平成 19 年度 ( 第 29 回 ) 数学入門公開講座テキスト ( 京都大学数理解析研究所, 平成 19 ~8 年月 72 月日開催 30 日 ) R = T, Fermat Wiles, Taylor-Wiles R = T.,,.,. 1. Fermat Fermat,. Fermat, 17

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1 R = T, Fermat Wiles, Taylor-Wiles R = T.,,.,. 1. Fermat Fermat,. Fermat, 17, Descartes ( ) Corneille ( ), Milton ( ), Velázquez ( ), Rembrandt van Rijn ( ),,,. Fermat, Fermat, Fermat, 1995 Wiles Fermat. Fermat Fermat., Fermat : 1.1. (Fermat, Wiles ) n 3, (1.1) a n + b n = c n , 2, 3,...,. 0 1, 2,.... 1/2 π, 1., Z.,,. Q, R, C., 0.,, R, C. n = Fermat. n = 3 Fermat 100 Euler. n, n 3 n, n n, (*) n 3 n 1 n.. 2 n 2 n 1. 2, 2, 3, 5, 7, 11,....., (*) n = 4 n 3,. n = 4, n p l. l, n = l l. n = 3 Euler Kummer, l 1.1 l. 1

2 Kummer,. Fermat Kummer, 1980, Frey [Fr]. Hellegauarch 1960, Frey. Frey, 1.1 l a l + b l = c l, (1.2) y 2 = x(x a l )(x + b l ). x, y. 2 x, y 1, 2 1 1, 1. 1 Frey. (1.2) E a,b. E a,b Frey. E a,b,. (1.2),, Frey Q. a l + b l = c l, a, b, c 1, a b. E a,b, Frey. E, E E E N E. E E, E = E a,b N E E. Szpiro 1, E E E a,b. Fermat, (1.1),, Szpiro E N E, 2 Szpiro. E E y 2 = f(x), f(x) = 0 16.,.., y 2 = f(x), y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 ( a 1, a 2, a 3, a 4, a 6, 2 )., - Wiles., ( ) Galois. Galois. Galois, Galois., L, -. Fermat Frey, 0 3 a, b, c a + b = c, y 2 = x(x a)(x + b), Szpiro, Masser Oesterlé abc 3 : 1 : ε > 0, C(ε) > 0, Q E E C(ε)N 6+ε E. 2 a5 a N 1+ɛ N 6/5+ɛ, y 2 = x(x a)(x + b) E Szpiro. 2

3 1.2. ε > 0 C(ɛ) > 0 : a, b, c a + b = c, a b 1, a, b, c N, (1.3) c < C(ɛ)N 1+ɛ. Fermat n. abc, (1.3),.. Fery, Fermat Szpiro, - Q. Frey, - Fermat Frey Serre [Se2], Serre. Ribet [R] Serre, Fermat -. -, Langlands, Langlands, Langlands. Langlands,,,. 2.,, Galois Frey Fermat -,,,, Galois.,.,. S, S S. x S x S., x S,. x S y S x = y, x y. T S T S S T. T S, S T S \ T.. S, S S. S S. S, S, S S S S. S S = {(x, x ) x S, x S }. S, S. S S S S. S S f f : S S. f x S x S, f x x ( ), f x x,. x x. x x f f x f(x). S g : S S, x S g(f(x)) S S S, f g g f : S S. f : S S. T S f(t ) = {f(x) x T }. S. f(t ) f T. T = S, f(s) f, Im f. x S, S {x S f(x) = x } f x, f 1 (x ). T S, S {x S f(x) T } f T, f 1 (T ). f : S S, f ( ) S 2 S 2. f : S S, Im f = S.,,, 0. Q, R, C. Z. 3

4 .. p. p.,, 3. Z, Q, R, C. xy = yx.,, 3, M 2 (Z), M 2 (R). 2 G, 2 G G G (g 1, g 2 ) g 1 g 2, 3 (1) : (g 1 g 2 )g 3 = g 1 (g 2 g 3 ), (2) : 1 G g1 = 1g = g g G, (3) : g G gh = hg = 1 h G,.. X, X X,. Aut(X)., R, R = {x R xy = 1, y R}. (M n (R)) GL n (R). ( R), det : M n (R) R. a b n = 2 det( ) = ad bc. GL n (R) = {x M n (R) det(x) R } c d.. X x X π 1 (X, x). x ( γ : [0, 1] X γ(0) = γ(1) = x ). G X, G Aut(X). G X, X, G X G. X G G\X. G, H G, H. K, R. K K, K K. H G, H G h (g hg). H\G, H G, H\G [G : H]. G, G 2, G G. G G G G. R, R 2, R R. R R R R. R R, R R. G, G 2, G G. R, R 2, R R (,, ). f : G G. Im f G. G f 1 (1) f, Ker f. Ker f G. G f Ker f G G. R, det GL n (R) M n (R) GL n (R) R. SL n (R). GL n (R). H G, H\G. G H. H\G G/H.,. Z, Q, R, C. 0., 0. R R,, 1.,. n, n nz Z. Z/nZ,. p Z/pZ 4

