1. Γ, R 2,, M R. M R. M M Map(M, M) 3, Aut R (M). ρ : Γ Aut R (M) Γ. M R n, R, R ρ : Γ Aut R (M) GL n (R) := {g M n (R) det(g) R } 4. ρ Γ R R M.,,.,,

I ( ) (i) l, l, l (ii) (Q p ) l, l, l (iii) Artin (iv). (i),(ii). (iii) 1. (iv),.. [9]. [4] L-,.. Contents 1. 2 2. 4 2.1. 4 2.2. l 5 2.3. l 9 2.4. l 10 2.5. 12 2.6. Artin 13 3. 15 3.1. l, l, l 15 3.2. Weil-Deligne 18 4. 21 5. 24 6. 27 References 27 1 Artin,,.,, Artin l. 1

1. Γ, R 2,, M R. M R. M M Map(M, M) 3, Aut R (M). ρ : Γ Aut R (M) Γ. M R n, R, R ρ : Γ Aut R (M) GL n (R) := {g M n (R) det(g) R } 4. ρ Γ R R M.,,.,, Γ K G K := Gal(K sep /K)., K,, R l Q l, l Z l, F l (Z l /l n Z l, Z l [[X]],...) Banach. G K... 1-1.. ρ 1, ρ 2 : Γ Aut R (M) Γ., ρ 1 ρ 2 (equivalent) t Aut R (M), ρ 1 (g) = tρ 2 (g)t 1, g Γ., ρ 1 ρ 2. R M, : trρ i : Γ ρ i tr Aut R (M) GL n (R) R, i = 1, 2. M R. ρ 1, ρ 2,, ( 2-2-6 2-4-1 ). 2,. X, X 1. 3 X, Y, Map(X, Y ) X Y. X A Y B, W (A, B) := {f Map(X, Y ) f(a) B}. W (A, B) Map(X, Y ). 4 GL n, M, Aut R (M) Aut R (M, ) = {f Aut R (M) f } G. G, GSp 2n, GO(n), GU(n, m)., 2004 ([17]). 2

1-2. Γ ρ : Γ Aut R (M) H Γ, M H M H := {m M h H, ρ(h)m = m} Γ/H R. ρ H : ρ H :.Γ/H Aut R (M H )., H M/M H Γ/H ρ H. 1-3.(1) ρ : Γ Aut R (M) Γ., R R, ρ R ρ R. m r M R R,., M R R, ρ R (m r ) = ρ(m) r (2) ρ : Γ Aut R (M) Γ. ρ, ρ M 0 M Γ R. (3) K, V K. ρ : Γ Aut K (V ) Γ. ρ, K L, ρ L. 1-4. (1) Γ ρ : Γ Aut R (M), R M m M, M, Sym r RM, r R M, r r r ρ, m ρ, Sym m ρ, r ρ., M R n, detρ := n ρ ρ., Aut R (M) GL n (R), ρ 1 Γ ρ Aut R (M) GL n (R) det R., det., M M := Hom R (M, R) ρ, (contragredient representation)., φ M, γ Γ,. γφ(m) := φ(γ 1 m), m M 1 χ : Γ R χ n, n Z ρ ρ χ n ρ χ n. (2) Γ ρ i : Γ Aut R (M i ), i = 1, 2, M 1 R M 2, M 1 R M 2 ρ 1 ρ 2, ρ 1 ρ 2., Γ ρ : Γ Aut R (M), End R (M) = M R M, adρ := ρ ρ, ρ (adjoint representation)., φ End R (M), γ Γ, γφ(m) := γφ(γ 1 m), m M 3

. 2. 2.1.. K, p l. Σ K K. L K ( ), O L. Gal(L/K), Krull (cf. [9]). v K, v L w., D L,v = {σ Gal(L/K) σ(w) = w} v. D L,v w, Gal(L/K)., D L,v Gal(F w /F v ), σ σ mod w., F w := O L /w w. I L,v, v : 1 I L,v D L,v Gal(F w /F v ) 1. Gal(F w /F v ) Frob v (x) = x F v, x F w Frob v, ( ) 5. D L,v Frob v., I L,v = {1} L v,, L v. L v Frob v. S K, K S S Σ K K (S ). L = K S, v S, v K S, Frob v D KS,v., D KS,v G KS, v K S w, Frob v w. 2-1-1. R, M R 6. v K, I v := I K,v. G K ρ : G K Aut R (M) v, ρ(i v ) = {1},, ρ v. I v (, w ). ρ K K Sρ := K Kerρ, K Sρ S ρ ρ. 6 {id M } Aut R (M), Kerρ., ( ) ([9] 1.12 ), G K /Kerρ Gal(K Sρ /K). 5. 6, Hausdorff., Aut R (M) Hausdorff (cf. [21] p.170 30.2 -(2)),, Aut R (M). 4

