q n/2 X H n (X Fq,et, Q l) Frobenius q 1/2 (Deligne, [D5]) X (I) (II) X κ open (proper )smooth X κ proper strictly semi-stable weight filtration cohom
|
|
|
- たかとし あると
- 5 years ago
- Views:
Transcription
1 Weight filtration on log crystalline cohomology ( ) 1 κ κ X cohomology X H n (X) Grothendieck, Deligne weight yoga ([G], [D1], [D4]) 1.1. H n (X) filtrationp k H n (X) (k Z) (1) gr P k Hn (X) := P k H n (X)/P k 1 H n (X) (k Z) κ proper, smooth k cohomology (2) filtration cohomology (pull back, push forward, base change, Künneth formula, Poincaré duality ) compatible (3) P k H n (X) smooth. ( proper, smooth Y k cohomology H k+2i (Y )(i) (i Z) (i) twist cohomology ) filtration P k H n (X) (k Z) H n (X) weight filtration Weight filtration (1) (proper, smooth )X cohomology proper, smooth cohomology proper, smooth cohomology proper, smooth cohomology 1.2. (1) X C proper, smooth Betti cohomology H n (X an, Q) pure Hodge X Betti cohomology H n (X an, Q) Hodge (Deligne, [D2],[D3]) ( X an X ) (2) X F q proper, smooth, l q X Fq l cohomology H n (X Fq,et, Q l) Frobenius 1
2 q n/2 X H n (X Fq,et, Q l) Frobenius q 1/2 (Deligne, [D5]) X (I) (II) X κ open (proper )smooth X κ proper strictly semi-stable weight filtration cohomology ( (II) log ) 1 Betti cohomology de Rham cohomology. κ = C X Betti cohomologyh n (X an, Q) weight filtration de Rham cohomology HdR n (X/C) := Hn (X, Ω X/C ) weight filtration (I) Deligne([D2]) (II) Steenbrink([St1]) Betti cohomology de Rham cohomology (de Rham ) H n (X an, Q) Q C = HdR n (X/k) filtration cohomology Hodge (II) Steenbrink-Zucker([St-Zu]), Steenbrink([St2]), F.Kato([Kf]), K.Kato-Nakayama([Kk-Ny]), Matsubara([Ma]), K.Kato-Matsubara- Nakayama([Kk-Ma-Ny]), Fujisawa-Nakayama ([F-Ny]) 2 l etale cohomology κ p (p l) κ X κ (X κ base change) l cohomology H n (X κ,et, Q l ) weight filtration (I) Deligne([D5]) (II) Rapoport-Zink([R-Zi]) (II) Nakayama ([Ny]) 3 log de Rham-Witt cohomology κ p > 0 X log de Rham-Witt cohomology weight filtration (I), (II) Mokrane([Mo1], [Mo2]), κ p > 0 X log crystalline cohomology weight filtration remark X log crystalline cohomology log de Rham-Witt cohomology (Illusie[I1], Hyodo-Kato[H-Kk]), 3 (Mokrane ) log crystalline cohomology weight filtration 1 2
3 Betti cohomology, de Rham cohomology ( )weight filtration log crystalline cohomology weight filtration Illusie, Hyodo- Kato log de Rham-Witt cohomology (Mokrane )weight filtration compatible log crystalline cohomology weight filtration (I) weight filtration smooth (Mokrane ), 1.1 (3) ( weight filtration ) Mokrane 2 (I) X open, smooth Betti de Rham cohomology weight filtration log crystalline cohomology weight filtration Betti de Rham cohomology log crystalline cohomology ( )Betti cohomology 2 3 (II) X proper strictly semi-stable l etale cohomology Rapoport-Zink 2 Open, smooth cohomology 2.1 Betti, de Rham cohomology weight filtration X C open, smooth j : X X compact D := X \ X X simple normal crossing divisor( SNCD ) ( j ) D = m i=1 D i D k 0 D (k) := 1 i 1 <i 2 < <i k m D i 1 D i2 D ik a (k) : D (k) X D ij X (k = 0 D (k) = X, a (0) = id X.) X an, X an, j an U, X, j X Betti cohomology H n (X an, Q) = H n (X an, Rj an Q) weight filtration 2.1 (purity). R k j an Q = a (k) an Q( k) ( exponential sequence cup ) τ k Rj an Q (k Z) 3
4 Rj an Q canonical filtration ( { H l H l (Rj an Q), (if l k), (τ k Rj an Q) = 0, if l > k). filtration) gr τ k Rj an Q = a (k) an Q( k) filtration E k,n+k 1 = H n k (D (k) an, Q( k)) = H n (X an, Q) weight H n (X an, Q) filtration weight filtration weight filtration filter filter object (Rj an Q, τ k ) filter H n (X an, ) H n (X an, (Rj an Q, τ k )) weight filtration Betti cohomology H n (X an, Q) filtration X de Rham cohomology HdR n (X/C) := Hn (X Zar, Ω X/C ) = Hn (X, Rj Ω X/C ) = H n (X, Ω X/C (log D)) weight filtration Ω (log D) filtration P k Ω (log D) X/C X/C P k Ω (log D) := X/C 0, (if k < 0), (log D) Ω k (log D)), (if 0 k ), Im(Ω k X/C Ω (log D), X/C X/C Ω X/C (if < k) gr P k Ω (log D) = a (k) X/C Ω k D (k) /C ( k) ( Poincaré residue ) filtration P k Ω X/C (log D) E k,n+k 1 = H n k (D (k), Ω ) = D (k) /C Hn (X, Ω (log D)), X/C E k,n+k 1 = H n k (D(k) /C) = HdR(X/C) n dr weight H dr n (X/C) filtration weight filtration filter filter object (Ω (log D), P X/C k) filter H n (X, ) H n (X, (Ω X/C (log D), P k )) weight filtration de Rham cohomology HdR n (X/C) de Rham H n (X an, Q) Q C = HdR n (X/C) 4
5 2.3. H n (X an, Q) Q C = HdR n (X/C) weight filtration Proof. X an X u filter (Ω X/C (log D), P k) Ru (Ω X an/c (log D an), P k ) Ru (Ω X an /C (log D an), τ k ) Ru (Rj an Ω X an/c, τ k ) Ru ((Rj an Q) Q C, τ k ) ((Ω (log D X an /C an), P k ) log de Rham Ω (log D X an /C an) P k filtration.) filter GAGA filter Ω (log D X an /C an) Rj an Ω X an /C filter Poincaré filter log de Rham Poincaré residue filter filter H n (X, ) 2.4., log de Rham Ω (log D X an/c an) = u (Ω X/C (log D)) 2.2 log crystalline cohomology weight filtration p S quasi-compact p scheme p-adic Noetherian formal scheme I O S p quasi-coherent PD-ideal S 0 := Spec S O S /I X S 0 smooth scheme X X S 0 compact D := X \ X S 0 SNCD S 0 = Spec κ X X (X, D) (Fontaine-Illusie-Kato )S fine log scheme crystalline cohomology proper scheme ( ) X S crystalline cohomology cohomology cohomology log scheme (X, D) S crystalline cohomology (X, D) S crystalline cohomology log scheme (X, D) 2.1 X k Z, k 0 D (k), a (k) : D (k) X 2.1 ((X, D)/S) crys, (X/S) crys, (D (k) /S) crys (X, D), X, D (k) S crystalline site 2.1 X an, X an, D an (k) 5