5 . F p. X, X, X X. X X, X X. F, (*): ( ) F C,, ( ). F C F Q, [F : Q].,, F, F Q, F Q. F Q [F : Q]. K K, K K [K : K ]. [K : K ] < K K. C Q. Q (*). Q = Q G Q, Q Galois. G Q. Galois Q F G Q H 1 1. F Q H G Q, H = {σ G Q σ(x) = x, x F }, F = {x Q σ(x) = x, σ H},. F Q H G Q, [F : Q] H\G Q. H F Galois G F. F, F Q, F F G F G F. G F G F, F F Galois. Galois, K, K K, K Galois G K, K Galois. k Galois G k Ẑ, Frobenius Frob k G k ( ). 3. Frey E a,b. E a,b Serre Q E a,b G Q. Q E 3 f(x) = x 3 + Ax 2 + Bx + C (A, B, C ), y 2 = f(x). f(x). f(x) 3 f(x) = 0, y 2 = f(x) Q. E(C) = { } {(x, y) x, y C, y 2 = f(x)}. Riemann. Riemann, C. E(C). γ : [0, 1] C,, γ π 1 (E(C), ) γ. π 1 (E(C), ) C ( C ). Λ. P E(C), P γ, γ dx 2y dx 2y. C/Λ E P, γ. P γ E(C) C/Λ,. C/Λ, E(C).. n {}}{ n E[n] = { } {(x, y) E(C) (x, y) + + (x, y) = }. E(C). E[n] n 2. (x, y) E[n], x, y Q 5 dx 2y

6 . E[n] G Q. G Q E[n]. l 3, a l +b l = c l (1) 1.1, E Frey E = E a,b. E a,b [l] G Q,., ,,, Riemann. Riemann E(C), Riemann.. Rimann f : X Y,. Y. f 1 (y) y Y,. f 1 (y) y Y f. F, F S F. F = Q S F {2, 3, 5, 7,...}. 2 F F S F S F. S F. v S F v, v v, v v, v v. F Q, F v, v Q w. w = v, v. v v k v. v v k v k v. v S F v v [k v : k v] = [F : F ]. [k v : k v ] [F : F ] (v k v, k v, [k v : k v ] = 1 ). F /F v. S Q lim F S F. F Q. G Q S Q. F Q, S Q S F. w S Q v S F w v. v S F w v w S Q, G F w G Fv. F v. G Fv w, w w. F v, F F v, G Fv F v Galois. F = Q, v = p F v = Q p p. 10,, 2 3. p. p, 0 p 1 p. p,, p.. p Q p. Z p. n 1 p n., n. p, p,, p Q p p 4 [ ], [ ], [ - - ]. 6

7 . F v p, F v Q p. p. v F v R C. F, O F F. F, F v O Fv F v. v p, O Fv Z p. O Fv F v k v. G Fv G kv ( v G kv = 1 ). I Fv G Fv. F v, I Fv = G Fv. F S, v S I F,v G F ( ) G F G F,S. v S, G Fv G F G F,S G Fv /I F,v = Gk.v. G kv G F,S Frob kv G kv Frob v. 5. Galois,. 5 C (1) Obj(C), (2) Obj(C) X, Y Hom C (X, Y ), (3) Obj(C) X id X Hom C (X, X), (4) Obj(C) X, Y, Z (5.1) Hom C (X, Y ) Hom C (Y, Z) Hom C (X, Z).. Obj(C) C, Hom C (X, Y ) X Y.,. f Hom C (X, Y ) f : X Y. (5.1), 2,, 2,. f : X Y g : Y Z g f : X Z.,., G, (1) Obj(C) = {X}, (2) Hom C (X, X) = G, (3) id X = 1, (4) Hom C (X, X) Hom C (X, X) = G G G = Hom C (X, X) 4.. C f Hom C (X, Y ), g Hom C (Y, X), f g = id Y, g f = id X.. C X, Y, Hom C (X, Y ) Isom C (X, Y ). X = Y, Hom C (X, X) = End C (X), Isom C (X, X) = Aut C (X). Hom C, End C C Hom, End. G C X G Aut C (X).,. Obj(C), Hom C (X, Y ) X Y, (5.1). Obj(C), Hom C (X, Y ) X Y, (5.1).,,.,,,.,. X, Y, Hom(X, Y ). X End(X). R, X, R End(X) (X, R End(X)) ( ) R. (X, R End(X)) X. X = R, a R x ax X X R End(X), X = R R 5 [ ], [ ]. 7

8 . X, Y R, X Y R. X X = X 2, X 2 X = X 3,..., n 0 X n. X 0 = {0}. X, Y R, f : X Y, R End(X) End(Y ) Hom(X, Y ) X Y R. R R, R. X Y R Hom R (X, Y ). Hom(X, Y ). R, Hom R (X, Y ) R. n 0. R X, n R, R n X R. K, K, R K, X, Y K, X Y K X Y K. n K n K. 6. Galois G, X G ρ : G Aut(X) (X, ρ) G. (X, ρ) X ρ. R, R X G G Aut(X) R[G] G R. K, Galois G K, ( ) R ( ) K R Galois., K R Galois (X, ρ : G K Aut(X)), X R n R, ρ K R n Galois. l, K, l Galois K mod l Galois. F, v F. F v R M G Fv Aut R (M) G Fv G kv. F R M v, M G Fv. S F, M S v S M. M S G F End R (M) G F G F,S. F, K l, V F K n Galois. v F, I F,v V I F,v = {x V σ(x) = x, σ I F,v } F v K n Galois. F v q v. l F w V F w de Rham., V L s L(V, s) Euler L(V, s) = v det(1 Frob v q s v ; V I F,v ) 1 w l L w (V, s). v F l. L w (V, s). Galois V, Euler s ( ), s. Galois V L(V, s),,, L(V, s) L. 8

