wiles05.dvi

Size: px

Start display at page:

Download "wiles05.dvi"

あきたけかいじ
5 years ago
Views:

1 Andrew Wiles 1953, 20 Fermat.. Fermat 10,. 1 Wiles , Fermat 1. (Fermat). p 3 x p + y p =1 xy 0 x, y 2., n- t n =1 ζ n Q Q(ζ n ). Q F,., F = Q( 5) 6=2 3 = (1 + 5)(1 5) 2. Kummer Q(ζ p ), p Fermat Fermat p., F Cl(F ). Cl(F ). O F F F O F,. Cl(F ), Cl(F )= {F O F } {F O F }, Cl(F ). F Cl(F )={0}, Cl(Q( 5)) = Z/(2). F Cl(F ), F Cl(F ) 3. 1 Wiles. 2 n 3 x n + y n =1 n =4 n 3. 3 Cl(F) ={0} 2 F. 1

2 , F O F P ζ F (s) = (1 N(P) s ) 1 (N(P) N(P) = O F /P ). ζ F (s) s =1 1.. (Dirichlet ). F Cl(F ) : Cl(F ) = lim s 1 (s 1)ζ F (s) w F D F 1/2 2 r 1 (2π) r 2RF (w F F 1, D F, R F. r 1, r 2 F ). F Q 0. Serre Q d X Hasse-Weil ζ X (s) GQ = Gal(Q/Q) Hét d (X Q, Q l) ζ X (s) = (det(1 p: Frob p t; Hét d (X Q, Q l) I p ) t=p s) 1 4., Sel(X) Hét d (X Q, Q/Z) H 1 (GQ,Hét d (X Q, Q/Z)). ζ X (s) Sel(X) ζ F (s) Cl(F ) ζ X (s) Sel(X) Wiles. Wiles. 1. F 20,. 2 Kronecker-Weber. 4 Frob p p, I p

3 2. Grothendieck Q X GQ Hét (X Q, Q l). 3. p- ( ) mod p n Diophantine. 20 ( ) p-, p-. Hilbert,, Artin,., ( ). Wiles. Wiles 3 : A. ( ) BSD ( 2 ) B. ( 3 ) C. Fermat ( 4 ) 2 BSD E Q. 3 x 3 + ax + b y 2 = x 3 + ax + b (a, b Q), 1. 0., F Q E(F ) Mordell-Weil. Birch and Swinnerton-Dyer (, BSD ). E Q., ζ E (s) s =1 ord s=1 ζ E (s) E(Q). E EndQ (E) n Z End Q (E). EndQ (E) = Z, EndQ (E) 3

4 E., EndQ (E) 2 K = Q( d) O K, E., E D : y 2 = x 3 + Dx, (x, y) ( x, 1y) n 6. Wiles Coates : 1 (Coates-Wiles/1977). E Q 1 2 K EndQ (E) = O K. ζ E (1) 0 E(Q). ζ E (s) K.. u Z p [ζ p ] f u (ζ p 1) = u f u (t) Z p [[t]], g (r) u (t) = ( (1 + t) d dt) r log(fu (t)). r ϕ (r) : Z p [ζ p ] Z/(p)Z ϕ (r) (u) =g (r) u (t) t=0 Z p mod p (1). u f u (t) g u (r) (t)., g u (r) (t) g u (r) (t) t=0 p, ϕ (r) well-defined., u =1+ζ p Z[ζ p ]., f u (t) =2+t. g u (r) (t) t=0 ζ(s) (1 2 r )ζ(1 r) 7. ( ), E(Q) ζ E (1) = 0. K p = ππ p 8.[π r ] EndQ (E) π r O K, Ker([π r ]) E(Q) K K(E π r ), r 1 K(E π r ) K(E π ). P E(Q). [π n ]Q n = P Q n E(Q) K(E π )(Q n )/K(E π ). 9, : g (r) u K(E π )(Q n )/K(E π ) n π. (2) 6, EndQ (E D) = Z[ 1]. 7 t = T 1 g u (r) (T )=(T d dt )r 1 T 1+T dx )r 1 ex 1+e x x=0. xex = (2x)e2x 1+e x e 2x 1, T = exp(x) (t) t=0 =( d r (r 1)! ζ(1 r) Coates-Wiles.. xex Taylor e x 1 4