6 O (X,D/S), O X/S, O D (k) /S ((X, D)/S) crys, (X/S) crys, (D (k) / S) crys X an, X an, D an (k) C j : (X, D) X log scheme crystalline site ((X, D)/S) crys (X/S) crys j crys 2.1 j, j an a (k) crystalline site (D (k) /S) crys (X/S) crys a crys (k). (X, D) S 0 site f (X,D)/S : ((X, D)/S) crys S Zar R n f (X,D)/S O (X,D)/S (X, D) S crystalline cohomology H n S(((X, D)/S) crys, O (X,D)/S ) X, D (k) S crystalline cohomology H n S((X/S) crys, O X/S ), H n S((D (k) /S) crys, O D /S) (k) crystalline cohomology H n S(((X, D)/S) crys, O (X,D)/S ) = H n S((X/S) crys, Rj crys O (X,D)/S ) weight filtration 2.5 (purity). R k j crys O (X,D)/S = a crys O (k) D /S( k) (k) log de Rham Poincaré residue Rj crys O (X,D)/S canonical filtration τ k Rj crys O (X,D)/S E k,n+k 1 = H n k S ((D (k) /S) crys, O D (k) /S )( k) = Hn S(((X, D)/S) crys, O (X,D)/S ) (2.1) weight H n S(((X, D)/S) crys, O (X,D)/S ) filtration weight filtration filter filter object (Rj crys O (X,D)/S, τ k ) filter H n S((X/S) crys, ) H n S((X/S) crys, (Rj crys O (X,D)/S, τ k )) weight filtration crystalline cohomologyh n S(((X, D)/S) crys, O (X,D)/S ) crystalline cohomology weight filtration cohomology weight filtration 1 de Rham cohomology S 0 S, (X, D) (X, D) S (X, D) X := X \ D X S de Rham cohomology H n dr(x /S) := H n S(X, Ω X/S ) = H n S(X, Ω (log D)) ( X /S Hn S(?, )? S ) 2.1 filtration P k Ω (log D) weight X /S filtration Berthelot([B]), K.Kato([Kk]), H n dr(x /S) = H n S(((X, D)/S) crys, O (X,D)/S ) [Nj-Sh] 6
7 2.6. weight filtration 2.3 log de Rham filter u : (X/S) crys X Zar = X Zar Zariski site ( 2.1 u ) (X/S) crys X X crystalline site ϕ : (X/S) crys X X Zar, ψ : (X/S) crys X (X/S) crys site O X - M L(M) := ψ ϕ M ([Nj-Sh]) 2.7 (Rj O (X,D)/S Poincaré ). (1) L(Ω (log D)) O X /S X/S - ( crystal.) (2) Rj crys O (X,D)/S L(Ω (log D)) X /S L(Ω (log D)) log de Rham ( site X /S ϕ = u ψ u site u L = ψ ϕ 2.4 ) filter (Ω X /S (log D), P k) Ru (L(Ω X /S (log D)), P k) Ru (L(Ω X /S (log D)), τ k) Ru (Rj crys Ω (X,D)/S, τ k). filter L(Ω (log D)) u X /S - (GAGA ) filter L(Ω (log D)) X /S Poincaré residue. (.) filter filter H n S(X, ) 2 log de Rham-Witt cohomology (Mokrane filtration) κ p S 0 = Spec κ, S = Spf W m (κ) X, (X, D) crystalline cohomology H n S Hn log de Rham-Witt complex X W m Ω (log D) X cohomology H n (X, W m Ω (log D)) ( log de Rham-Witt cohomology X ) crystalline cohomologyh n (((X, D)/S) crys, O (X,D)/S ) (Illusie[I1], Hyodo-Kato[H-Kk]. Nakkajima [Nj1] ) Mokrane([Mo1],[Mo2]) W m Ω (log D) filtration P X kw m Ω (log D) X (log de Rham ) H n (X, W n Ω (log D)) weight X filtration [Nj-Sh] 7
8 2.8. Illusie, Hyodo-Kato H n (X, W m Ω (log D)) = H n (((X, D)/S) X crys, O (X,D)/S ) weight filtration S proper, smooth scheme X X S SNCD D (X, D) Frobenius (X, D) (X, D) (X, D) (X, D) PD envelope Y Ru Rj crys O (X,D)/S O Y OX Ω (log D) W X /S m Ω (log D) X (Illusie, Hyodo-Kato ) filtration filter global weight V (0, p) S (Ogus[O1] )p formal V -scheme, S 0 := Spec S O S /po S X, (X, D) 2.9. crystalline cohomology H n S(((X, D)/S) crys, O (X,D)/S ) weight (2.1) modulo torsion E 2 Betti, de Rham, l etale cohomology V = W (κ), S = Spf W (κ) κ (log de Rham-Witt cohomology ) Mokrane κ Nakkajima ([Nj1]). Step 1: S = Spf W (κ)(κ ) Mokrane crystalline cohomology Weil (Katz-Messing[Kz-Me], Chiarellotto-Le Stum[C-L1]) Step 2: S = Spf A Spf W (κ)(κ ) formally smooth S Frobenius A W (A/m) weight pa m A m: ideal specialize Step 1 Step 3: S = Spf W (κ)(κ ) model Step 2 Step 4: S = Spf A A Q := A Z Q Artin m A Q ideal B := Im(A A Q /m), C := B C V κ C 8
9 Spf B Spf A Ogus formal V -scheme A Spf A Spf A Spf A Spf B A A A Q A Q implication Step 4 Step 3 = (S = Spf W (κ ) ) = ( ) (S = Spf C ) = (S = Spf B ) = (S = Spf A ) = (S = Spf A ). ( ) log crystalline cohomology (modulo torsion) (Berthelot-Ogus[B-O2]) weight filtration Berthelot-Ogus Step 5: S = Spf A A Q ideal m m A (m) := Im(A A Q /m m ) Step 4 S = Spf A (m) (A Q ) m lim ma Q /m m = lim ma (m)q open, smooth crystalline cohmology weight filtration (1) S 0, S 2.9 crystalline cohomology H n S(((X, D)/S) crys, O (X,D)/S ) S S/V convergent isocrystal ([O1]) (2) Weight filtration pull-back strictly compatible (Compatibility strictness crystalline cohomology Weil.) (3) Compact support crystalline cohmology weight filtration (4) crystalline cohomology base change theorem, Künneth formula, Poincaré duality ([B],[B-O1],[Kk],[O1],[Tj] ) weight filtration compatible (5) S = Spf W (κ) ( κ ) crystalline cohomology H n (((X, D)/ S) crys, O (X,D)/S ) X rigid cohomology Hrig(X) n ([Sh2]). Hrig(X) n weight filtration ( Hrig(X) n 9
10 weight filtration Chiarellotto-Le Stum([C-L2]) ) (6) κ (smooth ) scheme X Nakkajima([Nj2]) Hrig(X) n weight filtration (5) Tsuzuki([Tz]) rigid cohomology cohomological descent 3 Strictly semi-stable cohomology p V (0, p) κ K V K K S = Spec V, Ŝ := Spf V, S 0 := Spec κ X S proper scheme strictly semi-stable reduction X S proper, flat scheme X S generic fiber X K smooth, special fiber X S S 0 =: Y SNCD Y = m i=1 D i k 0 Y (k) := 1 i 0 <i 1 < <i k m D i 0 D i1 D ik a (k) : Y (k) Y D ij Y ( (k) 2 ) SNCD Y X X log structure M, S 0 S S log structure N M Y N Ŝ, S 0, Y M, N f : Y S 0 ( X S special fiber) log smooth (Y, M) (S 0, N) f log 3.1 l etale cohomology weight filtration V ur V Y κ Y i X V ur i X j X K j X K Y κ Y κ base change, X? X? base change, i, j i, j i, j l p l nearby cycle RψQ l RψQ l := i Rj Q l l etale cohomology H n (X K,et, Q l ) = H n (Y κ,et, RψQ l ) weight filtration ([R-Zi], [I2]) 10
11 3.1. (1) (purity) i R k+1 j Q l (k + 1) = a (k) Q l (2) 1 a (0) Q l = i R 1 j Q l (1) cup i R k j Q l i R k+1 j Q l (1)[1] θ k 0 R k ψq l ( 1) α k i R k+1 β k+1 j Q l R k+1 ψq l 0 θ k = α k β k i Rj Q l L 1 a (0) Q l = i Rj Q l (1) cup i Rj Q l i Rj Q l (1)[1] θ : L L(1)[1] RψQ l A := [(τ 1 L(1))[1] (τ 2 L(2))[2] (τ i+1 L(i + 1))[i + 1] ] ((τ 1 L(1))[1]) 0 (0, 0) ) associate Q τ i Q canonical filtration ( { H l H l (Rj an Q), (if l i), (τ i Q) = 0, if l < i). filtration) A filtration P k A (k Z) P k A := [(τ [1,k+1] L(1))[1] (τ [2,k+3] L(2))[2] (τ [i+1,k+2i+1] L(i + 1))[i + 1] ] Q τ [i,j] Q := τ j τ i Q gr P k A gr P k A = i R k+2i+1 j Q l (i + 1)[ 2i k] i 0,i k = a (2i+k) Q l ( i k)[ 2i k] i 0,i k filtration P k A (k Z) E k,n+k = H 2n i k (Y (2i+k) κ, Q l ( i)) = H n (X K, Q l ) i 0,i k weight H n (X K, Q l ) filtration weight filtration 3.2. H n (X K, Q l ) l log etale cohomology H n ((Y, M) κ,tl, Q l ) ((Y, M) κ,tl [Ny] ) H n ((Y, M) κ,tl, Q l ) weight filtration. H n ((Y, M) κ,tl, Q l ) weight filtration f log : (Y, M) (S, N) (Nakayama [Ny]) 3.3. RψQ l Y κ,et perverse (Illusie [I2]), filtration P k ( ) (T.Saito [Sa]). 11