9 7. H = {x + 1y x, y R, y > 0}.. ( Γ ) SL 2 (Z) a b. k 0, H f, Γ c d f((az + b)/(cz + d)) = (cz + d) 2k f(z),, Γ k. Γ k f S k (Γ). C. Γ, Γ SL 2 (Z) Γ Γ, S k (Γ) S k (Γ ). N 1. {( ) } a b Γ 1 (N) = SL 2 (Z) N c, N (d 1) c d ( N c c N,. N (d 1) ). SL 2 (Z). k 0, f S k (Γ 1 (N)). f(z + 1) = f(z), f(z) = n 1 a n(f)q n. q = e 2π 1z, n a n (f) C. f q. C R, Γ 1 (N) k f, q f(z) = n 1 a n(f)q n, n 1 a n (f) R S k (Γ 1 (N), R). R. f S k (Γ 1 (N), R). N p, H T p f p 1 (T p f)(z) = f(pz) + f((z + i)/p), T p f S k (Γ 1 (N), R). d (Z/NZ), γ = c mod n = 0, d mod n = d, H d f i=0 ( a c ) b SL 2 (Z) d ( d f)(z) = (cz + d) 2k f((az + b)/(cz + d)), d f d, S k (Γ 1 (N), R). T p (p N ) d (d (Z/NZ) ) End(S k (Γ 1 (N), Z)) Hecke.. S k (Γ 1 (N), C) N, k,. f(x) 3 f(x) = 0. E, y 2 = f(x) Q. {(x, y) x, y {0, 1,..., p 1}, y 2 f(x) p }.. p a p (E). E f(x), a p (E) f(x), f(x) a p (E) p Wiles [W], Taylor-Wiles [TW], [BCDT]. 9

10 7.1. E Q. N 1 6, N, 2 f = a n (f)q n, p, a p (f) = a p (E). 8. Galois N, k 1, f = n 1 a n(f)q n S k (Γ 1 (N), C) N, k. K(f), a n (f) C. K(f). K(f) λ, O f,λ K(f) λ. f Q O f,λ Galois (V f,λ, ρ f,λ : G Q Aut(V f,λ )). Galois (1) V f,λ 2 Galois, O f,λ V f,λ 2 O f,λ. (2) S f N, (V f,λ, ρ f,λ ) S. (3) p S, Frob p V f,λ V f,λ a p (f). f Galois V f,λ k = 2 Eichler [E] - [Sh1], k 2 [Sh2], Deligne [De], k = 1 Deligne-Serre [DS]. Deligne k 2. Riemann Γ 1 (N)\H...,. [ C, C op. C., Obj(C op ) = Obj(C), Hom C op(x, Y ) = Hom C (Y, X). C, C 2. C C, (1) Obj(C) X Obj(C ) F (X), (2) Obj(C) X, Y F : Hom C (X, Y ) Hom C (F (X), F (Y )) 2.. g f F (g) F (f) = F (g f). C C F F : C C. C op C C C. G F (G) = G, f : G G F (f) = f, F. C, X 0 C, C X F (X) = Hom C (X 0, X), C f : X Y g : X 0 X f g : X 0 Y Hom C (X 0, X) Hom C (X 0, Y ), C. h X0. C C 2 F, F, Obj(C) X F (X) F (X), C f : X Y F (X) F (Y ) F (X) F (Y ). 2 C, C, F : C C F : C C F F C, F F C.,, {0}, {0, 1}, {0, 1, 2},..., 6 N E NE. 10

11 .,. C F, C X, F h X. X. C, C, F : C C. F : C C F, C X, C Y Hom C (X, F (Y )) = Hom C (F (X), Y ),. C op C 2. F. F : C C F, C X, C Y Hom C (F (X), Y ) = Hom C (X, F (Y )),. F. R, R, R R. R (M, R End(M)), R R End(M) R. R R.. R M R R M. R R M,. R R n, R n. R, R R R M M R R..,,. 7. R, Spec R. S, X f : X S (X, f) S. (X, f) X., f(x) 3 f(x) = 0 y 2 = f(x) Q. S, S. S = Spec Q, Spec Q Q., S, S E, S S E (E, S E). (E, S E) S E S E. S, S. S E. (E, S E) S. S T, S T E E(T ). E(T ). N 1. N S, S E E(S) x, (1) x N, (2) 1 N < N S S x E(S ) N, (E, x) N. N 5. N Y 1 (N). Y 1 (N)(C) = Γ 1 (N)\H. f : E univ Y 1 (N). Y 1 (N) X 1 (N). S k (Γ 1 (N)) X 1 (N) Spec C. l, k 2, Y 1 (N) Z[1/Nl] l Sym k 2 (R 1 f Q l ). X 1 (N) Q V 1 (N) Hecke T G Q. 7 [ ], [ ]. 11

