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
|
|
- つづる はぎにわ
- 6 years ago
- Views:
Transcription
1 Galois ) Galois Galois Galois TaylorWiles Fermat [W][TW] Galois Galois Galois 1 Noether 2 1
2 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 C Λ Λ Λ /m lim /a a a /a rtin /a C Λ Λ 2
3 Ob C Λ m /a rtin Ob C Λ n M M n n M M n G V n k ρ : G ut k V ) 1.1. ρ Ob C Λ n M G ρ : G ut M) k[g] ψ : M k V ρ, ψ ) D ρ : C Λ Sets Ob C Λ D ρ ) = { ρ } ρ, ψ ) ρ, ψ ) C Λ D ρ 1 ) D ρ 2 ) ρ 1, ψ 1 ) D ρ 1 ) ρ 1 1 2, ψ ) D ρ R ρ Ob C Λ R ρ ρ ρ R ρ ρ V V k ut k V ) = GL n k) G ρ ut k V ) = GL n k) ρ Ob C Λ p : k Hom ρ G, GLn ) ) = { ρ : G GL n ) ρ GLn p ) ρ = ρ } ρ Hom ρ G, GLn ) ) G ρ GL n ) = ut n ) n k = k n Hom ρ G, GLn ) ) D ρ ) 1.1) ρ, ρ Hom ρ G, GLn ) ) ρ, ρ 1.1) ρ = Hρ H 1 H Ker GL n p ) ρ ρ, ψ ) 1.1) ρ Hom ρ G, GLn ) ) ρ, ψ ) ρ ρ, ψ ) V k β 3
4 1.2. ρ Ob C Λ n M G ρ : G ut M) k[g] ψ : M k V ψ β M β ρ, ψ, β ) D ρ : C Λ Sets Ob C Λ D ρ ) = { ρ } D ρ R ρ Ob C Λ R ρ ρ ρ R ρ ρ V k β ut k V ) = GL n k) G ρ ut k V ) = GL n k) ρ Ob C Λ ρ Hom ρ G, GLn ) ) G ρ GL n ) = ut n ) n k = k n n Hom ρ G, GLn ) ) D ρ ) 1.2) ρ : G ut k V ) V k[g] End k[g] V = k ρ R ρ ρ R ρ. 1.2) ρ R ρ Ob C Λ ρ univ Hom ρ G, GLn R ρ ) ) Ob C Λ Hom CΛ R ρ, ) Hom ρ G, GLn ) ) ; f f ρ univ) 2.1) f ρ univ) G ρ univ GL n R ρ ) GL nf) GL n ) 4
5 G Λ[G, n] Λ X g ij g G, 1 i, j n. { 1 i = j Xij e = 0 i j n X gh ij = X g il Xh lj g, h G, 1 i, j n. e G Λ l=1 Hom Λ lg Λ[G, n], ) Hom G, GL n ) ) ; f ρ f 2.2) ρ f g G ρ f g) = fx g ij )) i,j 2.2) ρ : G GL n k) Λ Λ[G, n] k m ρ m ρ Λ[G, n] Λ[G, n] m ρ R ρ R ρ Noether C Λ 2.2) Λ[G, n] R ρ ρ univ : G GL n R ρ ) ρ univ Hom ρ G, GLn R ρ ) ) 2.1) 2.1) Ob C Λ ρ Hom ρ G, GLn ) ) 2.2) ρ f ρ Hom Λ lg Λ[G, n], ) ρ m ρ f ρ m ρ ) m /a rtin a Λ[G, n] f ρ R ρ f ρ,a /a Λ f ρ,a : R ρ /a a f ρ,a Λ ˆf ρ : R ρ Hom ρ G, GLn ) ) Hom CΛ R ρ, ); ρ ˆf ρ 2.1) G = lim H H G ρ GL n k) ρ H H 5
6 ρ H : H GL n k) G H H ρ H R ρ H Ob C Λ ρ H,univ Hom ρ H, GLn R ρ H ) ) G 2 G H H H H ρ H,univ GL n R ρ H ) C Λ R ρ H R ρ H C Λ R H ) H R ρ = lim H R ρ H G H ρ H,univ GL n R ρ H ) H ρ univ : G GL n R ρ ) R ρ H rtin R ρ H Λ[H, n] R ρ H R ρ H G 2 G H H Λ[H, n] R ρ H Λ[H, n] R ρ H Λ[H, n] R ρ H R ρ R ρ H R ρ H R ρ rtin R ρ rtin R ρ H rtin R ρ Ob C Λ C Λ rtin = i lim i Hom CΛ R ρ, ) = lim i Hom CΛ R ρ, i ) = lim i lim H Hom CΛ R ρ H, i ) = lim i lim Hom ρh H H, GLn i ) ) = lim i Hom ρ G, GLn i ) ) = Hom ρ G, GLn ) ) 2.1) 2.1 End k[g] V = k 2.1 V k ut k V ) = GL n k) G ρ ut k V ) = GL n k) ρ G g 1,..., g r ρg i ) M n k) ρg i ) E i M n Λ) 6
7 C Λ M n Λ) M n ) E i E i M n ) M 0 n) = M n )/ i : M 0 n) M n ) r ; M mod ME i E i M) r i=1 π Λ i Λ = id M 0 n ) Λ π Λ : M n Λ) r M 0 nλ) π : M n ) r M 0 n) M n ) r = Mn Λ) r Λ π Λ id M 0 n λ) Λ = M 0 n) Ob C Λ Hom ρ G,GLn ) ) M n ) r π M 0 n ) 2.3) ρ ρg i ) ) r i=1 ρ Hom ρ G, GLn ) ) π E 1,..., E r ) ρ Hom ρ G, GLn ) ) Hom ρ,well G, GLn ) ) 2.3. Ob C Λ ρ Hom ρ G, GLn ) ) m GL n k) H GL n ) Hρ H 1 H GL n ) 1 + m. m m m = 0 m m = 1 m 2 m m 1 M0 n) L m m 1 M n) L mod m m 1 L mod m m 1 m m 1 M0 n) Hom ρ G, GLn ) ) Hom ρ G, GLn ) ) Hom ρ G, GLn ) ) ; ρ 1 + L)ρ 1 + L) 1 L ρ m m L)ρ 1 + L) 1 L mod m m 1 m m 1 M0 n) L mod m m 1 M 0 n) M 0 n) M 0 n); M mod M + L mod m m 1 M0 n) 2.3) m m 1 M0 n) m m = /m m ρ m m ρ m : G GL n m ) 7
8 m m GL n k) H m GL n m ) H m ρ Hm 1 H m 1 + m m m 1 > m 2 H m1 m m 2 H m2 H m m H m GL n ) H H GL n ) 1 + m Hom ρ,well G, GLn ) ) Hom ρ G, GLn ) ) 1.1) 2.3 Hom ρ,well G, GLn ) ) D ρ ) 2.4) 2.1. ρ ρ univ 2.3 H GL n Runiv) ρ well = Hρ univh 1 g G ρ well g) Runiv Λ R ρ R ρ C Λ ρ well Hom ρ,well G, GLn R ρ ) ) ρ univ 2.4) Ob C Λ Hom CΛ R ρ, ) Hom ρ,well G, GLn ) ) ; f f ρ univ ) 2.5) f ρ univ ) G ρ univ GL n R ρ ) GL nf) GL n ) 2.5) R ρ 2.5) ρ Hom ρ,well G, GLn ) ) 2.1) ρ f Hom CΛ R ρ, ) f R ρ f f ρ univ ) = f ρ well) = f Hρ univh 1 ) = GL n f)h)ρ GLn f)h) ) 1 f ρ univ ) ρ 2.3 f ρ univ ) = ρ n = 1 End k[g] V = k n = 1 k p Λ m Λ Λ G ab,p G bel p ρ R ρ Λ G ab,p Λ[[G ab,p ]] 8
