17 Θ Hodge Θ Hodge Kummer Hodge Hodge
|
|
- しおり なかきむら
- 5 years ago
- Views:
Transcription
1 Teichmüller ( ) Teichmüller 8 4 Diophantus
2 17 Θ Hodge Θ Hodge Kummer Hodge Hodge
3 0,, Teichmüller., Teichmüller, Diophantus Teichmüller, Teichmüller Hodge,., 2. (a),, Teichmüller, ( ) Teichmüller,,,,,. (b) Teichmüller. ( ),,,.,,,.,,,.,,,,,,.,,,.,., : 1 3: Teichmüller,. 4 12: Diophantus,,,., Teichmüller :,, / Hodge :,, / Hodge. 26: Hodge.,,.,,,,, 3
4 .,, 1.,.,,. Teichmüller,,,. Teichmüller., Teichmüller,,,, ,, Teichmüller.,., JSPS KAKENHI Grant Number 15K04780,. 1, 1 3, Teichmüller,,., Teichmüller,,,,,. 1, (cyclotome).. Tate Ẑ(1)., Ẑ(1),, (Q/Z)(1).,.,. Ẑ(1), Ẑ(1)., Ẑ(1) : (a) ( 0 ) Ω Λ(Ω) = lim µ n n (Ω), n 1, µ n (Ω) Ω, Ω 1 n. (b) ( 0 ) Ω C Λ(C) ( = HomẐ H 2ét (C, Ẑ), Ẑ), i 0, Het í, i. (c) ( 0 ) Ω C c C I c = πét 1 ( Spec ( (OC,c ) ) \ {c} ), πét 1,. (,, Ω 1 Ω((t)) Galois.) ( ),, Ẑ(1). 4
5 ,,,,,,, /., (a) (b) /. 1 Chern Pic C Hét 2 (C, Λ(Ω)) (Pic C/Pic 0 C) Z Ẑ HomẐ(Λ(C), Λ(Ω))., 1 Z Pic C/Pic 0 C 1, / Λ(C) Λ(Ω)., 1, /,,.,,.,,.,.., 0 2 K K, K K. K/ K, K/ K,, K K K K.,, (Gal( K/ K), Gal( K/ K)) Λ( K) Λ( K).,, K/ K, K/ K,, Gal( K/ K) Gal( K/ K).,,, (Gal( K/ K), Gal( K/ K)) Λ( K) Λ( K)., Gal( K/ K) Gal( K/ K)., Gal( K/ K) Gal( K/ K),, 1 (Gal( K/ K), Gal( K/ K)) Λ( K) Λ( K)., Λ( K) Λ( K). ( (c) ), (,, ).,,,.,,, Ẑ(1).,,., (cyclotomic synchronization isomorphism),, (cyclotomic rigidity isomorphism),,. (b) (c),. (b) (c) 5
6 . U = C \ {c}., C,, 2. (I c πét 1 (U) πét 1 (C))., (H G Q).,, G Q G E Q 1 : H G E H Ker(E Q)., E Q H, [E] H 2 (Q, H). (H I c,,.),, [E] H 2 (Q, H) 1 Ẑ = H2 (Q, HomẐ(H 2 (Q, Ẑ), Ẑ)) 1 H HomẐ(H 2 (Q, Ẑ), Ẑ)., (H G Q),, (b) ( ) HomẐ(H 2 (Q, Ẑ), Ẑ) (Q πét 1 (C) ) (c) H (H I c ) H HomẐ(H 2 (Q, Ẑ), Ẑ).,, (H G Q).,,, (I c πét 1 (U) πét 1 (C)), I c Λ(C). H HomẐ(H 2 (Q, Ẑ), Ẑ) ,. p, k p (, Q p ), k k, G k = Gal(k/k), : k R (a a ) k p = p 1 p, O k = { a k a = 1 } O k = { a k 0 < a 1 } O k = { a k a 1 } k, O k = (O k )G k O k = (O k )G k O k = O G k k k = k G k. G G k ( ) (isomorph). (.), O k, O k, k, Λ(k) ( ) G k, G k, G k, G k O k (G), O k (G), k (G), Λ(G), (, ) G.,, [9] Introduction, I2, Remark Remark.,, [1], 1,,., O k O k k Λ(k) G k,, [1], 2,,. (, [1], 2,, O k O k 6
7 ., O k O k, G H k (H) k (G), O k O k O k O k.) G G O k (G), O k (G), k (G), Λ(G) (, G k O k, O k, k, Λ(k) )., G k O G M. G M, k (Frobenioid cf. [6], Definition 1.3). (,, ), G (étale-like cf., e.g., [6], Introduction, I4),, M Frobenius (Frobenius-like cf., e.g., [6], Introduction, I4). ( ),, Galois,,,, (,, )., ( ) Frobenius,,,, (,, )., G M,, (G G k ) G G Λ(G) /., M O, n M[n] = Ker(n: M M) µ k n (k), n, Λ(M) = lim M[n] Λ(k),,. G Λ(G) n, G Λ(M) Frobenius Frobenius., 1 G M, G Λ(G) Frobenius G Λ(M) 2. ( ) 2,. ([9], Remark 3.2.1,.) G M,, G Λ(M) Λ(G), Frobenius.,, G M,.,,,. G k O k, Frobenius O k, O k k,, Frobenius.,, O k ( ). O k ( k ),,, Λ(M) Λ(G) Ẑ ( {±1}). (, [9], Proposition 3.3, (i),.), O k ( k ), Ẑ 7
8 ( {±1}),. (G M) {1} = (G k O ) : Λ(M) Λ(G) :, k (G M) Ẑ = (G k O ) : Λ(M) Λ(G) : k Ẑ, (G M) = (G k k ) : Λ(M) {±1} Λ(G) : {±1}. 3 Teichmüller 3, 1 2.,, J, J A, i 0, H i (J, A) = lim H i (H, A) H J, H J. 2 (G k O ) G M. k, Kummer. G 1 M[n] n M gp 1 (M gp ) G /((M gp ) G ) n H 1 (G, M[n]) M gp., n, M G ((M gp ) G ) H 1 (G, Λ(M))., G, M H(G, Λ(M)) ( Kummer )., [1], 2,, G Kmm(G): k (G) G H 1 (G, Λ(G))., G Λ(G) G,, Kmm(G) G. O k (G) k (G) H 1 (G, Λ(G)), 2 Λ(M) Λ(G), : ( ): Λ(M) H 1 (G, Λ(G)), Λ(G) G H 1 (G, Λ(M)) M O k (G). G Kmm(G M) : Kmm(G M): M O k (G). 8
9 ,,, ( ) 1 Λ(M) Λ(G).,, G Kmm(G M): M O (G) k.,, Frobenius (, ), Teichmüller, Kummer (Kummer isomorphism). Kummer, Frobenius. (, [11], Introduction,.), 2 (G k O ) 2 k G M, G M., G M G M, 1 /. p,, G M G M., 2 /,,, α: G G. 2 / G M, G M α: G G,,.,,. Kummer M Kmm( G M) O k ( G) O k (α) O k ( G) Kmm( G M) 1, ( G, G) Frobenius M M.,, G G,, Frobenius M,.,,, Frobenius, Kummer, (,, H 1 (G, Λ(G)),, O (G) ),, k, Frobenius.,, Frobenius..,,, ( G M) ( G M)., Teichmüller,,,,., M 9
10 , Teichmüller.,.,, 2 : (a) Kummer Frobenius. (, Kmm(G M),.) (b),. (, α,.) Teichmüller, (a) Kummer (Kummer-detachment indeterminacy cf. [12], Remark 1.5.4), (b) (étale-transport indeterminacy cf. [12], Remark 1.5.4).,, Kummer (, ),., 2 Frobenius O k ( k ).,, Frobenius, Ẑ ( {±1})., Kummer, M O (G) k ( M k (G)) Ẑ ( {±1})., α: G G, M M Ẑ ( {±1}). Ẑ ( {±1}), Kummer : M, M = O k : M {1} O k ( G) O k ( G) {1} M, M, M = O k : M Ẑ O k ( G) O k ( G) Ẑ M, M, M = k : M {±1} k ( G) k ( G) {±1} M., ( ) α: G G. G M G M G k O k., G G Isom( G, G) 6, G G.,, O k ( G) O k ( G) Aut( G) = Aut( G)., M M Aut( G) = Aut( G). : M {1} O k ( G) Aut( G) =Aut( G) O k ( G) {1} M. 10
11 4 Diophantus 4 12, ( Teichmüller ) Diophantus,,,.. F (, Q ), F F, G F = Gal(F /F ), V(F ) F, E F, X E E F., v V(F ), F v F v, F v F F v, G v = Gal(F v /F v ) G F, : F v R (a a ) F v v p p = p 1, v 1 = 1, O v = { a F v a = 1 } O v = { a F v 0 < a 1 } O v = { a F v a 1 } F v, O F v = (O v ) G v O F v = (O v ) G v O Fv = O G v v F v = F G v v, E v = E F F v, X v = X F F v., v V(F ), E v O Fv. v V(F ), q v OF v E v q ( 1)., q q E = (q v ) v V(F ) v V(F ) O F v F L = {q v O Fv } v V(F ) (, L q 1 E ), ( ) deg L = [F : Q] 1 v V(F ) log ( (O Fv /q v O Fv ) ) ( 0) (, S S S )., Teichmüller Diophantus, Szpiro,,, L deg L (= deg L).,,,. deg L,. 2 N C deg L N deg L + C., deg L N = N deg L,, deg L C/(N 1) 11
12 deg L.,, deg L N deg L + C,,, deg L N = deg L,,. deg L N = deg L,. deg L N = deg L,, deg L N deg L., deg L ( deg L N ), q E ( qe N = (qv N ) v V(F ) v V(F ) O F v ) L ( L N ),.,, q E = q N E,., E/F, q E = qe N. (, q E = qe N E.), E/F, q E = qe N,., (, F E ) 2 S, S, S q N E ( q N E ) S q E ( q E ) S S (, ) S S. (, S, Teichmüller,. [10], [11], [12], [13], S. Situation, Setting, Settei S.),.,, Q 7 = , Q 2 Q, Q, 49 Q 7 Q Q Q,.,,. Teichmüller (link).,,,, / (,, ).,, 3 2 G M, G M α: G G, 2 G M, G M ( G M) ( G M). ( Teichmüller,,.),, q N E q E S S S S ( ).,,,, q N E q E S S, 12
13 deg L = deg L N. 49 Q 7 Q Q Q,, 7 = Q 7 Q, 7 = 49., deg L = deg L N., deg L ( deg L N ), q E ( qe N ) L ( L N )., q N E q E deg L N deg L,., (, q N E q E ) S S,, deg L = deg L N., ( ),, Teichmüller : Teichmüller : ( E/F ) qe N q E S S,, qe N q E ( ) Q Q,, nz log( (Z/nZ))., ( ) ( ): ϕ: Q Q, Z Z,,, log( ( Z/ n Z)) = log( (ϕ( Z)/ϕ( n)ϕ( Z)))., 49 7 Q Q, 7 = 49., ( ), log 49 = log( ( Z/ 49 Z)) = log( (ϕ( Z)/ϕ( 49)ϕ( Z))) = log( ( Z/ 7 Z)) = log 7, 7 = 49. (,,.) 5,, S S. qe N q E,., S S,,.,,,,,, 2., 2, 13
14 ,,., S S.,., Teichmüller.,, Teichmüller, ( )., 5 6,, S S,.,, (, ) ( ) / / (coric) / /.,,.,,, S S, S S., qe N q E,, qe N q E ( E/F ) Q 7 Q Q Q,, 49 Q 7 Q Q Q., qe N q E, S S Q Q, , Q Q, , Q Q,., qe N q E,,, /.,,,.,, (,, ) Teichmüller,,,,, (holomorphic)., ( ) ( ) (mono-analytic).,, S S. S S,, 2 S, S F, F F v, F v, E E v, X X v 14
15 .,. 6,,.,, v V(F )., q E, qe N, (, E v v ),,. p v. Galois G v.,.,, qe N q E, G v ( S G v ) qv N G v ( S G v ) q v,, qv N q v ( G v, G v )., G v qv N q v, ( ), G v,., G v., G v, G v, G v Galois. Galois G v, F v., Galois G v, G v Aut(F v ) (, )., 5,, Galois,, G v, G v G v,,., G v,, G v G v, G v G v, 2., G v G v, G v G v Aut( G v ) = Aut( G v ),. G v G v, 1,,,,. G v G v Aut( G v ) = Aut( G v ), 3, ( 3 ). ( 10.),, Teichmüller, 2 A, B, A B ( ) (poly-morphism), A B ( ) (poly-isomorphism), A B (full poly-isomorphism).,,,,.,, G v G v G v G v., 4 Diophantus,,., ( ) 15
16 , 4 /., Teichmüller. 1,. ( ), 5,.,.,, Galois (, G v ) ( ).,,,,, Galois,.,. (,. Galois,, GL n ( ) ( ).,,, ( ).,,,,,,.), Teichmüller 1.,. F v 5,, F v ( ) G v (F v ) +.,.,, (F v ) +, O v (G v ) (O v ) + (F v ) +,,., [1], 2,, (O v ) + (F v ) +, F v,., (O v ) + (F v ) +,., (F v ) +,.,, G v (F v ) +,, G v O µ v,,.,. G v F v G v O v.,. qe N q E,, qe N q E., qv N q v F v O v.,,. (, G v O v.) ( G v O v ) ( G v O v ),, O F v O F v.,,, O F v / O F v O F v / O F v., 16
