17 Θ Hodge Θ Hodge Kummer Hodge Hodge

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17 Θ 50 18 55 19 57 20 Hodge 59 21 63 22 67 23 Θ Hodge 69 24 Kummer 72 25 Hodge 77 26 Hodge 78 27 80 2

0,, Teichmüller., Teichmüller, Diophantus Teichmüller, Teichmüller Hodge,., 2. (a),, Teichmüller, ( ) Teichmüller,,,,,. (b) Teichmüller. ( ),,,.,,,.,,,.,,,,,,.,,,.,., : 1 3: Teichmüller,. 4 12: Diophantus,,,., Teichmüller. 13 20:,, / Hodge. 21 25:,, / Hodge. 26: Hodge.,,.,,,,, 3

.,, 1.,.,,. Teichmüller,,,. Teichmüller., 2013 100 Teichmüller,,,,. 1 3 2015 3,, 1 8 2015 6 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

,,,,,,, /., (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

. 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) 1. 2 3,. 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 1.9.8 3 Remark.,, [1], 1,,., O k O k k Λ(k) G k,, [1], 2,,. (, [1], 2,, O k O k 6

., 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

( {±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

,,, ( ) 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

, 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

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

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 = 49 7 49., 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

deg L = deg L N. 49 Q 7 Q Q Q,, 7 = 49. 49 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 ( ). 49 7 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

,,., S S.,., Teichmüller.,, Teichmüller, ( )., 5 6,, S S,.,, (, ) ( ) / / (coric) / /.,,.,,, S S, S S., qe N q E,, qe N q E ( E/F ). 4 49 Q 7 Q Q Q,, 49 Q 7 Q Q Q., qe N q E, S S. 49 7 Q Q, 1 49 7, Q Q, 1 49 7, 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

.,. 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

, 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

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

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

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 1.9.2.),. (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

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

GL 2 (R) : C C GL 2 (R) R 2 / GL 2 (R) R 2 = R 2. 8 4, 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

, 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

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

, 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

.,,., ( ), : 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

,,., (, 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

.,,,, 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

,. 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

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 3.12.2). 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

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

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

.,,.,, 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

., 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

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

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 1.12.4, 3.6.2), Kummer,,, ((Ind 1 ), (Ind 2 ), (Ind 3 ) )., / (, 8 Fmod ). κ, / (, 8 Fmod ), (, 8 F MOD ).,, ( ),,.,,, deg L 35

. (, [12], Corollary 3.12,.) (iii).,, q q E ( L ) ( ) q (q-pilot object cf. [12], Definition 3.8, (i)). ([12], Remark 3.12.1, (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, 2. 13 11, Teichmüller, qe N /,. Teichmüller,,. 13, 14 15 36

,. ( 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

,, 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

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

.,,, : 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

., 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] 1 2 5.) 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

. 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

,, 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 14,. 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

(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

,.,,. (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

. 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

( 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. 16 15 3,,. (,, [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

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

(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

. 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, 13 16 Teichmüller., Teichmüller Θ 17. 4. 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,, ( )., 16. 16,, 13 Θ, Θ l ( ).,, 13 (m) 1 (q v ) q v = q 1/2l v ( ).,, (, 8, q N E, ) 50

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

. (, 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

(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

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

., 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. 18 17 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

. 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

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. 19 13 16 Archimedes. 19 20,, Teichmüller Θ ±ell Hodge, D-Θ ±ell Hodge. 14,. Π tp X,, D. D 57

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

. 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

/. 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

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

., 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},. 19 20, : 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

. 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 13 20,. 21 25,. 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,. 19 20,,.,, 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

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

.,, LabCusp, 19 20 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

/., 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

,, 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 21, 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

(,, 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

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

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

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

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

,, 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

. 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

, : 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

(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

.,, /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

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

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 6.12.2,.),, 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

( 20 ϕ Θ± ± ψ± Θ± ) F D, D, F., F, Θ ±ell Hodge ΘNF Hodge. ([10], Remark 6.12.2,.), 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 ( ). ( 4 8 12.) 80