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.,,.,,
|
|
- ともみ こびき
- 5 years ago
- Views:
Transcription
1 I ( ) (i) l, l, l (ii) (Q p ) l, l, l (iii) Artin (iv). (i),(ii). (iii) 1. (iv),.. [9]. [4] L-,.. Contents l l l Artin l, l, l Weil-Deligne References 27 1 Artin,,.,, Artin l. 1
2 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 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,. 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
3 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) 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
4 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 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 (2)),, Aut R (M). 4
5 , 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 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 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. ( ). 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
6 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 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., 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) ( ). 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
7 .,, 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]) l G K., G K,, l., K, G K l. Grothendieck ( [7] )., l, ( 2-2-2)., l 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 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
8 , 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 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 ) S ρ... S ρ 0... ( [4] 1., [20] ) 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
9 {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 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 ) 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 V E-, ρ : G K Aut E (V )., ρ, l V G K , 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
10 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) E/K, T l (E) l (cf )., ρ l : G K Aut Zl (T l (E)) GL 2 (Z l ) l l χ l : G K Z l. ( ) l X/K K., T l := H í et(x K, Z l )/(torsion) (0 i 2dimX) Z l (cf. ). G K T l l 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 ( ). 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
11 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 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., 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 ) 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
12 G K E l E[l](K). ρ E,l : G K Aut Z/lZ (E[l](K)) GL 2 (F l ) l, l K., 1 ρ : G K F l l ( : G K Imρ,, ρ (Z/NZ). N = l t M, l M. l t = 1 ) 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
13 . R(ρ) (rigid analytic space)x, E X(E), (E Q p ) ρ E p. p ρ X,., 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 , 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, ρ : G K Aut C (V )., ρ K ρ : G K Aut C (V ) S ρ ( 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
14 . 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 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 ι(σ) =, ι(τ) = 0 ζ , 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]) 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 (2) A =, B =, C = 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 ( ) f(τ) := θ A (τ) θ B (τ) 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 ( ) Gal(H/Q) Artin ρ. ρ 1 0 f ρ f (cf. [4], [5]), F p f p. 14
15 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 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
16 ,. 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] (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 G F = Gal(K ur /K) K ur Gal(K tm /K ur ). 16
17 . 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= 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 ) (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) 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
18 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
19 , 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 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
20 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) ρ (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 (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 Fontaine D pst, (cf. [6]). 1:
21 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) ) (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 (1) Weil-Deligne (r, N), N = 0, r, K L, r WL. (2) l Weil-Deligne., l, Weil-Deligne. [7]. 4. K, Σ K K. l ρ v Σ K, P v,ρ (T ) := det(1 ρ(frob v )T ) 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) 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
22 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) (a) l (χ l ) l,. (b) (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 ) (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
23 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, 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) ) ([14] I-12 3) 4-6, l ρ l, ρ l l ρ l (l l) ( l. [12] 5.1 ). 23
24 (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 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
25 ., 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 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 , 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 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= 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 , (a i ) Mi A End A (M i ) a i A. 25
26 . M ((a i ) Mi ) i End A (M n ) , 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 = , 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
27 , 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), [2] P. Deligne, Résumé des premiers exposés de A. Grothendieck, Groupes de monodromie en geometrie algebrique. I. Seminaire de Geometrie Algebrique du Bois-Marie (SGA 7 I). pp [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 [6] J-M. Fontaine, Représentations l-adiques potentiellement semi-stables, Exposé VIII, Astérisque 1994, 224, périodes p-adiques, seminaire de Bures, [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, [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, xxiv+184 pp. [15] J-P. Serre and J. Tate, Good reduction of abelian varieties. Ann. of Math. (2) [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), [19] Y. Taniyama, L-functions of number fields and zeta functions of abelian varieties. J. Math. Soc. Japan [20], p,. 27
28 [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
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 informationSAMA- 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 information( ) 1., ([SU] ): F K k., Z p -, (cf. [Iw2], [Iw3], [Iw6]). K F F/K Z p - k /k., Weil., K., K F F p- ( 4.1).,, Z p -,., Weil..,,. Weil., F, F projectiv
( ) 1 ([SU] ): F K k Z p - (cf [Iw2] [Iw3] [Iw6]) K F F/K Z p - k /k Weil K K F F p- ( 41) Z p - Weil Weil F F projective smooth C C Jac(C)/F ( ) : 2 3 4 5 Tate Weil 6 7 Z p - 2 [Iw1] 2 21 K k k 1 k K
More informationMazur [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
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 information16 B
16 B (1) 3 (2) (3) 5 ( ) 3 : 2 3 : 3 : () 3 19 ( ) 2 ax 2 + bx + c = 0 (a 0) x = b ± b 2 4ac 2a 3, 4 5 1824 5 Contents 1. 1 2. 7 3. 13 4. 18 5. 22 6. 25 7. 27 8. 31 9. 37 10. 46 11. 50 12. 56 i 1 1. 1.1..