17 O F v / O F v, O F v / O F v N, 1, O F v / O F v 1 (, 1 N ) OF v / O F v 1 (, 1 N )., q v 1 (, q v OF v /O F v 0 ), ( G v O v ) ( G v O v ) q N E,, q E. q N E q E O F v /O F v F v /O F v,, G v F v G v O v,,., G v O v,., G v O v.,,, q N E q E,., ( 7 ),., 3,,., S S, S S,., G v, G v O v,,,,.,, G v O v Ẑ, O v Oµ v = (O v ) tor O v. G v O v, Ẑ., 3,., Ẑ,. ( Λ, Ẑ, Ẑ., Ẑ,.) 3, Kummer G v O v. G v O v, O v O µ v O v G v O µ v = O v /O µ v..., : F v G v (F v ) +, G v (O v ) + (F v ) +,. F v G v F v G v O v,,. G v O v G v O v,,,,. 17
18 G v O v G v O µ v,,. G v (F v ) +, G v O µ v., p G v O µ v (F v ) + G v (F v ) +, G v O µ v G v (F v ) +,. (, 8 9.),,, O µ v Q p., ( (F v ) + ),, Q p, Q p. O µ v, G v H G v = Im ( (O v ) H O v O µ ) µ (O ) H I κ H, G v O µ v, H IH κ. (, IH κ Z p.), µ-kummer ( µ-kummer structure cf. [11], Definition 4.9, (i)). 6,.,. G v, G v O v, (G v O µ v ; H IH κ (O µ v ) H ) ( ), Teichmüller, Dv, Fv, Fv µ., X v, E v X v X v, E v X v X v. (, 13,, 17.), X v (tempered fundamental group) π temp 1 (X v ),, X v πét 1 (X v ) Π v, Π v ( ), D v v v., Π v Π v G v O v Π v O v ( ), F v : D v G v F v G v O v F µ v (G v O µ v ; H IH κ (O µ v ) H ) D v Π v F v Π v O v. 18
19 6 v V(F ), Dv ( Fv ; Fv µ ; D v ; F v ), {Dv } v V(F ), ( {Fv } v V(F ) ; {Fv µ } v V(F ) ; {D v } v V(F ) ; {F v } v V(F ) ), D ( F ; F µ ; D; F) (D - (respectively, F -; F µ -; D-; F-) prime-strip cf. [10], Definition 4.1, (iii) (respectively, [11], Definition 4.9, (vii); [11], Definition 4.9, (vii); [10], Definition 4.1, (i); [10], Definition 5.2, (i)). (, F,,, v,,, 17.), F ( ), D ( ).,, Teichmüller,. 7 Teichmüller,. 8 Teichmüller, 7,. (,, [11] Example 1.7 Remark ),. (radial data cf. [11], Example 1.7, (i))., ( ) (coric data cf. [11], Example 1.7, (i)). (radial environment cf. [11], Example 1.7, (ii)).,, : (a), 1 C ( ),, C ( ) 2 R 2 ( ). (b) p p k C.,, πét 1 (C) (πét 1 (C) ) G k. ([4], Lemma 1.1.4, (ii),, [4], Lemma 1.1.5,.), C πét 1 (C) Π., Π, Π Π G(Π). (, Π G(Π) πét 1 (C) G k.), G(Π) G, G(Π) Π G ( = G(Π))., (, 2, ), Ψ. Ψ, Ψ (corically ined algorithm/coric algorithm cf. [11], Example 1.7, (iv))., Ψ ( ), Ψ (multiradially ined algorithm/multiradial algorithm cf. [11], Example 1.7, (iv)),,, Ψ (uniradially 19
20 ined algorithm/uniradial algorithm cf. [11], Example 1.7, (iv)). (a),., Ψ, C. Ψ C,, R 2,, Ψ., Ψ C, C GL 2 (R) (,,,,, ), R 2., Ψ,.,, Ψ C, R 2, Ψ, / (, )., R 2 (, ), /, Ψ.,, ( ).,,, /.,,. (a),, (, ),,. C, C, C,,, : C C R 2 / R 2 =, (a) (, ), R 2. 20
21 GL 2 (R) : C C GL 2 (R) R 2 / GL 2 (R) R 2 = R , qe N q E ( 6 F v O v ) S S, qe N q E,., 5, S S S S., qe N q E., S S, ( ),.,, S S.,,.,,,,,., 5 6, S S, (, ), D,, F µ.,,,, q E, F µ. 21
22 , F µ Frobenius ( v ) O µ v., q v. 6, O µ v,.,, ( q E F µ ) F µ,, 6 O µ v (F v ) +. 6, ( 6 ) p O µ v (F v ) +., (F v ) + F v,.,, F µ,., 6 O µ v. v, O µ v IG κ v O µ v., O µ v,. ( 6 ) F v (F v ) + = ((F v ) + ) G v I v = (2p) 1 Im(IG κ v O µ p v (F v ) + ) (F v ) + (log-shell cf. [12], Definition 1.1, (i)). 6, Z p.,, (O Fv ) + I v, Im(O F v p (F v ) + ) I v, I v Z Q = (F v ) +. ([12], Proposition 1.2, (v),.) (, (O Fv ) + 6 (F v ) +, (2p) 1.) v,,. ([12], Definition 1.1, (ii), [12], Proposition 1.2, (v),.),,,,, (, ) /,, : (O Fv ) + (F v ) + : I v (F v ) + :. (O Fv ) + I v ( ), q E q N E v V(F ) I v., (O Fv ) + = I v (, q v (O Fv ) + = q v I v ),, I v,, (, I v /(O Fv ) + ), I v Z Q (= (F v ) + ) (,, G v ). ([9], Proposition 5.8, (iii),.), 22
23 F µ ( 9 ), q v q N v ) (, ). (, v V(F ) v (OF v ) gp /O F v, F, F ( v (O Fv ) gp /O ) F v v( O Fv /O ) F v.,, F ( v (O Fv ) gp /O ) F v v( O Fv /O ) F v (, Fv ), F. (, = H 1 (O ).), F MOD. ([12], Example 3.6, (i),.) ( 22.) F MOD, F ( v (O Fv ) gp /O ) F v v( O Fv /O ) F v,,., (, ),., ( v (O Fv ) gp /O ) F v,,, v( (O Fv ) gp /O F v ),,.,, F, F v (F v) + v (O F v ) + v (F v) +.,,, F. (, 1 O.), F mod. ([12], Example 3.6, (ii),.) ( 22.) Fmod,,,,., F MOD, (, )., v (O F v ) + v (F v) +,. F MOD F mod (, [12], Fig. I.7,.),, ( F MOD F mod ) F, 2 F MOD, F mod., F, 2 F MOD, F mod : F mod F F MOD F MOD F MOD...,, F, F MOD Fmod.,, F 23
24 , 2,.,, I v. I v (F v ) + I v Z Q = (F v ) +., I v, v I v v (I v Z Q) = v (F v) +, Fmod., F µ (, 9 ),., qe N q E, F ( ) qe N v (I v Z Q),,, v (I v Z Q), : qe N I v, F (I v Z Q). v v, Teichmüller,,.,,,.,, ( ) : Teichmüller : ( E/F ) ( F E ) S, {I v } v, qe N I v, F (I v Z Q) v v 3 ( ).,, q N E q E S S., q N E q E S S, Θ (Θ-link) (, Θ µ LGP-link [12], Definition 3.8, (ii), )., (, Θ )., (, ),, qe N F,.,, ( ) ( ) 24
25 .,,., ( ), : 1 O F v F v F v /O F v 1 = O F v ( ) I v ( ) F v /O F v..,,,. ( 12,.) S, S 2.,, qe N q E S S., deg L N = deg L. (., 49 7 ϕ: Q Q ϕ( 49) = 7.),, deg L N = deg L N., deg L = deg L N ( S, ).,,, deg L N deg L + C. ( 10.), 4, deg L.,,, q E qe N.,, q E qe N. qe N, (, qn E, 11 ), (, )., q E. 9 8, Teichmüller. 9,, 10,, (, )., 9, v V(F )., 8. F µ IG κ v O µ v,. (,.), I v (F v ) +,.,,.,, a n 1 (1 a)n /n,, 25
26 ,,., (, Im ( I κ G v O µ v ) 1/2p, ) q N E, I v = (2p) 1 Im ( IG κ v O µ p ) v (F v ) + F, /,.,.,,. F F = { F v } v V(F ), v F v = ( Π v O v ). (, Π v O v Π v O v.), O v O v O v O µ v O v O µ v = O v / O µ v, Π v O µ v Π v O µ v.,, Π v,,.,, Π v O µ v,, O µ v (F v ) + F v,, ( O µ v, O µ v ) Fv /. (, Π v Fv Π v F v.),, F v log( F v ) = ( Π v Õ v = { a Fv 0 < a 1 }). (, [12], Definition 1.1, (i),.) ( [12], Definition 1.1, (ii), ) F log( F) = {log( F v )} v., S 2 0 S, 1 S. F 0 F, 1 F, F log( 0 F) 1 F 0 S 1 S (log-link cf. [12], Definition 1.1, (iii)). 0 S 1 S,, ( ) 0 O v 0 O v 0 O µ v = 0 Fv 0 Õ v 1 O v. ( Hodge, 26.) 0 S 1 S., 1 S 1 q N E 1 O v, 1 q N E 0 Õ v 0 Fv (= 0 O µ v ),, 1 q N E 0 I v ( ( 0 Fv ) + ),.,, qe N,,,.,.,.,, 26
27 .,,,, 1 S log 1 S log 0 S 0 S,,.,,,..,,,.,, ( ).,,.,. 1 0 O v 0 O v 0 O µ v = 0 Fv 0 Õ v 1 O v., log 2 S log 1 S log 0 S log 1 S log 2 S log,, 1 O v 1 O v 1 O µ v 1 Õ v 0 O v 0 O v 0 O µ v 0 Õ v 1 O v 1 O v 1 O µ v 1 Õ v. (, Z n Z +1, Z +1.), I v, 1 I v 0 O F v 0 I v 1 O F v 1 I v 2 O F v,.,. 27
28 ,. 8,, (, ).,, n Z m Z,..,., log v : O F v (F v ) + A O F v, A log v (A), A log v (A). ([12], Proposition 1.2, (iii),.),. 8, qe N F.,, n Z qe N F,..,. qe N, 15, qe N 1, 1 0,. ([12], Proposition 3.5, (ii),.) F, F F v V(F ) O v,, F 1, 1 0,. ([12], Proposition 3.10, (ii),.) 10 9, 10, ( 8 ) (, ).,, v V(F ). 3. 1, G v. 6, G v, G v., G v,,.,, G v,. Galois, 1 (Ind 1 ). 2, Kummer. 3,, Frobenius (,, F µ Frobenius (O µ v ; H IH κ (O µ v ) H ) ) (,, F µ G v ) Kummer., Kummer, Ism = { µ-kummer G v O µ v } ([11], Example 1.8, (iv), ). ([12], Proposition 1.2, (vi), (vii), (viii),.) 2 (Ind 2 ). 28
29 3, 1, 2 Kummer : Frobenius (Ind 2): Ism (Ind 1 ): Aut(G v ) (Ind 2 ): Ism Frobenius. 3 (Ind 3 ) ( 9 ), 3, Kummer. 9,, D,, D. (,, log( 0 F) 1 F 0 Π v 1 Π v,, 0 G v 1 G v.), 9 log 2 S log 1 S log 0 S log 1 S log 2 S log, D ( ) D 2 D 1 D 0 D 1 D 2 D 2 D 1 D 0 D 1 D 2 D. D, D D, D D, D., 6, D D qe N q E.,, D., 9, n Z Kummer n I v I D v, n I v n Z Frobenius, I D D v D v ( ), n Z Kummer n I v I D v 1 I v 0 O F v 0 I v 1 O F v 1 I v 2 O F v n I v 2 I v 1 I v 0 I v 1 I v 2 I v Kmm Kmm Kmm Kmm Kmm = I D v = I D v = I D v = I D v = I D v =. Kummer (log-kummer correspondence cf., e.g., [12], Introduction, Remark ). Kummer Kummer n I v I D v, n Z.,,, (upper semi-compatibility cf., e.g., [12], Proposition 3.5, (ii))., Kummer, Kummer. 3 (Ind 3 ). 29
30 9,, 3, : I v ( I v Z Q),,., q N E q E,, q N E q E. qe N q E,,.,, Kummer, Kummer. Kummer, 3 (Ind 3 ). Kummer ( ), (, ),.,., (O Fv ) +, 10. (O Fv ) +,, : (Ind 3 ) (O Fv ) +,. (O Fv ) +,,., (O Fv ) +,, (O Fv ) + ( ) ((O Fv ) + ) I v ( F v ) + G v,. (Ind 3 ), ( F v ) + O µ v (IG κ v ) 1/2p O µ v G v,. (Ind 2 ),, O µ v ( µ-kummer ) M M ( = O µ v ) G v,. (Ind 1 ),, Galois G M G ( = G v ),. M G F µ ( v )., 6,,.,,. 30