9 . ρ ut k V ) = k G ρ ut k V ) = k ρ ρg) k 0 k 0 k p k 0 Witt W k 0 ) Λ W k 0 ) Teichmüller k 0 W k 0 ) W k 0 ) Λ s Λ : k 0 Λ Ob C Λ GL 1 ) π : G G ab,p ρ univ : G GL 1 Λ[[G ab,p ]]); g s Λ ρg) ) πg) rtin Ob C Λ Hom CΛ Λ[[G ab,p ]], ) Hom ρ G, GL1 ) ) ; f f ρ univ ) 2.6) f ρ univ ) G ρ univ GL 1 Λ[[G ab,p ]]) GL 1f) GL 1 ) 2.6) rtin C Λ Λ s Λ : k Λ Λ s : k ρ Hom ρ G, GL1 ) ) h ρ : G ; g ρg)s ρg) ) 1 G h ρ π h ρ G ab,p h ρ : G ab,p Λ[G ab,p ] Λ G ab,p Λ f ρ : Λ[G ab,p ] g G ab,p Λ[G ab,p ] g h ρ g) f ρ Λ ˆf ρ : Λ[[G ab,p ]] Hom ρ G, GL1 ) ) Hom CΛ Λ[[G ab,p ]], ); ρ ˆf ρ 2.6) 9
10 3 k[ϵ] ϵ ϵ 2 = 0 k k[ϵ] k x, y x + yϵ k[ϵ] C Λ R C Λ R Zariski t R t R = Hom CΛ R, k[ϵ]) t R k k k[ϵ] k k[ϵ] k[ϵ] k k[ϵ] = { x + y 1 ϵ, x + y 2 ϵ) k[ϵ] k[ϵ] x, y1, y 2 k } h : Hom CΛ R, k[ϵ] k k[ϵ]) = Hom CΛ R, k[ϵ]) Hom CΛ R, k[ϵ]) = t R t R k f s : k[ϵ] k k[ϵ] k[ϵ]; x + y 1 ϵ, x + y 2 ϵ) x + y 1 + y 2 )ϵ t R t R t R h 1 Hom CΛ R, k[ϵ] k k[ϵ]) Hom C Λ R,f s ) Hom CΛ R, k[ϵ]) = t R a k k t R a m a : k[ϵ] k[ϵ]; x + yϵ x + ayϵ t R = Hom CΛ R, k[ϵ]) Hom C Λ R,m a) Hom CΛ R, k[ϵ]) = t R d ρ) End k V ) g G End k V ) End k V ); ϕ ρg)ϕ ρg) 1 G G d ρ) 3.1. End k[g] V = k R ρ ρ k t R ρ = H 1 G, d ρ) ). V k End k V ) M n k) ρ G GL n k) Hom ρ G, GLn k[ϵ]) ) D ρ k[ϵ]) = t R ρ 3.1) 10
11 Z 1 G, d ρ) ) G d ρ) 1 Z 1 G, d ρ) ) c : G M n k) Z 1 G, d ρ) ) Hom ρ G, GLn k[ϵ]) ) ; c 3.2) 3.1) g 1 + cg)ϵ ) ρg) ) 3.2) Z 1 G, d ρ) ) t R ρ 3.3) k 3.3) R C Λ k Hom k-cont mr /m 2 R + m Λ R), k ) = tr m R /m 2 R + m ΛR) k k m R /m 2 R + m ΛR) R. m 2 k[ϵ] = 0 Hom CΛ R/m 2 R + m Λ R), k[ϵ] ) Hom CΛ R, k[ϵ]) = t R k m R /m 2 R + m Λ R) R/m 2 R + m Λ R) m R /m 2 R + m ΛR) Hom CΛ R/m 2 R + m Λ R), k[ϵ] ) Hom k-cont mr /m 2 R + m Λ R), k ) R ρ Noether 3.3. End k[g] V = k R ρ ρ R ρ Noether H 1 G, d ρ) ) k. 3.1 H 1 G, d ρ) ) k t R ρ k R ρ rtin lim i R i 3.2 t R ρ = Hom CΛ R ρ, k[ϵ]) = lim i Hom CΛ R i, k[ϵ]) = lim i Hom k mri /m 2 R i + m Λ R i ), k ) 11
12 t R ρ k dim k mri /m 2 R i + m Λ R i ) ) i Λ Noether dim k m Λ /m 2 Λ ) dim k mri /m 2 R i + m Λ R i ) ) dim k m Ri /m 2 R i ) dim k m Λ /m 2 Λ ) dim k mri /m 2 R i +m Λ R i ) ) i dim k m Ri /m 2 R i ) i H 1 G, d ρ) ) k 1. K p G = G K H 1 G, d ρ) ) k 2. K S K K G S K Galois H 1 G, d ρ) ) k , 1 C Λ p 1,0 : 1 0 C Λ I = Ker p 1,0 Im 1 = 0 I k ρ 0, ψ 0 ) ρ 0 ρ 0, ψ 0 ) p 1,0 Oρ 0, ψ 0 ) H 2 G, d ρ) k I ) 1. Oρ 0, ψ 0 ) 2. ρ 1 ρ 1, ψ 1 )D ρ p 1,0 ) : D ρ 1 ) D ρ 0 )ρ 0, ψ 0 ). ρ 0, ψ 0 ) ρ 0 Hom ρ G, GLn 0 ) ) γ 1 : G GL n 1 ) GL n p 1,0 ) γ 1 = ρ 0 c : G G d ρ) k I g 1, g 2 G cg 1, g 2 ) = γ 1 g 1 g 2 )γ 1 g 2 )γ 1 g 1 ) 1 + M n k) k I = d ρ) k I c G d ρ) k I 2 c H 2 G, d ρ) k I ) Oρ 0, ψ 0 ) Oρ 0, ψ 0 ) ρ 0 γ End k[g] V = k R ρ ρ i = 1, 2 d i = dim k H i G, d ρ) ) d 1 Krull dimr ρ /m Λ R ρ ) d 1 d 2 d 2 = 0 R ρ Λ d 1 12
13 . d R ρ Noether Zariski k f 0 : k[[x 1,..., X d1 ]] R ρ /m Λ R ρ m k[[x 1,..., X d1 ]] J = Ker f 0 f 0 : k[[x 1,..., X d1 ]]/mj R ρ /m Λ R ρ f 0 k 0 J/mJ k[[x 1,..., X d1 ]]/mj f 0 R ρ /m Λ R ρ 0 ρ R ρ R ρ /m Λ R ρ ρ 0, ψ 0 ) f 0 ρ 0, ψ 0 ) Oρ 0, ψ 0 ) H 2 G, d ρ) k J/mJ ) rtin- Rees J/mJ f Hom k J/mJ, k) T f : d ρ) k J/mJ G id f d ρ) k k = d ρ) Φ : Hom k J/mJ, k) H 2 G, d ρ) ) ; f H 2 G, T f ) Oρ 0, ψ 0 ) ) Φ f Hom k J/mJ, k) Φf) = 0 f 0 k[[x 1,..., X d1 ]]/mj Ker f R f 0 : R R ρ /m Λ R ρ f 0 k 0 k R f 0 R ρ /m Λ R ρ 0 Φf) = 0 f 0 ρ 0, ψ 0 ) ρ R ρ, ψ ) D ρ f 0) ρ 0, ψ 0 ) R k ρ, ψ ) R ρ R g 0 : R ρ /m Λ R ρ R f 0 g 0 = id R ρ /m Λ R ρ g 0 Zariski g 0 f 0 Φ k[[x 1,..., X d1 ]] J dim k J/mJ Krull dimr ρ /m Λ R ρ ) d 1 dim k J/mJ Φ dim k J/mJ d 2 d 2 = 0 Φ J = 0 f 0 Krull dimr ρ /m Λ R ρ ) = d 1 Λ[[X 1,..., X d1 ]] f R ρ k[[x 1,..., X d1 ]] 13 f 0 R ρ /m Λ R ρ
14 f : Λ[[X 1,..., X d1 ]] R ρ f Zariski C Λ g 1 : R ρ R ρ /m Λ R ρ f 1 0 k[[x 1,..., X d1 ]] k[x 1,..., X d1 ]/X 1,..., X d1 ) 2 g 1 Zariski d 2 = 0 g 1 k[x 1,..., X d1 ]/X 1,..., X d1 ) 2 Λ[[X 1,..., X d1 ]]/m Λ, X 1,..., X d1 ) 2 k[x 1,..., X d1 ]/X 1,..., X d1 ) 2 g 1 g 2 : R ρ Λ[[X 1,..., X d1 ]]/m Λ, X 1,..., X d1 ) 2 m 2 g m : R ρ Λ[[X 1,..., X d1 ]]/m Λ, X 1,..., X d1 ) m g m+1 : R ρ Λ[[X 1,..., X d1 ]]/m Λ, X 1,..., X d1 ) m+1 g m ) m 2 g : R ρ Λ[[X 1,..., X d1 ]] g g 1 Zariki g f : Λ[[X 1,..., X d1 ]] Λ[[X 1,..., X d1 ]] Ker f 0 Kerg f) m) m 1 Λ[[X 1,..., X d1 ]] Λ[[X 1,..., X d1 ]] Noether f k 4.2 d 1 d 2 Euler-Poincaré 5 S D ρ S R) 1. Sk) = D ρ k). 2. ρ, ψ ) Ob C Λ ρ, ψ ) S) a a ρ /a, ψ /a) S/a) 14