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 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 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 information( ),, ( [Ka93b],[FK06]).,. p Galois L, Langlands p p Galois (, ) p., Breuil, Colmez([Co10]), Q p Galois G Qp 2 p ( ) GL 2 (Q p ) p Banach ( ) (GL 2 (Q
2017 : msjmeeting-2017sep-00f006 p Langlands ( ) 1. Q, Q p Q Galois G Q p (p Galois ). p Galois ( p Galois ), L Selmer Tate-Shafarevich, Galois. Dirichlet ( Dedekind s = 0 ) Birch-Swinnerton-Dyer ( L s
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 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 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 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 information2 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)
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 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 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 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 informationS K(S) = T K(T ) T S K n (1.1) n {}}{ n K n (1.1) 0 K 0 0 K Q p K Z/pZ L K (1) L K L K (2) K L L K [L : K] 1.1.
() 1.1.. 1. 1.1. (1) L K (i) 0 K 1 K (ii) x, y K x + y K, x y K (iii) x, y K xy K (iv) x K \ {0} x 1 K K L L K ( 0 L 1 L ) L K L/K (2) K M L M K L 1.1. C C 1.2. R K = {a + b 3 i a, b Q} Q( 2, 3) = Q( 2
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 information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
R = T, Fermat Wiles, Taylor-Wiles R = T.,,.,. 1. Fermat Fermat,. Fermat, 17, 400.. Descartes ( ) Corneille ( ), Milton ( ), Velázquez ( ), Rembrandt van Rijn ( ),,,. Fermat, Fermat, Fermat, 1995 Wiles
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 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 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 informationnewmain.dvi
数論 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/008142 このサンプルページの内容は, 第 2 版 1 刷発行当時のものです. Daniel DUVERNEY: THÉORIE DES NOMBRES c Dunod, Paris, 1998, This book is published
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 informationTOP URL 1
TOP URL http://amonphys.web.fc.com/ 3.............................. 3.............................. 4.3 4................... 5.4........................ 6.5........................ 8.6...........................7
More informationC p (.2 C p [[T ]] Bernoull B n,χ C p p q p 2 q = p p = 2 q = 4 ω Techmüller a Z p ω(a a ( mod q φ(q ω(a Z p a pz p ω(a = 0 Z p φ Euler Techmüller ω Q
p- L- [Iwa] [Iwa2] -Leopoldt [KL] p- L-. Kummer Remann ζ(s Bernoull B n (. ζ( n = B n n, ( n Z p a = Kummer [Kum] ( Kummer p m n 0 ( mod p m n a m n ( mod (p p a ( p m B m m ( pn B n n ( mod pa Z p Kummer
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,2,4
2005 12 2006 1,2,4 iii 1 Hilbert 14 1 1.............................................. 1 2............................................... 2 3............................................... 3 4.............................................
More informationÎã³°·¿¤Î¥·¥å¡¼¥Ù¥ë¥È¥«¥êto=1=¡á=1=¥ë¥�¥å¥é¥¹
(kaji@math.sci.fukuoka-u.ac.jp) 2009 8 10 R 3 R 3 ( wikipedia ) (Schubert, 19 ) (= )(Ehresmann, 20 ) (Chevalley, 20 ) G/P: ( : ) W : ( : ) X w : W X w W G: B G: Borel P B: G/P: 1 C n ( ) Fl n := {0 V
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 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 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 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 informationI A A441 : April 15, 2013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida )
I013 00-1 : April 15, 013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida) http://www.math.nagoya-u.ac.jp/~kawahira/courses/13s-tenbou.html pdf * 4 15 4 5 13 e πi = 1 5 0 5 7 3 4 6 3 6 10 6 17
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 informationI A A441 : April 21, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) Google
I4 - : April, 4 Version :. Kwhir, Tomoki TA (Kondo, Hirotk) Google http://www.mth.ngoy-u.c.jp/~kwhir/courses/4s-biseki.html pdf 4 4 4 4 8 e 5 5 9 etc. 5 6 6 6 9 n etc. 6 6 6 3 6 3 7 7 etc 7 4 7 7 8 5 59
More informationR R P N (C) 7 C Riemann R K ( ) C R C K 8 (R ) R C K 9 Riemann /C /C Riemann 10 C k 11 k C/k 12 Riemann k Riemann C/k k(c)/k R k F q Riemann 15
(Gen KUROKI) 1 1 : Riemann Spec Z 2? 3 : 4 2 Riemann Riemann Riemann 1 C 5 Riemann Riemann R compact R K C ( C(x) ) K C(R) Riemann R 6 (E-mail address: kuroki@math.tohoku.ac.jp) 1 1 ( 5 ) 2 ( Q ) Spec
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 information006 11 8 0 3 1 5 1.1..................... 5 1......................... 6 1.3.................... 6 1.4.................. 8 1.5................... 8 1.6................... 10 1.6.1......................