31 11 8, q N E q E, I v, q N E, F 3 ( ), Diophantus. 11,,., 9 10,,,. qe N F,,, q N E F,, q N E q E,., q N E q E,.,.,,, ( 10 ) ( ).,, Z. Z,, Z.,,,, Z Z +,, {±1}.,,, ( ) : : Z {1} : Z, Z + {±1}., qe N F,,. qe N q E,,. (,,.) qe N F,, qn E F ( / ) ( / ),,.,,, Z 7 Z,. (, 1 Z 7.), Z, Z, 7 Z, ( ) ±.,, 7 31
32 .,,.,, qe N F.,,,. qe N F., qe N F (orbicurve cf. [5], Definition 2.2, (i)),,, qe N F., qe N F, /.. Q P 1,, Q(t).,, Q : 1 Q Q(t) a P 1 ord a Q a P 1 Q Z, ord a a.,, ( ) Q(t) M ( ) Q A.,,,, ( ):, a P 1 Q ord a : Q(t) Z o a : M Z 2 P 1 Q, ev 2 : Ker(ord 2 ) Q ; ev 2 (f) = f(2) e 2 : Ker(o 2 ) A /. (,, (a).),, ( ): 14 P 1 Q, ev 14 : Ker(ord 14 ) Q ; ev 14 (f) = f(14) e 14 : Ker(o 14 ) A /. (,, (b).), (, ) Q(t) Q,, ( ) ( ), (,, ), 32
33 ., Q, ( ) 7 Q, 7 Q., /. ( ), ϕ M 1 : a {0, } ϕ Ker(o a ),, o 0 (ϕ) = 1, o (ϕ) = 1., e 2 (ϕ) = 1. ϕ e 2 (ϕ) = 1., Q,, 1 (= e 2 (ϕ)) ( ),. ϕ M 7 Q., 2 ( ) e 14 ϕ e 14 (ϕ) A 7 Q (ϕ )., ϕ, 7 Q e 14 (ϕ) A.,, qe N F,.,,, (a) ( Kummer ),,.,,,, (b) ( ) ( ).,,, (c),.,,, Teichmüller, qe N, F κ (κ-coric cf. [10], Remark 3.1.7, (i)), κ ( κ-coric cf. [10], Remark 3.1.7, (ii)), κ ( κ -coric cf. [10], Remark 3.1.7, (ii)) κ. ( 13, 24,, κ.),. 33
34 12 8 Teichmüller., 9 11, Teichmüller.,. (, [12], Theorem 3.11,.) Teichmüller : E/F., R 3 : {I v } v V(F )., (= (q j2 /2l E ) j=1,...,(l 1)/2 ) v V(F ) I v. κ v V(F ) (I v Z Q)., : (i) ( ) log 2 S log 1 S log 0 S log 1 S log 2 S log, (Ind 1 ), 3 R (D ). (ii) ( Kummer ) log 2 S log 1 S log 0 S log 1 S log 2 S log, (i) R n Z Frobenius 3 n R, (Ind 2 ) Kummer n R R., Kummer, (Ind 3 ), n. (iii) (Θ ) log 2 S log 2 S log 1 S log 1 S log 0 S Θ log 0 S log 1 S log 1 S log 2 S log 2 S log log,, Θ, 0 F µ 0 F µ, 0 (= ( 0 q j2 /2l E ) j=1,...,(l 1)/2 ) 0 q E, 0 0 q E 0 / ( ) 34
35 0 / ( ), (i) (ii) 0 R. (ii) (i) (ii) R R 0 R Θ 6, (iii) Θ (, ), F µ. ((iii) 0 F µ 0 F µ.), Teichmüller,, Θ,, (iii) 90.,,, Teichmüller, 4 E/F, ( ) E/F., 17 Θ E/F. 11,,,,, κ. Teichmüller ( ),,, : /Frobenius κ, Kummer,. κ, Galois (Galois evaluation cf., e.g., [11], Remarks , 3.6.2), Kummer,,, ((Ind 1 ), (Ind 2 ), (Ind 3 ) )., / (, 8 Fmod ). κ, / (, 8 Fmod ), (, 8 F MOD ).,, ( ),,.,,, deg L 35
36 . (, [12], Corollary 3.12,.) (iii).,, q q E ( L ) ( ) q (q-pilot object cf. [12], Definition 3.8, (i)). ([12], Remark , (ii),.), (= (q j2 /2l E ) j=1,...,(l 1)/2 ) (,, L N ) ( ) Θ (Θ-pilot object cf. [12], Definition 3.8, (i)).,,,. deg(q ), Θ, Θ q : Θ q.,, (Ind 1 ), (Ind 2 ), (Ind 3 ), Θ Θ : Θ (Ind 1, 2, 3) Θ., 2,, q Θ (Ind 1 ), (Ind 2 ), (Ind 3 ) (,, (holomorphic hull cf. [12], Remark 3.9.5)) : q ( (Ind 1, 2, 3) Θ )., q, Θ (Ind 1 ), (Ind 2 ), (Ind 3 ) (,, ) : deg(q ) (= deg(q )) vol(θ )., deg L deg L N C 4., deg(q ) vol(θ ),, ( ) E/F,,, [13] 1, , Teichmüller, qe N /,. Teichmüller,,. 13,
37 ,. ( 13,, [7] 1 2.) Archimedes. 2, p, k p, k k, G k = Gal(k/k), : k R (a a ) k p = p 1 p, O k = { a k a = 1 } O k = { a k 0 < a 1 } O k = { a k a 1 } k, O k = (O k )G k O k = (O k )G k O k = O G k k k = k G k., p l. E k O k, q q E Ok \ O k. (, q E, 12 q E q v.), E ( ) k X log.,, k, O k., X log, k X log, X log O k X log., {±1} E X log k C log : X log C log = [X log /{±1}] ( ). (, Teichmüller ),,.,, p, k, k C log k (k-core cf. [5], Remark 2.1.1).,, C log : C log (, C log Kummer Kummer ), k Kummer., (elliptic cuspidalization cf. [8], Example 3.2, Corollaries 3.3, 3.4)., k,.,, C log.,, C log (, X log ),. (,, 1,.,., ([5], Remark 2.5.1, ), (arithmetic cf. [5], Definition 2.1, (i); Remark 2.1.1).) 37
38 ,, log (1) Ÿ Ÿ log (2) (3) Y log (4) Y log (5) (6) X log (7) X log (8) X log (9) (10) C log (11) C log C log Kummer. ((10) C log 2.),,, : (a) 3. (b) Kummer. (5) (6) (Kummer )., l, 2, 2, l,,, l, l, 2, 2, l. (c) (6) (8) Y log X log X log ( ) Kummer.,, Galois Z. Galois Z.,, Y log Z, Z. (d) (8) Z Y log X log l Z Z ( ).,, Galois Z/l Z = F l. Galois F l.,, X log l, l.,, (l ) X log 1 X log ( ), X log X log., X log 1, (zero cusp cf., e.g., [11], Remark 1.4.1, (i)).,, X log.,, {±1} X log. (e) (9) (d) {±1} X log ( ) 2. (9) C log. (f) (11) Galois. Galois, µ l, µ 2, µ 2, µ l, l Z, l Z, µ l, F l, {±1}, {±1}. (g) (7) X log (, l ) Kummer., X log l, l. (7) X log ( 1 ). (h) (3) Y log (, Z ) 2., Ÿ log 38
39 Z, Z 2. (i) (j) C log, C log k. X log, X log, X log k Gal(X log /C log ) = {±1}, Gal(X log /C log ) = Gal(X log /X log ) Gal(X log /C log ) = F l {±1}, Gal(X log /C log ) = Gal(X log /X log ) Gal(X log /C log ) = µ l {±1}. ( ) Vert( ), Cusp( )., Vert(Y log ) (, Y log ), X log 1. (, Vert(Y log ) 1 l Z.), (c),, Z Vert(Y log ) Cusp(Y log ) Z.,,, Vert(Ÿ log (1) ) Vert(Ÿ log ) (2) (3) Vert(Y log (4) ) Vert(Y log ) (5) (6) Vert(X log (7) ) Vert(X log (8) ) Vert(X log ) (9) (10) Vert(C log (11) ) Vert(C log ) Cusp(Ÿ log (1) ) Cusp(Ÿ log ) (2) (3) Cusp(Y log (4) ) Cusp(Y log ) (5) (6) Cusp(X log (7) ) Cusp(X log (8) ) Cusp(X log ) (9) (10) Cusp(C log (11) ) Cusp(C log ) Z (2) (3) Z (1) Z (4) Z (8) F l Z/Z = {1} (9) (10) (5) (6) F l (7).,, F l /{±1} (11) Z/Z = {1} (1) Z Z Z Z (2) (3) (4) Z Z (5) (6) (7) (8) F l F l Z/Z = {1} (9) (10) F l /{±1} (11) Z/Z = {1} 1, 1 F l Vert(X log ), X log,, ( ) 1 F l /{±1} Vert(C log ), C log 39
40 .,,, : X log : E. C log : ( ). Y log X log : X log ( ).. Ÿ log ( Y log ):. Ÿ log ( Ÿ log ): l. Y log ( Y log ):. X log X log : Y log X log.,, Z F l. ( Teichmüller, 13 16,, Teichmüller,.,,,, Y log X log., X log X log.) X log X log : X Ÿ log Ÿ log (, Y log Y log )., 0 Z Vert(Ÿ log ) (, Y log Ÿ log 1 ) Θ(ü) = q 1 8 E n Z ( 1) n q 1 2 (n+ 1 2 )2 E ü 2n+1, ü Ÿ log Θ 1. ([7], Proposition 1.4,.) (, l ( ) ) Teichmüller. 1, (k) Θ, Ÿ 1,, j Z ( ) (j 2 /2) ord k (q E ),,.,. X log ((d) ), 0 F l Vert(X log ), 2 ( 1 k ) µ., j Z, j mod l F l Cusp(X log ) µ X log (k ), j (, j mod l) (evaluation point cf. [10], Example 4.4, (i)). (,, 0 µ.), j Z Vert(Ÿ log ),, X log j Ÿ log (k ), j 40
41 ., X log ( Ÿ log ) (k ), X log ( Ÿ log ) j, j., (l) j Ÿ log ξ j, Θ(ξ j ) 1 = ± Θ(ξ 0 ) 1 q j2 /2 E,, (m) j Ÿ log ξ j, Θ(ξ 0 ) 1 Θ ( ) ξ j ±q j2 /2 E. (m) (of standard type cf. [7], Definition 1.9, (ii)). (m), ( ) ξ j ±q j2 /2 E qe N.., 14 14,. ( 14,, [7] ) 13 ( ) log, Π tp ( ),, tp ( ) = Ker(Π tp ( ) G k)., N, µ N = µ N (k)., Π tp ( ) G k G k µ N, Π tp ( ) µ N.., Πtp ( ) [µ N] = µ N Π tp ( ) X log ( ), tp X X 2., ab X = X /[ X, X ] 2 Ẑ,, Θ = [ X, X ]/[ X, [ X, X ]] ab X ab X 1 Ẑ.,, X log 1 (, 1 (c) ) Θ, X X /[ X, [ X, X ]]., Θ X log. ( 1 ) Λ(k) Θ. η Θ H 1 (Π tp Ÿ, Λ(k)) H 1 (Π tp Ÿ, Θ) O k Θ Kummer. ( 13 (k).) (étale theta function cf. [7], Proposition 1.4, (iii))., l η Θ H 1 (Π tp Ÿ, l Θ)., H 1 (Π tp Ÿ, l Θ) H 1 (Π tp Ÿ, Θ) η Θ Ÿ H 1 (Π tp Ÿ, Θ) 41
42 . l X log, η Θ Gal(Ÿ log /X log ) = Π tp X /Πtp Ÿ = (l Z) µ 2.. N.,, η Θ,l Z µ 2 H 1 (Π tp Ÿ, l Θ) η Θ,l Z µ 2 mod N H 1 (Π tp Ÿ, (l Θ) Z (Z/NZ)) Π tp (l Ÿ Θ) Z (Z/NZ), (l Θ ) Z (Z/NZ) µ N Π tp µ Ÿ N.,, Π tp Ÿ [µ N ] Π tp Ÿ. Π tp Ÿ s Θ Ÿ : Πtp Ÿ Πtp Ÿ [µ N ] Π tp Ÿ [µ N ] Π tp Y [µ N] Π tp Y [µ N ] N (mod N theta section cf. [7], Definition 2.13, (i)).,, Π tp Y [µ N ] µ N ( Π tp Y [µ N ]),, (Π tp Y [µ N] ) Π tp Y Out(Π tp Y [µ N]), H 1 (Π tp Y, µ N ).,, H 1 (Π tp Y, µ N ) Out(Π tp Y [µ N])., Kummer k /(k ) N H 1 (G k, µ N ), k /(k ) N Out(Π tp Y [µ N ]). D Y Out(Π tp Y [µ N ]), k /(k ) N Out(Π tp Y [µ N ]) l Z = Gal(Y log /X log ) = Π tp X [µ N ]/Π tp Y [µ N] Out(Π tp Y [µ N ]). Π tp Y [µ N], ( ) D Y Out(Π tp Y [µ N ]), ( ) {γ Im(s Θ) Ÿ γ 1 Π tp Y [µ N]} γ µn 3 N (mod N model mono-theta environment cf. [7], Definition 2.13, (ii))., 3 M Θ N = (Π, D Π, s Θ Π ) (, Π, D Π Out(Π), s Θ Π Π ) N (mod N mono-theta environment cf. [7], Definition 2.13, (ii)),,, (mono-theta environment)., Teichmüller., 13 (k),.,, ( N)., Kummer,, Θ (l ) ( )., 2,,. 42