15 3. ρ, ψ ) Ob C Λ a b a, b ρ /a, ψ /a) S/a)ρ /b, ψ /b) S/b) ρ /a b), ψ /a b) ) S /a b) ) 4. ρ, ψ ) Ob C Λ rtin C Λ Λ ρ, ψ ) S) ρ, ψ ) S ) 5.1. End k[g] V = k R ρ ρ D ρ S R) R ρ a S R ρ /a S Ob C Λ Ob C Λ Hom CΛ R ρ, ) D ρ ) Hom CΛ R ρ /a S, ) S). ρ R ρ, ψ R ρ ) R ρ X S ρ R ρ R ρ R ρ /a, ψ R ρ R ρ R ρ /a) SR ρ /a) R ρ a a S = a X S a.3 R ρ lim R ρ a XS /a R ρ /a S lim R ρ a XS /a a S C Λ n M G ρ : G ut M) ρ n G ut M) = det ρ δ : G Λ C Λ δ : G δ Λ Λ Λ D ρ D ρ,δ C Λ { } D ρ,δ ) = ρ : G ut M), ψ ) D ρ ) det ρ = δ 5.2. End k[g] V = k D ρ,δ k) = D ρ k) D ρ,δ R ρ,δ Ob C Λ. D ρ,δ R) 5.1 I G n = 2 C Λ ρ ρ : G ut M), ψ ) I M I M I M 1 I D ρ D ord ρ 15
16 5.3. End k[g] V = k D ord ρ k) = D ρ k) D ord ρ Ob C Λ R ord ρ. ρ : G ut M), ψ ) ρ C Λ ρ k I ρ, ψ ) I ψ x 1) 0 x M I V g 0 I [G] G g I M g 1) g 0 det ρ g 0 ) ) [G] C I ) ρ, ψ ) I C I ) C I ) y 0, g 0 det ρ g 0 ) ) y 0 ) k y 0 V ψ y 1) = y 0 y M x = g 0 det ρ g 0 ) ) y) x M I ψ x 1) 0 ρ, ψ ) I ρ, ψ ) I C I ) D ord ρ R) 5.1 n K p G = G K k p rtin C Λ ρ, ψ ) ρ O K G K C Λ ρ, ψ ) rtin ρ, ψ ) ρ, ψ ) D ρ Dfl ρ 5.4. End k[g] V = k D fl ρk) = D ρ k) D fl ρ Rfl ρ Ob C Λ. Dfl ρ R) C Λ rtin lim i i 1 j 3 M j i ) i M j i M 1 i ) M 2 i ) M 3 i ).1) i ) i i 1 j 3 M j i M j i i.1) lim Mi 1 lim i Mi 2 lim i i M 3 i 16
17 . i 1 j 3 M j i rtin rtin Mittag-Leffler.2. C Λ rtin lim i i 1. Noether 2. dim k m i /m 2 i ) i m Noether m m m m lim i m m i m m = 0 m m m lim i m m i m m+1 lim i m m+1 i 0 m m+1 i m m i m m i /m m+1 i 0.2) N = i lim m m i /m m+1 i 2 dim k m m i /m m+1 i ) i i m m i+1 /m m+1 i+1 m m i /m m+1 i N k.2) i.1 0 lim i m m+1 i m m N 0.3) l = dim k N N k m m a 1,..., a l a j i a j,i i l i m m i ; x 1,... x l ) x 1 a 1,i + + x l a l,i i.1 m m a 1,..., a l l dim k m m /m m+1 ) dim kn) = l m m+1.3) m m N mm+1 lim i mm+1 i Noether m b 1,..., b h m Λ Λ[[X 1,..., X h ]] ; X j b j 17
18 Noether m m m m m m 0 m m i i i /m m i 0.1 m m = lim i mm i /m m = lim i i /m m i 2 ) dim k i /m m i i i i+1 /m m i+1 i /m m i i /m m i m m.3. Ob C Λ X F = a X a lim a X /a /F lim a X /a. a i a i a 0 a 0 i a 0,i a i aa 0,i a 0,i rtin i a 0,i lim a a a 0,i.1 a 0 = lim i a 0,i lim i lim a a 0,i = lim lim a a 0,i = lim a0 + a)/a ) i a a a 0 = lim a /a a 0 lim a /a lim a0 a + a)/a ).1 a 0 lim a0 + a)/a ) lim /a lim /a 0 + a) 0 a a lim a /a 0 + a) rtin lim a0 a + a)/a ) lim /a lim /a a a a [Ma1] [Ma2] B. Mazur, Deforming Galois representations, Galois groups over Q Berkeley, C, 1987), , Math. Sci. Res. Inst. Publ. 16, Springer, New York, B. Mazur, n introduction to the deformation theory of Galois representations, Modular forms and Fermat s last theorem Boston, M, 1995), , Springer, New York, [Sch] M. Schlessinger, Functors on rtin rings, Trans. mer. Math. Soc ),
19 [SL] [TW] [W] B. de Smit, H. W. Lenstra, Jr. Explicit construction of universal deformation rings, Modular forms and Fermat s last theorem Boston, M, 1995), , Springer, New York, R. Taylor,. Wiles, Ring-theoretic properties of certain Hecke algebras nn. of Math. 2) ), no. 3, Wiles, Modular elliptic curves and Fermat s last theorem nn. of Math. 2) ), no. 3, [],,,
SAMA- SUKU-RU Contents p-adic families of Eisenstein series (modular form) Hecke Eisenstein Eisenstein p T
SAMA- SUKU-RU Contents 1. 1 2. 7.1. p-adic families of Eisenstein series 3 2.1. modular form Hecke 3 2.2. Eisenstein 5 2.3. Eisenstein p 7 3. 7.2. The projection to the ordinary part 9 3.1. The ordinary
More information2.1 H f 3, SL(2, Z) Γ k (1) f H (2) γ Γ f k γ = f (3) f Γ \H cusp γ SL(2, Z) f k γ Fourier f k γ = a γ (n)e 2πinz/N n=0 (3) γ SL(2, Z) a γ (0) = 0 f c
GL 2 1 Lie SL(2, R) GL(2, A) Gelbart [Ge] 1 3 [Ge] Jacquet-Langlands [JL] Bump [Bu] Borel([Bo]) ([Ko]) ([Mo]) [Mo] 2 2.1 H = {z C Im(z) > 0} Γ SL(2, Z) Γ N N Γ (N) = {γ SL(2, Z) γ = 1 2 mod N} g SL(2,
More informationk + (1/2) S k+(1/2) (Γ 0 (N)) N p Hecke T k+(1/2) (p 2 ) S k+1/2 (Γ 0 (N)) M > 0 2k, M S 2k (Γ 0 (M)) Hecke T 2k (p) (p M) 1.1 ( ). k 2 M N M N f S k+
1 SL 2 (R) γ(z) = az + b cz + d ( ) a b z h, γ = SL c d 2 (R) h 4 N Γ 0 (N) {( ) } a b Γ 0 (N) = SL c d 2 (Z) c 0 mod N θ(z) θ(z) = q n2 q = e 2π 1z, z h n Z Γ 0 (4) j(γ, z) ( ) a b θ(γ(z)) = j(γ, z)θ(z)
More informationMorse ( ) 2014
Morse ( ) 2014 1 1 Morse 1 1.1 Morse................................ 1 1.2 Morse.............................. 7 2 12 2.1....................... 12 2.2.................. 13 2.3 Smale..............................