12 3.2 Log crystalline cohomology weight filtration ((Y, M)/(Ŝ, N)) crys (Y, M) (Ŝ, N) log crystalline site O (Y,M)/(Ŝ,N) (Y, M) (Ŝ, N) log crystalline cohomology Hn (((Y, M)/(Ŝ, N)) crys, O (Y,M)/( Ŝ,N) ) Q ( Q Z Q ) H n ((Y, M) κ,tl, Q l ) H n (X K, Q l ) H n (((Y, M)/(Ŝ, N)) crys, O (Y,M)/( Ŝ,N) ) Q weight filtration ( ) Weight filtration 3.1 i Rj Q l nearby cycle RψQ l (Y, L) := (Y, M) Y (Y, N) j : (Y, L) (Y.N) j : (Y, M) (Y, N) f log (f j = f log ) j, j 2.1 j, j. j crys : ((Y, L)/(Ŝ, N)) crys ((Y, N)/(Ŝ, N)) crys = (Y/Ŝ) crys, j crys : ((Y, M)/(Ŝ, N)) crys ((Y, N)/(Ŝ, N)) crys = (Y/Ŝ) crys crystalline site Rj crys O (Y,L)/( Ŝ,N), Rj crys O (Y,M)/( Ŝ,N) 2.1 Rj Q l, RψQ l 3.4. (R 1 j O (Y,L)/( Ŝ,N) (1)) Q (a (0) crys O ((Y (0),N)/(Ŝ,N)) Q ( Q (Y/Ŝ) crys Hom Z Q ) U Y f smooth locus affine open sub log scheme U T S 0 Ŝ PD T affine log formal V -scheme T n := Spec T O T /p n O T U T n ((Y, M)/(Ŝ, N)) crys object R := Γ(T, O T ) (R 1 j O (Y,L)/( Ŝ,N) (1)), (a(0) crys O ((Y (0),N)/(Ŝ,N)) U T n P n, Q n (lim np n ) Q = (R t /tr t ) Q, (lim nq n ) Q = R Q ( R t R PD p ) 3.4 log crystalline site 3.1 weight filtration 3.1 log crystalline cohomology singularity scheme Y log crystalline site log convergent site ((Y, M)/(Ŝ, N)) conv ([O1], [O2], [Sh1], [Sh2]) Log convergent site log crystalline site p Log convergent site ([Sh2]) 12
13 3.5. K (Y,M)/( Ŝ,N) log convergent site ((Y, M)/(Ŝ, N)) conv Q H n (((Y, M)/(Ŝ, N)) crys, O (Y,M)/( Ŝ,N) ) Q = H n (((Y, M)/(Ŝ, N)) conv, K (Y,M)/( Ŝ,N) ) weight filtration cohomology log convergent cohomology log convergent site Rj conv K (Y,L)/( Ŝ,N), Rj conv K (Y,M)/( Ŝ,N) 3.1 Rj Q l, RψQ l ( j conv, j conv j crys, j crys log convergent site ) 3.6. (1) (purity) R k+1 j conv K (Y,L)/( Ŝ,N) (k + 1) = a (k) conv K (Y (k),n)/(ŝ,n) (2) 1 a (0) conv K (Y (0),N)/(Ŝ,N) = R 1 j conv K (Y,L)/( Ŝ,N) cup R k j conv K (Y,L)/( Ŝ,N) i R k+1 j conv K (Y,L)/( Ŝ,N) [1] θ k 0 R k j conv K (Y,M)/( Ŝ,N) ( 1) α k R k+1 β k+1 j conv K (Y,L)/( Ŝ,N) R k+1 j conv K (Y,M)/( Ŝ,N) 0 θ k = α k β k Crystalline site 3.4 (lim np n ) Q R Q t /tr Q t = R Q ( R Q t R Q [[t]] ) 3.1 log convergent cohomology H n (((Y, M)/(Ŝ, N)) conv, K (Y,M)/( Ŝ,N) ) weight filtration (1) Mokrane 3.2 (Y, M) (Spf W (κ), N) (N Spf W (κ) ) log de Rham-Witt cohomology ( Illusie, Hyodo-Kato ([I1],[H-Kk]) log crystalline cohomology ) weight filtration log de Rham-Witt complex Hyodo-Steenbrink complex (3.1 A ) filtration (2) Nakkajima log crystalline cohomology weight filtration Mokrane Hyodo-Steenbrink complex crystalline site ( ) 13
14 (1) (3) 3.3 Rj conv K (Y,M)/( Ŝ,N) perverse (4) 3.2 p nearby cycle Gros [B] [B-O1] Berthelot, P. Cohomologie cristalline des schémas de caractéristique p > 0. Lecture Notes in Math. 407, Springer-Verlag, Berlin-New York, (1974). Berthelot, P., Ogus, A. Notes on crystalline cohomology. Princeton Univ. Press, (1978). [B-O2] Berthelot, P., Ogus, A. F -isocrystals and de Rham cohomology. I. Invent. Math. 72, (1983), [C-L1] Chiarellotto, B., Le Stum, B. Sur la pureté de la cohomologie cristalline. C. R. Acad. Sci. Paris, Série I, 326, (1998), [C-L2] Chiarellotto, B., Le Stum, B. A comparison theorem for weights. J. reine angew. Math. 546, (2002), [D1] Deligne, P. Théorie de Hodge I. Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, pp Gauthier-Villars, Paris, (1971). [D2] Deligne, P. Théorie de Hodge, II. IHES Publ. Math. 40, (1971), [D3] Deligne, P. Théorie de Hodge, III. IHES Publ. Math. 44, (1974),
15 [D4] Deligne, P. Poids dans la cohomologie des variétés algébriques. Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, pp [D5] Deligne, P. La conjecture de Weil, II. IHES Publ. Math. 52, (1980), [F-Ny] [G] [H-Kk] [H] Fujisawa, T., Nakayama, C., Mixed Hodge structures on log deformations. Rend. Sem. Mat. Univ. Padova 110 (2003), Grothendieck, A. Récoltes et s les: Réflexions et témoinage sur un passé de mathématicien I, II, IV. Gendai-Sugaku-sha, Japanese translation by Y. Tsuji, (1989), (1990), unpublished. Hyodo, O., Kato, K. Semi-stable reduction and crystalline cohomology with logarithmic poles. Périodes p-adiques (Bures-sur-Yvette, 1988). Astérisque 223, (1994), Hyodo, O. On the de Rham-Witt complex attached to a semi-stable family. Comp. Math. 78, (1991), [I1] Illusie, L. Complexe de de Rham-Witt et cohomologie cristalline. Ann. Scient. Éc. Norm. Sup. 4e série 12, (1979), [I2] [Kf] Illusie, L., Autour du theoreme de monodromie locale. In: Periodes p-adiques (Bures-sur-Yvette, 1988). Asterisque No. 223 (1994), Kato, F., The relative log Poincare lemma and relative log de Rham theory. Duke Math. J. 93 (1998), [Kk] Kato, K. Logarithmic structures of Fontaine-Illusie. In: Algebraic analysis, geometry, and number theory, Johns Hopkins Univ. Press, (1989), [Kk-Ny] Kato, K., Nakayama, C. Log Betti cohomology, log étale cohomology, and log de Rham cohomology of log schemes over C. Kodai Math. J. 22, (1999), [Kk-Ma-Ny] Kato, K., Matsubara, T., Nakayama, C., Log C -functions and degenerations of Hodge structures. In: Algebraic geometry 2000, Azumino (Hotaka), , Adv. Stud. Pure Math., 36, Math. Soc. Japan, Tokyo,