, ρ : G K π ρ Aut R (M) eρ Gal(K Sρ /K), π σ σ KS ρ. v S ρ, Frob v D KS ρ,v ρ(frob v ). Frob v Gal(K Sρ /K), G K., ρ, ρ(frob v ) ρ(frob v )., ρ(frob v ) v K Sρ w. w, g G K, gfrob v g 1. ρ ( ρ ), ρ(gfrob v g 1 ) = ρ(g)ρ(frob v )ρ(g) 1 ρ(frob v )., ρ(frob v ) w. ρ. 2.2. l. E Q l, O E, V E. V 0 V O E 7, V., ρ : G K Aut E (V ) l., Aut E (V ) V Map(V, V ). V Aut E (V ) GL n (E), (n = dim E V ), GL n (E) E,. E Q l, ρ Q l ( 2.2.8)., l E V Q l. l. 2-2-1. 1 l µ l n(k) := {x K x ln = 1} Z/l n Z {µ l n+1(k) l µ l n(k)} n Z l (1) := lim l µ l n(k) Z l. G K µ l n(k) (l ), G K Z l (1). χ l : G K Aut Zl (Z l (1)) Z l l. ( 2-3-2 ). χ l Q l χ l. i, Z l, G K χ i l Z l (i) i Tate (i-th Tate Twist)., 7 V O E T, T OE E = V. 5

Q l (i) := Z l (i) Zl Q l. Frob p, p l Z l (i) Q l (i) p i., χ i l (Frob p) = p i. i l (ρ, V ), V (i) V, G K ρ χ i l, V i Tate., µ l 1 (Q l ) Z l., Z l µ l 1 (1 + lz l ), l χ l χ l = ω l χ l,1, ω l : G K µ l 1, χ l,1 : G K 1 + lz l. ω l Teichmüller. l Teichmüller lift. 2-2-2. log l : (1 + lz l ) lz l, x (1 x) n l., n n 1 ( ) 1 log ρ : G K GL 2 (Q l ), g ρ(g) = l χ l,1 (g) 0 1.,. 2-2-3. K, E P 2 K [x : y : z] Weierstrass zy 2 + a 1 xyz + a 3 z 2 y = x 3 + a 2 zx 2 + a 4 z 2 x + a 6 z 3, a 1, a 3, a 2, a 4, a 6 K, K. O K. K L, E(L) := {[x : y : z] P 2 (L) zy 2 + a 1 xyz + a 3 z 2 y = x 3 + a 2 zx 2 + a 4 z 2 x + a 6 z 3 } O E := [0 : 1 : 0] ( [16] )., l n - E[l n ](K) = {P E(K) l n P = O} (Z/l n Z) 2 {E[l n+1 ](K) l E[l n ](K)} n T l (E) := lim E[l n ](K) Z 2 l l l (l-adic Tate module). V l (E) = T l (E) Zl Q l, l ( l-adic rational Tate module). V l (E) Q l 2. l K, G K E[l n ] G K., G K T l (E) V l (E)., ρ E,l : G K Aut Ql (V l (E)) GL 2 (Q l ). V l (E) G K T l (E) ( 2-3-2 ). g A l Tate V l (A), Q l 2g. p l K v, F v. v E D E ( O K ) ρ E,l v (cf. [16] 5.1)., ρ E,l (Frob v ), ( ) 6

.,, det(ρ E,l (Frob v )) = χ l (Frob v ) = F v, tr(ρ E,l (Frob v )) = F v + 1 Ẽ(F v) ([16] )., Ẽ E v. l. 4 ([7]). 2-2-4. l G K., G K,, l., K, G K l. Grothendieck ( [7] )., l, ( 2-2-2)., l. 2-2-5. E V G K., V E[G K ] V 0 = V V 1 V t = {0} V i /V i+1, i = 0,..., t 1 E[G K ] ( - ). {V i /V i+1 } t 1 i=0., E[G K ] t 1 V ss := V i /V i+1 i=0 V. l (ρ, V ) ρ ss., ρ v, trρ(frob v ) = trρ ss (Frob v ), detρ(frob v ) = detρ ss (Frob v )., ρ ρ ss = ρ.,. [4] Chebotarev. 2-2-6. ρ : G K Aut E (V ) S ρ., ρ ρ ss trρ(frob v ), v Σ K \ S ρ.. ρ, ρ : G K Aut E (V ) tr(ρ(frob v )) = tr(ρ (Frob v )), v Σ K \ S, S := S ρ S ρ 7