12 N, k 1, f = n 1 a n(f)q n S k (Γ 1 (N)) N, k, λ l K(f). Hecke V f,λ,q = Hom T K(f) (K(f)f, V 1 (N) Ql K(f) λ ) = Q K(f) λ 2 Galois. K(f) λ ι : V f,λ,q K(f) 2 λ, ι 1 (O 2 f,λ ) Q O f,λ 2 Galois. Galois (V f,λ, ρ f,λ ). k f,λ O f,λ. V f,λ Of,λ k f,λ G Q k f,λ 2 Galois. Galois ρ f,λ. 8.1(3),. Hecke T p, Frob p,,. 8.1(3), Galois V f,λ L L(V f,λ, s), f L L(f, s), Nl Euler,. L(V f,λ, s) L(f, s), Nl Euler E Q. n 1, E[n] Q Z/nZ 2 Galois (E[n], ρ E,n ). l, E[l n ] Q Z l 2 Galois (T l E, ρ E,l ), E l Tate. - ( 7.1) E. f E N, 2., K(f) = Q. T l E Zl Q l V f,l Zl Q l. K l, O. (V, ρ : G Q Aut(V )) Q O 2 Galois. ρ, k 1, N 1, k, N f, K(f) λ, ( ) O f,λ O, V O K G Q K V f,λ Of,λ K. κ K. (V, ρ : G Q Aut(V )) Q κ 2 Galois. ρ, k 1, N 1, k, N f, K(f) λ, ( ) O f,λ O, V O k G Q κ ρ f,λ F O κ 2 Galois. F, F C R. Serre [Se2] : 9.2. κ, S. (V, ρ : G Q Aut(V )) Q κ 2 Galois, : (1) ρ, (2) ρ S, (3) c G Q det(ρ(c)) = 1, ρ. Serre 9.2 (1) (3) ρ, k(ρ), N(ρ), κ. S. (V, ρ : G Q Aut(V )) Q κ Galois, 9.2 (1), (2), (3). N(ρ), k(ρ) f, K(f) λ, O f,λ κ, ρ V f,λ Of,λ κ. Ribet [R] Jacobian Neron Mazur, 2, N f, N p, N K(f) λ, ρ f,λ p, ρ f,λ = ρf,λ N/p 8 k(ρ), N(ρ), Serre ε : (Z/N(ρ)Z) C,. 12

13 f K(f ) λ., ρ, ρ 9.2 ρ 9.3..,, ρ, ρ 9.2 ρ , Khare Wintenbeger ([KW1], [KW2])., Fermat - Ribet. - ( 7.1) 1.1. l, a l + b l = c l (1) 1.1, Frey E a,b. - E a,b E a,b. Q F l Galois E a,b [l] ρ, ρ. ρ Ribet, 9.3 ρ. N(ρ) 1 2, k(ρ) , 2. S 2 (Γ 1 (1)) = S 2 (Γ 1 (2)) = {0} , - ( 7.1). (MLT), Langlands-Tunnell, (3, 5) trick 3. MLT R = T. MLT, ρ E,l ρ E,l G Q( l ) ( l = ( 1) l 1 2 l) ρ E,l., Kisin [Ki1] MLT ( 14.1). Langlands-Tunnell, ρ E,3,. MLT, ρ E,3 G Q( 3) E. ρ E,3 G Q( 3),, E, ρ E,5 G Q( 5). E E, ρ E,5 G Q( 5) E. MLT, ρ E,5 E. (3, 5) trick, Q E ρ E,5 = ρe,5 ρ E,3 G Q( 3). E, ρ E, Langlands R = T Langlands. Langlands.. 9., A A Q, R A f. Q A. F, A F = A Q F, F. A F, F A F. A F A F F.., F F \A F 1 G F 1. G F.. A F F. G(F )\G(A F ) C G(A F ) 9 [ - - ], [ ]. 13

14 . G(A F ) C, G(A F ) C. G(A F ) C G(A F ). Langlands G(A F ) F Galois. C Lie Dynkin, Q C,. 4 (X, Φ, X, Φ ). (X, Φ, X, Φ ), (X, Φ, X, Φ). (X, Φ, X, Φ ). F p. F Weil. G F, W F G F.,. G F. G F F (X, Φ, X, Φ ). Ĝ (X, Φ, X, Φ) C. Ĝ G F. W F W F Ĝ. Ĝ(C) W F L G, G L. 10 F, n 1, GL n,f F., GL n,f L GL n (C) W F. Langlands, W F L F. F L F F Langlands. F p L F F Weil-Deligne. F p L F W F, F L F, L F, L F. L F W F. F, G(A F ) A(G). F p, G(F ) A(G). Langlands, A(G) ( ), L F L G, L F W F. L G,. Langlands, G, H F, L H L G, W F L H L G W F, A(H) A(G). 1 1,.,. Langlands, Jacquet-Langlands. p F GL n, inner form F ( L GL n W F ), F /F, χ : G F Q/Z, G F = {σ G F χ(σ) = 0} F F, F 2 GL n,f Res F /F GL 11 n,f. 10 H G, G H GoH : (1) GoH = G H, (2) H G α : H Aut(G), G o H (g 1, h 1 )(g 2, h 2 ) = (g 1 α(h 1 )(g 2 ), h 1 h 2 ). 11 ResF /F Weil. 14 W F