More informationSiegel Hecke 1 Siege Hecke L L Fourier Dirichlet Hecke Euler L Euler Fourier Hecke [Fr] Andrianov [An2] Hecke Satake L van der Geer ([vg]) L [Na1] [Yo
Siegel Hecke 1 Siege Hecke L L Fourier Dirichlet Hecke Euler L Euler Fourier Hecke [Fr] Andrianov [An2] Hecke Satake L van der Geer ([vg]) L [Na1] [Yo] 2 Hecke ( ) 0 1n J n =, Γ = Γ n = Sp(n, Z) = {γ GL(2n,
More informationZ[i] Z[i] π 4,1 (x) π 4,3 (x) 1 x (x ) 2 log x π m,a (x) 1 x ϕ(m) log x 1.1 ( ). π(x) x (a, m) = 1 π m,a (x) x modm a 1 π m,a (x) 1 ϕ(m) π(x)
3 3 22 Z[i] Z[i] π 4, (x) π 4,3 (x) x (x ) 2 log x π m,a (x) x ϕ(m) log x. ( ). π(x) x (a, m) = π m,a (x) x modm a π m,a (x) ϕ(m) π(x) ϕ(m) x log x ϕ(m) m f(x) g(x) (x α) lim f(x)/g(x) = x α mod m (a,
More information( ) ( ) 1729 (, 2016:17) = = (1) 1 1
1729 1 2016 10 28 1 1729 1111 1111 1729 (1887 1920) (1877 1947) 1729 (, 2016:17) 12 3 1728 9 3 729 1729 = 12 3 + 1 3 = 10 3 + 9 3 (1) 1 1 2 1729 1729 19 13 7 = 1729 = 12 3 + 1 3 = 10 3 + 9 3 13 7 = 91
More information1 (Contents) (1) Beginning of the Universe, Dark Energy and Dark Matter Noboru NAKANISHI 2 2. Problem of Heat Exchanger (1) Kenji
8 4 2018 6 2018 6 7 1 (Contents) 1. 2 2. (1) 22 3. 31 1. Beginning of the Universe, Dark Energy and Dark Matter Noboru NAKANISHI 2 2. Problem of Heat Exchanger (1) Kenji SETO 22 3. Editorial Comments Tadashi
More informationλ n numbering Num(λ) Young numbering T i j T ij Young T (content) cont T (row word) word T µ n S n µ C(µ) 0.2. Young λ, µ n Kostka K µλ K µλ def = #{T
0 2 8 8 6 3 0 0 Young Young [F] 0.. Young λ n λ n λ = (λ,, λ l ) λ λ 2 λ l λ = ( m, 2 m 2, ) λ = n, l(λ) = l {λ n n 0} P λ = (λ, ), µ = (µ, ) n λ µ k k k λ i µ i λ µ λ = µ k i= i= i < k λ i = µ i λ k >
More information[Oc, Proposition 2.1, Theorem 2.4] K X (a) l (b) l (a) (b) X [M3] Huber adic 1 Huber ([Hu1], [Hu2], [Hu3]) adic 1.1 adic A I I A {I n } 0 adic 2
On the action of the Weil group on the l-adic cohomology of rigid spaces over local fields (Yoichi Mieda) Graduate School of Mathematical Sciences, The University of Tokyo 0 l Galois K F F q l q K, F K,
More informationE1 (4/12)., ( )., 3,4 ( ). ( ) Allen Hatcher, Vector bundle and K-theory ( HP ) 1
E1 (4/12)., ( )., 3,4 ( ). ( ) Allen Hatcher, Vector bundle and K-theory ( HP ) 1 (4/12) 1 1.. 2. F R C H P n F E n := {((x 0,..., x n ), [v 0 : : v n ]) F n+1 P n F n x i v i = 0 }. i=0 E n P n F P n
More information1.2 (Kleppe, cf. [6]). C S 3 P 3 3 S 3. χ(p 3, I C (3)) 1 C, C P 3 ( ) 3 S 3( S 3 S 3 ). V 3 del Pezzo (cf. 2.1), S V, del Pezzo 1.1, V 3 del Pe
3 del Pezzo (Hirokazu Nasu) 1 [10]. 3 V C C, V Hilbert scheme Hilb V [C]. C V C S V S. C S S V, C V. Hilbert schemes Hilb V Hilb S [S] [C] ( χ(s, N S/V ) χ(c, N C/S )), Hilb V [C] (generically non-reduced)
More information1
1 Borel1956 Groupes linéaire algébriques, Ann. of Math. 64 (1956), 20 82. Chevalley1956/58 Sur la classification des groupes de Lie algébriques, Sém. Chevalley 1956/58, E.N.S., Paris. Tits1959 Sur la classification
More information2 (2016 3Q N) c = o (11) Ax = b A x = c A n I n n n 2n (A I n ) (I n X) A A X A n A A A (1) (2) c 0 c (3) c A A i j n 1 ( 1) i+j A (i, j) A (i, j) ã i
[ ] (2016 3Q N) a 11 a 1n m n A A = a m1 a mn A a 1 A A = a n (1) A (a i a j, i j ) (2) A (a i ca i, c 0, i ) (3) A (a i a i + ca j, j i, i ) A 1 A 11 0 A 12 0 0 A 1k 0 1 A 22 0 0 A 2k 0 1 0 A 3k 1 A rk
More informationwiles05.dvi
Andrew Wiles 1953, 20 Fermat.. Fermat 10,. 1 Wiles. 19 20., 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
More information20 9 19 1 3 11 1 3 111 3 112 1 4 12 6 121 6 122 7 13 7 131 8 132 10 133 10 134 12 14 13 141 13 142 13 143 15 144 16 145 17 15 19 151 1 19 152 20 2 21 21 21 211 21 212 1 23 213 1 23 214 25 215 31 22 33
More informationd ϕ i) t d )t0 d ϕi) ϕ i) t x j t d ) ϕ t0 t α dx j d ) ϕ i) t dx t0 j x j d ϕ i) ) t x j dx t0 j f i x j ξ j dx i + ξ i x j dx j f i ξ i x j dx j d )
23 M R M ϕ : R M M ϕt, x) ϕ t x) ϕ s ϕ t ϕ s+t, ϕ 0 id M M ϕ t M ξ ξ ϕ t d ϕ tx) ξϕ t x)) U, x 1,...,x n )) ϕ t x) ϕ 1) t x),...,ϕ n) t x)), ξx) ξ i x) d ϕi) t x) ξ i ϕ t x)) M f ϕ t f)x) f ϕ t )x) fϕ
More informationDynkin Serre Weyl
Dynkin Naoya Enomoto 2003.3. paper Dynkin Introduction Dynkin Lie Lie paper 1 0 Introduction 3 I ( ) Lie Dynkin 4 1 ( ) Lie 4 1.1 Lie ( )................................ 4 1.2 Killing form...........................................
More information2 2 1?? 2 1 1, 2 1, 2 1, 2, 3,... 1, 2 1, 3? , 2 2, 3? k, l m, n k, l m, n kn > ml...? 2 m, n n m
2009 IA I 22, 23, 24, 25, 26, 27 4 21 1 1 2 1! 4, 5 1? 50 1 2 1 1 2 1 4 2 2 2 1?? 2 1 1, 2 1, 2 1, 2, 3,... 1, 2 1, 3? 2 1 3 1 2 1 1, 2 2, 3? 2 1 3 2 3 2 k, l m, n k, l m, n kn > ml...? 2 m, n n m 3 2
More informationII R n k +1 v 0,, v k k v 1 v 0,, v k v v 0,, v k R n 1 a 0,, a k a 0 v 0 + a k v k v 0 v k k k v 0,, v k σ k σ dimσ = k 1.3. k
II 231017 1 1.1. R n k +1 v 0,, v k k v 1 v 0,, v k v 0 1.2. v 0,, v k R n 1 a 0,, a k a 0 v 0 + a k v k v 0 v k k k v 0,, v k σ kσ dimσ = k 1.3. k σ {v 0,...,v k } {v i0,...,v il } l σ τ < τ τ σ 1.4.
More informationK 2 X = 4 MWG(f), X P 2 F, υ 0 : X P 2 2,, {f λ : X λ P 1 } λ Λ NS(X λ ), (υ 0 ) λ : X λ P 2 ( 1) X 6, f λ K X + F, f ( 1), n, n 1 (cf [10]) X, f : X
2 E 8 1, E 8, [6], II II, E 8, 2, E 8,,, 2 [14],, X/C, f : X P 1 2 3, f, (O), f X NS(X), (O) T ( 1), NS(X), T [15] : MWG(f) NS(X)/T, MWL(f) 0 (T ) NS(X), MWL(f) MWL(f) 0, : {f λ : X λ P 1 } λ Λ NS(X λ
More informationMilnor 1 ( ), IX,. [KN].,. 2 : (1),. (2). 1 ; 1950, Milnor[M1, M2]. Milnor,,. ([Hil, HM, IO, St] ).,.,,, ( 2 5 )., Milnor ( 4.1)..,,., [CEGS],. Ω m, P
Milnor 1 ( ), IX,. [KN].,. 2 : (1),. (2). 1 ; 1950, Milnor[M1, M2]. Milnor,,. ([Hil, HM, IO, St] ).,.,,, ( 2 5 )., Milnor ( 4.1)..,,., [CEGS],. Ω m, PC ( 4 5 )., 5, Milnor Milnor., ( 6 )., (I) Z modulo
More information4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5.