, ρ ρ., ( ) trρ(g) = trρ (g), g G K. H = G K /(Kerρ Kerρ ), ρ, ρ H., : F := {h H v Σ K \ S such that h = Frob v } {h H trρ(h) = trρ (h)} H H 8. H S, H Chebotarev, F H,, ( ). A = E[G K ] M = V, ρ ρ. 2-2-7 k p 0, A k-, M, M k A. p > 0, p > max{dim k (M), dim k (M )}., tr M (a) = tr M (a), a A, M M A., tr M (a) k M a.. (5 ). 2-2-7-1 2-2-6 S ρ... S ρ 0... ( [4] 1., [20] ). 2-2-8. ρ : G K Aut Ql (V ) Aut E (V E )., E Q l V E G K V E dim E V E = dim Ql V.., V {e i }, Aut Ql (V ) GL n (Q l ). Imρ = Imρ GL n (E ) E /Q l :, Imρ GL n (E ),., Imρ,., (Baire category theorem) 9 E, Imρ GL n (E )., H := Imρ GL n (E ). G K /ρ 1 (H), 8 H E E, h (trρ(h), trρ (h)) E = {(x, x) E E}. 9 X X X. ( ),,,. 8

{g i } r i=1, ρ(g i), 1 i r E E E Q l., Imρ GL n (E). V E = i Ee i. 2.3. l. E Q l, O E, π O. T O n, O 0 T, T., ρ : G K Aut O (T ) l., Aut O (T ) Map(T, T )., O GL n (O)( M n (O) O n2 ). 2-3-1. G K E- V, V G K -,, O T T O E V.. {e λ } λ V, T 0 = λ Oe λ. T 0 V., G K -. ρ : G K Aut E (V ) G K V. V Aut O (T 0 ) Aut E (V )., ρ T 0 H = {g G K ρ(g)t 0 = T 0 } G K, [G K : H]. G K /H {γ i } t i=1. T := t i=1 γ it 0,., G K = t i=1 γ ih, g G K γ i, 1 i t, gγ i = γ ki h, h H, 1 k i t ({k 1,..., k t } = {1,..., t})., ρ(g)t = t ρ(gγ i )T 0 = i=1 t ρ(γ ki )ρ(h)t 0 = i=1 t ρ(γ ki )T 0 = T. i=1, l. 2-3-2. V E-, ρ : G K Aut E (V )., ρ, l V G K -.. 2-3-1., G K - T. ρ : G K Aut OE (T ), ρ OE E = ρ, Imρ Imρ, Imρ Aut OE (T ). π O E, n 1 mod πn U n := Ker(Aut OE (T ) Aut OE /π n O E (T/π n T )). U n Aut OE (T ), 1 = id T., G K /ρ 1 (U n ) Aut OE (T )/U n,., G K ( ) ρ 1 (U n ) G K. Aut OE (T ) gu n, g Aut OE (T ), n 1,., ρ. 2-3-1. 9

2-3-3. Rep E (G K ) G K E, Rep O (G K ) G K O., Rep O (G K ) Rep E (G K ), T T Zl Q l (essentially surjective) 10. 2-3-4. E/K, T l (E) l (cf. 2-2-3)., ρ l : G K Aut Zl (T l (E)) GL 2 (Z l ) l. 2-2-1 l χ l : G K Z l. ( ) l 2-3-5. X/K K., T l := H í et(x K, Z l )/(torsion) (0 i 2dimX) Z l (cf. ). G K T l l. 2.4. l. F l F l (F F l ). F. V F n., ρ : G K Aut F (V ) l. Aut F (V ),., Aut F (V ) GL n (F),. l, l l. ρ : G K Aut E (V ) l., 2-3-1, V G K O E T., ρ : G K ρ Aut OE (T ) mod m E Aut OE /m E (T/m E T ) l. ρ ρ. ρ V ρ ss ( 2-4-1 ). l ρ : G K Aut O (T ), O m, {ρ n : G K Aut O/m n(v n )} n., V n = V O O/m n,. n = 1, l 10, Rep E (G K ) Rep O (G K ). 10

ρ = ρ 1 : G K Aut F (V 1 ) : ρ G K ρ n ρ:=ρ 1 Aut O (T ) mod m n Aut O/m n(v n ) mod m Aut F (V 1 ), F := O/m., {ρ n : G K Aut O/m n(v n )} n, l., l : { } { } l ρ : G K Aut O (V ) {(ρ n, V n )} n ρ = lim n ρ n {ρ O O/m n } n = {ρ n } n. l [20]. l. 2-4-1. l ρ : G K Aut F (V ) S. S., ρ ρ ss tr i ρ(frob v ), v Σ K \ S, i = 1,..., n., ρ l > dim F (V ), ρ.. ρ, ρ l, tr i ρ(frob v ) = tr i ρ (Frob v ), v Σ K \ S ρ S ρ, i = 1,..., n, Chebotarev, G K.,. 2-4-2. k p > 0, A k-, M, M k A. dim k (M) = dim k (M ) =: n., tr i M(a) = tr i M (a), a A, i = 1,..., n ( a A ), M M A.. (5 ). 2-4-3. G K µ l (K) Z/lZ = F l 1 l χ l : G K Aut Z/lZ (µ l (K)) F l l. l χ l. 11