16 [Kz-Me] [Ma] [Mo1] Katz, N., Messing, W. Some consequences of the Riemann hypothesis for varieties over finite fields. Invent. Math. 23, (1974), Matsubara, T., On log Hodge structures of higher direct images. Kodai Math. J. 21 (1998), no. 2, Mokrane, A. La suite spectrale des poids en cohomologie de Hyodo- Kato. Duke Math. J. 72, (1993), [Mo2] Mokrane, A. Cohomologie cristalline des variétés ouvertes. Rev. Maghrebine Math. 2, (1993), [Ny] Nakayama, C., Degeneration of l-adic weight spectral sequences. Amer. J. Math. 122 (2000), [Nj1] Nakkajima, Y. p-adic weight spectral sequences of log varieties. Preprint. [Nj2] [Nj-Sh] Nakkajima, Y. Weight filtration and slope filtration on the rigid cohomology of a variety in characteristic p > 0. Preprint. Nakkajima, Y., Shiho, A. Weight filtrations on log crystalline cohomologies of families of open smooth varieties of characteristic p > 0. Preprint. [O1] Ogus, A. F -isocrystals and de Rham cohomology. II. Convergent isocrystals. Duke Math. J. 51, (1984), [O2] Ogus, A. F -crystals on schemes with constant log structure. Compositio Math. 97, (1995), [R-Zi] Rapoport, M., Zink, Th., Uber die lokale Zetafunktion von Shimuravarietaten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik. Invent. Math. 68 (1982), [Sa] Saito, T., Weight spectral sequences and independence of l. J. Inst. Math. Jussieu 2 (2003), [Sc] Schneiders, J.-P., Quasi-abelian categories and sheaves. Mém. Soc. Math. Fr. (N.S.) 76, (1999). [Sh1] Shiho, A. Crystalline fundamental groups I Isocrystals on log crystalline site and log convergent site. J. Math. Sci. Univ. Tokyo 7, (2000),
17 [Sh2] Shiho, A. Crystalline fundamental groups II Log convergent cohomology and rigid cohomology. J. Math. Sci. Univ. Tokyo 9, (2002), [St1] Steenbrink, J. H. M., Limits of Hodge structures. Invent. Math. 31 (1975/76), [St2] [St-Zu] [Tj] [Tz] Steenbrink, J. H. M., Logarithmic embeddings of varieties with normal crossings and mixed Hodge structures. Math. Ann. 301 (1995), Steenbrink, J. H. M., Zucker, S., Variation of mixed Hodge structure. I. Invent. Math. 80 (1985), Tsuji, T. Poincaré duality for logarithmic crystalline cohomology. Compositio Math. 118, (1999), Tsuzuki, N. Cohomological descent of rigid cohomology for proper coverings. Invent. Math. 151 (2003),
日本数学会・２０１１年度年会（早稲田大学）・総合講演
日本数学会 2011 年度年会 ( 早稲田大学 ) 総合講演 2011 年度日本数学会春季賞受賞記念講演 MSJMEETING-2011-0 ( ) 1. p>0 p C ( ) p p 0 smooth l (l p ) p p André, Christol, Mebkhout, Kedlaya K 0 O K K k O K k p>0 K K : K R 0 p = p 1 Γ := K k
[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
![[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](/thumbs/49/25561854.jpg "[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,
K 2 X = 4 MWG(f), X P 2 F, υ 0 : X P 2 2,, {f λ : X λ P 1 } λ Λ NS(X λ ), (υ 0 ) λ : X λ P 2 ( 1) X 6, f λ K X + F, f ( 1), n, n 1 (cf [10]) X, f : X
![K 2 X = 4 MWG(f), X P 2 F, υ 0 : X P 2 2,, {f λ : X λ P 1 } λ Λ NS(X λ ), (υ 0 ) λ : X λ P 2 ( 1) X 6, f λ K X + F, f ( 1), n, n 1 (cf [10]) X, f : X](/thumbs/91/106734137.jpg "K 2 X = 4 MWG(f), X P 2 F, υ 0 : X P 2 2,, {f λ : X λ P 1 } λ Λ NS(X λ ), (υ 0 ) λ : X λ P 2 ( 1) X 6, f λ K X + F, f ( 1), n, n 1 (cf [10]) X, f : X")
2 E 8 1, E 8, [6], II II, E 8, 2, E 8,,, 2 [14],, X/C, f : X P 1 2 3, f, (O), f X NS(X), (O) T ( 1), NS(X), T [15] : MWG(f) NS(X)/T, MWL(f) 0 (T ) NS(X), MWL(f) MWL(f) 0, : {f λ : X λ P 1 } λ Λ NS(X λ
( ) 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- ( 4.1).,, Z p -,., Weil..,,. Weil., F, F projectiv](/thumbs/102/156263797.jpg "( ) 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
k + (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)
.5.1. G K O E, O E T, G K Aut OE (T ) (T, ρ). ρ, (T, ρ) T. Aut OE (T ), En OE (F ) p..5.. G K E, E V, G K GL E (V ) (V, ρ). ρ, (V, ρ) V. GL E (V ), En
p 1. 1.1., 01 8 3, 57,,.. 1.., Gal(Q p /Q p ), 1. Wach,,. 1.3. Part I,,. Part II, Part III. 1.4.., Paé. Part 1. p.. p p.1. p Q p p (Q p p )... E Q p, E p Z p E, O E. O E E. E Q p, O E. v p : E Q Q E, v
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, 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, P](/thumbs/94/118231293.jpg "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, 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
Noether [M2] l ([Sa]) ) ) ) ) ) ( 1, 2) ) ( 3) K F = F q O K K l q K Spa(K, O K ) adc adc [Hu1], [Hu2], [Hu3] K A Spa(A, A ) Sp A A B X A X B = X Spec
![Noether [M2] l ([Sa]) ) ) ) ) ) ( 1, 2) ) ( 3) K F = F q O K K l q K Spa(K, O K ) adc adc [Hu1], [Hu2], [Hu3] K A Spa(A, A ) Sp A A B X A X B = X Spec](/thumbs/92/108053130.jpg "Noether [M2] l ([Sa]) ) ) ) ) ) ( 1, 2) ) ( 3) K F = F q O K K l q K Spa(K, O K ) adc adc [Hu1], [Hu2], [Hu3] K A Spa(A, A ) Sp A A B X A X B = X Spec")
l Wel (Yoch Meda) Graduate School of Mathematcal Scences, The Unversty of Tokyo 0 Galos ([M1], [M2]) Galos Langlands ([Ca]) K F F q l q K, F K, F Fr q Gal(F /F ) F Frobenus q Fr q Fr q Gal(F /F ) φ: Gal(K/K)
1.2 (Kleppe, cf. [6]). C S 3 P 3 3 S 3. χ(p 3, I C (3)) 1 C, C P 3 ( ) 3 S 3( S 3 S 3 ). V 3 del Pezzo (cf. 2.1), S V, del Pezzo 1.1, V 3 del Pe
![1.2 (Kleppe, cf. [6]). C S 3 P 3 3 S 3. χ(p 3, I C (3)) 1 C, C P 3 ( ) 3 S 3( S 3 S 3 ). V 3 del Pezzo (cf. 2.1), S V, del Pezzo 1.1, V 3 del Pe](/thumbs/86/93804893.jpg "1.2 (Kleppe, cf. [6]). C S 3 P 3 3 S 3. χ(p 3, I C (3)) 1 C, C P 3 ( ) 3 S 3( S 3 S 3 ). V 3 del Pezzo (cf. 2.1), S V, del Pezzo 1.1, V 3 del Pe")
3 del Pezzo (Hirokazu Nasu) 1 [10]. 3 V C C, V Hilbert scheme Hilb V [C]. C V C S V S. C S S V, C V. Hilbert schemes Hilb V Hilb S [S] [C] ( χ(s, N S/V ) χ(c, N C/S )), Hilb V [C] (generically non-reduced)
Mazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R M End R M) M R ut R M) M R R G R[G] R G Sets 1 Λ Noether Λ k Λ m Λ k C Λ
![Mazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R M End R M) M R ut R M) M R R G R[G] R G Sets 1 Λ Noether Λ k Λ m Λ k C Λ](/thumbs/78/78261740.jpg "Mazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R M End R M) M R ut R M) M R R G R[G] R G Sets 1 Λ Noether Λ k Λ m Λ k C Λ")
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
2018 : msjmeeting-2018mar-02i002 ( ) 1. 1:. X (= ), X, X., X Z, 1 π1 ab (X) 0 Chow ( CH 0 (X) := Coker div X : κ(x) ) Z x X 1 x X 0 2., x X, x
2018 :2018 21 msjmeeting-2018mar-02i002 () 1. 1:. (= ),,., Z, 1 π1 ab () 0 Chow ( CH 0 () := Coker iv : κ(x) ) Z x 1 x 0 2., x, x Frobenius ϱ : CH 0 () π ab 1 (), ϱ, π ab 1 () 3 (, Artin, Lang [23], Bloch