15 12. GL 2 (A) Γ 1 (N) SL 2 (Z), GL 2 (A f ) K 1 (N), K 1 (N) SL 2 (Q) = Γ 1 (N).. GL 2 (A),. GL 2 (A).,., GL 2 (A) = GL 2 (Q)GL 2 (R) + K 1 (N). GL 2 (R) + = {g GL 2 (R) det(g) > 0}. GL 2 (A f ) (C \ R). GL 2 (Q), γ GL 2 (Q), g GL 2 (A F ), z C \ R, γ (g, z) = (γg, az+b cz+d ). GL 2(A f ) (C \ R) C f, 3 (1) g GL 2 (A f ) C \ R z f(g, z) z, (2) K GL 2 (A f ), ( ) (g, z) GL 2 (A f ) (C \ R), h K f(gh, z) = f(g, a b z), (3) γ = GL 2 (Q), (g, z) GL 2 (A f ) (C \ R) c d f(γ (g, z)) = (cz + d) k f(g, z), A 0 (k), A 0 (k) GL 2 (A f ). f S k (Γ), GL 2 (A f ( ) (C ) \ R) C f, (g, z) GL 2 (A f ) (C \ R), (g, z) = γ (h, z ) a b γ = GL 2 (Q), h K 1 (N), z H, f(g, h) = det(γ) k 2 (cz + d) k f(z ) c d, well-defined, f A 0 (k). f f C- S k (Γ 1 (N)) A 0 (k), S k (Γ 1 (N)) A 0 (k) K 1 (N)- A 0 (k) K. f A 0 (k), GL 2 (A) C φ ef, g GL 2 (A f ), h = φ ef (gh) = det(h) k 2 (c 1 + d) k f ( a c ( g, a ) 1 + b c 1 + d ) b GL 2 (R) d, well-defined, φ ef GL 2 (Q)\GL 2 (A). f φ ef A 0 (k) GL 2 (A). G H, Ind, G H, G H. Z H. Z, H H Aut(Z) = {±1} h H 1. End(IndZ). X G, X H X H = {x X h(x) = x, h H} End(IndZ). X X H, End(IndZ), G 1 1. G = GL 2 (A f ), H = K 1 (N), S k (Γ 1 (N)) G A 0 (k) H, End(IndZ). 7 Hecke End(IndZ) End C (S k (Γ 1 (N))). 13. f Galois V f,λ, l, Galois. 15

16 ,, Q G R h : S G Q R (G, h),. S Deligne R., h. [HT] Kottwitz, Galois., Lefschetz Selberg, Selberg. [I], Langlands [L].,. Ngo., Galois. 14. MLT R = T, (MLT). MLT. F. K l, O K, κ K. ρ F O 2 Galois, ρ = ρ O κ. MLT, ρ, ρ ρ,. Taylor-Wiles [TW], Wiles [W]. Wiles F = Q, ρ. MLT,. MLT, [Ki1] F = Q, l > 2, ρ, ρ : (1) ρ Q( ( 1) l 1 2 l), (2) ρ, (3) ρ G Ql Barsotti-Tate, 12. ρ ρ. Khare-Wintenberger Serre l = 2. (3), ρ ρ O K = ρ f,λ Of,λ K f 2, Serre f ρ MLT. ρ f. F Q Galois Gal(F/Q) := G Q /G F., ρ, ρ G F. F G F, 2 : (1) l ρ, l ρ. (2) Skinner-Wiles f ρ F, ρ F, MLT R = T. 15. Galois R = T R Galois ρ. Galois Mazur, Galois R. Mazur R, R T. R = T Wiles Taylor-Wiles 12 Ql K ρ, K Barsotti-Tate Galois. Barsotti-Tate, l p, Barsotti-Tate p. 16

17 , R T, Mazur. Galois, Galois.,.,,.,,, S S. Galois Galois. Galois,, K l, O K, κ K., Artin O κ AR. F. (V, ρ) F κ 2 Galois. AR A, F A 2 Galois (W, ρ) G F W A κ = V. ρ ( ). R. F. κ V = κ 2, ρ G F GL 2 (κ). AR A, G F GL 2 (A) GL 2 (A) GL 2 (κ) ρ. ρ. R., AR A, F A Galois (W, ρ) G F W A κ = V A W = A 2, W A κ = κ 2 W A κ = V = κ, 3. F, S F l. ρ F κ Galois, l F. Σ S l., v S \ Σ, (1 q v )((1 + q v ) 2 det ρ(frob v ) q v (trρ(frob v )) 2 ) κ. Σ. AR A, S F A 2 Galois (W, ρ) G F W A κ = V, v Σ A W = A 2, W A κ = κ 2 W A κ = V = κ. ρ. RF,S.. A Obj(AR) 2 Galois (W, ρ) det ρ O ψ. ψ R ψ, R,ψ, R,ψ F,S.. Galois. G G M, i H i (G, M). i, M H i (G, M) G. H G,