43 ,, Π tp X. (,, [7], 2,.) Π Π tp X.,, Π, Π tp Ÿ Πtp X Πtp X Πtp(Π Ÿ ) Π Π tp X (Π ), Π tp X l Θ Θ Π tp X (Π ) (l Θ )(Π ) Θ (Π )., O k Θ l Kummer (O k ηθ )(Π ) H 1 (Π tp (Π Ÿ ), (l Θ )(Π )) Π /(Π tp Ÿ )(Π ),, Gal(Ÿ log /X log )..,,,. Π, Π tp X Πtp Y Π Π tp Y (Π ) /, Π(Π ) = ( (l Θ (Π )) Z/NZ ) Π tp Y (Π ), Π tp Y [µ N]., D Y Out(Π tp Y [µ N ]) {γ Im(s Θ Ÿ ) γ 1 Π tp Y [µ N]} γ µn,, Π.,. (,, [7], 5,.) Π tp X. F F. F,, Π tp X M = { i i, X }. (, F,.) F, Θ(ξ 0 ) 1 Θ M Πtp Ÿ (, ξ 0, 0 Ÿ log ) l Θ 1 M Πtp Ÿ Gal(Ÿ log /X log ) µ l..,,,,., F : Π tp X = = F ,. 15,. ( 15,, [7], 2,.) (cyclotomic rigidity). ([7], Corollary 2.19, (i),.) N M Θ N = (Π, D Π, s Θ Π ) Π Πtp Y [µ N ],, 2. 1 µ N = µ N {1} µ N Π tp Y Π µ (M Θ N ), 1 Πtp = Πtp Y [µ N] Π Y [µ N ] l Θ Π (l Θ )(M Θ N ). Π µ (M Θ N ) (exterior cyclotome cf. [11], Definition 1.1, (ii)), 43
44 (l Θ )(M Θ N ) (interior cyclotome cf. [11], Definition 1.1, (ii))., 2.,. M Θ N = (Π, D Π, s Θ Π ),, (,, ) (l Θ )(M Θ N ) Z/NZ Π µ (M Θ N ) F = (Π tp X M),, Πtp X l Θ ( ),,, M µ N (, ) (Frobenius ).,, l Θ Z/NZ µ N,., 2. G k O k, Frobenius,.,, G k O, Frobenius k, Ẑ.,, O k, O k., 6, Diophantus,.. O k,,, 6 1., O,, k.,, O k. 1,.,.,,., Teichmüller (, ),,,., X /[ X, [ X, X ]] X,. 13 (k) (m), ( ) Z.,,, 1 Chern., 1 Chern Z.,, Z. (, [11], Remark 1.1.1,.),, ( ) Z, Z,,., 20, ( Z ), 44
45 ,.,,. (discrete rigidity). ([7], Corollary 2.19, (ii),.) N (Π, D Π, s Θ Π ) N M, Π mod M (, µ N µ M ) M., (N N 0 ).,,, : N N, Π, N M, N mod M M.,,, (N N 0 ) (, ).,,,. M Θ = (M Θ N ) N., N, N M Θ N, /.,, M Θ.,,, /.,,,, / Ẑ,.,, l Z,, l Z l Ẑ,. Z/NZ Ẑ,,., N, N (Π tp Y [µ N], D Y Out(Π tp Y [µ N ]), {γ Im(s Θ) Ÿ γ 1 Π tp Y [µ N]} γ µn ). s Θ {γ Im(s Θ Ÿ ) γ 1 } 1., t Θ Π tp Y [µ N], D Y s Θ Π tp Y [µ N].,, Π tp Ÿ [µ N] α : α(s Θ ) = t Θ,, α µ N ( Π tp Ÿ [µ N ]),, (Π tp Ÿ [µ N] ) Π tp Ÿ.,,, : α, D Y Out(Π tp Y [µ N]) Π tp Y [µ N].,., 13 (k) (m). (k) (m), (, l ), 13 (j) Vert(Y log ) ( = Vert(Y log ) = Vert(Ÿ log ))., (, l ), ( ). ( 13 ) Z = Vert(Y log ),, 0 Z, Z., α D Y l Z = Gal(Y log /X log ) 45
46 . 0 Z Z,, 0 Z, l Z,.,,,,.,. M Θ = (M Θ N ) N, N Π µ (M Θ N ) MΘ Π µ (M Θ ) = lim Π µ (M Θ N N )., MΘ, N (l Θ )(M Θ N ), (l Θ )(M Θ ).,,. (l Θ )(M Θ ) Π µ (M Θ ) (constant multiple rigidity). ([7], Corollary 2.19, (iii),.), Π tp X Π,, O k Θ l Kummer (l Z µ 2 ) (O k ηθ,l Z µ 2 )(Π ) H 1 (Π tp Ÿ (Π ), (l Θ )(Π )) /., O k, O k,,.,,, 1.,,.,,, 13, Π, Π tp Ÿ Πtp Ÿ ( = G k ) Π tp Ÿ (Π ) Π tp Ÿ (Π ) ( = G k (Π ), G k (Π ), Π tp X G k Π /., O k ( Kummer ), (, )., 13 (m), O k 1. (, 11 ϕ e 2 (ϕ) = 1.), l (O k ηθ,l Z µ 2 )(Π ) H 1 (Π tp Ÿ (Π ), (l Θ )(Π )), l Z O k l Z µ 2l : θ = O k ηθ,l Z µ 2 = ( O k ( Θ )) l Z µ 2 ( µ l ( Θ(ξ 0 ) 1 Θ Θ(ξ 0 ) Θ 1 l Θ )) l Z µ 2 46
47 ( µ 2l µ 2 Gal(Ÿ log /Y log ),, µ l Gal(Ÿ log /Ÿ log ), l.,, µ 2l.) 9,, q N E, 1, Kummer ,,. (,, [11], 1,.),, (a), (b) : (a) Π tp X Π. (b) Π,, N N M Θ N (Π ).,, (a ), (b ) : (a ) F F. (b ) F,, N N M Θ N ( F).,,,,.,,, : (c) ((b) (b ),, ) N {M Θ N } N M Θ = (M Θ N ) N,.,, Π tp Ÿ Π tp Ÿ (MΘ (Π )) tp Ÿ Πtp Ÿ tp Ÿ (MΘ (Π )) Π tp Ÿ (MΘ (Π )) /. (d) ( (c) ) M Θ, l ( ) θ(m Θ ) H 1 (Π tp Ÿ (MΘ ), (l Θ )(M Θ )) l Z µ 2l /., Kummer, θ(m Θ ), θ(m Θ ) H 1 (Π tp Ÿ (MΘ ), (l Θ )(M Θ )) (,, θ(m Θ ) ) /. ( 3, 47
48 Kummer Kummer,,.) (e) Kummer Frobenius, (d)., M Θ θ(m Θ ) θ(m Θ ). (l Θ )(M Θ ) Π µ (M Θ ) θ env (M Θ ) θ env (M Θ ) H 1 (Π tp Ÿ (MΘ ), Π µ (M Θ )),., 15 ( ),.,. ( 9 ) Π.,, Π, Π tp X k Kummer Π k(π ) k(π ) H 1 (G k (Π ), (l Θ )(Π )) /.,,, O µ k(π ) O k(π ) O k(π ) ( H 1 (G k (Π ), (l Θ )(Π )) ) (Π tp X ) Oµ k O k O k., O k(π ) (, O ), k. (f) (e), Π O µ k(π ) O O k(π ) k(π ) ( H 1 (G k (Π ), (l Θ )(Π )) ),., (e) O µ (M Θ (Π )) O (M Θ (Π )) O (M Θ (Π )) H 1 (G k (Π ), Π µ (M Θ (Π ))) H 1 (Π tp Ÿ (MΘ (Π )), Π µ (M Θ (Π )))., (e) ( ) (O θ env )(M Θ (Π )) = O (M Θ (Π )) + θ env (M Θ (Π )) H 1 (Π tp Ÿ (MΘ (Π )), Π µ (M Θ (Π ))) (, a + η, a O (M Θ (Π )), η θ env (M Θ (Π )) ). 48
49 (g) 13 (j), X log 1 2 k.,, X log Ÿ log Π tp Ÿ (MΘ (Π )) ι,, : l Z, tp Ÿ (MΘ (Π )),, µ 2, µ 2 Gal(Ÿ log /Y log )., ι, θ env (M Θ (Π ))/O µ (M Θ (Π )) (O θ env )(M Θ (Π ))/O µ (M Θ (Π )),, l Z,. (O θ env )(M Θ (Π )) θ env (M Θ (Π )) ι (O θ env )(M Θ (Π )) ι ( (O θ env )(M Θ (Π )) )., Ÿ log D Π tp Ÿ (MΘ (Π )) µ 2, ι ( ). ( (ι, D) (pointed inversion automorphism cf. [11], Remark 1.4.1, (ii); [11], Corollary 1.12, (i)).) (h) (, ), (g) ι D (, 0 Galois ) (O θ env )(M Θ (Π )) ι H 1 (Π tp Ÿ (MΘ (Π )), Π µ (M Θ (Π ))) H 1 (D, Π µ (M Θ (Π ))) O (M Θ (Π )),, : (O θ env )(M Θ (Π )) ι O (M Θ (Π )).,. (O θ env )(M Θ (Π )) ι /O µ (M Θ (Π )) = O µ (M Θ (Π )) ( ) θ env (M Θ (Π )) ι /O µ (M Θ (Π )),,,, O µ (M Θ (Π )),. ( q j2 /2 E O µ., 9., 9, O µ,.), /, (h),, 49
50 . 8 Teichmüller,.,,., 8, ( )., ( ) (O θ env )(M Θ (Π )) ι /O µ (M Θ (Π )) = O µ (M Θ (Π )) ( ) θ env (M Θ (Π )) ι /O µ (M Θ (Π )),. 17 Θ 17, Teichmüller., Teichmüller Θ F, F F, G F = Gal(F /F ), V(F ) F, E F, X E E F., v V(F ), F v F v, F v F F v, G v = Gal(F v /F v ) G F, : F v R (a a ) F v v p p = p 1, v 1 = 1, O v = { a F v a = 1 } O v = { a F v 0 < a 1 } O v = { a F v a 1 } F v, O F v = (O v ) G v OF v = (O v ) G v O Fv = O G v v F v = F G v v, E v = E F F v, X v = X F F v., v V(F ), E v O Fv. v V(F ), q v OF v E v q ( 1), q E = (q v ) v V(F ) v V(F ) O F v., 13, l. 8, ( ), qe N., 11,, ( )., ,, 13 Θ, Θ l ( ).,, 13 (m) 1 (q v ) q v = q 1/2l v ( ).,, (, 8, q N E, ) 50
51 N q N E = (q N v ) v V(F ).,,.,, ( ) q N E. qn E (q N v ) v V(F ), q v N v.,,.,,, q N E., 13 (m),, ( 13 Z ) j q j2 Gauss, q j j 2., q v N v,,.,,, 13 X log X log. K E[l](F ) G F (, ) F Galois, X K = X F K, X K X K K, Galois F l. (, F,,, K.), ( ), X K X K X K, X K l K. X K LabCusp ±., 13 ( X log ) (, 13 F l Cusp(X log ) ). 13 (d), LabCusp ± 1 X K., X K X K {±1} ( ), X K X K C K C K K., 13 (e), X K C K.,, 13, K C K K 51
52 . (, C K, 13 C log.), 13 (i) (j), C K C K K, Aut K (X K ) = Gal(X K /C K ) = {±1}, Aut K (X K ) = Gal(X K /C K ) = Gal(X K /X K ) Gal(X K /C K ) = F l {±1}. 13, Y log X log. Galois Z ( = Z), ±1 Z ( 2 )., Z ±1 Z, X log (, ) (, 17, LabCusp ± ). ( 13.), ±1 Z, LabCusp ±, Gal(X K /X K ) = F l ±, C K (, ) 1, ϵ.,, ϵ, F l Gal(XK /X K ) ( ) F l LabCusp ± {±1}.,, F ± l (F ± l -group cf. [10], Definition 6.1, (i)), T T F l {±1}.,, ϵ, Gal(X K /X K ) LabCusp ± F ± l 1.,,, F ± l = F l {±1} = Gal(X K /C K ) = Aut K (X K ) LabCusp ± = Fl.,, 13 X log X log, X K X K 1.,, 13 ±1 F l Cusp(X log ), 1 F l /{±1} Cusp(C log ), ϵ C K 1.,, (X K X K, ϵ),.,, w V(K) K E K = E F K, X K X K w X w X w 13 X log X log / (, l ).,, ϵ ϵ w X w 13 ±1 F l Cusp(X log ) (, ).,, E K w V(K) 52
53 (X K X K, ϵ) E/F,, G F E[l](F ), (X K X K, ϵ). (,, Hodge-Arakelov, [2], [3] Szpiro, (X K X K, ϵ), [2], 1.5.1,.),,., F mod F X (, j E E j, F mod = Q(j E ))., X F mod X Fmod,, F mod., K/F mod, F/F mod Galois. ([10], Remark 3.1.5,.), Galois, F mod K w 1, w 2, X w1 X w2 ( w 1 w 2 Gal(K/F mod ) ),, q w1 = q w2., V(K), V(K) V mod = V(F mod ) V(K), V(K) V mod V V(K) (, V V(K) V mod )., E K V(K) (X K X K, ϵ) E/F, E/F., V V(K), E K V (X K X K, ϵ) E/F., V(K), V V(K),.,,., E K V B V V mod., B, B ( ) V bad mod B V mod,. V bad mod V mod V V mod V V bad., B \ V bad, ( B ) E K (, V bad mod ), (X K X K, ϵ) E/F. ( 13,, 2., 2 B, V bad mod.) V bad mod V mod V V V(K)., V V(K) V(K) V mod,, V bad mod V mod V,, v V V bad, E K v, 53
54 X v X v 13 X log X log / (, l ),, ϵ v 13 1 F l /{±1} Cusp(C log ) X w,. 7 (F /F, X, l, C K V, V bad mod, ϵ) Θ (initial Θ-data cf. [10], Definition 3.1). Teichmüller.,, C K, X K X K (, ).,, C K X K X K X K ( )., ( ) C K, Θ., Θ,., 1 F ( 1, µ ) l 2 V bad mod ( 13,, l 2 ),, G F E[l](F ) SL 2 (F l ) GL 2 (F l ) = Aut Fl (E[l](F )) (,,, [10], Definition 3.1, ). ( 21, Galois.), Θ 1. 6,., V(F ) V bad V,. v V(F ), v V v V(F ). v V(F ) (, v V ), G v = Gal(F v /K v ), G v O v, (G v O µ v ; H IH κ (O µ v ) H ) ( ), Dv, Fv, Fv µ. v V bad, 13 X log X log X v = X K K K v X v X v, π temp 1 (X v ) Π v., v V good = V \ V bad, X v X v X v X v (, [10], Definition 1.1, ), v V πét 1 (X v) Π v. v V (, V bad ), Π v ( ) D v, Π v O v ( ) F v 54