A 1. Boltzmann Planck u(ν, T )dν = 8πh ν 3 c 3 kt 1 dν h 6.63 10 34 J s Planck k 1.38 10 23 J K 1 Boltzmann u(ν, T ) T ν e hν c = 3 10 8 m s 1 2. Planck λ = c/ν Rayleigh-Jeans u(ν, T )dν = 8πν2 kt dν c
More information平成 30 年度 ( 第 40 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 30 ~8 年月 72 月日開催 30 日 [6] 1 4 A 1 A 2 A 3 l P 3 P 2 P 1 B 1 B 2 B 3 m 1 l 3 A 1, A 2, A 3 m 3 B 1,
[6] 1 4 A 1 A 2 A 3 l P 3 P 2 P 1 B 1 B 2 B 3 m 1 l 3 A 1, A 2, A 3 m 3 B 1, B 2, B 3 A i 1 B i+1 A i+1 B i 1 P i i = 1, 2, 3 3 3 P 1, P 2, P 3 1 *1 19 3 27 B 2 P m l (*) l P P l m m 1 P l m + m *1 A N
More informationSiegel modular forms of middle parahoric subgroups and Ihara lift ( Tomoyoshi Ibukiyama Osaka University 1. Introduction [8] Ihara Sp(2, R) p
Siegel modular forms of middle parahoric subgroups and Ihara lift ( Tomoyoshi Ibukiyama Osaka University 1. Introduction [8] Ihara 80 1963 Sp(2, R) p L holomorphic discrete series Eichler Brandt Eichler
More informationmeiji_resume_1.PDF
β β β (q 1,q,..., q n ; p 1, p,..., p n ) H(q 1,q,..., q n ; p 1, p,..., p n ) Hψ = εψ ε k = k +1/ ε k = k(k 1) (x, y, z; p x, p y, p z ) (r; p r ), (θ; p θ ), (ϕ; p ϕ ) ε k = 1/ k p i dq i E total = E
More informationA11 (1993,1994) 29 A12 (1994) 29 A13 Trefethen and Bau Numerical Linear Algebra (1997) 29 A14 (1999) 30 A15 (2003) 30 A16 (2004) 30 A17 (2007) 30 A18
2013 8 29y, 2016 10 29 1 2 2 Jordan 3 21 3 3 Jordan (1) 3 31 Jordan 4 32 Jordan 4 33 Jordan 6 34 Jordan 8 35 9 4 Jordan (2) 10 41 x 11 42 x 12 43 16 44 19 441 19 442 20 443 25 45 25 5 Jordan 26 A 26 A1
More informationSO(3) 7 = = 1 ( r ) + 1 r r r r ( l ) (5.17) l = 1 ( sin θ ) + sin θ θ θ ϕ (5.18) χ(r)ψ(θ, ϕ) l ψ = αψ (5.19) l 1 = i(sin ϕ θ l = i( cos ϕ θ l 3 = i ϕ
SO(3) 71 5.7 5.7.1 1 ħ L k l k l k = iϵ kij x i j (5.117) l k SO(3) l z l ± = l 1 ± il = i(y z z y ) ± (z x x z ) = ( x iy) z ± z( x ± i y ) = X ± z ± z (5.118) l z = i(x y y x ) = 1 [(x + iy)( x i y )
More informationD 24 D D D
5 Paper I.R. 2001 5 Paper HP Paper 5 3 5.1................................................... 3 5.2.................................................... 4 5.3.......................................... 6
More informationuntitled
8- My + Cy + Ky = f () t 8. C f () t ( t) = Ψq( t) () t = Ψq () t () t = Ψq () t = ( q q ) ; = [ ] y y y q Ψ φ φ φ = ( ϕ, ϕ, ϕ,3 ) 8. ψ Ψ MΨq + Ψ CΨq + Ψ KΨq = Ψ f ( t) Ψ MΨ = I; Ψ CΨ = C; Ψ KΨ = Λ; q
More informationQ p G Qp Q G Q p Ramanujan 12 q- (q) : (q) = q n=1 (1 qn ) 24 S 12 (SL 2 (Z))., p (ordinary) (, q- p a p ( ) p ). p = 11 a p ( ) p. p 11 p a p
.,.,.,..,, 1.. Contents 1. 1 1.1. 2 1.2. 3 1.3. 4 1.4. Eisenstein 5 1.5. 7 2. 9 2.1. e p 9 2.2. p 11 2.3. 15 2.4. 16 2.5. 18 3. 19 3.1. ( ) 19 3.2. 22 4. 23 1. p., Q Q p Q Q p Q C.,. 1. 1 Q p G Qp Q G
More information20 6 4 1 4 1.1 1.................................... 4 1.1.1.................................... 4 1.1.2 1................................ 5 1.2................................... 7 1.2.1....................................
More information1. 2 P 2 (x, y) 2 x y (0, 0) R 2 = {(x, y) x, y R} x, y R P = (x, y) O = (0, 0) OP ( ) OP x x, y y ( ) x v = y ( ) x 2 1 v = P = (x, y) y ( x y ) 2 (x
. P (, (0, 0 R {(,, R}, R P (, O (0, 0 OP OP, v v P (, ( (, (, { R, R} v (, (, (,, z 3 w z R 3,, z R z n R n.,..., n R n n w, t w ( z z Ke Words:. A P 3 0 B P 0 a. A P b B P 3. A π/90 B a + b c π/ 3. +
More information量子力学 問題
3 : 203 : 0. H = 0 0 2 6 0 () = 6, 2 = 2, 3 = 3 3 H 6 2 3 ϵ,2,3 (2) ψ = (, 2, 3 ) ψ Hψ H (3) P i = i i P P 2 = P 2 P 3 = P 3 P = O, P 2 i = P i (4) P + P 2 + P 3 = E 3 (5) i ϵ ip i H 0 0 (6) R = 0 0 [H,
More informationi Version 1.1, (2012/02/22 24),.,..,.,,. R-space,, ( R- space),, Kahler (Kähler C-space)., R-space,., R-space, Hermite,.
R-space ( ) Version 1.1 (2012/02/29) i Version 1.1, (2012/02/22 24),.,..,.,,. R-space,, ( R- space),, Kahler (Kähler C-space)., R-space,., R-space, Hermite,. ii 1 Lie 1 1.1 Killing................................
More informationMacdonald, ,,, Macdonald. Macdonald,,,,,.,, Gauss,,.,, Lauricella A, B, C, D, Gelfand, A,., Heckman Opdam.,,,.,,., intersection,. Macdona
Macdonald, 2015.9.1 9.2.,,, Macdonald. Macdonald,,,,,.,, Gauss,,.,, Lauricella A, B, C, D, Gelfand, A,., Heckman Opdam.,,,.,,., intersection,. Macdonald,, q., Heckman Opdam q,, Macdonald., 1 ,,. Macdonald,
More information非可換Lubin-Tate理論の一般化に向けて
Lubin-Tate 2012 9 18 ( ) Lubin-Tate 2012 9 18 1 / 27 ( ) Lubin-Tate 2012 9 18 2 / 27 Lubin-Tate p 1 1 ( ) Lubin-Tate 2012 9 18 2 / 27 Lubin-Tate p 1 1 Lubin-Tate GL n n 1 Lubin-Tate ( ) Lubin-Tate 2012
More information17 Θ Hodge Θ Hodge Kummer Hodge Hodge
Teichmüller ( ) 2015 11 0 3 1 4 2 6 3 Teichmüller 8 4 Diophantus 11 5 13 6 15 7 19 8 21 9 25 10 28 11 31 12 34 13 36 14 41 15 43 16 47 1 17 Θ 50 18 55 19 57 20 Hodge 59 21 63 22 67 23 Θ Hodge 69 24 Kummer
More information1 G K C 1.1. G K V ρ : G GL(V ) (ρ, V ) G V 1.2. G 2 (ρ, V ), (τ, W ) 2 V, W T : V W τ g T = T ρ g ( g G) V ρ g T W τ g V T W 1.3. G (ρ, V ) V W ρ g W
Naoya Enomoto 2002.9. paper 1 2 2 3 3 6 1 1 G K C 1.1. G K V ρ : G GL(V ) (ρ, V ) G V 1.2. G 2 (ρ, V ), (τ, W ) 2 V, W T : V W τ g T = T ρ g ( g G) V ρ g T W τ g V T W 1.3. G (ρ, V ) V W ρ g W W G- G W
More informationIII 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F
III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F F 1 F 2 F, (3) F λ F λ F λ F. 3., A λ λ A λ. B λ λ
More information2019 1 5 0 3 1 4 1.1.................... 4 1.1.1......................... 4 1.1.2........................ 5 1.1.3................... 5 1.1.4........................ 6 1.1.5......................... 6 1.2..........................