1
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
1 M = (M, g) m Riemann N = (N, h) n Riemann M N C f : M N f df : T M T N M T M f N T N M f 1 T N T M f 1 T N C X, Y Γ(T M) M C T M f 1 T N M Levi-Civi
1 Surveys in Geometry 1980 2 6, 7 Harmonic Map Plateau Eells-Sampson [5] Siu [19, 20] Kähler 6 Reports on Global Analysis [15] Sacks- Uhlenbeck [18] Siu-Yau [21] Frankel Siu Yau Frankel [13] 1 Surveys
xia2.dvi
Journal of Differential Equations 96 (992), 70-84 Melnikov method and transversal homoclinic points in the restricted three-body problem Zhihong Xia Department of Mathematics, Harvard University Cambridge,
2 (March 13, 2010) N Λ a = i,j=1 x i ( d (a) i,j x j ), Λ h = N i,j=1 x i ( d (h) i,j x j ) B a B h B a = N i,j=1 ν i d (a) i,j, B h = x j N i,j=1 ν i
1. A. M. Turing [18] 60 Turing A. Gierer H. Meinhardt [1] : (GM) ) a t = D a a xx µa + ρ (c a2 h + ρ 0 (0 < x < l, t > 0) h t = D h h xx νh + c ρ a 2 (0 < x < l, t > 0) a x = h x = 0 (x = 0, l) a = a(x,
( ),, ( [Ka93b],[FK06]).,. p Galois L, Langlands p p Galois (, ) p., Breuil, Colmez([Co10]), Q p Galois G Qp 2 p ( ) GL 2 (Q p ) p Banach ( ) (GL 2 (Q
![( ),, ( [Ka93b],[FK06]).,. p Galois L, Langlands p p Galois (, ) p., Breuil, Colmez([Co10]), Q p Galois G Qp 2 p ( ) GL 2 (Q p ) p Banach ( ) (GL 2 (Q](/thumbs/91/106303752.jpg "( ),, ( [Ka93b],[FK06]).,. p Galois L, Langlands p p Galois (, ) p., Breuil, Colmez([Co10]), Q p Galois G Qp 2 p ( ) GL 2 (Q p ) p Banach ( ) (GL 2 (Q")
2017 : msjmeeting-2017sep-00f006 p Langlands ( ) 1. Q, Q p Q Galois G Q p (p Galois ). p Galois ( p Galois ), L Selmer Tate-Shafarevich, Galois. Dirichlet ( Dedekind s = 0 ) Birch-Swinnerton-Dyer ( L s
Donaldson Seiberg-Witten [GNY] f U U C 1 f(z)dz = Res f(a) 2πi C a U U α = f(z)dz dα = 0 U f U U P 1 α 0 a P 1 Res a α = 0. P 1 Donaldson Seib
![Donaldson Seiberg-Witten [GNY] f U U C 1 f(z)dz = Res f(a) 2πi C a U U α = f(z)dz dα = 0 U f U U P 1 α 0 a P 1 Res a α = 0. P 1 Donaldson Seib](/thumbs/78/77523767.jpg "Donaldson Seiberg-Witten [GNY] f U U C 1 f(z)dz = Res f(a) 2πi C a U U α = f(z)dz dα = 0 U f U U P 1 α 0 a P 1 Res a α = 0. P 1 Donaldson Seib")
( ) Donaldson Seiberg-Witten Witten Göttsche [GNY] L. Göttsche, H. Nakajima and K. Yoshioka, Donaldson = Seiberg-Witten from Mochizuki s formula and instanton counting, Publ. of RIMS, to appear Donaldson
k k Q (R )Z k X 1. X Q Cl (X) 2. nef cone Nef (X) nef semi-ample ( ) 3. 2 f i : X X i X i 1 2 movable cone Mov (X) fi (Nef (X i)) 3 movable
1 (Mori dream space) 2000 [HK] toric toric ( 2.1) toric n n + 1 affine 1 torus ( GIT ) [Co] toric affine ( )torus GIT affine torus ( 2.2) affine Cox ( 3) Cox 2 okawa@ms.u-tokyo.ac.jp Supported by the Grant-in-Aid
2.1 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,
' , 24 :,,,,, ( ) Cech Index theorem 22 5 Stability 44 6 compact 49 7 Donaldson 58 8 Symplectic structure 63 9 Wall crossing 66 1
1998 1998 7 20 26, 44. 400,,., (KEK), ( ) ( )..,.,,,. 1998 1 '98 7 23, 24 :,,,,, ( ) 1 3 2 Cech 6 3 13 4 Index theorem 22 5 Stability 44 6 compact 49 7 Donaldson 58 8 Symplectic structure 63 9 Wall crossing
SAMA- SUKU-RU Contents p-adic families of Eisenstein series (modular form) Hecke Eisenstein Eisenstein p T
SAMA- SUKU-RU Contents 1. 1 2. 7.1. p-adic families of Eisenstein series 3 2.1. modular form Hecke 3 2.2. Eisenstein 5 2.3. Eisenstein p 7 3. 7.2. The projection to the ordinary part 9 3.1. The ordinary
( ) (, ) ( )
( ) (, ) ( ) 1 2 2 2 2.1......................... 2 2.2.............................. 3 2.3............................... 4 2.4.............................. 5 2.5.............................. 6 2.6..........................
2018/10/04 IV/ IV 2/12. A, f, g A. (1) D(0 A ) =, D(1 A ) = Spec(A), D(f) D(g) = D(fg). (2) {f l A l Λ} A I D(I) = l Λ D(f l ). (3) I, J A D(I) D(J) =
2018/10/04 IV/ IV 1/12 2018 IV/ IV 10 04 * 1 : ( A 441 ) yanagida[at]math.nagoya-u.ac.jp https://www.math.nagoya-u.ac.jp/~yanagida 1 I: (ring)., A 0 A, 1 A. (ring homomorphism).. 1.1 A (ideal) I, ( ) I
[AI] G. Anderson, Y. Ihara, Pro-l branched cov erings of P1 and higher circular l-units, Part 1 Ann. of Math. 128 (1988), 271-293 ; Part 2, Intern. J. Math. 1 (1990), 119-148. [B] G. V. Belyi, On Galois
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
Design of highly accurate formulas for numerical integration in weighted Hardy spaces with the aid of potential theory 1 Ken ichiro Tanaka 1 Ω R m F I = F (t) dt (1.1) Ω m m 1 m = 1 1 Newton-Cotes Gauss
第5章 偏微分方程式の境界値問題
October 5, 2018 1 / 113 4 ( ) 2 / 113 Poisson 5.1 Poisson ( A.7.1) Poisson Poisson 1 (A.6 ) Γ p p N u D Γ D b 5.1.1: = Γ D Γ N 3 / 113 Poisson 5.1.1 d {2, 3} Lipschitz (A.5 ) Γ D Γ N = \ Γ D Γ p Γ N Γ
D-brane K 1, 2 ( ) 1 K D-brane K K D-brane Witten [1] D-brane K K K K D-brane D-brane K RR BPS D-brane
![D-brane K 1, 2 ( ) 1 K D-brane K K D-brane Witten [1] D-brane K K K K D-brane D-brane K RR BPS D-brane](/thumbs/94/118614691.jpg "D-brane K 1, 2 ( ) 1 K D-brane K K D-brane Witten [1] D-brane K K K K D-brane D-brane K RR BPS D-brane")
D-brane K 1, 2 E-mail: sugimoto@yukawa.kyoto-u.ac.jp (2004 12 16 ) 1 K D-brane K K D-brane Witten [1] D-brane K K K K D-brane D-brane K RR BPS D-brane D-brane RR D-brane K D-brane K D-brane K K [2, 3]
Q p G Qp Q G Q p Ramanujan 12 q- (q) : (q) = q n=1 (1 qn ) 24 S 12 (SL 2 (Z))., p (ordinary) (, q- p a p ( ) p ). p = 11 a p ( ) p. p 11 p a p
.,.,.,..,, 1.. Contents 1. 1 1.1. 2 1.2. 3 1.3. 4 1.4. Eisenstein 5 1.5. 7 2. 9 2.1. e p 9 2.2. p 11 2.3. 15 2.4. 16 2.5. 18 3. 19 3.1. ( ) 19 3.2. 22 4. 23 1. p., Q Q p Q Q p Q C.,. 1. 1 Q p G Qp Q G
0. 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
untitled
18 1 2,000,000 2,000,000 2007 2 2 2008 3 31 (1) 6 JCOSSAR 2007pp.57-642007.6. LCC (1) (2) 2 10mm 1020 14 12 10 8 6 4 40,50,60 2 0 1998 27.5 1995 1960 40 1) 2) 3) LCC LCC LCC 1 1) Vol.42No.5pp.29-322004.5.