55 ., Dv, Fv, Fv µ, D v, F v. (, [10], Definition 4.1, (i), (iii); [10], Definition 5.2, (i); [11], Definition 4.9, (vii),. v V, D v, X v Riemann Aut (Aut-holomorphic space cf. [9], Definition 2.1, (i)) Riemann,.) {Dv } v V, {Fv } v V, {Fv µ } v V, {D v } v V, {F v } v V ( V(F ) V(K) V ), D, F, F µ, D, F LabCusp ±., X K,,. 17, ( ),., 17, LabCusp ± F ± l = Aut K (X K ),, Θ ϵ, F ± l (, {±1} F l LabCusp ± )., 17 ( ) D. (,, [10], Definition 6.1, (iii),.) D = { D v } v V D, v V., 17, v, D v D v Π v ( ).,, D v πét 1 (X v ) ( Π v ),,, X v /. LabCusp ± ( D v ).,, 13.,, D v, ϵ v LabCusp ± ( D v ) /,, LabCusp ± ( D v ) F ± l. v, v. T F ± l, T F l, F ± l Aut(T ), F ± l. (F ± l =) Aut ± (T ) Aut(T ),, F l F ± l Aut ± (T ) (F l =) Aut(T )+ Aut ± (T ). ([10], Definition 6.1, (i),.) (,, {±1}.),, Aut( D v ) Aut ± (LabCusp ± ( D v )) Aut ± (LabCusp ± ( D v ))/Aut + (LabCusp ± ( D v )) = {±1} 55
56 . Aut + ( D v ) Aut( D v ). ([10], Definition 6.1, (iii),.), 20 Hodge., LabCusp ±,,,.,,, LabCusp ±. 17,, t { (l 1)/2,..., 0,..., (l 1)/2} = F l = LabCusp ±, q t2., F l = LabCusp ± {±1} F l /{±1},, F l = LabCusp ± \ {0} {±1} F l /{±1}. ( ) q N.,, E qn E,.,,,,. 17,, LabCusp ±.,.,,, F l /{±1} = LabCusp ± /{±1}, F l /{±1} = (LabCusp ± \ {0})/{±1}, C K, C K., F l /{±1} = LabCusp ± /{±1},, 4.,, C K LabCusp.,,,. LabCusp ±,,, LabCusp,, : { } = LabCusp ± = Fl F l F l /{±1} = LabCusp = { }. 56
57 17, LabCusp ± Aut K (X K ) ( = F ± l ).,,,, F l F l F l /{±1}, F ± l,, LabCusp ± F ± l LabCusp F ± l.,, LabCusp F l ( ),, LabCusp ± F l LabCusp F l. F l F l /{±1},, LabCusp F l /{±1}., F l /{±1} LabCusp, : E[l](F ) Q, X K X K E[l](F ) (1 F l )., X K,, LabCusp (Q \ {0})/{±1}., F l Q F l, (Q \ {0})/{±1} F l /{±1},, F l /{±1} LabCusp. LabCusp F l /{±1} F l : F l = F l /{±1}. LabCusp ± F ± l, LabCusp F l. F ± l LabCusp ± F l LabCusp LabCusp ±, LabCusp D. (,, [10], Definition 4.1, (ii),.), D = { D v } v V D, v V., 17, v, D v D v Π v ( ).,, D v πét 1 (C v ) ( Π v ),,, C v /. LabCusp( D v ).,, LabCusp.,, D v, F l ( F l ) LabCusp( D v ) ϵ v LabCusp( D v ) /. v, v Archimedes ,, Teichmüller Θ ±ell Hodge, D-Θ ±ell Hodge. 14,. Π tp X,, D. D 57
58 D = { D,v } v V 1., 17 18, X K πét 1 (X K ) ( ) D ±, D ± 1. ([10], Definition 6.1, (v),.),, D ±, LabCusp ± LabCusp ± ( D ± ) /. ([10], Definition 6.1, (vi),.), v V X v, X v (X v ) X K {D v } v V D ± ( = {D v D ± } v V ) D D ±., v V, LabCusp ± ( D,v ) LabCusp ± ( D ± ).,, v V, v V LabCusp ± ( D,v ). v V LabCusp ± ( D,v ) LabCusp ± ( D ).,, LabCusp ± ( D ) LabCusp ± ( D ± )., ( ). ( Hodge, 20.), LabCusp ± Aut K (X K ) ( = F ± l ) LabCusp ± ( = F l ),. (,, [10], Definition 6.1, (v),.), D ±, LabCusp ± LabCusp ± ( D ± ) /,, Aut csp (X K ) = Ker(Out(πét 1 (X K )) Aut(X K ) Aut(LabCusp ± )) ( ),, Aut csp ( D ± ) = Ker(Aut( D ± ) Aut(LabCusp ± ( D ± ))) Aut( D ± ).,, D ±, Out(πét 1 (X K )) Aut(X K ) (E[l](F ) Q), E[l](F ) Q, X K X K E[l](F ) (1 F l ),, Aut + (X K ) = Ker(Aut(X K ) Aut Fl (Q)) Aut ± (X K ) = Ker(Aut(X K ) Aut Fl (Q)/{±1}) (Aut csp ( D ± ) ) Aut + ( D ± ) Aut ± ( D ± ) Aut( D ± ) 58
59 . Aut ± ( D ± ) LabCusp ± ( D ± ) Aut ± ( D ± )/Aut csp ( D ± )., Aut ± ( D ± )/Aut csp ( D ± ) LabCusp ± ( D ± ), LabCusp ± Aut K (X K ) ( = F ± l ) LabCusp ± ( = F l ) (, ). ( 17 Aut K ( ) Aut K (X K ) Aut ± (X K ) Aut ± (X K )/Aut csp (X K ).), D D ± Aut( D ± ) LabCusp ± ( D ) LabCusp ± ( D ± ) Aut ± ( D ± )/Aut csp ( D ± ) Aut ± ( D ± )., Hodge,, D-Θ ±ell Hodge., v V bad, D v D,v 16 (a) (e), θ env (M Θ ( D,v )) θ env (M Θ ( D,v )). F l = { (l 1)/2,..., 0,..., (l 1)/2} LabCusp ± ( D ± ) = F l j F l Π tp Ÿ ( D,v ) θ env (M Θ ( D,v )), 13 (m), µ 2l q j2 v. 20 Hodge, LabCusp ± ( D ± ), 1 q N E.,,. 4, 1 q N E µ 2l = µ 2l q 02,,, v LabCusp ± ( D ± ).,, j F l,, D j Π tp Ÿ ( D,v ) H 1 (D j, Π µ (M Θ ( D,v ))).,,,, j F l., 16 (h). 16 (h), 59
60 /. 16 (h),,,,., 6, 16 (h)., 16 (h) ( ), ( ), 0, ( j F l.,, ) 0 F l,.,. F l {0}, F ± l.., LabCusp ± (,, ). (, 4 S S 9 ),, LabCusp ±, D. F ± l T, D D T = { D t } t T., D t t D. T D, F ± l,. (,, [10], Example 6.3,.), 19 X v, X v (X v ) X K D 0,v = D v D ±., 19 F ± l LabCusp ± Aut ± (D ± )/Aut csp (D ± ) LabCusp ± (D ± )., F ± l Aut ± (D ± )/Aut csp (D ± ) Aut csp (D ± )., D 0,v D ± Aut csp (D ± ).,,, Aut csp (D ± ).,,, D 0,v,., D 0,v D ± Aut + (D 0,v ).,, D 0,v D ± (Aut + (D 0,v ), Aut csp (D ± )) ϕ Θell 0,v., t F l, ϕ Θell 0,v t F l F ± l Aut ± (D ± )/Aut csp (D ± ) 60
61 D t,v = D v D ± ϕ Θell t,v., t F l v V ϕ Θell t, t F l, = {ϕ Θell t,v } v V : D t = {D t,v } v V D ± ϕ Θell ± = {ϕ Θell t } t Fl : {D t } t Fl D ± ( = {ϕ Θell t : D t D ± } t Fl )., D ± D T = { D t } t T, ϕ Θell ± ϕ Θell ± : D T = { D t } t T D ± D-Θ ell (D-Θ ell -bridge cf. [10], Definition 6.4, (ii)). D-Θ ell ( [10], Definition 6.4, (ii), ) Aut ± ( D ± )/Aut csp ( D ± ) ( = F ± l ). ([10], Proposition 6.6, (ii),.),, D-Θ ell, T D, F ± l. t T. D-Θ ell,,,, v V LabCusp ± ( D t,v ). LabCusp ± ( D t )., D-Θ ell,, LabCusp ± ( D t ) LabCusp ± ( D ± ) /. (, [10], Proposition 6.5, (i),.), D,. t, D 1 D t D {Aut + ( D t,v )} v (, {Aut + ( D,v )} v ) ϕ Θ± t. ϕt Θ± ϕ Θ± ± = { ϕ Θ± t } t T : D T = { D t } t T D ( = { ϕ Θ± t : D t D } t T ) D-Θ ± (D-Θ ± -bridge cf. [10], Definition 6.4, (i))., D-Θ ell D-Θ ± ϕ Θ± ± D D T = { ϕ Θ ± D t } t T D ± D-Θ ±ell Hodge (D-Θ ±ell -Hodge theater cf. [10], Definition 6.4, (iii)). D-Θ ±ell Hodge, 19,, D D ± ( D 0 ), D v LabCusp ±.,, LabCusp ± ( D ) ell 61
62 ., D-Θ ±ell Hodge, t T LabCusp ± ( D t ) LabCusp ± ( D ) LabCusp ± ( D ± ) T /. (, [10], Proposition 6.5, (ii), (iii),.), D-Θ ±ell Hodge ( [10], Definition 6.4, (iii), ) {±1}. ([10], Proposition 6.6, (iii),.), {±1}, , : 17,., D D ±.., ( ),,., F ± l.,,,,.,,,.,., D-Θ ±ell Hodge Θ ±ell Hodge., F t (t T ), F, D D t, D F. F F T = { F t } t T D ±, F T = { F t } t T D { D t } t T D ± D-Θ ell, Θ ell (Θ ell -bridge cf. [10], Definition 6.11, (ii)),., ψ Θell ± : F T = { F t } t T D ± F D ([10], Corollary 5.3, (ii),, 3 ), D-Θ ± ϕ Θ± ± : D T = { D t } t T D, ψ Θ± ± : F T = { F t } t T F ( = {ψ Θ± t : F t F } t T ) 62
63 . ψ Θ± ± Θ± (Θ ± -bridge cf. [10], Definition 6.11, (i))., D-Θ ±ell Hodge, Θ ell Θ ± ψ Θ± ± F F T = { ψ Θ ± F t } t T D ± Θ ±ell Hodge (Θ ±ell -Hodge theater cf. [10], Definition 6.11, (iii)). ell , ,. 21, D-ΘNF Hodge.,.. F, E l F K, E F mod. 8,,.,,. F, K, F mod ( ) F mod,, ( 8 F, ) F mod., 17 K/F mod Galois, 8 12,, F mod., 17, (X K X K, ϵ), V(F ) V(K), V ( V mod ).,, F K, F mod., 8,, F mod, ,,.,, Gauss {q j2} j, 8 E qe N.,,, Gauss.,.,. 17,,, (F F mod ) K., F mod.,,, K/F mod (, Gal(K/F mod ),,, V(K) V mod ). 2,, 63
64 F ± l,,,.,,, 0 LabCusp ±. j q j2, 0 LabCusp ±,, (, LabCusp ± \ {0} ),., 20,,.,,, Gauss, F mod Gauss,,. ( ),. (,, [11], Remark 4.7.3,.),. (,, [10], Example 4.3, (i),.) 17, Aut K (C K ) = {1},, F mod E, Aut(C K ) Aut(K) Gal(K/F mod ) Aut(K),, Aut(C K ) Gal(K/F mod ),, C K /, K/F mod. Aut ϵ (C K ) Aut(C K ) C K ( Θ ) ϵ.,, ({±1} ) Aut(C K ) Aut(E[l](F ))/{±1} ({±1} ) Aut(C K ) Aut(Q)/{±1} = F l, E[l](F ) Q, X K X K E[l](F ) (1 F l ) Aut ϵ (C K ) Aut(C K )., 17 G F E[l](F ) SL 2 (F l ) GL 2 (F l ) = Aut Fl (E[l](F )), Aut(C K ) F l., Aut(C K )/Aut ϵ (C K ) F l., Aut(C K )/Aut ϵ (C K ) F l LabCusp LabCusp F l. ( 18.), 18,, X K C K LabCusp LabCusp ±,. 18, LabCusp F ± l 64