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 informationtomocci ,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p.
tomocci 18 7 5...,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p. M F (M), X(F (M)).. T M p e i = e µ i µ. a a = a i
More information医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 第 2 版 1 刷発行時のものです.
医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/009192 このサンプルページの内容は, 第 2 版 1 刷発行時のものです. i 2 t 1. 2. 3 2 3. 6 4. 7 5. n 2 ν 6. 2 7. 2003 ii 2 2013 10 iii 1987
More information等質空間の幾何学入門
2006/12/04 08 tamaru@math.sci.hiroshima-u.ac.jp i, 2006/12/04 08. 2006, 4.,,.,,.,.,.,,.,,,.,.,,.,,,.,. ii 1 1 1.1 :................................... 1 1.2........................................ 2 1.3......................................
More information( ) (, ) ( )
( ) (, ) ( ) 1 2 2 2 2.1......................... 2 2.2.............................. 3 2.3............................... 4 2.4.............................. 5 2.5.............................. 6 2.6..........................
More information( 9 1 ) 1 2 1.1................................... 2 1.2................................................. 3 1.3............................................... 4 1.4...........................................
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 information入試の軌跡
4 y O x 4 Typed by L A TEX ε ) ) ) 6 4 ) 4 75 ) http://kumamoto.s.xrea.com/plan/.. PDF) Ctrl +L) Ctrl +) Ctrl + Ctrl + ) ) Alt + ) Alt + ) ESC. http://kumamoto.s.xrea.com/nyusi/kumadai kiseki ri i.pdf
More information2011de.dvi
211 ( 4 2 1. 3 1.1............................... 3 1.2 1- -......................... 13 1.3 2-1 -................... 19 1.4 3- -......................... 29 2. 37 2.1................................ 37
More informationii
ii iii 1 1 1.1..................................... 1 1.2................................... 3 1.3........................... 4 2 9 2.1.................................. 9 2.2...............................
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 informationver Web
ver201723 Web 1 4 11 4 12 5 13 7 2 9 21 9 22 10 23 10 24 11 3 13 31 n 13 32 15 33 21 34 25 35 (1) 27 4 30 41 30 42 32 43 36 44 (2) 38 45 45 46 45 5 46 51 46 52 48 53 49 54 51 55 54 56 58 57 (3) 61 2 3
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 information,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,.