λ 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 >
Twist knot orbifold Chern-Simons
Twist knot orbifold Chern-Simons 1 3 M π F : F (M) M ω = {ω ij }, Ω = {Ω ij }, cs := 1 4π 2 (ω 12 ω 13 ω 23 + ω 12 Ω 12 + ω 13 Ω 13 + ω 23 Ω 23 ) M Chern-Simons., S. Chern J. Simons, F (M) Pontrjagin 2.,
1980年代半ば，米国中西部のモデル 理論，そして未来－モデル理論賛歌
2016 9 27 RIMS 1 2 3 1983 9-1989 6 University of Illinois at Chicago (UIC) John T Baldwin 1983 9-1989 6 University of Illinois at Chicago (UIC) John T Baldwin Y N Moschovakis, Descriptive Set Theory North
Gauss Fuchs rigid rigid rigid Nicholas Katz Rigid local systems [6] Fuchs Katz Crawley- Boevey[1] [7] Katz rigid rigid Katz middle convolu
![Gauss Fuchs rigid rigid rigid Nicholas Katz Rigid local systems [6] Fuchs Katz Crawley- Boevey[1] [7] Katz rigid rigid Katz middle convolu](/thumbs/93/114422882.jpg "Gauss Fuchs rigid rigid rigid Nicholas Katz Rigid local systems [6] Fuchs Katz Crawley- Boevey[1] [7] Katz rigid rigid Katz middle convolu")
rigidity 2014.9.1-2014.9.2 Fuchs 1 Introduction y + p(x)y + q(x)y = 0, y 2 p(x), q(x) p(x) q(x) Fuchs 19 Fuchs 83 Gauss Fuchs rigid rigid rigid 7 1970 1996 Nicholas Katz Rigid local systems [6] Fuchs Katz
E1 (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
24 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. 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](/thumbs/94/118744578.jpg "24 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),
References tll A. Hurwitz, IJber algebraischen Gebilde mit eindeutige Transformationen Ann. in sich, Math. L27 A. Kuribayashi-K. Komiya, On Weierstrass points of non-hyperelliptic compact Riemann surfaces
0 17 l l Grothendieck Weil Grothendieck SGA (Séminaire de Géométrie Algébrique du Bois-Marie) [Del2], [Del3] Grothendieck Weil Ramanujan Deligne [Del1
![0 17 l l Grothendieck Weil Grothendieck SGA (Séminaire de Géométrie Algébrique du Bois-Marie) [Del2], [Del3] Grothendieck Weil Ramanujan Deligne [Del1](/thumbs/49/25561069.jpg "0 17 l l Grothendieck Weil Grothendieck SGA (Séminaire de Géométrie Algébrique du Bois-Marie) [Del2], [Del3] Grothendieck Weil Ramanujan Deligne [Del1")
l 0 2 1 4 1.1 Tate.......................... 4 1.2........................ 6 1.3...................... 9 1.4.................... 21 2 Galois 31 2.1 Galois.......... 31 2.2.................... 31 3 Galois
Siegel Hecke 1 Siege Hecke L L Fourier Dirichlet Hecke Euler L Euler Fourier Hecke [Fr] Andrianov [An2] Hecke Satake L van der Geer ([vg]) L [Na1] [Yo
![Siegel Hecke 1 Siege Hecke L L Fourier Dirichlet Hecke Euler L Euler Fourier Hecke [Fr] Andrianov [An2] Hecke Satake L van der Geer ([vg]) L [Na1] [Yo](/thumbs/96/128156654.jpg "Siegel Hecke 1 Siege Hecke L L Fourier Dirichlet Hecke Euler L Euler Fourier Hecke [Fr] Andrianov [An2] Hecke Satake L van der Geer ([vg]) L [Na1] [Yo")
Siegel Hecke 1 Siege Hecke L L Fourier Dirichlet Hecke Euler L Euler Fourier Hecke [Fr] Andrianov [An2] Hecke Satake L van der Geer ([vg]) L [Na1] [Yo] 2 Hecke ( ) 0 1n J n =, Γ = Γ n = Sp(n, Z) = {γ GL(2n,
Banach-Tarski Hausdorff May 17, 2014 3 Contents 1 Hausdorff 5 1.1 ( Unlösbarkeit des Inhaltproblems) 5 5 1 Hausdorff Banach-Tarski Hausdorff [H1, H2] Hausdorff Grundzüge der Mangenlehre [H1] Inhalte
newmain.dvi
数論 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/008142 このサンプルページの内容は, 第 2 版 1 刷発行当時のものです. Daniel DUVERNEY: THÉORIE DES NOMBRES c Dunod, Paris, 1998, This book is published
Chern-Simons Jones 3 Chern-Simons 1 - Chern-Simons - Jones J(K; q) [1] Jones q 1 J (K + ; q) qj (K ; q) = (q 1/2 q
![Chern-Simons Jones 3 Chern-Simons 1 - Chern-Simons - Jones J(K; q) [1] Jones q 1 J (K + ; q) qj (K ; q) = (q 1/2 q](/thumbs/94/118766307.jpg "Chern-Simons Jones 3 Chern-Simons 1 - Chern-Simons - Jones J(K; q) [1] Jones q 1 J (K + ; q) qj (K ; q) = (q 1/2 q")
Chern-Simons E-mail: fuji@th.phys.nagoya-u.ac.jp Jones 3 Chern-Simons - Chern-Simons - Jones J(K; q) []Jones q J (K + ; q) qj (K ; q) = (q /2 q /2 )J (K 0 ; q), () J( ; q) =. (2) K Figure : K +, K, K 0
, CH n. CH n, CP n,,,., CH n,,. RH n ( Cartan )., CH n., RH n CH n,,., RH n, CH n., RH n ( ), CH n ( 1.1 (v), (vi) )., RH n,, CH n,., CH n,. 1.2, CH n
( ), Jürgen Berndt,.,. 1, CH n.,,. 1.1 ([6]). CH n (n 2), : (i) CH k (k = 0,..., n 1) tube. (ii) RH n tube. (iii). (iv) ruled minimal, equidistant. (v) normally homogeneous submanifold F k tube. (vi) normally
kb-HP.dvi
Recent developments in the log minimal model program II II Birkar-Cascini-Hacon-McKernan 1 2 2 3 3 5 4 8 4.1.................. 9 4.2.......................... 10 5 11 464-8602, e-mail: fujino@math.nagoya-u.ac.jp
( ; ) C. H. Scholz, The Mechanics of Earthquakes and Faulting : - ( ) σ = σ t sin 2π(r a) λ dσ d(r a) =
1 9 8 1 1 1 ; 1 11 16 C. H. Scholz, The Mechanics of Earthquakes and Faulting 1. 1.1 1.1.1 : - σ = σ t sin πr a λ dσ dr a = E a = π λ σ πr a t cos λ 1 r a/λ 1 cos 1 E: σ t = Eλ πa a λ E/π γ : λ/ 3 γ =
R R P N (C) 7 C Riemann R K ( ) C R C K 8 (R ) R C K 9 Riemann /C /C Riemann 10 C k 11 k C/k 12 Riemann k Riemann C/k k(c)/k R k F q Riemann 15
(Gen KUROKI) 1 1 : Riemann Spec Z 2? 3 : 4 2 Riemann Riemann Riemann 1 C 5 Riemann Riemann R compact R K C ( C(x) ) K C(R) Riemann R 6 (E-mail address: kuroki@math.tohoku.ac.jp) 1 1 ( 5 ) 2 ( Q ) Spec
[bica]) our gmeff means Abel Milnor $K$ - motif (Mochizuki Satoshi) * Graduate School of Mathematical Sciences, the University o
![[bica]) our gmeff means Abel Milnor $K$ - motif (Mochizuki Satoshi) * Graduate School of Mathematical Sciences, the University o](/thumbs/89/100133180.jpg "[bica]) our gmeff means Abel Milnor $K$ - motif (Mochizuki Satoshi) * Graduate School of Mathematical Sciences, the University o")
Title 半 Abel 多様体に付随するMilnor $K$- 群のmotif 論的解釈 ( 代数的整数論とその周辺 ) Author(s) 望月, 哲史 Citation 数理解析研究所講究録 (2005), 1451: 155-164 Issue Date 2005-10 URL http://hdl.handle.net/2433/47730 Right Type Departmental
Siegel modular forms of middle parahoric subgroups and Ihara lift ( Tomoyoshi Ibukiyama Osaka University 1. Introduction [8] Ihara Sp(2, R) p
![Siegel modular forms of middle parahoric subgroups and Ihara lift ( Tomoyoshi Ibukiyama Osaka University 1. Introduction [8] Ihara Sp(2, R) p](/thumbs/98/138321532.jpg "Siegel modular forms of middle parahoric subgroups and Ihara lift ( Tomoyoshi Ibukiyama Osaka University 1. Introduction [8] Ihara Sp(2, R) p")