65 .,, LabCusp, F ± l., (F l LabCusp ) F l, LabCusp LabCusp ±.,,., t 1, t 2 LabCusp ±,, LabCusp, t 2 = ±t 1, ( 13 (m) ) t 1 t 2. ( µ 2l.) F l LabCusp, 20 D-Θ ell. (,, [10], Example 4.3,.), C K πét 1 (C K ) ( ) D,, Aut ϵ (C K ) Aut(C K ) Aut ϵ (D ) Aut(D ). X v, X v (C v ) C K D 1,v = D v D. LabCusp F l LabCusp, Aut(C K )/Aut ϵ (C K ) LabCusp, D-Θ ell, D 1,v D ± (Aut(D 1,v ), Aut ϵ (D )). ϕ NF 1,v., j F l, ϕ NF 1,v j F l Aut(D )/Aut ϵ (D ) D j,v = D v D ± ϕ NF j,v., j F l v V ϕ NF j = {ϕ NF j,v } v V : D j = {D j,v } v V D, j F l, ϕ NF = {ϕ NF j } j F : {D j } l j F l D ( = {ϕ NF j : D j D } j F ). v V, ϕ NF (j, v) ϕnf j,v, D j,v D ± (Aut(D j,v ), Aut ϵ (D )),, K, v V Aut ϵ (D ). (Aut ϵ (D ) D V(K).) V ±un : l V ±un = Aut ϵ (D ) V. D D, l = F l = (l 1)/2 J, J D D J = { D j } j J, ϕ NF ϕ NF : D J = { D j } j J D D-NF (D-NF-bridge cf. [10], Definition 4.6, (i)). D D, LabCusp LabCusp( D ) 65
66 /., D-NF ( [10], Definition 4.6, (i), ) Aut(C K )/Aut ϵ (C K ) ( = F l ). ([10], Proposition 4.8, (i),.),, D-NF, J D, F l., D-NF ( ), 13 20,,. (,, [10], Example 4.4,.) v V bad j F l, j t F l X v Π tp X G v, j F l (evaluation section cf. [10], Example 4.4, (i))., j t F l., X v ϕ Θ j,v : D j,v D >,v = D v, v V good, v V bad j F l (, Π tp X G v Π tp X j ) (Aut(D j,v ), Aut(D >,v )). ϕ Θ j = {ϕ Θ j,v} v V : D j D > = {D >,v } v V, D > D ϕ Θ = {ϕ Θ j } j J : { D j } j F l D > ( = {ϕ Θ j : D j D > } j F ) l ϕ Θ = { ϕ Θ j } j J : D J = { D j } j J D > D-Θ (D-Θ-bridge cf. [10], Definition 4.6, (ii))., D-Θ ±ell Hodge, D-NF D-Θ D ϕ NF D J = { ϕ Θ D j } j J D > D-ΘNF Hodge (D-ΘNF-Hodge theater cf. [10], Definition 4.6, (iii)). Hodge, D-ΘNF Hodge,, D-Θ ±ell LabCusp( D ) J. ([10], Proposition 4.7, (iii),.) D-ΘNF Hodge, Teichmüller,., D-ΘNF Hodge ( [10], Definition 4.6, (iii), ) 1. ([10], Proposition 4.8, (ii),.) 21, D-Θ ±ell Hodge D-ΘNF Hodge. D-Θ ±ell Hodge F ± l LabCusp ± 66
67 ,, F ± l., D-ΘNF Hodge F l LabCusp,, F l.,, /. D-Θ ±ell Hodge F ± l LabCusp ±, Aut K (X K ), X K., D-ΘNF Hodge F l LabCusp, Gal(K/F mod ), C K, V ±un Aut(C K ) (D-NF ) : (D-)Θ ±ell Hodge F ± l LabCusp ± F ± l : Aut K (X K ) :, (D-)ΘNF Hodge F l LabCusp F l : Gal(K/F mod ) :. Hodge, LabCusp ± LabCusp LabCusp ± = Fl F l F l = LabCusp, D-Θ ±ell NF Hodge. ( 26.),,, ( ), : Aut K (X K ) Gal(K/F mod ) , 22, F mod. 17, K C K K,, F mod X, C K F mod. C Fmod., C Fmod πét 1 (C Fmod ) ( ) D. ([10], Example 5.1, (i),.), C K C Fmod D D., S mod Spec(O K ) Gal(K/F mod ) ([10], Remark 3.1.5, ): S mod = [Spec(O K )/Gal(K/F mod )]. 17 K/F mod, F/F mod Galois, q (, ) E K, S mod., ([10], Example 5.1, (ii), (iii), ): F D,, C Fmod., S mod 67
68 (,, Frobenius (Frobenius degree cf. [6], Definition 1.1, (iii), (iv)),.),, D,, C Fmod D,, C K F,, F, D,, C Fmod F mod. ([10], Example 5.1, (iii),.), F F F mod,,. F mod S mod, 8 2 F MOD, F mod,., F MOD. v V, F mod K v /O K v ( = Z) β v., F MOD, (T, {t v} v V ) : F mod T, v V, β v T K v /O K v T v t v., Fmod., v V J v K v {J v } v V, v V J v = O Kv., J v, v V, K v O Kv, O Kv K v.,, v V F mod,, S mod, S mod (, F mod ), F mod T., v V T v,, F MOD.,, F mod F MOD., F MOD, F mod /,., 8, F mod / /., F MOD,., 8, / F mod F MOD, F mod /., F mod, v V J v = O Kv, F mod,, S mod.,, 68
69 Fmod F mod., F mod, F mod, F mod /,., 8, F mod / /, v V J v. /, F mod (realification cf. [6], Proposition 5.3) C mod. (,, [10], Example 3.5, (i),.), v V mod ( V v ), v V mod ( (Fmod ) v /O (F mod ) v ).,, Fmod,. C mod,, v V mod, (F mod ) v /O (F mod ) v ( = Z),, (F mod ) v /O (F mod ) v ( = Z) ((F mod ) v /O (F mod ) v ) Z R = (K v /O K v ) Z R ( = R). v V, O K v /O K v K v /O K v (K v /O K v ) Z R (K v /O K v ) Z R ( = R) R 0 R O K v R 0 (K v /O K v ) Z R.,, C mod,, ( (K v /O K v ) Z R ) V, v V O K v R 0. V Prime(C mod), v Prime(C mod ) O K v R 0 Φ C mod,v. 23 Θ Hodge, F (F -prime-strip cf. [10], Definition 5.2, (ii)). v V, p v. v V bad, G v (O v q N {1} v qn ) ( (split Frobenioid v cf. [10], Example 3.2, (v))), v V good, G v (O v p N {1} p N ) ( [10], Example 3.3, (ii), ) F v., F v, {F v } v V F. 69
70 F = { Fv } v V F. v, Fv Frobenius (, O v q N v O v p N ),, ( = R) ( = R 0 R) ( ( (O v q N v )/(O v {1}) ) gp Z R ( (O v p N )/(O v {1}) ) gp Z R )., v V, F v ( = R 0 R). ( = R 0 ). Φ rlf F v ( = R), F (F -prime-strip cf. [10], Definition 5.2, (iv)) 4 (C mod, Prime(C mod ) V, F, {ρ v } v V )., Cmod 22, Prime( C ) V, F = {Fv } v V 23, ρ v Φ C mod,v Φrlf F, v Prime( C ) Prime( C ) V v V v., v V, F v. v V bad ( v V ), F v, 14 F. 14, ( ), Π v { i i, X }., v V good, F v ( 17 ) F v., F v F v F v. ([10], Examples 3.2, (iii), (vi); 3.3, (iii); 3.4, (ii),.), Θ Hodge (Θ-Hodge theater cf. [10], Definition 3.6) ({ F v } v V, F )., v V, F v F v,, { F v } v V F F, F F, F (, F ), { F v } v V F F., Θ Hodge, {F v } v V, ( ) Cmod., 21 D-Θ ϕ Θ = { ϕ Θ j } j J : D J = { D j } j J D >. 23, D-Θ Θ. Θ,,. ([10], Remark 5.3.1,.) F, F F ; D, D F, F D ; D, D D, D D., ϕ: D D D, ϕ : D D.,, ϕ 70
71 F ψ : F F : v V ; F v = ( Π v O v ), F v = ( Π v O v ) F, F v ; D v = Π v, D v = Π v D, D v, Dv = G v ( Π v ), D v = G v ( Π v ) D, D v., ϕ v Π v Π v,, G v G v., Π v Π v Π v O v, F v ϕ ( F v ) = ( Π v O v ).,,, F ϕ ( F) = {ϕ ( F v )} v V.,, F ϕ ( F) D D,, ( 20 ϕ Θ± ± ψ Θ± ± ) F D, D F ϕ ( F) 1., ( ) F ϕ ( F) ϕ ( F) F, ψ : F F., Θ, F j (j J), F >, D D j, D > F., j J, D-Θ ϕ Θ j ϕ Θ j : D j D >, j (, ),, D ( ϕ Θ j ) : D j D >.,, j ϕ Θ j : D j D > F ψj Θ : F j F >. ψj Θ { ψ Θ j } j J : F J = { F j } j J F > ( = { ψ Θ j : F j F > } j J ) ψ Θ = { ψ Θ j } j J., D-Θ ϕ Θ, ψ Θ., HT Θ = ({ F v } v V, F ) Θ Hodge, { F v } v V F ( F v ) F >. F > HT Θ., D-Θ ϕ Θ ψ Θ,, F > HT Θ F J = { ψ Θ F j } j J F > HT Θ, Θ (Θ-bridge cf. [10], Definition 5.5, (ii))., Θ Hodge, {F v } v V C mod., Θ, Θ Hodge, (, ) (, ϕ Θ ψ ) Θ. 71
72 24 Kummer 24, Teichmüller Kummer. (, [10], 5,.), Kummer (, 11 ) κ, κ, κ. (,, [10], Remark 3.1.7,.) κ, C Fmod,, C (Fmod ) v (v V mod ).,, C Fmod,, 3 C Fmod : f f, f 1,, 2. E 2 f. E 2 f 1.,,,, κ., C Fmod ( ), κ ( ) κ,, C Fmod ( ), κ κ.,, C (Fmod ) v,, κ, κ, κ. (κ, κ, κ, 16 θ env (M Θ (Π )), θ env (M Θ (Π )), (O θ env )(M Θ (Π )).) κ 1, a F mod ( F ; (F mod ) v ; F v ), E 2 F mod ( F ; (F mod ) v ; F v ) c C Fmod ( C Fmod ; C (Fmod ) v ; C (Fmod ) v ) κ f, f c a., ( 11 ) κ C Fmod, F mod.,, Kummer. (,, [10], Example 5.1,.) D D., D πét 1 (C K ) ( ).,, (a) πét 1 (C K ) πét 1 (C Fmod ) D D /. (, D 22 D.), C K Belyi (of strictly Belyi type cf. [8], Definition 3.5), Belyi (Belyi cuspidalization cf. [8], Example 3.6, Corollaries 3.7, 3.8) 72
73 ,, D ( D ) / : (b) C Fmod πét 1 (C Fmod ) D. (c) πét 1 (C Fmod ) F D F ( D ). ( D D F ( D ), 24,, D F ( D ), D,.), F mod (= F πét 1 (C F mod ) ) Fmod ( D ) = F ( D ) D. (d) πét 1 (C Fmod ) (C Fmod Galois ) (C Fmod ) D Drat M rat ( D ) ( F ( D )), Kummer (F ( D ) ) M rat ( D ) H 1 ( D rat, Λ( D )), Λ( D ) D 1 (b). ( 1.), (c),,,, / : (e) F (πét 1 (C Fmod ) ) V(F ) D V( D ),, V(K), V mod V( D ) = V( D ) D, V mod ( D ) = V( D ) D. (f) ( 22 ) F F ( D ). (g) (h) ( 22 ) F F ( D ) = F ( D ) D. ( 22 ) F mod F mod ( D ) = F ( D ) D., (b) /, (d) C Fmod ( ),.,, (i) κ, κ, κ Drat ( M κ ( D ) M κ( D ) M κ ( D ) ) ( M rat ( D ) ) /. (.) (,, M κ ( D ) = M κ( D ) D rat.), Kummer., Kummer. (,, [10], Example 5.1,.) F F F, F.,, (j) F, F D, D D, D /., (h), (k) F mod F mod /.,, D D D D (, F F F F D ) D D 73
74 . F F. F, ( ), (l) πét 1 (C Fmod ) F D F ( F ),, F mod (= (F ) πét 1 (C F ) mod ) F mod ( F ) = F ( F ) D. /., 22 Cmod Prime(C mod ),, (m) F, F mod (πét 1 (C Fmod ) ) V(F ), V mod D Prime( F ), Prime( F mod ). /., 1. (j) D (d), D rat., Drat ( )., (i) D rat M κ( D ) ( (i) D rat M κ ( D ) ),, F κ ( κ-coric structure cf. [10], Example 5.1, (v)) ( κ ( κ -coric structure cf. [10], Example 5.1, (v)))., F 1 κ ( κ )., Kummer., /Frobenius Kummer. (,, [10], Example 5.1,.) Kummer F F F κ Drat M. M,, M lim (n n ) Drat Λ(M). (, 1 κ.), 3 G M, D rat M H 1 ( D rat, Λ(M))., (j) D (i), F ( ) κ (, κ ) D rat M κ( D ) ( H 1 ( D rat, Λ( D )) ) ( (d) )., Q Ẑ Z Q, Q >0 Ẑ = {1} 74