5 , K(E π ) K O K(Eπ ) (π) =p p 1. O K(Eπ ) p O p. 2 : E = {x O K(E π ) x 1modp}, U p = {x O p x 1modp}. U p K(E π ), E U p. U p O p -, (2) 10. E U p p U p (3) (1) ϕ (r), ζ p E π, log ((1 + t) d dt )r ψ (r) : U p O p /p = Z/(p). Θ p E, ψ (1) (Θ p ) ζ E (1)/Ω E mod p (4)., Ω E = E(R) ω E, ζ E (1)/Ω E Q. (3), (4) ζ E (1)/Ω E p. p, ζ E (1) = 0. ( ) K(E π )/K(E π ) Z p, Z p -.. Coates-Wiles K(E p )/K 2, Rubin 80. 2, 2 (ζ alg K,p (S, T )) = (ζanal K,p (S, T )), Coates-Wiles S = T =0 p. 2. E,. Kolyvagin, ζ E (1) 0 E(Q). 10 E Up p Gal(K(E π )/K). 5

6 BSD Coates-Wiles. BSD ord s=1 ζ E (s) 0 1 ranke(q) = ord s=1 ζ E (s)., ord s=1 ζ E (s) 2 ord s=1 ζ E (s) ranke(q). ord s=1 ζ E (s) 2. 3 Wiles, Mazur. K n = Q(ζ p n) K = 1 n< K n, Γ = Gal(K /K 1 ) Z p [[Γ]] = lim Z p [Γ/Γ pn ]. Γ χ cyc χ cyc :Γ 1+pZ p = Zp. p- A n = Cl(K n ){p} Gal(K n /Q). ω : Gal(K 1 /Q) (Z/(p)) A n = 0 i p 2 A ωi n, X (i) = lim na ωi n i Z p [[Γ]]-., R R- M M Char R (M). R = Z, M = 1 j r Z/(n j ), Char R (M) =(n 1 n r ). CharZ p[[γ]](x (i) ) Z p [[Γ]] L alg,(i) p p- L., -Leopoldt,, Coleman p- L L anal,(i) p Z p [[Γ]] : r i mod p 1 r χ r cyc(l anal,(i) p )=(1 p r )ζ( r) Mazur-Wiles. 2( =Mazur-Wiles /1984). 0 <i<p 1 i Z p [[Γ]] (L alg,(i) p )=(L anal,(i) p ). ( ) K n Dirichlet i (L alg,(i) p ) (L anal,(i) p ) i (L alg,(i) p )= (L anal,(i) p )., (L alg,(i) p ). Char R (M) R- M., 6

7 ( )X (i) L anal,(i) p., H n (p) p K n (p) Gal(K 1 /Q)- A n Gal(H n /K n )., n Gal(K 1 /Q) ω i Gal(Q/K n ) D p = r 1 Z/(p r ) L anal,(i) p. ( ) ρ (i) 1 b n : GQ GL 2 (Z/p rn n (g) Z), g 0 ω i (g). 1. ρ (i) n p. 2. Gal(Q/K n ) GQ p. 3. GQ Z/(p r n ), g b n (g)., g, g Gal(Q/K n )(n 1) ω i (g) = 1 b n (gg ) = b n (g)+b n (g ) b n : Gal(Q/K n ) D p. ρ (i) n, b n Gal(K 1 /Q). GQ ρ (i) n.. Ribet.. H SL 2 (R). N {( ) } a b Γ 1 (N) = SL c d 2 (Z) a d 1,c 0modN H/Γ 1 (N) X 1 (N). X 1 (N)(C) =X 1 (N) Q X 1 (N) N. X 1 (N). GL 2 n Het(X 1 1 (p n )Q,D p) ρ (i) n. Wiles, 11. g =[F : Q] g. Wiles 11 F F, 2 Q( d). 7