Siegel modular forms of middle parahoric subgroups and Ihara lift ( Tomoyoshi Ibukiyama Osaka University 1. Introduction [8] Ihara 80 1963 Sp(2, R) p L holomorphic discrete series Eichler Brandt Eichler
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
Feynman Encounter with Mathematics 52, [1] N. Kumano-go, Feynman path integrals as analysis on path space by time slicing approximation. Bull
![Feynman Encounter with Mathematics 52, [1] N. Kumano-go, Feynman path integrals as analysis on path space by time slicing approximation. Bull](/thumbs/94/118365979.jpg "Feynman Encounter with Mathematics 52, [1] N. Kumano-go, Feynman path integrals as analysis on path space by time slicing approximation. Bull")
Feynman Encounter with Mathematics 52, 200 9 [] N. Kumano-go, Feynman path integrals as analysis on path space by time slicing approximation. Bull. Sci. Math. vol. 28 (2004) 97 25. [2] D. Fujiwara and
(2) Fisher α (α) α Fisher α ( α) 0 Levi Civita (1) ( 1) e m (e) (m) ([1], [2], [13]) Poincaré e m Poincaré e m Kähler-like 2 Kähler-like
![(2) Fisher α (α) α Fisher α ( α) 0 Levi Civita (1) ( 1) e m (e) (m) ([1], [2], [13]) Poincaré e m Poincaré e m Kähler-like 2 Kähler-like](/thumbs/97/133596579.jpg "(2) Fisher α (α) α Fisher α ( α) 0 Levi Civita (1) ( 1) e m (e) (m) ([1], [2], [13]) Poincaré e m Poincaré e m Kähler-like 2 Kähler-like")
() 10 9 30 1 Fisher α (α) α Fisher α ( α) 0 Levi Civita (1) ( 1) e m (e) (m) ([1], [], [13]) Poincaré e m Poincaré e m Kähler-like Kähler-like Kähler M g M X, Y, Z (.1) Xg(Y, Z) = g( X Y, Z) + g(y, XZ)
Centralizers of Cantor minimal systems
Centralizers of Cantor minimal systems 1 X X X φ (X, φ) (X, φ) φ φ 2 X X X Homeo(X) Homeo(X) φ Homeo(X) x X Orb φ (x) = { φ n (x) ; n Z } x φ x Orb φ (x) X Orb φ (x) x n N 1 φ n (x) = x 1. (X, φ) (i) (X,
() n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (5) (6 ) n C + nc + 3 nc n nc n (7 ) n C + nc + 3 nc n nc n (
3 n nc k+ k + 3 () n C r n C n r nc r C r + C r ( r n ) () n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (4) n C n n C + n C + n C + + n C n (5) k k n C k n C k (6) n C + nc
62 Serre Abel-Jacob Serre Jacob Jacob Jacob k Jacob Jac(X) X g X (g) X (g) Zarsk [Wel] [Ml] [BLR] [Ser] Jacob ( ) 2 Jacob Pcard 2.1 X g ( C ) X n P P
![62 Serre Abel-Jacob Serre Jacob Jacob Jacob k Jacob Jac(X) X g X (g) X (g) Zarsk [Wel] [Ml] [BLR] [Ser] Jacob ( ) 2 Jacob Pcard 2.1 X g ( C ) X n P P](/thumbs/91/106734463.jpg "62 Serre Abel-Jacob Serre Jacob Jacob Jacob k Jacob Jac(X) X g X (g) X (g) Zarsk [Wel] [Ml] [BLR] [Ser] Jacob ( ) 2 Jacob Pcard 2.1 X g ( C ) X n P P")
15, pp.61-80 Abel-Jacob I 1 Introducton Remann Abel-Jacob X g Remann X ω 1,..., ω g Λ = {( γ ω 1,..., γ ω g) C g γ H 1 (X, Z)} Λ C g lattce Jac(X) = C g /Λ Le Abel-Jacob (Theorem 2.2, 4.2) Jac(X) Pcard
²ÄÀÑʬΥ»¶ÈóÀþ·¿¥·¥å¥ì¡¼¥Ç¥£¥ó¥¬¡¼ÊýÄø¼°¤ÎÁ²¶á²òÀÏ Asymptotic analysis for the integrable discrete nonlinear Schrödinger equation
Asymptotic analysis for the integrable discrete nonlinear Schrödinger equation ( ) ( ) 2016 12 17 1. Schrödinger focusing NLS iu t + u xx +2 u 2 u = 0 u(x, t) =2ηe 2iξx 4i(ξ2 η 2 )t+i(ψ 0 +π/2) sech(2ηx
TOP URL 1
TOP URL http://amonphys.web.fc.com/ 3.............................. 3.............................. 4.3 4................... 5.4........................ 6.5........................ 8.6...........................7
2017 : msjmeeting-2017sep-00f003 ( ) 1. 1 = A 1.1. / R T { } Λ R = a i T λ i a i R, λ i R, lim λ i = + i i=0 R Λ R v T ( a i T λ i ) = inf λ i, v T (0
2017 : msjmeeting-2017sep-00f003 ( ) 1. 1 = A 1.1. / R T { } Λ R = a i T λ i a i R, λ i R, lim λ i = + i i=0 R Λ R v T ( a i T λ i ) = inf λ i, v T (0) = + a i 0 i=0 v T Λ R 0 = {x Λ R v T (x) 0} R Remark
( ) ( ) (B) ( , )
() 2006 2 6 () 2006 2 6 2 7 7 (B) ( 574009, ) 2006 4 .,.. Introduction. [6], I. Simon (), J.-E. Pin. min-plus ().,,,. min-plus. (min-plus ). a, b R,, { a b := min(a, b), a b := a + b.. (R,, ) (, ). ( min-plus
21(2009) I ( ) 21(2009) / 42
21(2009) 10 24 21(2009) 10 24 1 / 21(2009) 10 24 1. 2. 3.... 4. 5. 6. 7.... 8.... 2009 8 1 1 (1.1) z = x + iy x, y i R C i 2 = 1, i 2 + 1 = 0. 21(2009) 10 24 3 / 1 B.C. (N) 1, 2, 3,...; +,, (Z) 0, ±1,
1 1.1 H = µc i c i + c i t ijc j + 1 c i c j V ijklc k c l (1) V ijkl = V jikl = V ijlk = V jilk () t ij = t ji, V ijkl = V lkji (3) (1) V 0 H mf = µc
013 6 30 BCS 1 1.1........................ 1................................ 3 1.3............................ 3 1.4............................... 5 1.5.................................... 5 6 3 7 4 8
(2004 ) 2 (A) (B) (C) 3 (1987) (1988) Shimono and Tachibanaki(1985) (2008) , % 2 (1999) (2005) 3 (2005) (2006) (2008)
,, 23 4 30 (i) (ii) (i) (ii) Negishi (1960) 2010 (2010) ( ) ( ) (2010) E-mail:fujii@econ.kobe-u.ac.jp E-mail:082e527e@stu.kobe-u.ac.jp E-mail:iritani@econ.kobe-u.ac.jp 1 1 16 (2004 ) 2 (A) (B) (C) 3 (1987)
compact compact Hermann compact Hermite ( - ) Hermann Hermann ( ) compact Hermite Lagrange compact Hermite ( ) a, Σ a {0} a 3 1
014 5 4 compact compact Hermann compact Hermite ( - ) Hermann Hermann ( ) compact Hermite Lagrange compact Hermite ( ) 1 1.1. a, Σ a {0} a 3 1 (1) a = span(σ). () α, β Σ s α β := β α,β α α Σ. (3) α, β
II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
![II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2](/thumbs/101/149159646.jpg "II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2")
II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh
1. R n Ω ε G ε 0 Ω ε B n 2 Ωε = with Bu = 0 on Ω ε i=1 x 2 i ε +0 B Bu = u (Dirichlet, D Ω ε ), Bu = u ν (Neumann, N Ω ε ), Ω ε G ( ) / 25
.. IV 2012 10 4 ( ) 2012 10 4 1 / 25 1. R n Ω ε G ε 0 Ω ε B n 2 Ωε = with Bu = 0 on Ω ε i=1 x 2 i ε +0 B Bu = u (Dirichlet, D Ω ε ), Bu = u ν (Neumann, N Ω ε ), Ω ε G ( ) 2012 10 4 2 / 25 1. Ω ε B ε t
Jacobi, 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
[2, 3, 4, 5] * C s (a m k (symmetry operation E m[ 1(a ] σ m σ (symmetry element E σ {E, σ} C s 32 ( ( =, 2 =, (3 0 1 v = x 1 1 +
![[2, 3, 4, 5] * C s (a m k (symmetry operation E m[ 1(a ] σ m σ (symmetry element E σ {E, σ} C s 32 ( ( =, 2 =, (3 0 1 v = x 1 1 +](/thumbs/92/108011164.jpg "[2, 3, 4, 5] * C s (a m k (symmetry operation E m[ 1(a ] σ m σ (symmetry element E σ {E, σ} C s 32 ( ( =, 2 =, (3 0 1 v = x 1 1 +")
2016 12 16 1 1 2 2 2.1 C s................. 2 2.2 C 3v................ 9 3 11 3.1.............. 11 3.2 32............... 12 3.3.............. 13 4 14 4.1........... 14 4.2................ 15 4.3................