75 , : 1 D rat Λ(M) Λ( D ), H 1 ( D rat, Λ(M)) H 1 ( D rat, Λ( D )),. M M κ( D ), κ Kummer ( D rat M) ( D rat M κ( D )). κ, κ, Kummer., Kummer 1,.,, F, 1 κ ( κ )., (l) D rat D F ( F ).,, lim n (n ) D rat, Λ(F ( F )).,, Kummer, D rat F ( F ) H 1 ( D rat, Λ(F ( F )))., (j) D (c), (d), D rat F ( D ) H 1 ( D rat, Λ( D )).,, : 1 D rat Λ(F ( F )) Λ( D ), H 1 ( D rat, Λ(F ( F ))) H 1 ( D rat, Λ( D )), F ( F ) F ( D ), F mod ( F ) F mod ( D ). Kummer F ( F ) F ( D ), F mod ( F ) F mod ( D ),, F ( F ) {0} F mod ( F ) {0}.,, 75
76 (n) (e) V mod ( D ) (m) Prime( F mod ) Prime( F mod ) V mod ( D ). F ( F ) {0} Archimedes p Prime( F )., F ( F ) {0} p,, D Prime( F ) p Prime( F ) (, [10], Example 5.4, (iv), ), (o) p Prime( F ) v V F v = (Π v O v )., F ( F ) {0} Archimedes p Prime( F ), (p) p Prime( F ) v V F v. Kummer. κ (, κ ), 24,, : 2 F F., ( 1 ) F 1 κ Drat M κ., ((i) ) M κ = M D rat κ M κ, κ. ( 3 G M ) D rat M κ M κ H 1 ( D rat, Λ(M κ)),, M κ ( ). Belyi ((b) ), M κ ( ), πét 1 (C Fmod ) F D F ( F ),, F mod (= (F ) πét 1 (C F ) mod ) F mod ( F ) = F ( F ) D /. ( 24 κ 1.) Q Ẑ Z Q, Q >0 Ẑ = {1} Λ(F ( F )) Λ( D ) Kummer F ( D ), F mod ( D ) F ( F ) {0} F mod ( F ) {0} /., Kummer,., D, F 76
77 .,, /Frobenius /Frobenius κ. (,, [10], Definition 5.2,.) 25 Hodge 25, 21 D-ΘNF Hodge ΘNF Hodge. 21 D-ΘNF Hodge,., 24 Kummer Kummer, 25 ΘNF Hodge. 21 D-NF ϕ NF = { ϕ NF j } j J : D J = { D j } j J D.,, 24 Kummer F F, F 24 (j) D D-NF D., δ LabCusp( D ).,, δ ( Θ ) ϵ LabCusp D D 1 Aut ϵ (D ) (, 1 D D ). 24 (e) V( D ), D D 21 V ±un ( = Aut ϵ (D ) V) δ (δ-valuation cf. [10], Example 5.4, (iii)). δ, D πét 1 (C K ),,., D-NF,,, D-NF, D-NF. ([10], Example 5.4, (iii),.) 25, F F ( δ ) δ v V 24 (o) (p), F. F F δ. ([10], Example 5.4, (iv),.), V ±un V(K) V ( ), v V V ±un V mod F F δ. j J, D D j F F j., j J, δ j LabCusp( D ) 1 : D-NF 77
78 ϕ NF j ϕ NF j V( D ) δ j. (, [10], Example 5.4, (iii),.), F j F δj ψ NF j : F j F,, ψ NF = { ψ NF j } j J : F J = { F j } j J F ( = { ψ NF j : F j F } j J ). ψ NF, F F F J = { ψ NF F j } j J F F NF (NF-bridge cf. [10], Definition 5.5, (i)). ψ NF,, F δj j ψ NF j : F j F,, F j F δj,,.,, j ψj NF : F j F, ( ) 24 κ., ψj NF : F j F, κ,, Kummer, κ,, Kummer. ([10], Example 5.4, (iv),.), NF, 24 Kummer ( 24 ) Kummer,., Θ ±ell Hodge, 23 Θ NF F F ψ NF F J = { ψ Θ F j } j J F > HT Θ ΘNF Hodge (ΘNF-Hodge theater cf. [10], Definition 5.5, (iii)). 26 Hodge 20 D-Θ ±ell Hodge ϕ Θ± ± D D T = { ϕ Θ ± D t } t T D ±., F ± l = Aut K (X K ) /,., 21 D-ΘNF Hodge D ϕ NF D J = { ϕ Θ D j } j J D >, Gal(K/F mod ) F l /,. 2 Hodge. ( 18 ) LabCusp ± LabCusp ell 78
79 D-Θ ±ell NF Hodge (D-Θ ±ell NF-Hodge theater cf. [10], Definition 6.13, (ii))., D-Θ ±ell NF Hodge, D-Θ ±ell Hodge, D-ΘNF Hodge,, 3 : ϕ Θ± ± D D T = { ϕ Θ ± D t } t T D ± ell glue D > ϕ Θ t T \ {0}, D-Θ ± D J = { ϕ NF D j } j J D. ϕ Θ ± t D t D ( ϕ Θ± t ) 1 D t, D t D t, D t., ϕ Θ ± 0 D 0 D, D 0 D, D 0., T = (T \ {0})/{±1}., D T = { D t } t T D, D 0 D.,, D T = { D t } t T D 0 D-Θ : ϕ Θ ( ϕ Θ± ± ): D T D 0. ([10], Proposition 6.7,.) D-Θ ϕ Θ ( ϕ Θ± ± ) D-Θ ϕ Θ,. ([10], Remark ,.),, D 0, D, D >., 21,, ( LabCusp ±, LabCusp ) Hodge F ± l LabCusp ±, F l LabCusp,., 20 Θ ±ell Hodge 25 ΘNF Hodge ψ Θ± ± F F T = { ψ Θ ± F t } t T D ± ell F F ψ NF F J = { ψ Θ F j } j J F > HT Θ. D-Θ ±ell Hodge D-ΘNF Hodge,, D., 79
80 ( 20 ϕ Θ± ± ψ± Θ± ) F D, D, F., F, Θ ±ell Hodge ΘNF Hodge. ([10], Remark ,.), Θ ±ell Hodge ΘNF Hodge 3 Θ ±ell NF Hodge (Θ ±ell NF-Hodge theater cf. [10], Definition 6.13, (i)) : ψ Θ± ± F F T = { ψ Θ ± F t } t T D ± ell glue HT Θ ψ Θ F > F J = { ψ NF F j } j J F F. HT Θ±ell NF, HT Θ±ell NF Θ ±ell NF Hodge, HT D-Θ±ell NF, HT D-Θ±ell NF D-Θ ±ell NF Hodge. D-Θ ±ell NF Hodge Ξ: HT D-Θ±ell NF HT D-Θ±ellNF, ( 20 ϕ Θ± ± ψ± Θ± ) F D, Ξ D D D, T J { } {>}, F log( F ) F, log( F ), 9. {log( F ) F } T J { } {>} Ξ., {Ξ }, Ξ HT D-Θ±ell NF HT D-Θ±ellNF, HT Θ±ellNF HT Θ±ellNF,, HT Θ±ell NF log HT Θ±ell NF. ([12], Proposition 1.3, (i),.), 9 ( 10) ( Teichmüller ) Hodge. 9 10, log 1 HT Θ±ell NF log 0 HT Θ±ell NF log 1 HT Θ±ell NF log, Kummer. 27,, Hodge, : Diophantus, (a) (b) q (1 ) (c) 3 ( ). ( ) 80
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 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 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 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 information(iii) 0 V, x V, x + 0 = x. 0. (iv) x V, y V, x + y = 0., y x, y = x. (v) 1x = x. (vii) (α + β)x = αx + βx. (viii) (αβ)x = α(βx)., V, C.,,., (1)
1. 1.1...,. 1.1.1 V, V x, y, x y x + y x + y V,, V x α, αx αx V,, (i) (viii) : x, y, z V, α, β C, (i) x + y = y + x. (ii) (x + y) + z = x + (y + z). 1 (iii) 0 V, x V, x + 0 = x. 0. (iv) x V, y V, x + y
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 information微分積分 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, 初版 1 刷発行時のものです.
微分積分 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. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
More information基礎数学I
I & II ii ii........... 22................. 25 12............... 28.................. 28.................... 31............. 32.................. 34 3 1 9.................... 1....................... 1............
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 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 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[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 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 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 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 informationuntitled
i ii iii iv v 43 43 vi 43 vii T+1 T+2 1 viii 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 a) ( ) b) ( ) 51
More information2
1 2 3 4 5 6 7 8 9 10 I II III 11 IV 12 V 13 VI VII 14 VIII. 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 _ 33 _ 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 VII 51 52 53 54 55 56 57 58 59
More informationD = [a, b] [c, d] D ij P ij (ξ ij, η ij ) f S(f,, {P ij }) S(f,, {P ij }) = = k m i=1 j=1 m n f(ξ ij, η ij )(x i x i 1 )(y j y j 1 ) = i=1 j
6 6.. [, b] [, d] ij P ij ξ ij, η ij f Sf,, {P ij } Sf,, {P ij } k m i j m fξ ij, η ij i i j j i j i m i j k i i j j m i i j j k i i j j kb d {P ij } lim Sf,, {P ij} kb d f, k [, b] [, d] f, d kb d 6..
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 informationAkito Tsuboi June 22, T ϕ T M M ϕ M M ϕ T ϕ 2 Definition 1 X, Y, Z,... 1
Akito Tsuboi June 22, 2006 1 T ϕ T M M ϕ M M ϕ T ϕ 2 Definition 1 X, Y, Z,... 1 1. X, Y, Z,... 2. A, B (A), (A) (B), (A) (B), (A) (B) Exercise 2 1. (X) (Y ) 2. ((X) (Y )) (Z) 3. (((X) (Y )) (Z)) Exercise
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 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 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 information(yx4) 1887-1945 741936 50 1995 1 31 http://kenboushoten.web.fc.com/ OCR TeX 50 yx4 e-mail: yx4.aydx5@gmail.com i Jacobi 1751 1 3 Euler Fagnano 187 9 0 Abel iii 1 1...................................
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 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 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 informationx 3 a (mod p) ( ). a, b, m Z a b m a b (mod m) a b m 2.2 (Z/mZ). a = {x x a (mod m)} a Z m 0, 1... m 1 Z/mZ = {0, 1... m 1} a + b = a +
1 1 22 1 x 3 (mod ) 2 2.1 ( )., b, m Z b m b (mod m) b m 2.2 (Z/mZ). = {x x (mod m)} Z m 0, 1... m 1 Z/mZ = {0, 1... m 1} + b = + b, b = b Z/mZ 1 1 Z Q R Z/Z 2.3 ( ). m {x 0, x 1,..., x m 1 } modm 2.4
More informationNote.tex 2008/09/19( )
1 20 9 19 2 1 5 1.1........................ 5 1.2............................. 8 2 9 2.1............................. 9 2.2.............................. 10 3 13 3.1.............................. 13 3.2..................................
