ) (j 1 ) 1 ( 1 ) 1 ( s ) j t j, j! (s 1 ) j t j. j!. Paé (1 t) s := ( s) n n 0 n! t n. Beukers- Tijeman [BT], f 1 (1 t) s f t K[[t]]. v p (a p ), z = u 1 a p f (k 1,,i ) 1 ( x/p), δ = p f (k 1,,i ) ( x/p), (δ, z) (x1), (x5). δ R. (x) r, f (k 1,,i ) 1, f (k 1,,i ) p. : Resultant. f (s,1,) 1 f (s,1,) resultant 1 ( ) j (s + 1 j) ± 1 j! , 1,. j=1 f 1 f (s, 1, 1) f f (s, 1, 1) 1 = ( 1) 1 1 ( ) 1 (s 1 ) 1 + t 1+ ( 1 + )! ( ) 1 f 1 f (s, 1 1, ) f f (s, 1 1, ) = ( 1) 1 1 (s 1 + 1) 1 + t 1+ ( 1 + )! References [Berg1] Berger, L. Limites e représentations cristallines. Compositio Math. 140 (004), [Berg] Berger, L. Équations ifferentielles p-aiques et (ϕ, N)-moules filtrés. Astérisque 319 (008), [Berg3] Berger, L. Représentations moulaires e GL (Q p ) et représentations galoisiennes e imension. Astérisque 330 (010), [Berg4] Berger, L. Local constancy for the reuction mo p of -imensional crystalline representations. Bull. Lonon Math. Soc. 44 (01), [BB] Berger, L., Breuil, C. Sur la réuction es représentations cristallines e imension en poi moyens. unpublishe note, which is containe in [Berg3]. [BLZ] Berger, L., Li, H., Zhu, H. J. Construction of some families of -imensional crystalline representations. Math. Ann. 39() (004), [Bert] Berthelot, P. Cohomologie cristalline es schémas e caractéristique p > 0. Lect. Notes Math. 407, Springer-Verlag (1974). [BT] Beukers, F., Tijeman, R. On the multiplicities of binary complex recurrences. Compos. Math. 51, no. (1984), [BM] Breuil, C., Mézar, A. Multiplicités moulaires et repréntations e GL (Z p ) et e Gal(Q p /Q p ) en l = p. Duke Math. J. 115, No. (00),

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