9 α ν β Ξ ξ Γ γ o δ Π π ε ρ ζ Σ σ η τ Θ θ Υ υ ι Φ φ κ χ Λ λ Ψ ψ µ Ω ω Def, Prop, Th, Lem, Note, Remark, Ex,, Proof, R, N, Q, C [a, b {x R : a x b} : a, b {x R : a < x < b} : [a, b {x R : a x < b} : a,
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 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 information日本数学会・2011年度年会(早稲田大学)・総合講演
日本数学会 2011 年度年会 ( 早稲田大学 ) 総合講演 2011 年度日本数学会春季賞受賞記念講演 MSJMEETING-2011-0 ( ) 1. p>0 p C ( ) p p 0 smooth l (l p ) p p André, Christol, Mebkhout, Kedlaya K 0 O K K k O K k p>0 K K : K R 0 p = p 1 Γ := K k
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 informationPart () () Γ Part ,
Contents a 6 6 6 6 6 6 6 7 7. 8.. 8.. 8.3. 8 Part. 9. 9.. 9.. 3. 3.. 3.. 3 4. 5 4.. 5 4.. 9 4.3. 3 Part. 6 5. () 6 5.. () 7 5.. 9 5.3. Γ 3 6. 3 6.. 3 6.. 3 6.3. 33 Part 3. 34 7. 34 7.. 34 7.. 34 8. 35
More informationGauss Fuchs rigid rigid rigid Nicholas Katz Rigid local systems [6] Fuchs Katz Crawley- Boevey[1] [7] Katz rigid rigid Katz middle convolu
rigidity 2014.9.1-2014.9.2 Fuchs 1 Introduction y + p(x)y + q(x)y = 0, y 2 p(x), q(x) p(x) q(x) Fuchs 19 Fuchs 83 Gauss Fuchs rigid rigid rigid 7 1970 1996 Nicholas Katz Rigid local systems [6] Fuchs Katz
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 information1 = = = (set) (element) a A a A a A a A a A {2, 5, (0, 1)}, [ 1, 1] = {x; 1 x 1}. (proposition) A = {x; P (x)} P (x) x x a A a A Remark. (i) {2, 0, 0,
2005 4 1 1 2 2 6 3 8 4 11 5 14 6 18 7 20 8 22 9 24 10 26 11 27 http://matcmadison.edu/alehnen/weblogic/logset.htm 1 1 = = = (set) (element) a A a A a A a A a A {2, 5, (0, 1)}, [ 1, 1] = {x; 1 x 1}. (proposition)
More informationALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University
ALGEBRA I 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 8 8 1 9 9 1 10 10 1 E-mail:hsuzuki@icu.ac.jp 0 0 1 1.1 G G1 G a, b,
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 information.2 ρ dv dt = ρk grad p + 3 η grad (divv) + η 2 v.3 divh = 0, rote + c H t = 0 dive = ρ, H = 0, E = ρ, roth c E t = c ρv E + H c t = 0 H c E t = c ρv T
NHK 204 2 0 203 2 24 ( ) 7 00 7 50 203 2 25 ( ) 7 00 7 50 203 2 26 ( ) 7 00 7 50 203 2 27 ( ) 7 00 7 50 I. ( ν R n 2 ) m 2 n m, R = e 2 8πε 0 hca B =.09737 0 7 m ( ν = ) λ a B = 4πε 0ħ 2 m e e 2 = 5.2977
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 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 information1 4 1 ( ) ( ) ( ) ( ) () 1 4 2
7 1995, 2017 7 21 1 2 2 3 3 4 4 6 (1).................................... 6 (2)..................................... 6 (3) t................. 9 5 11 (1)......................................... 11 (2)
More informationBanach-Tarski Hausdorff May 17, 2014 3 Contents 1 Hausdorff 5 1.1 ( Unlösbarkeit des Inhaltproblems) 5 5 1 Hausdorff Banach-Tarski Hausdorff [H1, H2] Hausdorff Grundzüge der Mangenlehre [H1] Inhalte
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 informationA Brief Introduction to Modular Forms Computation
A Brief Introduction to Modular Forms Computation Magma Supported by GCOE Program Math-For-Industry Education & Research Hub What s this? Definitions and Properties Demonstration H := H P 1 (Q) some conditions
More information1 α X (path) α I = [0, 1] X α(0) = α(1) = p α p (base point) loop α(1) = β(0) X α, β α β : I X (α β)(s) = ( )α β { α(2s) (0 s 1 2 ) β(2s 1) ( 1 2 s 1)
1 α X (path) α I = [0, 1] X α(0) = α(1) = p α p (base point) loop α(1) = β(0) X α, β α β : I X (α β)(s) = ( )α β { α(2s) (0 s 1 2 ) β(2s 1) ( 1 2 s 1) X α α 1 : I X α 1 (s) = α(1 s) ( )α 1 1.1 X p X Ω(p)
More information, = = 7 6 = 42, =
http://www.ss.u-tokai.ac.jp/~mahoro/2016autumn/alg_intro/ 1 1 2016.9.26, http://www.ss.u-tokai.ac.jp/~mahoro/2016autumn/alg_intro/ 1.1 1 214 132 = 28258 2 + 1 + 4 1 + 3 + 2 = 7 6 = 42, 4 + 2 = 6 2 + 8
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 informationTOP URL 1
TOP URL http://amonphys.web.fc.com/ 1 19 3 19.1................... 3 19.............................. 4 19.3............................... 6 19.4.............................. 8 19.5.............................
More information2 2 MATHEMATICS.PDF 200-2-0 3 2 (p n ), ( ) 7 3 4 6 5 20 6 GL 2 (Z) SL 2 (Z) 27 7 29 8 SL 2 (Z) 35 9 2 40 0 2 46 48 2 2 5 3 2 2 58 4 2 6 5 2 65 6 2 67 7 2 69 2 , a 0 + a + a 2 +... b b 2 b 3 () + b n a
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 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 information: , 2.0, 3.0, 2.0, (%) ( 2.