More informationi ii iii iv v vi vii ( ー ー ) ( ) ( ) ( ) ( ) ー ( ) ( ) ー ー ( ) ( ) ( ) ( ) ( ) 13 202 24122783 3622316 (1) (2) (3) (4) 2483 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) 11 11 2483 13
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<4D6963726F736F667420506F776572506F696E74202D208376838C835B83938365815B835683878393312E707074205B8CDD8AB78382815B83685D>
i i vi ii iii iv v vi vii viii ix 2 3 4 5 6 7 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
More informationSC-85X2取説
I II III IV V VI .................. VII VIII IX X 1-1 1-2 1-3 1-4 ( ) 1-5 1-6 2-1 2-2 3-1 3-2 3-3 8 3-4 3-5 3-6 3-7 ) ) - - 3-8 3-9 4-1 4-2 4-3 4-4 4-5 4-6 5-1 5-2 5-3 5-4 5-5 5-6 5-7 5-8 5-9 5-10 5-11
More informationn=1 1 n 2 = π = π f(z) f(z) 2 f(z) = u(z) + iv(z) *1 f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x
n= n 2 = π2 6 3 2 28 + 4 + 9 + = π2 6 2 f(z) f(z) 2 f(z) = u(z) + iv(z) * f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x f x = i f y * u, v 3 3. 3 f(t) = u(t) + v(t) [, b] f(t)dt = u(t)dt
More informationchap10.dvi
. q {y j } I( ( L y j =Δy j = u j = C l ε j l = C(L ε j, {ε j } i.i.d.(,i q ( l= y O p ( {u j } q {C l } A l C l
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 nakayama/print/ Def (Definition ) Thm (Theorem ) Prop (Proposition ) Lem (Lemma ) Cor (Corollary ) 1. (1) A, B (2) ABC
1 http://www.gem.aoyama.ac.jp/ nakayama/print/ Def (Definition ) Thm (Theorem ) Prop (Proposition ) Lem (Lemma ) Cor (Corollary ) 1. (1) A, B (2) ABC r 1 A B B C C A (1),(2),, (8) A, B, C A,B,C 2 1 ABC
More informationDirac 38 5 Dirac 4 4 γ µ p µ p µ + m 2 = ( p µ γ µ + m)(p ν γ ν + m) (5.1) γ = p µ p ν γ µ γ ν p µ γ µ m + mp ν γ ν + m 2 = 1 2 p µp ν {γ µ, γ ν } + m
Dirac 38 5 Dirac 4 4 γ µ p µ p µ + m 2 p µ γ µ + mp ν γ ν + m 5.1 γ p µ p ν γ µ γ ν p µ γ µ m + mp ν γ ν + m 2 1 2 p µp ν {γ µ, γ ν } + m 2 5.2 p m p p µ γ µ {, } 10 γ {γ µ, γ ν } 2η µν 5.3 p µ γ µ + mp
More information64 3 g=9.85 m/s 2 g=9.791 m/s 2 36, km ( ) 1 () 2 () m/s : : a) b) kg/m kg/m k
63 3 Section 3.1 g 3.1 3.1: : 64 3 g=9.85 m/s 2 g=9.791 m/s 2 36, km ( ) 1 () 2 () 3 9.8 m/s 2 3.2 3.2: : a) b) 5 15 4 1 1. 1 3 14. 1 3 kg/m 3 2 3.3 1 3 5.8 1 3 kg/m 3 3 2.65 1 3 kg/m 3 4 6 m 3.1. 65 5
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. ev=,604k m 3 Debye ɛ 0 kt e λ D = n e n e Ze 4 ln Λ ν ei = 5.6π / ɛ 0 m/ e kt e /3 ν ei v e H + +e H ev Saha x x = 3/ πme kt g i g e n
003...............................3 Debye................. 3.4................ 3 3 3 3. Larmor Cyclotron... 3 3................ 4 3.3.......... 4 3.3............ 4 3.3...... 4 3.3.3............ 5 3.4.........
More informationuntitled
( 9:: 3:6: (k 3 45 k F m tan 45 k 45 k F m tan S S F m tan( 6.8k tan k F m ( + k tan 373 S S + Σ Σ 3 + Σ os( sin( + Σ sin( os( + sin( os( p z ( γ z + K pzdz γ + K γ K + γ + 9 ( 9 (+ sin( sin { 9 ( } 4
More information18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α
18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α 2 ), ϕ(t) = B 1 cos(ω 1 t + α 1 ) + B 2 cos(ω 2 t
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 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 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 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 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 informationK E N Z U 2012 7 16 HP M. 1 1 4 1.1 3.......................... 4 1.2................................... 4 1.2.1..................................... 4 1.2.2.................................... 5................................
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 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 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 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 informationJacobi, Stieltjes, Gauss : :
Jacobi, Stieltjes, Gauss : : 28 2 0 894 T. J. Stieltjes [St94a] Recherches sur les fractions continues Stieltjes 0 f(u)du, z + u f(u) > 0, z C z + + a a 2 z + a 3 +..., a p > 0 (a) Vitali (a) Stieltjes
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 informationTOP URL 1
TOP URL http://amonphys.web.fc.com/ 3.............................. 3.............................. 4.3 4................... 5.4........................ 6.5........................ 8.6...........................7
More information(1) (2) (3) (4) 1
8 3 4 3.................................... 3........................ 6.3 B [, ].......................... 8.4........................... 9........................................... 9.................................
More informationIII III 2010 PART I 1 Definition 1.1 (, σ-),,,, Borel( ),, (σ-) (M, F, µ), (R, B(R)), (C, B(C)) Borel Definition 1.2 (µ-a.e.), (in µ), (in L 1 (µ)). T
III III 2010 PART I 1 Definition 1.1 (, σ-),,,, Borel( ),, (σ-) (M, F, µ), (R, B(R)), (C, B(C)) Borel Definition 1.2 (µ-a.e.), (in µ), (in L 1 (µ)). Theorem 1.3 (Lebesgue ) lim n f n = f µ-a.e. g L 1 (µ)
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 information構造と連続体の力学基礎
II 37 Wabash Avenue Bridge, Illinois 州 Winnipeg にある歩道橋 Esplanade Riel 橋6 6 斜張橋である必要は多分無いと思われる すぐ横に道路用桁橋有り しかも塔基部のレストランは 8 年には営業していなかった 9 9. 9.. () 97 [3] [5] k 9. m w(t) f (t) = f (t) + mg k w(t) Newton
More informationこれわかWord2010_第1部_100710.indd
i 1 1 2 3 6 6 7 8 10 10 11 12 12 12 13 2 15 15 16 17 17 18 19 20 20 21 ii CONTENTS 25 26 26 28 28 29 30 30 31 32 35 35 35 36 37 40 42 44 44 45 46 49 50 50 51 iii 52 52 52 53 55 56 56 57 58 58 60 60 iv
More informationパワポカバー入稿用.indd
i 1 1 2 2 3 3 4 4 4 5 7 8 8 9 9 10 11 13 14 15 16 17 19 ii CONTENTS 2 21 21 22 25 26 32 37 38 39 39 41 41 43 43 43 44 45 46 47 47 49 52 54 56 56 iii 57 59 62 64 64 66 67 68 71 72 72 73 74 74 77 79 81 84
More informationこれでわかるAccess2010
i 1 1 1 2 2 2 3 4 4 5 6 7 7 9 10 11 12 13 14 15 17 ii CONTENTS 2 19 19 20 23 24 25 25 26 29 29 31 31 33 35 36 36 39 39 41 44 45 46 48 iii 50 50 52 54 55 57 57 59 61 63 64 66 66 67 70 70 73 74 74 77 77
More informationZ: Q: R: C: 3. Green Cauchy
7 Z: Q: R: C: 3. Green.............................. 3.............................. 5.3................................. 6.4 Cauchy..................... 6.5 Taylor..........................6...............................
More information1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2
2005 9/8-11 2 2.2 ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1 1 (Berry,1975) 2-6
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 information1 (1) () (3) I 0 3 I I d θ = L () dt θ L L θ I d θ = L = κθ (3) dt κ T I T = π κ (4) T I κ κ κ L l a θ L r δr δl L θ ϕ ϕ = rθ (5) l
1 1 ϕ ϕ ϕ S F F = ϕ (1) S 1: F 1 1 (1) () (3) I 0 3 I I d θ = L () dt θ L L θ I d θ = L = κθ (3) dt κ T I T = π κ (4) T I κ κ κ L l a θ L r δr δl L θ ϕ ϕ = rθ (5) l : l r δr θ πrδr δf (1) (5) δf = ϕ πrδr
More information2016 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 1 16 2 1 () X O 3 (O1) X O, O (O2) O O (O3) O O O X (X, O) O X X (O1), (O2), (O3) (O2) (O3) n (O2) U 1,..., U n O U k O k=1 (O3) U λ O( λ Λ) λ Λ U λ O 0 X 0 (O2) n =
More informationS I. dy fx x fx y fx + C 3 C dy fx 4 x, y dy v C xt y C v e kt k > xt yt gt [ v dt dt v e kt xt v e kt + C k x v + C C k xt v k 3 r r + dr e kt S dt d
S I.. http://ayapin.film.s.dendai.ac.jp/~matuda /TeX/lecture.html PDF PS.................................... 3.3.................... 9.4................5.............. 3 5. Laplace................. 5....
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 information平成18年版 男女共同参画白書
i ii iii iv v vi vii viii ix 3 4 5 6 7 8 9 Column 10 11 12 13 14 15 Column 16 17 18 19 20 21 22 23 24 25 26 Column 27 28 29 30 Column 31 32 33 34 35 36 Column 37 Column 38 39 40 Column 41 42 43 44 45
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 informationW u = u(x, t) u tt = a 2 u xx, a > 0 (1) D := {(x, t) : 0 x l, t 0} u (0, t) = 0, u (l, t) = 0, t 0 (2)
3 215 4 27 1 1 u u(x, t) u tt a 2 u xx, a > (1) D : {(x, t) : x, t } u (, t), u (, t), t (2) u(x, ) f(x), u(x, ) t 2, x (3) u(x, t) X(x)T (t) u (1) 1 T (t) a 2 T (t) X (x) X(x) α (2) T (t) αa 2 T (t) (4)
More information43433 8 3 . Stochastic exponentials...................................... 3. Girsanov s theorem......................................... 4 On the martingale property of stochastic exponentials 5. Gronwall
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 information201711grade1ouyou.pdf
2017 11 26 1 2 52 3 12 13 22 23 32 33 42 3 5 3 4 90 5 6 A 1 2 Web Web 3 4 1 2... 5 6 7 7 44 8 9 1 2 3 1 p p >2 2 A 1 2 0.6 0.4 0.52... (a) 0.6 0.4...... B 1 2 0.8-0.2 0.52..... (b) 0.6 0.52.... 1 A B 2
More informationQCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1
QCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1 (vierbein) QCD QCD 1 1: QCD QCD Γ ρ µν A µ R σ µνρ F µν g µν A µ Lagrangian gr TrFµν F µν No. Yes. Yes. No. No! Yes! [1] Nash & Sen [2] Riemann
More informationTricorn
Triorn 016 3 1 Mandelbrot Triorn Mandelbrot Robert L DevaneyAn introdution to haoti dynamial Systems Addison-Wesley, 1989 Triorn 1 W.D.Crowe, R.Hasson, P.J.Rippon, P.E.D.Strain- Clark, On the struture
More informationG H J(g, τ G g G J(g, τ τ J(g 1 g, τ = J(g 1, g τj(g, τ J J(1, τ = 1 k g = ( a b c d J(g, τ = (cτ + dk G = SL (R SL (R G G α, β C α = α iθ (θ R
1 1.1 SL (R 1.1.1 SL (R H SL (R SL (R H H H = {z = x + iy C; x, y R, y > 0}, SL (R = {g M (R; dt(g = 1}, gτ = aτ + b a b g = SL (R cτ + d c d 1.1. Γ H H SL (R f(τ f(gτ G SL (R G H J(g, τ τ g G Hol f(τ
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 information第1章 微分方程式と近似解法
April 12, 2018 1 / 52 1.1 ( ) 2 / 52 1.2 1.1 1.1: 3 / 52 1.3 Poisson Poisson Poisson 1 d {2, 3} 4 / 52 1 1.3.1 1 u,b b(t,x) u(t,x) x=0 1.1: 1 a x=l 1.1 1 (0, t T ) (0, l) 1 a b : (0, t T ) (0, l) R, u
More informationIII
III 1 1 2 1 2 3 1 3 4 1 3 1 4 1 3 2 4 1 3 3 6 1 4 6 1 4 1 6 1 4 2 8 1 4 3 9 1 5 10 1 5 1 10 1 5 2 12 1 5 3 12 1 5 4 13 1 6 15 2 1 18 2 1 1 18 2 1 2 19 2 2 20 2 3 22 2 3 1 22 2 3 2 24 2 4 25 2 4 1 25 2
More informationiii iv v vi vii viii ix 1 1-1 1-2 1-3 2 2-1 3 3-1 3-2 3-3 3-4 4 4-1 4-2 5 5-1 5-2 5-3 5-4 5-5 5-6 5-7 6 6-1 6-2 6-3 6-4 6-5 6 6-1 6-2 6-3 6-4 6-5 7 7-1 7-2 7-3 7-4 7-5 7-6 7-7 7-8 7-9 7-10 7-11 8 8-1
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 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 information平成 29 年度 ( 第 39 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 29 ~8 年月 73 月日開催 31 日 Riemann Riemann ( ). π(x) := #{p : p x} x log x (x ) Hadamard de
Riemann Riemann 07 7 3 8 4 ). π) : #{p : p } log ) Hadamard de la Vallée Poussin 896 )., f) g) ) lim f) g).. π) Chebychev. 4 3 Riemann. 6 4 Chebychev Riemann. 9 5 Riemann Res). A :. 5 B : Poisson Riemann-Lebesgue
More information