More information,, Andrej Gendiar (Density Matrix Renormalization Group, DMRG) 1 10 S.R. White [1, 2] 2 DMRG ( ) [3, 2] DMRG Baxter [4, 5] 2 Ising 2 1 Ising 1 1 Ising
,, Andrej Gendiar (Density Matrix Renormalization Group, DMRG) 1 10 S.R. White [1, 2] 2 DMRG ( ) [3, 2] DMRG Baxter [4, 5] 2 Ising 2 1 Ising 1 1 Ising Model 1 Ising 1 Ising Model N Ising (σ i = ±1) (Free
More information) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4
1. k λ ν ω T v p v g k = π λ ω = πν = π T v p = λν = ω k v g = dω dk 1) ) 3) 4). p = hk = h λ 5) E = hν = hω 6) h = h π 7) h =6.6618 1 34 J sec) hc=197.3 MeV fm = 197.3 kev pm= 197.3 ev nm = 1.97 1 3 ev
More information数学Ⅱ演習(足助・09夏)
II I 9/4/4 9/4/2 z C z z z z, z 2 z, w C zw z w 3 z, w C z + w z + w 4 t R t C t t t t t z z z 2 z C re z z + z z z, im z 2 2 3 z C e z + z + 2 z2 + 3! z3 + z!, I 4 x R e x cos x + sin x 2 z, w C e z+w
More information2018/10/04 IV/ IV 2/12. A, f, g A. (1) D(0 A ) =, D(1 A ) = Spec(A), D(f) D(g) = D(fg). (2) {f l A l Λ} A I D(I) = l Λ D(f l ). (3) I, J A D(I) D(J) =
2018/10/04 IV/ IV 1/12 2018 IV/ IV 10 04 * 1 : ( A 441 ) yanagida[at]math.nagoya-u.ac.jp https://www.math.nagoya-u.ac.jp/~yanagida 1 I: (ring)., A 0 A, 1 A. (ring homomorphism).. 1.1 A (ideal) I, ( ) I
More informationAI n Z f n : Z Z f n (k) = nk ( k Z) f n n 1.9 R R f : R R f 1 1 {a R f(a) = 0 R = {0 R 1.10 R R f : R R f 1 : R R 1.11 Z Z id Z 1.12 Q Q id
1 1.1 1.1 R R (1) R = 1 2 Z = 2 n Z (2) R 1.2 R C Z R 1.3 Z 2 = {(a, b) a Z, b Z Z 2 a, b, c, d Z (a, b) + (c, d) = (a + c, b + d), (a, b)(c, d) = (ac, bd) (1) Z 2 (2) Z 2? (3) Z 2 1.4 C Q[ 1] = {a + bi
More information1 Abstract 2 3 n a ax 2 + bx + c = 0 (a 0) (1) ( x + b ) 2 = b2 4ac 2a 4a 2 D = b 2 4ac > 0 (1) 2 D = 0 D < 0 x + b 2a = ± b2 4ac 2a b ± b 2
1 Abstract n 1 1.1 a ax + bx + c = 0 (a 0) (1) ( x + b ) = b 4ac a 4a D = b 4ac > 0 (1) D = 0 D < 0 x + b a = ± b 4ac a b ± b 4ac a b a b ± 4ac b i a D (1) ax + bx + c D 0 () () (015 8 1 ) 1. D = b 4ac
More information20 4 20 i 1 1 1.1............................ 1 1.2............................ 4 2 11 2.1................... 11 2.2......................... 11 2.3....................... 19 3 25 3.1.............................
More information. Mac Lane [ML98]. 1 2 (strict monoidal category) S 1 R 3 A S 1 [0, 1] C 2 C End C (1) C 4 1 U q (sl 2 ) Drinfeld double. 6 2
2014 6 30. 2014 3 1 6 (Hopf algebra) (group) Andruskiewitsch-Santos [AFS09] 1980 Drinfeld (quantum group) Lie Lie (ribbon Hopf algebra) (ribbon category) Turaev [Tur94] Kassel [Kas95] (PD) x12005i@math.nagoya-u.ac.jp
More informationhttp://www.ike-dyn.ritsumei.ac.jp/ hyoo/wave.html 1 1, 5 3 1.1 1..................................... 3 1.2 5.1................................... 4 1.3.......................... 5 1.4 5.2, 5.3....................
More informationprime number theorem
For Tutor MeBio ζ Eite by kamei MeBio 7.8.3 : Bernoulli Bernoulli 4 Bernoulli....................................................................................... 4 Bernoulli............................................................................
More information2000年度『数学展望 I』講義録
2000 I I IV I II 2000 I I IV I-IV. i ii 3.10 (http://www.math.nagoya-u.ac.jp/ kanai/) 2000 A....1 B....4 C....10 D....13 E....17 Brouwer A....21 B....26 C....33 D....39 E. Sperner...45 F....48 A....53
More information( ) (, ) ( )
( ) (, ) ( ) 1 2 2 2 2.1......................... 2 2.2.............................. 3 2.3............................... 4 2.4.............................. 5 2.5.............................. 6 2.6..........................
More informationX G P G (X) G BG [X, BG] S 2 2 2 S 2 2 S 2 = { (x 1, x 2, x 3 ) R 3 x 2 1 + x 2 2 + x 2 3 = 1 } R 3 S 2 S 2 v x S 2 x x v(x) T x S 2 T x S 2 S 2 x T x S 2 = { ξ R 3 x ξ } R 3 T x S 2 S 2 x x T x S 2
More information第5章 偏微分方程式の境界値問題
October 5, 2018 1 / 113 4 ( ) 2 / 113 Poisson 5.1 Poisson ( A.7.1) Poisson Poisson 1 (A.6 ) Γ p p N u D Γ D b 5.1.1: = Γ D Γ N 3 / 113 Poisson 5.1.1 d {2, 3} Lipschitz (A.5 ) Γ D Γ N = \ Γ D Γ p Γ N Γ
More informationv v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) 3 R ij R ik = δ jk (4) i=1 δ ij Kronecker δ ij = { 1 (i = j) 0 (i
1. 1 1.1 1.1.1 1.1.1.1 v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) R ij R ik = δ jk (4) δ ij Kronecker δ ij = { 1 (i = j) 0 (i j) (5) 1 1.1. v1.1 2011/04/10 1. 1 2 v i = R ij v j (6) [
More information24 I ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x
24 I 1.1.. ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x 1 (t), x 2 (t),, x n (t)) ( ) ( ), γ : (i) x 1 (t),
More information1. 1 A : l l : (1) l m (m 3) (2) m (3) n (n 3) (4) A α, β γ α β + γ = 2 m l lm n nα nα = lm. α = lm n. m lm 2β 2β = lm β = lm 2. γ l 2. 3
1. 1 A : l l : (1) l m (m 3) (2) m (3) n (n 3) (4) A 2 1 2 1 2 3 α, β γ α β + γ = 2 m l lm n nα nα = lm. α = lm n. m lm 2β 2β = lm β = lm 2. γ l 2. 3 4 P, Q R n = {(x 1, x 2,, x n ) ; x 1, x 2,, x n R}
More informationII 2 3.,, A(B + C) = AB + AC, (A + B)C = AC + BC. 4. m m A, m m B,, m m B, AB = BA, A,, I. 5. m m A, m n B, AB = B, A I E, 4 4 I, J, K
II. () 7 F 7 = { 0,, 2, 3, 4, 5, 6 }., F 7 a, b F 7, a b, F 7,. (a) a, b,,. (b) 7., 4 5 = 20 = 2 7 + 6, 4 5 = 6 F 7., F 7,., 0 a F 7, ab = F 7 b F 7. (2) 7, 6 F 6 = { 0,, 2, 3, 4, 5 },,., F 6., 0 0 a F
More information8.1 Fubini 8.2 Fubini 9 (0%) 10 (50%) Carathéodory 10.3 Fubini 1 Introduction 1 (1) (2) {f n (x)} n=1 [a, b] K > 0 n, x f n (x) K < ( ) x [a
% 100% 1 Introduction 2 (100%) 2.1 2.2 2.3 3 (100%) 3.1 3.2 σ- 4 (100%) 4.1 4.2 5 (100%) 5.1 5.2 5.3 6 (100%) 7 (40%) 8 Fubini (90%) 2007.11.5 1 8.1 Fubini 8.2 Fubini 9 (0%) 10 (50%) 10.1 10.2 Carathéodory
More informationma22-9 u ( v w) = u v w sin θê = v w sin θ u cos φ = = 2.3 ( a b) ( c d) = ( a c)( b d) ( a d)( b c) ( a b) ( c d) = (a 2 b 3 a 3 b 2 )(c 2 d 3 c 3 d
A 2. x F (t) =f sin ωt x(0) = ẋ(0) = 0 ω θ sin θ θ 3! θ3 v = f mω cos ωt x = f mω (t sin ωt) ω t 0 = f ( cos ωt) mω x ma2-2 t ω x f (t mω ω (ωt ) 6 (ωt)3 = f 6m ωt3 2.2 u ( v w) = v ( w u) = w ( u v) ma22-9
More information0. Intro ( K CohFT etc CohFT 5.IKKT 6.