2-4-4. 2-2-3 G K E l E[l](K). ρ E,l : G K Aut Z/lZ (E[l](K)) GL 2 (F l ) l, 2-3-4 l. 2-4-5. K., 1 ρ : G K F l l ( : G K Imρ,, ρ (Z/NZ). N = l t M, l M. l t = 1 ). 2.5.. h Z p Λ = Z p [[T 1,..., T h ]]. Λ (p, T 1,..., T h ). Λ.. S K, G K,S K S. p ρ : G K,S GL n (F p ). F p (A, m A ) ρ : G K,S GL n (A) (ρ, A), ρ (lift) : ρ G K,S GLn (A) ρ mod m A GL n (F p ) A ρ, R(ρ), ρ univ., A ρ, ι R(ρ) A, : ρ univ GL n (R(ρ)) G K,S ρ ρ ι GL n (A) mod m A GL n (F p ) Mazur ( ), Mazur., ρ. Λ/I (I Λ ), Krull h ρ.,. [10]. 12

. R(ρ) (rigid analytic space)x, E X(E), (E Q p ) ρ E p. p ρ X,.,. 2.6. Artin. K Q, V C., ρ : G K Aut C (V ) Artin.,., V, Aut C (V ) GL n (C) M n (C) C n2., Aut C (V ) C n2,.,, ρ(c), (c )., 2 Artin ρ : Gal(Q/Q) GL 2 (C)., ρ (odd),, det(ρ(c)) = 1, 1, Neben-type 11. 2-6-1.., ρ : G K Aut C (V ).. G K., G K. V, Aut C (V ) GL n (C). GL n (C) B I n ( ), 1 2., ρ 1 (B ),, G K H ρ(h) B., ρ(h) = {I n } ( [G : H]<, ). ρ(h) T I n. M n (C) = End(C n ), GL n (C).. T 1, Jordan, T N I n > 1 N., T 2 α, α N 1 > 1 2 N., T N B. Artin,. 2-6-2. ρ : G K Aut C (V )., ρ K. 2-6-3. ρ : G K Aut C (V ) S ρ ( 2-6-2 S ρ < )., ρ trρ(frob v ), v Σ K \ S ρ., trρ(frob v ) ρ(frob v ). 11 K = Q, ρ : G Q GL 2 (C) Artin Khare Wintenberger Serre,. 13

. ρ, ρ : G K Aut C (V ) tr(ρ(frob v )) = tr(ρ (Frob v )), v Σ K \ S ρ S ρ, ρ ρ. ρ, ρ K L, L/K S := S ρ S ρ. L/K Chebotarev, σ Gal(L/K), v Σ K \ S, Frob v = σ., tr(ρ(σ)) = tr(ρ (σ)), σ Gal(L/K), (cf. [13], p.17 3), ρ ρ. 2-6-3. L F (x) = x 3 + ax + b, a, b Q, 3 S 3 = σ, τ σ 3 = τ 2 = 1, τστ = σ 1. ι : S 3 GL 2 (C) ( ) ( ) ζ 3 0 0 1 ι(σ) =, ι(τ) = 0 ζ3 1 1 0., ζ 3 = e 2π 1 3., ρ : G Q L Gal(L/Q) S 3 ι GL 2 (C) 2 ( )Artin. F (x), detρ(c) = detι(τ) = 1. F (x), ρ Maass (Maass form) (cf. [18]). 2-6-4. K = Q( 47) Hilbert H. (1) K 5,, H F (x) = x 5 x 4 + x 3 + x 2 2x + 1 K F ( ) ([22] ( ). ), Gal(H/Q) ( ) D 5. 1 1 3 1 2 1 2 2 2 (2) A =, B =, C = 1 1 1 12 4 6 2 2 2 θ A, θ B, θ C : θ A (τ) = q m2 +mn+12n 2, θ B (τ) = q 3m2 +mn+4n 2, θ C (τ) = m,n Z q 2m2 +mn+6n 2., m,n Z ( 1 + 5 ) f(τ) := θ A (τ) θ B (τ) 2 ( 1 5 2 m,n Z ) θ C (τ) S 1 (Γ 0 (47), χ), χ = ( 47 )., f Hecke ( [11] 6 Hecke [1] p.204 ). ( ) ζ (3) D 5 = σ, τ σ 5 = τ 2 = 1, τστ = σ 1 5 0, σ, τ 0 ζ5 1 ( ) 0 1 2-6-3 Gal(H/Q) Artin ρ. ρ 1 0 f ρ f (cf. [4], [5]), F p f p. 14

3. l,. K. p v Σ K K K v., G Kv := Gal(K v /K v ) G K, σ σ K. G K G Kv G K v D K,v (cf. 2.1 )., G K., G K l ρ : G K Aut E (V ), G Kv ρ GK, v.,. v l. v l [8] p Hodge. 3.1. l, l, l. l, p l., K Q p, E Q l. K F. 2-1 G K G F I K : 1 I K G K G F 1 [9] I K = Gal(K/K ur ), K ur = K(ζ n )., p n ζ n K 1 n., G K I K,. K π, K tm := K ur (π 1 n ) K p n (maximal tamely ramified extension ), K K ur K tm K G K I K P K {1}. I K p P K := Gal(K/K tm ) (wild inertia), I t := I K /P K = Gal(K tm /K ur ) (tame inertia group). ( ): 1 I K G K G F 1, 1 I t G K /I P = Gal(K tm /K) G F 1 I t = lim Gal(K ur (π 1 n )/K ur ) lim Z/nZ(1) = p n p n r p Z r (1), I t τ G F G K /P K σ στσ 1 = τ χ l(τ) (G K /P K ). 15