18 H i (G, M) H i (H, M).,. 14,. f 2 f 1 f 0 f 1 M 1 M0 M1, n Im f n 1 = Kerf n.. 0 M m M n 0, M m,..., M n, n i=m ( M i) ( 1)i = 1. F Galois G F G F,S, v S F G Fv. G, G. G, G X G G\X G\X. Galois, G F X G Fv X,, H i (F, X), H i (F v, X). G F,S X Galois H i (G F,S, X). 15 M = {x End κ (V ) tr(x) = 0}. ρ, v S L v H 1 (F v, M), R F,S O Galois [ (15.1) Ker H 1 (G F,S, M) ] H 1 (F v, M)/L v v S. G F,S M, (15.1) M Selmer. (15.1).,. 16, 3,.. K p. M G K,. i 3 H i (K, M) = 0. i = 0, 1, 2, H i (K, M). Q/Z(1) 1 G K. Brauer H 2 (K, Q/Z(1)) Q/Z, M (1) = Hom(M, Q/Z(1)). Hom G K, M (1) G K., H i (K, M (1)) Hom(H 2 i (K, M), Q/Z).. Tate-Poitou. F, S F. M G F,S,, M F S., i H i (G F,S, M), 0 H 0 (G F,S, M) v S H0 (F v, M) H 2 (G F,S, M (1)) H 1 (G F,S, M) v S H1 (F v, M) H 1 (G F,S, M (1)) H 2 (G F,S, M) v S H2 (F v, M) H 0 (G F,S, M (1)) 0 14, [ ]. 15, Galois. 16, [NSW]. 18

19 . A A = Hom(A, Q/Z). Tate-Poitou. 16. R = T Mazur R, R.,, R T,. Wiles [W] Taylor-Wiles [TW], R T, -., Diamond [Di], [Fu] Taylor-Wiles, T Taylor-Wiles. [Fu] 2 Galois Wiles, Taylor-Wiles. R = T Kisin [Ki1]. R = T. F. R, F Galois ρ., R R F,S, R F,S,, R. R = T T Hecke. R N, 2 f Galois ρ f,λ ( ), T Hecke End(S 2 (Γ 1 (N), Z)) ρ f,λ. T, F 4 F G., Diamond [Di], [Fu] Taylor-Wiles, T. G, G(F R), G(A F ), T. G T Jacquet-Langlands ρ, F F Galois. F T Galois, R R T. T Galois, l. F 4 F G. R T ( ). R T 2., T ρ, R = T Langlands,., Hecke T L, Bloch-, R Galois Selmer Galois L, R T. R T ( ), Taylor-Wiles. Kisin [Ki1], Taylor-Wiles, Wiles [W], R = T. [Ki1],, GL 2, 19

20 , GL 2 R = T., -Tate. Taylor-Wiles. Spec R. R T,, R, T, R T., R. R, Selmer. Bloch-, Selmer Galois L. R, L. Tate-Poitou. 17 Taylor-Wiles, R, T, R T,, Spec R,, R T., Taylor-Wiles O, Kisin [Ki1], R Σ,. Kisin R = T,,., l, l. Breuil [Br],, Frobenius. ρ. Galois. ρ l F Galois. l p. G G G G, S G G S G G, S. p p Dieudonne., p p Fontaine Breuil [Br]. p p Galois, ρ f,λ., 2, p, p. p Fontaine. Wiles [W] Fontaine-Laffaille [FL]. Fontaine p Hodge. Fontaine Q p Galois, p Galois, 2 R = T., Fontaine (ϕ, Γ) [Fo], p Galois, p Langlands. 17 mod l ρ. 20

21 Serre R = T Serre -Tate. 18 Serre. 9.3., Khare-Wintenberger.. 2 (A), (B). (A) MLT. Khare-Wintenberger Kisin [Ki1] MLT ( 14.1), 2 l + 1 ( 14.1 ) MLT l 2, ρ l = 2 MLT, Kisin [Ki2] l = 2 MLT. (B), ( ) mod l compatible system,., Khare-Wintenberger [KW2]. Böckle [Bö] Taylor [T2], ρ O Galois ρ. ρ, Taylor, Q Galois F, ρ G F. Carayol [Ca], Taylor [T1], [Sa], Kisin [Ki5] Hilbert l Langlands,, ρ G F compatible system. Brauer ρ compatible system. (A), (B) 9.2. ρ. (B), ρ compatible system. l, compatible system mod l mod l. l, l.. (B), (B),.., (A). (A) p p Galois p Hodge.,,. Khare-Wintenberger MLT, [Ki3] [Ki4],,, Serre Tate E Q. E y 2 = f(x), Λ C, p, a p. a p 2 p < a p < 2 p. {x C xλ Λ} Z E. E,. -Tate E.. {( x SU(2) = y ) y x, y C, x 2 + y 2 = 1} x 18 R = T Fontaine-Mazur, Artin, GL2 type. 21