2017 1 2 1.1...................................... 2 1.2......................................... 4 1.3........................................... 10 1.4................................. 14 1.5..........................................
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 informationII A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )
II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11
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 informationI , : ~/math/functional-analysis/functional-analysis-1.tex
I 1 2004 8 16, 2017 4 30 1 : ~/math/functional-analysis/functional-analysis-1.tex 1 3 1.1................................... 3 1.2................................... 3 1.3.....................................
More information(1) (2) (3) (4) 1
8 3 4 3.................................... 3........................ 6.3 B [, ].......................... 8.4........................... 9........................................... 9.................................
More informationDecember 28, 2018
e-mail : kigami@i.kyoto-u.ac.jp December 28, 28 Contents 2............................. 3.2......................... 7.3..................... 9.4................ 4.5............. 2.6.... 22 2 36 2..........................
More information?
240-8501 79-2 Email: nakamoto@ynu.ac.jp 1 3 1.1...................................... 3 1.2?................................. 6 1.3..................................... 8 1.4.......................................
More informationArmstrong culture Web
2004 5 10 M.A. Armstrong, Groups and Symmetry, Springer-Verlag, NewYork, 1988 (2000) (1989) (2001) (2002) 1 Armstrong culture Web 1 3 1.1................................. 3 1.2.................................
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 informationI
I 6 4 10 1 1 1.1............... 1 1................ 1 1.3.................... 1.4............... 1.4.1.............. 1.4................. 1.4.3........... 3 1.4.4.. 3 1.5.......... 3 1.5.1..............
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 informationB [ 0.1 ] x > 0 x 6= 1 f(x) µ 1 1 xn 1 + sin sin x 1 x 1 f(x) := lim. n x n (1) lim inf f(x) (2) lim sup f(x) x 1 0 x 1 0 (
. 28 4 14 [.1 ] x > x 6= 1 f(x) µ 1 1 xn 1 + sin + 2 + sin x 1 x 1 f(x) := lim. 1 + x n (1) lim inf f(x) (2) lim sup f(x) x 1 x 1 (3) lim inf x 1+ f(x) (4) lim sup f(x) x 1+ [.2 ] [, 1] Ω æ x (1) (2) nx(1
More information21 2 26 i 1 1 1.1............................ 1 1.2............................ 3 2 9 2.1................... 9 2.2.......... 9 2.3................... 11 2.4....................... 12 3 15 3.1..........
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 information(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t
6 6.1 6.1 (1 Z ( X = e Z, Y = Im Z ( Z = X + iy, i = 1 (2 Z E[ e Z ] < E[ Im Z ] < Z E[Z] = E[e Z] + ie[im Z] 6.2 Z E[Z] E[ Z ] : E[ Z ] < e Z Z, Im Z Z E[Z] α = E[Z], Z = Z Z 1 {Z } E[Z] = α = α [ α ]
More information128 3 II S 1, S 2 Φ 1, Φ 2 Φ 1 = { B( r) n( r)}ds S 1 Φ 2 = { B( r) n( r)}ds (3.3) S 2 S S 1 +S 2 { B( r) n( r)}ds = 0 (3.4) S 1, S 2 { B( r) n( r)}ds
127 3 II 3.1 3.1.1 Φ(t) ϕ em = dφ dt (3.1) B( r) Φ = { B( r) n( r)}ds (3.2) S S n( r) Φ 128 3 II S 1, S 2 Φ 1, Φ 2 Φ 1 = { B( r) n( r)}ds S 1 Φ 2 = { B( r) n( r)}ds (3.3) S 2 S S 1 +S 2 { B( r) n( r)}ds
More informationmain.dvi
SGC - 70 2, 3 23 ɛ-δ 2.12.8 3 2.92.13 4 2 3 1 2.1 2.102.12 [8][14] [1],[2] [4][7] 2 [4] 1 2009 8 1 1 1.1... 1 1.2... 4 1.3 1... 8 1.4 2... 9 1.5... 12 1.6 1... 16 1.7... 18 1.8... 21 1.9... 23 2 27 2.1
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 information25 7 18 1 1 1.1 v.s............................. 1 1.1.1.................................. 1 1.1.2................................. 1 1.1.3.................................. 3 1.2................... 3
More information