,. G K ρ : G K = Gal(Q l /K) Aut E (V ) l. l l. [7], l V X (, 2-2-3), V X. Grothendieck SGA 7-I K A l ρ : G K Aut Ql (V l (A)). l ρ Ip.. [15] Appendix.,, SGA 7-I Deligne [2]. 3-1-1.(Grothendieck 12 ) v l, ρ(i K ) (quasi-unipotent matrix) 13.. O E E, π. D v, ρ Imρ., a 1,..., a r 0 x 1,..., x r GL n (E), Imρ = r i=1 (x i + π a i M n (O E )). GL n (O E ), K L, ρ GL GL n (O E ). k 1, I n + π k M n (O E ) GL n (O E ),, L M, g ρ(g M ) g I n mod π k., Imρ g g I n + π k M n (O E )., Imρ l., Imρ. P K I K p, ρ(p K ) = {I n }, ρ IK I t := I K /P K. F K, 1 I t = Gal(K tm /K ur ) Gal(K tm /K) Gal(K ur /K) = G F 1, G F t s Gal(K tm /K ur ) 14, Gal(K tm /K) l χ l : G F Z l, tst 1 = s χ l(t) (cf. [9])., ρ(tst 1 ) = ρ(s χ l(t) ) = ρ(s) χ l(t) ( ρ ). X = log ρ(s), X ρ(t)xρ(t) 1 = log ρ(s) χ l(t) = χ l (t)x. a i (X) X i, a i (X) = a i (χ l (t)x) = χ l (t) i a i (X) 12 Grothendieck. 13 A, m, n 1, (A m I) n = 0., I. 1. 14 G F = Gal(K ur /K) K ur Gal(K tm /K ur ). 16

. K F, χ l., i χ l (t) i 1 t., a i (X) = 0, i 0., X 0, X n = 0. k exp log ρ(s) = ρ(s) n 1 X j, ρ(s) = expx =,. j! j=0 3-1-2. p l, ρ(p K ).. G K, K L, Imρ GL l., ρ GL (P K G L ) p,. [G K, G L ]<, ρ(p K ). 3-1-3.(1) p = l,, ρ(p K ) (quasi-unipotent matrix)., ρ., P K Z p, Z p γ 1, log p ρ(γ a ) a log p ρ(γ a ), ρ Hodge-Tate ( log p χ p (γ a ) [8] ). (2) 3-1-1 l,, ( )., {X z } z D := {z C z < 1}, z D \ {0} 1. z = 0 X 0 ( ) π top 1 (D \ {0}, z) = γ 0 Z 1 H 1 (X z, Z) Z 2., γ 0 z, 0., z = 0 ( ) ρ top z : π top 1 (D \ {0}, z) Aut Z, (H 1 (X z, Z)) = SL 2 (Z).. γ 0 x = 0 (cf. [3] II 4 ). X 0 1,, X 0 ( ). ρ top z. E Qp Q p, E Q ur p Qur p p. E Q ur p C ). SpecZ ur p. Qur p. E SpecZ ur p SpecZur p Zur p. Néron E (cf. [16] Appendix {(p), (0)},, p 0 = SpecF p, p 1 = SpecQ ur p ( ), Spec Z ur p \ {p 0 } = {p 1 } E Q ur p π 1 (SpecZ ur p. p 0 \ {p 0 }) = π 1 (SpecQ ur p ) = Gal(Q p /Q ur p ) = I Qp 17

T l (E p1 ) = T l (E Q ur p ) : ρ l : I Qp Aut Ql (T l (E Q ur p ))., ρ l Q ur p I Q p l Z l (1), γ. D = {z C z < 1} SpecZ ur p 0 p 0 = SpecF p π top 1 (D \ {0}, z) Z γ 0 π 1 (SpecZ ur p \ {p 0 }) l Z l (1) γ {X z } z D E Spec Z ur p H 1 (X z, Z) Z 2 T l (E p1 ) = T l (E Q ur p Z 2 l 3.2. Weil-Deligne. Grothendieck, Weil Weil-Deligne. Langlands. Langlands.. l p, K/Q p, E/Q l. F K, q := F., 1 I K G K ι G F 1. Frob q G F Ẑ, Z-span FrobZ q = {Frob n q n Z} ι W K, K Weil : 1 I K W K ι Frob Z q Z 1. Z G F Ẑ, W K G K. 1 Z W K Φ, Weil W K W K = n Z Φn I K. I K G K, (W K I K )W K. E V, l ρ : G K Aut(V ) W K., t l : I K Q l. I K c Q l, c t l (I K ) = Z l., Imt l l, P K K p, p l t l I t := I K /P K., I K /P K r p Z r (1) ([9]), t l l Z l (1), t l Hom cont (Z l (1), Q l ) = Q l.. 18