22 . x, y x, y. SL 2 (C). SU(2) Haar, SU(2). M 2 (C) C α : SU(2) [ 2, 2] E Q., I = [a, b] [ 2, 2],,. a p, a p / p I N lim N N α 1 (I) SU(2) = 1 2π b a 4 t2 dt K E, E j j(e) K. E y 2 = x 3 +Ax 2 +Bx+C, j(e) = 32(A 2 3B) 3 /(4A 3 C A 2 B 2 18ABC +4B 3 +27C 2 ). Harris-Shepherd-Barron-Taylor [HSBT], Taylor [T3] E Q, j(e). ( E ) E. -Tate Serre L ([ ] ): E Q. l, V Q Q l 2 Galois T l E Zp Q p., n 1, Q Q l n + 1 Galois Sym n V L L(Sym n V, s) Re(s) 1 + n/2,, E F CM, (1) F R, (2) F F +, F + [F : F + ] = 2,. F + F. F ( CM ). [CHT] GL n (A F ) RAESDC ( RACSDC ). GL n (A F ) RAESDC ( RACSDC ) π = v π v, v π v 2, F Q l n Galois ρ π F l n Galois ρ π, 13 Clozel [Cl], [CL], Harris-Taylor [HT]. ρ F Q l n Galois. ρ, GL n (A F ) RAESDC ( RACSDC ) π (, v π v 2 ), ρ ρ π. F F l n Galois ρ. Harris-Shepherd-Barron-Taylor [HSBT], ( Brauer C, ) E Q, j(e), l, n. F, Q Galois, m n, Sym m T l E G F. [CHT] R = T Kisin [Ki1], Taylor [T3] MLT. 22

23 18.5. F CM, n 1, l > n F/Q, ρ F Q l n Galois ESD: ρ σ = ρ ɛ 1 n, RAM: ρ. FL: l F, ρ G Fv Fontaine-Laffaille [FL], Hodge-Tate 1. DS: l, F v, ρ v Frobenius,., ρ mod l ρ, ρ, ρ., Galois MLT F, n 1, l > n F/Q l, ρ F Q l n Galois ESD: G F χ, χ G Fv F, ρ σ = ρ ɛ 1 n, RAM: ρ. FL: l F v, ρ G Fv Fontaine-Laffaille [FL], Hodge-Tate 1. DS: l, F v, ρ v Frobenius,.., ρ mod l ρ, ρ ρ Kisin [Ki1],, G(F + Q R) F + G, Hecke, R = T T G(F + Q R), Taylor-Wiles T. R T, T Galois. T 1. T, Kottwitz,, T, F Galois. Galois ([CHT, Chapter 1]), T Galois. R R T. Kisin [Ki1] Taylor-Wiles R = T, 18.5., Q Galois F l, m n Q Galois Sym m ρ E,l F,, 18.4, (3, 5)., F, Moret-Baily [MB]... 23

24 19.,. 19 References, [Bö] Böckle, G.: A local-global principle for deformations, J. Reine Angew. Math. 509, (1999) [Br] Breuil, C.: Groupes p-divisibles, groupes finis et modules filtrés, Ann. of Math. 152, (2000) [BCDT] Breuil, C., Conrad, B., Diamond, F., Taylor, R.: On the modularity of elliptic curves over Q: wild 3-adic [Ca] exercises, J. Amer. Math. Soc. 14, (2001) Carayol, H.: Sur les représentations p-adiques associées aux formes modulaires de Hilbert, Ann. Sci. École Norm. Sup. 19, (1986) [Cl] Clozel, L.: On the cohomology of Kottwitz s arithmetic varieties, Duke Math. J. 72, (1993) [Co] [CHT] [CL] Colmez, P.: Sur la correspondence de Langlands p-adique pour GL 2 (Q p ), preprint Clozel, L., Harris, M., Taylor, R.: Automorphy for some l-adic lifts of automorphic modl Galois representations, preprint (2006) Clozel, L., Labesse, J.-P.: Changement de base pour les représentations cohomologiques des certaines groupes unitaires; appendix to Cohomologie, stabilizations et changement de base, Astérisque 257 (1999) [De] Deligne, P.; Formes modulaires et représentations l-adiques, Semin. Bourbaki 1968/69, no. 355, (1971) [Di] Diamond, F.: Taylor-Wiles construction and multiplicity one, Invent. Math. 128, (1997) [DDT] Darmon, H., Diamond, F., Taylor, R.: Fermat s last theorem, in R. Bott, A. Jaffe, S. T. Yau (ed.): Current developments in Mathematics, 1995, International Press (1995) [DS] Deligne, P., Serre, J.-P.: Formes modulaires de poids 1, Ann. Sci. École Norm. Sup. 7, (1974) [DT] Diamond, F., Taylor, R.: Nonoptimal levels of mod l modular representations, Invent. Math. 115, (1994) [E] Eichler, M.: Quaternäre quadratische Formen und die Riemannsche Vermutung für die Kongruenzzetafunktion, Arch. Math. 5, (1954) [Fo] Fontaine, J.-M.: Representations p-adiques des corps locaux, Grothendieck Feshschrift II, Progress in Math. 87, [Fr] (1991) Frey, G.: Links between stable elliptic curves and certain Diophantine equations, Ann. Univ. Sarav. Ser. Math. 1, no. 1, 1 40 (1986) [Fu] Fujiwara, K.: Deformation rngs and Hecke rings in totally real case, preprint (1997) [FL] Fontaine, J.-M., Laffaille, G.: Construction de représentations p-adiques, Ann. Sci. École Norm. Sup. 15, [HSBT] (1982) Harris, M., Shepherd-Barron, N., Taylor, R.: A family of Calabi-Yau varieties and potential automorphy, preprint (2006) [HT] Harris, M., Taylor, R.: The geometry and cohomology of some simple Shimura varieties, Ann. Math. Stud. 151, Princeton University Press, Princeton, NJ (2001) [I] Ihara, Y.: Hecke Polynomials as congruence ζ functions in elliptic modular case, Ann. Math. (2) 85, [Ki1] [Ki2] [Ki3] [Ki4] [Ki5] (1967) Kisin, M.: Moduli of finite flat group schemes and modularity, preprint Kisin, M.: Modularity of 2-adic Barsotti-Tate representations, preprint Kisin, M.: Modularity for some geometric Galois representations, preprint Kisin, M.: The Fontaine-Mazur conjecture for GL 2, preprint Kisin, M.: Potentially semi-stable deformation rings, preprint [KW1] Khare, C., Wintenberger, J.-P.: Serre s modularity conjecture: The case of odd conductor (I), preprint (2006) [KW2] Khare, C., Wintenberger, J.-P.: Serre s modularity conjecture: The case of odd conductor (II), preprint (2006) [Ma] Mazur, B.: Deforming Galois representations, in Y. Ihara, K. Ribet, J.-P. Serre (ed.): Galois group over Q, MSRI Publications 16, Springer Verlag (1989) 19,.. 24