8 . G, G = GL 1 (F ). O[[Γ]] (O Z p ), G = GL 2 (F ) ρ : G F GL 2 (O[[Γ]]). I =(p- ) I O[[Γ]] mod I ρ I ρ I : G F GL 2 (O[[Γ]]/I), g ( 1 B(g) 0 D(g) D(g) Z p - F, ρ I B., n B : Gal(F/ F ) O[[Γ]]/I,., F 2, (pseudo represenation)..,... Skinner-Urban BSD Mazur-Wiles. ) 4 - Fermat, N X 1 (N) E E. Q E, ζ E (s) ( )ζ E (s) ζ E (2 s) Weil 12,. 12 ζ E (s) twist. 8

9 3 (Wiles, Taylor-Wiles/1994). E 13, E Ribet.. Fermat p 3. 3 Wiles 7., 3 {Q } {Q }... Wiles,., d X p- Hét d (X Q, Z p) H p (X) X, X H p (X)., E H p (E) E Faltings, E H p (E ) = H p (E) 14 E. {Q }/. Wiles,,. H p (E) mod p n H p (E)/(p n )H p (E) (Z/(p n )) 2. GQ GL 2 (Z/(p n )) ρ E,p n., p =3 GL 2 (F 3 ).. Langlands-Tunnell, ρ E,3 ρ E,3 = ρe,3 E. p 5 GL 2 (F p ) 13 f(x) =x 3 +ax+b mod p p F p f p (x) F p [x]. p 5 f p(x) 2, p =2, 3 y 2 = x 3 +ax+b. 14 H p (E ) Z p Q p = Hp (E) Z p Q p. 9

10 ,. : R n = {ρ : GQ GL 2 (Z/(3 n )) ρ ρ E,3 mod (3) + ( )} T n = {ρ R n E ρ = ρ E,3 n} Langlands-Tunnell R 1 = T 1 n T n. n Z/(3 n+1 ) Z/(3 n ), R n+1, T n+1 R n, T n. R n, n ρ E,3 n R n. Faltings n ρ E,3 n T n R n E. T n,r n T n+1,r n+1 η(r n ),η(t n ) Z η(r n ) η(t n ). Wiles, η(r n ), n η(r n ) η(t n ). T n = R n 15.., Wiles η(r n ) η(t n ) Wiles - Fermat. Wiles - : - (=Breuil-Conrad-Diamond-Taylor /2001). Q E. 15, Langlands-Tunnell R n ρ E,3. ρ E,3, p =5 Wiles. 16 Wiles,. 10

11 Wiles. 5 Wiles p-,., Wiles.,.. Wiles.., Wiles. BSD. Wiles,.. Wiles.. ( ).,,,, p-.. 11

12 , II (,, ). 1 ( 2 )( ). Elementary theory of L-functions and Eisenstein series ( ) 7.,, 3 Wiles. Fermat, Wiles. 12

2 Riemann Im(s) > 0 ζ(s) s R(s) = 2 Riemann [Riemann]) ζ(s) ζ(2) = π2 6 *3 Kummer s = 2n, n N ζ( 2) = 2 2, ζ( 4) =.3 2 3, ζ( 6) = ζ( 8)

2 Riemann Im(s) > 0 ζ(s) s R(s) = 2 Riemann [Riemann]) ζ(s) ζ(2) = π2 6 *3 Kummer s = 2n, n N ζ( 2) = 2 2, ζ( 4) =.3 2 3, ζ( 6) = ζ( 8) (Florian Sprung) p 2 p * 9 3 p ζ Mazur Wiles 4 5 6 2 3 5 2006 http://www.icm2006.org/video/ eighth session [ ] Coates [Coates] 2 735 Euler n n 2 = p p 2 p 2 = π2 6 859 Riemann ζ(s) = n n s = p p s s ζ(s)