, t l : I K Q l, c Q l c t l (I K ) = Z l 15., γ c I K 1 Z l = c t l (I K ). p l., Grothendieck ( 3-1-1) P K ρ ( 3-1-2), I K I ρ(i ) l., ρ I I K /P K l σ I σ = γ c tl(σ)., γ c tl(γ) = γ 1 = γ., ρ(σ) = ρ(γ c tl(σ) ) = ρ(γ) c tl(σ) = exp(t l (σ)n), N = c log(ρ(γ)). 3-1, N., W K Φ n σ, n Z, σ I K., ρ, W K r r(φ n σ) := ρ(φ n σ) exp( t l (σ)n). σ I, r(σ) = ρ(σ) exp( t l (σ)n) = ρ(σ) exp( log ρ(γ) ctl(σ) ) = ρ(σ) exp( log ρ(σ)) = 1, r(i). g W K, σ I K, gσg 1 = σ χl(g) mod P K, t l (gσg 1 ) = χ l (g)t l (σ). g W K, ρ(g)nρ(g) 1 = ρ(g)(c log(ρ(γ)))ρ(g) 1 = log(ρ(gγ c g 1 ))) = log ρ(γ ct l(gγ c g 1) ) = log ρ(γ cχl(g) ) = χ l (g)n., r(g)nr(g) 1 = χ l (g)n, g W K., r Φ t l. 3-2-1.K/Q p, q. Ω 0, V Ω., Weil-Deligne /Ω K Weil W K r : W K g = Φ n σ, n Z, σ I K 17. Aut Ω (V ) 16 N End Ω (V ), r(g)nr(g) 1 = q n N 15 K π, π l n {π 1 l n } n I t σ σ(π l 1 n ), π l 1 n I t Z l (1) Z l. t l. 16 V v, {σ W K r(σ)v = v} W K. 17 Frob geom q : x x 1 q, r(g)nr(g) 1 = q n N q. 19

3-2-2. l p, : { } { Weil W K /Q l 1:1 Weil-Deligne /Q l ρ : W K Aut Ql (V ) (r, N) }. σ W K, ρ(σ) = r(σ)exp(t l (σ)n). 3-2-1 ρ. 3-2-3.(1) Ω 0. n Z, ω n (Φ) = q n, ω n (I K ) = 1 ω n : W K Ω 1 Weil. (2) Ω 0. V = Ω n {e i } n 1 i=0., r(φ)e i = ω i (Φ)e i, Ne i = e i+1, i = 0,..., n 2, Ne n 1 = 0, Φ n σ, σ I K, r(φ n σ) = r(φ n )exp(t l (σ)n) r sp(n), (special representation). Weil-Deligne (r, N) Im(r) I K Φ., Im r., ι : Q l Ω, { } { } 1:1 Weil-Deligne /Q l Weil-Deligne /Ω, V V ι Ω. 3-2-4.(1) W K G K { } { } l /Q l Weil W K /Q l, ρ ρ WK 1:1. W K r G K r(φ) l. (2) W K 18,., W K r., ω s, r ωs 1., ω s ω s (I K ) = 1, ω s (Φ) = q s, s C W K (q s C, Q l ). (3) l = p 3-2-2 Fontaine D pst, (cf. [6]). 1:1. 18. 20

3-2-5. Weil-Deligne (r, N) L L(r, s) := det(1 q s ρ(φ) (KerN) I K ) 1., s., L(ω n, s) = (1 q (s+n) ) 1, L(sp(n), s) = (1 q (s+n 1) ) 1. 3-2-6. (r, N) W K Weil-Deligne., r (Frobenius semisimplification) r ss : r(φ), r(φ) T U., g = Φ n σ W K, σ I K, n Z, r ss (g) := T n r(σ). r = r ss, r. 3-2-7.(1) Weil-Deligne (r, N), N = 0, r, K L, r WL. (2) l Weil-Deligne., 2-2-3 l, Weil-Deligne. [7]. 4. K, Σ K K. l ρ v Σ K, P v,ρ (T ) := det(1 ρ(frob v )T ). 4-1. l ρ (rational), Σ K S, : (i) ρ Σ K \ S (ii) v S ( ), P v,ρ (T ) Q, (ii) P v,ρ (T ) Z, ρ (integral). 4-2. l, l, ρ : G K Aut Ql (V ) ρ : G K Aut (V Ql ) l, l,., ρ, ρ (compatible) S Σ K, ρ, ρ S P v,ρ (T ) = P v,ρ (T ), v Σ K \ S. 21