25 [MB] Moret-Bailly, L.: Groupes de Picard et problèmes de Skolem, II, Ann. Sci. École Norm. Sup. 22, (1989) [L] Langlands, R. P.: Modular forms and l-adic representations, in Modular functions of one variable II, Proc. internat. Summer School, Univ. Antwerp, 1972, Lect. Notes Math. 349, Springer-Verlag, Berlin-Heidelberg- New York (1973) [R] Ribet, K.: On modular representations of Gal(Q/Q) arising from modular forms. Invent. Math. 100, [Sa] (1990) Saito, T.: Hilbert modular forms and p-adic Hodge theory, preprint [Se1] Serre, J.-P.: Points d order fini des courbes elliptiques, Invent. Math. 15, (1972) [Se2] Serre, J.-P.: Sur les representations modulaires de degré 2 de Gal(Q/Q), Duke Math. J. 54, (1987) [Sh1] Shimura, G.: Correspondances modulaires et les fonctions ζ de courbes algébriques, J. Math. Soc. Japan 10, [Sh2] 1 28 (1958) Shimura, G.: An l-adic method in the theory of automorphic forms, the text of a lecture at the conference Automorphic functions for arithmetically defined groups, Oberwolfach, Germany, July 28-August 3, 1968, in Collected Papers Vol. II, [T1] Taylor, R.: Galois representations associated to Siegel modular forms of low weight, Duke Math. J. 63, (1991) [T2] Taylor, R.: On the meromorphic continuation of degree 2 L-functions, preprint (2001) [T3] Taylor, R.: Automorphy for some l-adic lifts of automorphic modl Galois representations II, preprint (2006) [TW] Taylor, R., Wiles, A.: Ring theoretic properties of certain Hecke algebras, Ann. of Math. 141, (1995) [W] Wiles, A.: Modular elliptic curves and Fermat s last theorem, Ann. of Math. 141, (1995) [Book] ( ) Cornell, G., Silverman, J. H., Stevens, G. (ed.): Modular forms and Fermat s last theorem, [NSW] Neukirch, J., Schmidt, A., Wingberg, K.: Cohomology of number fields, Grundlehren der mathematischen Wissenschaften 323. Springer-Verlag, Berlin Heiderberg (2000) ( ) [ ] I, II,. (1976, 1977) [ - - ],, : 2, 10. (2000) [ ], N.: (, ). (2006) [ ] : Fermat 1, 11. (2000) [ ] J. H. :, ( ). (2003) [ - ] J. H., J. : (,,, ). (1995) [ ] J.-P. : l ( ), (1999) [ ] : 2. (1971) [ ] J. : (, ). (2003) [ ] R. : (, ). (2004, 2005) [ - - ],, :. (2004) [ ], S.: ( ). (2005) [ ], D.: ( ). (2006) [ ], (2007) 25

Mazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R M End R M) M R ut R M) M R R G R[G] R G Sets 1 Λ Noether Λ k Λ m Λ k C Λ

Mazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R M End R M) M R ut R M) M R R G R[G] R G Sets 1 Λ Noether Λ k Λ m Λ k C Λ Galois ) 0 1 1 2 2 4 3 10 4 12 5 14 16 0 Galois Galois Galois TaylorWiles Fermat [W][TW] Galois Galois Galois 1 Noether 2 1 Mazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R

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