E-mail: sako@math.keio.ac.jp 0. Intro ( K 1. 2. CohFT etc 3. 4. CohFT 5.IKKT 6. 1 µ, ν : d (x 0,x 1,,x d 1 ) t = x 0 ( t τ ) x i i, j, :, α, β, SO(D) ( x µ g µν x µ µ g µν x ν (1) g µν g µν vector x µ,y
More information0. I II I II (1) linear type: GL( ), Sp( ), O( ), (2) loop type: loop current Kac-Moody affine, hyperbolic (3) diffeo t
e-mail: koyama@math.keio.ac.jp 0. I II I II (1) linear type: GL( ), Sp( ), O( ), (2) loop type: loop current Kac-Moody affine, hyperbolic (3) diffeo type: diffeo universal Teichmuller modular I. I-. Weyl
More information( ) ) ) ) 5) 1 J = σe 2 6) ) 9) 1955 Statistical-Mechanical Theory of Irreversible Processes )
( 3 7 4 ) 2 2 ) 8 2 954 2) 955 3) 5) J = σe 2 6) 955 7) 9) 955 Statistical-Mechanical Theory of Irreversible Processes 957 ) 3 4 2 A B H (t) = Ae iωt B(t) = B(ω)e iωt B(ω) = [ Φ R (ω) Φ R () ] iω Φ R (t)
More informationII No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh
More informationn (1.6) i j=1 1 n a ij x j = b i (1.7) (1.7) (1.4) (1.5) (1.4) (1.7) u, v, w ε x, ε y, ε x, γ yz, γ zx, γ xy (1.8) ε x = u x ε y = v y ε z = w z γ yz
1 2 (a 1, a 2, a n ) (b 1, b 2, b n ) A (1.1) A = a 1 b 1 + a 2 b 2 + + a n b n (1.1) n A = a i b i (1.2) i=1 n i 1 n i=1 a i b i n i=1 A = a i b i (1.3) (1.3) (1.3) (1.1) (ummation convention) a 11 x
More informationchap1.dvi
1 1 007 1 e iθ = cos θ + isin θ 1) θ = π e iπ + 1 = 0 1 ) 3 11 f 0 r 1 1 ) k f k = 1 + r) k f 0 f k k = 01) f k+1 = 1 + r)f k ) f k+1 f k = rf k 3) 1 ) ) ) 1+r/)f 0 1 1 + r/) f 0 = 1 + r + r /4)f 0 1 f
More information( 3) b 1 b : b b f : a b 1 b f = f (2.7) g : b c g 1 b = g (2.8) 1 b b (identity arrow) id b f a b g f 1 b b c g (2.9) 3 C C C a, b a b Hom C (a, b) h
2011 9 5 1 Lie 1 2 2.1 (category) (object) a, b, c, a b (arrow, morphism) f : a b (2.1) f a b (2.2) ( 1) f : a b g : b c (composite) g f : a c ( 2) f f a b g f g c g h (2.3) a b c d (2.4) h (g f) = (h
More informationN cos s s cos ψ e e e e 3 3 e e 3 e 3 e
3 3 5 5 5 3 3 7 5 33 5 33 9 5 8 > e > f U f U u u > u ue u e u ue u ue u e u e u u e u u e u N cos s s cos ψ e e e e 3 3 e e 3 e 3 e 3 > A A > A E A f A A f A [ ] f A A e > > A e[ ] > f A E A < < f ; >
More information1 1.1 R (ring) R1 R4 R1 R (commutative [abelian] group) R2 a, b, c R (ab)c = a(bc) (associative law) R3 a, b, c R a(b + c) = ab + ac, (a + b)c = ac +
ALGEBRA II Hiroshi SUZUKI Department of Mathematics International Christian University 2004 1 1 1 2 2 1 3 3 1 4 4 1 5 5 1 6 6 1 7 7 1 7.1....................... 7 1 7.2........................... 7 4 8
More informationy π π O π x 9 s94.5 y dy dx. y = x + 3 y = x logx + 9 s9.6 z z x, z y. z = xy + y 3 z = sinx y 9 s x dx π x cos xdx 9 s93.8 a, fx = e x ax,. a =
[ ] 9 IC. dx = 3x 4y dt dy dt = x y u xt = expλt u yt λ u u t = u u u + u = xt yt 6 3. u = x, y, z = x + y + z u u 9 s9 grad u ux, y, z = c c : grad u = u x i + u y j + u k i, j, k z x, y, z grad u v =
More information1 1.1 H = µc i c i + c i t ijc j + 1 c i c j V ijklc k c l (1) V ijkl = V jikl = V ijlk = V jilk () t ij = t ji, V ijkl = V lkji (3) (1) V 0 H mf = µc
013 6 30 BCS 1 1.1........................ 1................................ 3 1.3............................ 3 1.4............................... 5 1.5.................................... 5 6 3 7 4 8
More information() n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (5) (6 ) n C + nc + 3 nc n nc n (7 ) n C + nc + 3 nc n nc n (
3 n nc k+ k + 3 () n C r n C n r nc r C r + C r ( r n ) () n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (4) n C n n C + n C + n C + + n C n (5) k k n C k n C k (6) n C + nc
More information.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
p 1. 1.1., 01 8 3, 57,,.. 1.., Gal(Q p /Q p ), 1. Wach,,. 1.3. Part I,,. Part II, Part III. 1.4.., 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
More informationI. (CREMONA ) : Cremona [C],., modular form f E f. 1., modular X H 1 (X, Q). modular symbol M-symbol, ( ) modular symbol., notation. H = { z = x
I. (CREMONA ) : Cremona [C],., modular form f E f. 1., modular X H 1 (X, Q). modular symbol M-symbol, ( ). 1.1. modular symbol., notation. H = z = x iy C y > 0, cusp H = H Q., Γ = PSL 2 (Z), G Γ [Γ : G]
More informationSO(3) 49 u = Ru (6.9), i u iv i = i u iv i (C ) π π : G Hom(V, V ) : g D(g). π : R 3 V : i 1. : u u = u 1 u 2 u 3 (6.10) 6.2 i R α (1) = 0 cos α
SO(3) 48 6 SO(3) t 6.1 u, v u = u 1 1 + u 2 2 + u 3 3 = u 1 e 1 + u 2 e 2 + u 3 e 3, v = v 1 1 + v 2 2 + v 3 3 = v 1 e 1 + v 2 e 2 + v 3 e 3 (6.1) i (e i ) e i e j = i j = δ ij (6.2) ( u, v ) = u v = ij
More information2011 8 26 3 I 5 1 7 1.1 Markov................................ 7 2 Gau 13 2.1.................................. 13 2.2............................... 18 2.3............................ 23 3 Gau (Le vy
More informationChap9.dvi
.,. f(),, f(),,.,. () lim 2 +3 2 9 (2) lim 3 3 2 9 (4) lim ( ) 2 3 +3 (5) lim 2 9 (6) lim + (7) lim (8) lim (9) lim (0) lim 2 3 + 3 9 2 2 +3 () lim sin 2 sin 2 (2) lim +3 () lim 2 2 9 = 5 5 = 3 (2) lim
More informationkoji07-01.dvi
2007 I II III 1, 2, 3, 4, 5, 6, 7 5 10 19 (!) 1938 70 21? 1 1 2 1 2 2 1! 4, 5 1? 50 1 2 1 1 2 2 1?? 2 1 1, 2 1, 2 1, 2, 3,... 3 1, 2 1, 3? 2 1 3 1 2 1 1, 2 2, 3? 2 1 3 2 3 2 k,l m, n k,l m, n kn > ml...?