4-3. l (ρ l ) l (compatible system) 2 l, l, ρ l, ρ l., S Σ K, (ρ l ) l (strictly compatible system) : (i) v Σ K \ S {v Σ K v l}, ρ l v P v,ρl (T ). (ii) l, l, P v,ρl (T ) = P v,ρl (T ), v Σ K \ S {v Σ K v ll } (i),(ii) S (ρ l ) l (exceptional set). 4-4. (a) l (χ l ) l,. (b) 2-2-3 (ρ E,l ) l. E. Neron-Ogg- Shafarevich (cf. [16] 7.1 ).. (c) X Q p. X, SpecZ p X, X (generic fiber) X Spec Zp SpecQ p X, X (special fiber) X Spec Zp SpecF p F p, X l (good reduction). X Q. X p X Qp := X Spec Q SpecQ p. X/Q, SpecZ X/SpecZ X., SpecZ ( ) U X U U (cf. [7] 3.26)., 0 i 2 dim X, V i := H í et (X Q, Q l), G Q Q l., ρ i,l : G Q Aut Ql (V i ),, (ρ i,l ) l. Weil (cf. [7])., SpecZ \ U.. V i GQ p Hí et (X Q p, Q l ) H í et (X F p, Q l ) (cf. [7] 3.25 ). 4-5. (a) [19]. Weil (A 0 ) G ab K l (A 0 )., Weil, [19] Math.review Weil At this point the author takes a step involving what is perhaps the most original idea of the whole paper; he considers any system (M l ) of l-adic representations of g, all of the same degree (l ranging over all 22

primes) satisfying the same set of conditions.. Weil, Weil. (b) l ρ (ρ l ) l., ρ l ρ. l l., Wiles (, ). ( l. 1, 1 ) 4-6. H = = Q 1 + Q i + Q j + Q ij, i 2 = j 2 = 1, ij = ji 2 Q Q 4. H l := H Q Q l l = 2,, M 2 (Q l ). G = {±1, ±i, ±j, ±ij} 4 (quaternion group). K = Q( (2 + 2)(3 + 3)), Gal(K/Q) G. Dedekind., l > 2 ρ l : G Q K Gal(K/Q) G H (H Q Q l ) GL 2 (Q l ) l (. Q l C, ρ l C, Artin., ). 2 ρ 2 : G Q GL 2 (Q 2 ), {ρ l } l. 2 ρ 2, ρ 2 ρ l (l 2). Q 2 C, Q l C, ρ 2, ρ l C ρ 2,C, ρ l,c. Artin. ρ 2 ρ l, 2-6-3 ρ 2,C ρ l,c., G,, r : G Imρ 2 GL 2 (Q 2 ). r M 2 (Q 2 ), r Q 2 Q 2 [G] M 2 (Q 2 ) Q 2 r. r Q 2., r H Q Q 2, M 2 (Q 2 ) Q 2, H Q Q 2 M 2 (Q 2 )., H 2. ρ l, l > 2. Artin ( 4-5 (b) ). 4-7. ([14] I-12 3) 4-6, l ρ l, ρ l l ρ l (l l) ( l. [12] 5.1 ). 23

5. 2-2-7 2-4-2. 3(p.136).. A, A M, End(M) M, End A (M) M A-., B M = {f End(M) fg = gf, g End A (M)} 19. A M A-, A B M. A M,, a A a M. 5-1. A, F i, i I A, (F i F j, i, j I). M = i F i, i, F = F i.,. B M B F, b b F.., well-defined. F M, F p : M M,, p A-., b B M,, bp = pb, b F (F ) = bp(m) = pb(m) p(m) = F, b F End(F ). f End A (F ),g : M M g F = f, 0, x F, b F f(x) = bg(x) = gb(x) = fb F (x)., b F B F., j I, f j : F F j M. b B F, b j = f j b f 1 j,. B Fj., g End A (F j ) g = f i gf 1 j, g End A (F ) b j g = (f j b f 1 j )(f i gf 1 j ) = f j b gf 1 j = f j gb f 1 j = f j gf 1 j f i b f 1 j = g b j, b End(M) b(x) = b j (x), x F j, b., b B M. c End A (M), 1 M End A (M) F j p j, 1 M = p j. j, x F k, x k = f k (y) y k F, cb(x) = cb k (x) = cb k f k (y) = cf k b (y), b = b 1 M = j b j p j, bc(x) = j b j p j cf k (y) = j f j b f 1 j p j cf k (y) 19 B M M. 24

., f 1 j p j cf k End A (F ), b., bc(x) = j p j cf k b (y) = cf k b (y) = cb(x)., x M,, b B M. 5-2. A, M A, b B M., M x 1,..., x n, ax i = b(x i ), i = 1,..., n a A.. i, M M n M n i f i : M M n, M i. b B M, b i := f i bf 1 i B Mi., b End(M n ) b (x) = b i (x), x M i, 5-1, b B M n, M i b i,, b f i = f i b i. M A M n., x := (x i ) i = f i x i M n, Ax i M n. 5-1 1 5, Ax B M n., b (Ax) Ax, a A, ax = b (x), (ax i ) i = ax = b (x) = i b f i x i = i f i bx i = (bx i ) i., ax i = bx i, i = 1,..., n. 5-3. M A, End A (M), M., B M = A M.. b B M. End A (M) M x 1,..., x n, 5-2, a M A M, a M x i = b(x i ), i = 1,..., n., x M x = n i=1 g i(x i ), g i End A (M), a M, b B M, n n n n a M (x) = a M g i (x i ) = g i a M (x i ) = g i b(x i ) = b g i (x i ) = b(x). i=1 i=1, B M A M. i=1 i=1 5-4. M 1,..., M n A,. M = i M i, M i End A (M i )., A n a 1,..., a n, a A, a M Mi = (a i ) Mi 20. 20, (a i ) Mi A End A (M i ) a i A. 25

. M ((a i ) Mi ) i End A (M n ) 5-3. 2-2-7 2-4-2. 2-2-7., A k, M, M k A, k, char(k)> max{dim k (M), dim k (M )}., tr M (a) = tr M (a), a A, M M. S A (S ). S λ, M A λ (λ ) M λ, n λ Z 0. M, M λ, n λ., S H, M = n λ M λ, M = n λm λ λ H λ H. M(resp. M ) A, End A (M) (resp. End A (M ) ). λ H, A N λ., n λ 0 n λ 0, M λ N λ M λ,. c Nµ tr Mλ (a) = tr Nλ (a) = tr M λ (a), a A, 1 A λ H N λ, 5-4, c A c Nλ = 1 = 0, µ λ., tr M (c) = tr M (c), (n λ n λ)tr Nλ (1) = 0., N λ 0,, k 0, tr Nλ (1) = dim k (M λ ) = dim k (M λ ) k., n λ = n λ., k p,, p > d := max{dim k (M), dim k (M )}, tr Nλ (1) k., n λ n λ 0 mod p., 0 n λ n λ d, n λ n λ = 0. 2-4-2.., M, M M = L 1 M p 1, M = L 1 M 1 p., L 1, L 1, M 1, M 1 A, A L 1 L p 1. a A M1, M 1 p, F (a, T ) p, F 1(a, T ) p k[t ] 26

, F (a, T ) p = F 1(a, T ) p. k p,, F (a, T ) = F 1(a, T ). M 1 0, F (a, T ) = F 1(a, T ), a A, dim k (M 1 ) = dim k (M 1), M, M M = L r M pr r, M = L r M r p r, L r L r., M, M k, r, M r = 0, M r = 0.,. 6.,,.,,,,.. References [1] J. A. Antoniadis, Diedergruppe und Reziprozitatsgesetz, J. Reine Angew. Math. 377 (1987), 197 209. [2] P. Deligne, Résumé des premiers exposés de A. Grothendieck, Groupes de monodromie en geometrie algebrique. I. Seminaire de Geometrie Algebrique du Bois-Marie 1967 1969 (SGA 7 I). pp. 1 24. [3],,,,. [4], II,. [5] P. Deligne and J-P. Serre, Formes modulaires de poids 1. Annales scientifiques de l Ecole Normale Superieure, Ser. 4, 7 no. 4 (1974), p. 507-530. [6] J-M. Fontaine, Représentations l-adiques potentiellement semi-stables, Exposé VIII, Astérisque 1994, 224, périodes p-adiques, seminaire de Bures, 1988. [7],,. [8], p-,. [9], - -,. [10],,. [11] A. Ogg, Modular forms and Dirichlet series, 1969, Benjamin New York. [12] A. Pizer, An algorithm for computing modular forms on Γ 0 (N). J. Algebra 64 (1980), no. 2, 340 390. [13] J-P. Serre, Linear representations of finite groups. Translated from the second French edition by Leonard L. Scott. Graduate Texts in Mathematics, Vol. 42. Springer-Verlag, New York-Heidelberg. [14] J-P. Serre, Abelian l-adic representations and elliptic curves. With the collaboration of Willem Kuyk and John Labute. Second edition. Advanced Book Classics. Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989. xxiv+184 pp. [15] J-P. Serre and J. Tate, Good reduction of abelian varieties. Ann. of Math. (2) 88 1968 492 517. [16] J. Silverman, Arithmetic of Elliptic curves, GTM 106. [17] 12. [18] P. Sarnak, Maass cusp forms with integer coefficients. A panorama of number theory or the view from Baker s garden (Zurich, 1999), 121 127. [19] Y. Taniyama, L-functions of number fields and zeta functions of abelian varieties. J. Math. Soc. Japan 9 1957 330 366. [20], p,. 27

[21],,. [22], 2,,. [23] A. Weil, On a certain type of characters of the idele-class group of an algebraic number-field. Proceedings of the international symposium on algebraic number theory, Tokyo and Nikko, 1955, pp. 1-7. 28