More informationNo δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2
No.2 1 2 2 δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j
More informationSO(2)
TOP URL http://amonphys.web.fc2.com/ 1 12 3 12.1.................................. 3 12.2.......................... 4 12.3............................. 5 12.4 SO(2).................................. 6
More informationkeisoku01.dvi
2.,, Mon, 2006, 401, SAGA, JAPAN Dept. of Mechanical Engineering, Saga Univ., JAPAN 4 Mon, 2006, 401, SAGA, JAPAN Dept. of Mechanical Engineering, Saga Univ., JAPAN 5 Mon, 2006, 401, SAGA, JAPAN Dept.
More informationTOP URL 1
TOP URL http://amonphys.web.fc.com/ 3.............................. 3.............................. 4.3 4................... 5.4........................ 6.5........................ 8.6...........................7
More information62 Serre Abel-Jacob Serre Jacob Jacob Jacob k Jacob Jac(X) X g X (g) X (g) Zarsk [Wel] [Ml] [BLR] [Ser] Jacob ( ) 2 Jacob Pcard 2.1 X g ( C ) X n P P
15, pp.61-80 Abel-Jacob I 1 Introducton Remann Abel-Jacob X g Remann X ω 1,..., ω g Λ = {( γ ω 1,..., γ ω g) C g γ H 1 (X, Z)} Λ C g lattce Jac(X) = C g /Λ Le Abel-Jacob (Theorem 2.2, 4.2) Jac(X) Pcard
More informationII Time-stamp: <05/09/30 17:14:06 waki> ii
II waki@cc.hirosaki-u.ac.jp 18 1 30 II Time-stamp: ii 1 1 1.1.................................................. 1 1.2................................................... 3 1.3..................................................
More information平成 15 年度 ( 第 25 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 ~8 15 月年 78 日開催月 4 日 ) X 2 = 1 ( ) f 1 (X 1,..., X n ) = 0,..., f r (X 1,..., X n ) = 0 X = (
1 1.1 X 2 = 1 ( ) f 1 (X 1,..., X n ) = 0,..., f r (X 1,..., X n ) = 0 X = (X 1,..., X n ) ( ) X 1,..., X n f 1,..., f r A T X + XA XBR 1 B T X + C T QC = O X 1.2 X 1,..., X n X i X j X j X i = 0, P i
More informationJacobson Prime Avoidance
2016 2017 2 22 1 1 3 2 4 2.1 Jacobson................. 4 2.2.................... 5 3 6 3.1 Prime Avoidance....................... 7 3.2............................. 8 3.3..............................
More informationJuly 28, H H 0 H int = H H 0 H int = H int (x)d 3 x Schrödinger Picture Ψ(t) S =e iht Ψ H O S Heisenberg Picture Ψ H O H (t) =e iht O S e i
July 8, 4. H H H int H H H int H int (x)d 3 x Schrödinger Picture Ψ(t) S e iht Ψ H O S Heisenberg Picture Ψ H O H (t) e iht O S e iht Interaction Picture Ψ(t) D e iht Ψ(t) S O D (t) e iht O S e ih t (Dirac
More information: : : : ) ) 1. d ij f i e i x i v j m a ij m f ij n x i =
1 1980 1) 1 2 3 19721960 1965 2) 1999 1 69 1980 1972: 55 1999: 179 2041999: 210 211 1999: 211 3 2003 1987 92 97 3) 1960 1965 1970 1985 1990 1995 4) 1. d ij f i e i x i v j m a ij m f ij n x i = n d ij
More informationi I II I II II IC IIC I II ii 5 8 5 3 7 8 iii I 3........................... 5......................... 7........................... 4........................ 8.3......................... 33.4...................
More information2016 Course Description of Undergraduate Seminars (2015 12 16 ) 2016 12 16 ( ) 13:00 15:00 12 16 ( ) 1 21 ( ) 1 13 ( ) 17:00 1 14 ( ) 12:00 1 21 ( ) 15:00 1 27 ( ) 13:00 14:00 2 1 ( ) 17:00 2 3 ( ) 12
More information1 R n (x (k) = (x (k) 1,, x(k) n )) k 1 lim k,l x(k) x (l) = 0 (x (k) ) 1.1. (i) R n U U, r > 0, r () U (ii) R n F F F (iii) R n S S S = { R n ; r > 0
III 2018 11 7 1 2 2 3 3 6 4 8 5 10 ϵ-δ http://www.mth.ngoy-u.c.jp/ ymgmi/teching/set2018.pdf http://www.mth.ngoy-u.c.jp/ ymgmi/teching/rel2018.pdf n x = (x 1,, x n ) n R n x 0 = (0,, 0) x = (x 1 ) 2 +
More information四変数基本対称式の解放
The second-thought of the Galois-style way to solve a quartic equation Oomori, Yasuhiro in Himeji City, Japan Jan.6, 013 Abstract v ρ (v) Step1.5 l 3 1 6. l 3 7. Step - V v - 3 8. Step1.3 - - groupe groupe
More informationn ξ n,i, i = 1,, n S n ξ n,i n 0 R 1,.. σ 1 σ i .10.14.15 0 1 0 1 1 3.14 3.18 3.19 3.14 3.14,. ii 1 1 1.1..................................... 1 1............................... 3 1.3.........................
More informationii 3.,. 4. F. (), ,,. 8.,. 1. (75%) (25%) =7 20, =7 21 (. ). 1.,, (). 3.,. 1. ().,.,.,.,.,. () (12 )., (), 0. 2., 1., 0,.
24(2012) (1 C106) 4 11 (2 C206) 4 12 http://www.math.is.tohoku.ac.jp/~obata,.,,,.. 1. 2. 3. 4. 5. 6. 7.,,. 1., 2007 (). 2. P. G. Hoel, 1995. 3... 1... 2.,,. ii 3.,. 4. F. (),.. 5... 6.. 7.,,. 8.,. 1. (75%)
More information211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,
More informationm dv = mg + kv2 dt m dv dt = mg k v v m dv dt = mg + kv2 α = mg k v = α 1 e rt 1 + e rt m dv dt = mg + kv2 dv mg + kv 2 = dt m dv α 2 + v 2 = k m dt d
m v = mg + kv m v = mg k v v m v = mg + kv α = mg k v = α e rt + e rt m v = mg + kv v mg + kv = m v α + v = k m v (v α (v + α = k m ˆ ( v α ˆ αk v = m v + α ln v α v + α = αk m t + C v α v + α = e αk m
More informationOn a branched Zp-cover of Q-homology 3-spheres
Zp 拡大と分岐 Zp 被覆 GL1 表現の変形理論としての岩澤理論 SL2 表現の変形理論 On a branched Zp -cover of Q-homology 3-spheres 植木 潤 九州大学大学院数理学府 D2 December 23, 2014 植木 潤 九州大学大学院数理学府 D2 On a branched Zp -cover of Q-homology 3-spheres
More informationx V x x V x, x V x = x + = x +(x+x )=(x +x)+x = +x = x x = x x = x =x =(+)x =x +x = x +x x = x ( )x = x =x =(+( ))x =x +( )x = x +( )x ( )x = x x x R
V (I) () (4) (II) () (4) V K vector space V vector K scalor K C K R (I) x, y V x + y V () (x + y)+z = x +(y + z) (2) x + y = y + x (3) V x V x + = x (4) x V x + x = x V x x (II) x V, α K αx V () (α + β)x
More informationall.dvi
29 4 Green-Lagrange,,.,,,,,,.,,,,,,,,,, E, σ, ε σ = Eε,,.. 4.1? l, l 1 (l 1 l) ε ε = l 1 l l (4.1) F l l 1 F 30 4 Green-Lagrange Δz Δδ γ = Δδ (4.2) Δz π/2 φ γ = π 2 φ (4.3) γ tan γ γ,sin γ γ ( π ) γ tan
More information