takei.dvi
0 Newton Leibniz ( ) α1 ( ) αn (1) a α1,...,α n (x) u(x) = f(x) x 1 x n α 1 + +α n m 1957 Hans Lewy Lewy 1970 1 1.1 Example 1.1. (2) d 2 u dx 2 Q(x)u = f(x), u(0) = a, 1 du (0) = b. dx Q(x), f(x) x = 0
Z: Q: R: C: sin 6 5 ζ a, b
Z: Q: R: C: 3 3 7 4 sin 6 5 ζ 9 6 6............................... 6............................... 6.3......................... 4 7 6 8 8 9 3 33 a, b a bc c b a a b 5 3 5 3 5 5 3 a a a a p > p p p, 3,
Auerbach and Kotlikoff(1987) (1987) (1988) 4 (2004) 5 Diamond(1965) Auerbach and Kotlikoff(1987) 1 ( ) ,
,, 2010 8 24 2010 9 14 A B C A (B Negishi(1960) (C) ( 22 3 27 ) E-mail:fujii@econ.kobe-u.ac.jp E-mail:082e527e@stu.kobe-u.ac.jp E-mail:iritani@econ.kobe-u.ac.jp 1 1 1 2 3 Auerbach and Kotlikoff(1987) (1987)
T rank A max{rank Q[R Q, J] t-rank T [R T, C \ J] J C} 2 ([1, p.138, Theorem 4.2.5]) A = ( ) Q rank A = min{ρ(j) γ(j) J J C} C, (5) ρ(j) = rank Q[R Q,
![T rank A max{rank Q[R Q, J] t-rank T [R T, C \ J] J C} 2 ([1, p.138, Theorem 4.2.5]) A = ( ) Q rank A = min{ρ(j) γ(j) J J C} C, (5) ρ(j) = rank Q[R Q,](/thumbs/93/114160334.jpg "T rank A max{rank Q[R Q, J] t-rank T [R T, C \ J] J C} 2 ([1, p.138, Theorem 4.2.5]) A = ( ) Q rank A = min{ρ(j) γ(j) J J C} C, (5) ρ(j) = rank Q[R Q,")
(ver. 4:. 2005-07-27) 1 1.1 (mixed matrix) (layered mixed matrix, LM-matrix) m n A = Q T (2m) (m n) ( ) ( ) Q I m Q à = = (1) T diag [t 1,, t m ] T rank à = m rank A (2) 1.2 [ ] B rank [B C] rank B rank
p *2 DSGEDynamic Stochastic General Equilibrium New Keynesian *2 2
2013 1 nabe@ier.hit-u.ac.jp 2013 4 11 Jorgenson Tobin q : Hayashi s Theorem : Jordan : 1 investment 1 2 3 4 5 6 7 8 *1 *1 93SNA 1 p.180 1936 100 1970 *2 DSGEDynamic Stochastic General Equilibrium New Keynesian
untitled
Lie L ( Introduction L Rankin-Selberg, Hecke L (,,, Rankin, Selberg L (GL( GL( L, L. Rankin-Selberg, Fourier, (=Fourier (= Basic identity.,,.,, L.,,,,., ( Lie G (=G, G.., 5, Sp(, R,. L., GL(n, R Whittaker
1 = = = (set) (element) a A a A a A a A a A {2, 5, (0, 1)}, [ 1, 1] = {x; 1 x 1}. (proposition) A = {x; P (x)} P (x) x x a A a A Remark. (i) {2, 0, 0,
![1 = = = (set) (element) a A a A a A a A a A {2, 5, (0, 1)}, [ 1, 1] = {x; 1 x 1}. (proposition) A = {x; P (x)} P (x) x x a A a A Remark. (i) {2, 0, 0,](/thumbs/91/105754557.jpg "1 = = = (set) (element) a A a A a A a A a A {2, 5, (0, 1)}, [ 1, 1] = {x; 1 x 1}. (proposition) A = {x; P (x)} P (x) x x a A a A Remark. (i) {2, 0, 0,")
2005 4 1 1 2 2 6 3 8 4 11 5 14 6 18 7 20 8 22 9 24 10 26 11 27 http://matcmadison.edu/alehnen/weblogic/logset.htm 1 1 = = = (set) (element) a A a A a A a A a A {2, 5, (0, 1)}, [ 1, 1] = {x; 1 x 1}. (proposition)
2 K = f (x) K[[x]] = r f (x) r D = D (0, r) a D f (x) a D Figure X d : X X R 0 d(x, z) max{d(x, y), d(y, z)} x, y, z X (X, d) clopen 1.1. (X,
![2 K = f (x) K[[x]] = r f (x) r D = D (0, r) a D f (x) a D Figure X d : X X R 0 d(x, z) max{d(x, y), d(y, z)} x, y, z X (X, d) clopen 1.1. (X,](/thumbs/53/32082263.jpg "2 K = f (x) K[[x]] = r f (x) r D = D (0, r) a D f (x) a D Figure X d : X X R 0 d(x, z) max{d(x, y), d(y, z)} x, y, z X (X, d) clopen 1.1. (X,")
2008. 1. 1.1.. 4 affinoids analytic reduction Raynaud visualization Zariski-Riemann 1.2.. 1905 K. Hensel p- 1918 A. Ostrowski Q 1930 W. Schöbe 1940 M. Krasner 1 2 K = f (x) K[[x]] = r f (x) r D = D (0,
1 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)
9 Feb 2008 NOGUCHI (UT) HDVT 9 Feb / 33
9 Feb 2008 NOGUCHI (UT) HDVT 9 Feb 2008 1 / 33 1 NOGUCHI (UT) HDVT 9 Feb 2008 2 / 33 1 Green-Griffiths (1972) NOGUCHI (UT) HDVT 9 Feb 2008 2 / 33 1 Green-Griffiths (1972) X f : C X f (C) X NOGUCHI (UT)
,,,17,,, ( ),, E Q [S T F t ] < S t, t [, T ],,,,,,,,
![,,,17,,, ( ),, E Q [S T F t ] < S t, t [, T ],,,,,,,,](/thumbs/91/105754403.jpg ",,,17,,, ( ),, E Q [S T F t ] < S t, t [, T ],,,,,,,,")
14 5 1 ,,,17,,,194 1 4 ( ),, E Q [S T F t ] < S t, t [, T ],,,,,,,, 1 4 1.1........................................ 4 5.1........................................ 5.........................................
December 28, 2018
e-mail : kigami@i.kyoto-u.ac.jp December 28, 28 Contents 2............................. 3.2......................... 7.3..................... 9.4................ 4.5............. 2.6.... 22 2 36 2..........................
,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,.
9 α ν β Ξ ξ Γ γ o δ Π π ε ρ ζ Σ σ η τ Θ θ Υ υ ι Φ φ κ χ Λ λ Ψ ψ µ Ω ω Def, Prop, Th, Lem, Note, Remark, Ex,, Proof, R, N, Q, C [a, b {x R : a x b} : a, b {x R : a < x < b} : [a, b {x R : a x < b} : a,
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
* 1 1 (i) (ii) Brückner-Hartree-Fock (iii) (HF, BCS, HFB) (iv) (TDHF,TDHFB) (RPA) (QRPA) (v) (vi) *
* 1 1 (i) (ii) Brückner-Hartree-Fock (iii) (HF, BCS, HFB) (iv) (TDHF,TDHFB) (RPA) (QRPA) (v) (vi) *1 2004 1 1 ( ) ( ) 1.1 140 MeV 1.2 ( ) ( ) 1.3 2.6 10 8 s 7.6 10 17 s? Λ 2.5 10 10 s 6 10 24 s 1.4 ( m
Part () () Γ Part ,
Contents a 6 6 6 6 6 6 6 7 7. 8.. 8.. 8.3. 8 Part. 9. 9.. 9.. 3. 3.. 3.. 3 4. 5 4.. 5 4.. 9 4.3. 3 Part. 6 5. () 6 5.. () 7 5.. 9 5.3. Γ 3 6. 3 6.. 3 6.. 3 6.3. 33 Part 3. 34 7. 34 7.. 34 7.. 34 8. 35
30
3 ............................................2 2...........................................2....................................2.2...................................2.3..............................
7 π L int = gψ(x)ψ(x)φ(x) + (7.4) [ ] p ψ N = n (7.5) π (π +,π 0,π ) ψ (σ, σ, σ )ψ ( A) σ τ ( L int = gψψφ g N τ ) N π * ) (7.6) π π = (π, π, π ) π ±
![7 π L int = gψ(x)ψ(x)φ(x) + (7.4) [ ] p ψ N = n (7.5) π (π +,π 0,π ) ψ (σ, σ, σ )ψ ( A) σ τ ( L int = gψψφ g N τ ) N π * ) (7.6) π π = (π, π, π ) π ±](/thumbs/91/106462075.jpg "7 π L int = gψ(x)ψ(x)φ(x) + (7.4) [ ] p ψ N = n (7.5) π (π +,π 0,π ) ψ (σ, σ, σ )ψ ( A) σ τ ( L int = gψψφ g N τ ) N π * ) (7.6) π π = (π, π, π ) π ±")
7 7. ( ) SU() SU() 9 ( MeV) p 98.8 π + π 0 n 99.57 9.57 97.4 497.70 δm m 0.4%.% 0.% 0.8% π 9.57 4.96 Σ + Σ 0 Σ 89.6 9.46 K + K 0 49.67 (7.) p p = αp + βn, n n = γp + δn (7.a) [ ] p ψ ψ = Uψ, U = n [ α
2 R U, U Hausdorff, R. R. S R = (S, A) (closed), (open). (complete projective smooth algebraic curve) (cf. 2). 1., ( ).,. countable ( 2 ) ,,.,,
15, pp.1-13 1 1.1,. 1.1. C ( ) f = u + iv, (, u, v f ). 1 1. f f x = i f x u x = v y, u y = v x.., u, v u = v = 0 (, f = 2 f x + 2 f )., 2 y2 u = 0. u, u. 1,. 1.2. S, A S. (i) A φ S U φ C. (ii) φ A U φ
Affine Hecke ( A ) Irreducible representations of affine Hecke algebras (survey talk with emphasis on type A) (Syu Kato) Recently, there are
Affine Hecke ( A ) Irreducible representations of affine Hecke algebras (survey talk with emphasis on type A) (Syu Kato) 20 10 29 Recently, there are several successful attempts on the classification of
C 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
![C 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](/thumbs/77/75469487.jpg "C 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
Langlands 1 1. Langlands p GL n Langlands [HT] The local Langlands conjecture is one of those hydra-like conjectures which seems to grow as it gets pr
![Langlands 1 1. Langlands p GL n Langlands [HT] The local Langlands conjecture is one of those hydra-like conjectures which seems to grow as it gets pr](/thumbs/98/136029119.jpg "Langlands 1 1. Langlands p GL n Langlands [HT] The local Langlands conjecture is one of those hydra-like conjectures which seems to grow as it gets pr")
Langlands 1 1. Langlands p GL n Langlands [HT] The local Langlands conjecture is one of those hydra-like conjectures which seems to grow as it gets proved. ([HT], p.1) hydra [KP] Langlands Langlands Langlands
Keiji Matsumoto (Hokkaido Univ.) Jan. 08, ,
Keiji Mtsumoto (Hokkido Univ.) Jn. 08, 009 009, . > b > 0 { n }, {b n } ( 0, b 0 ) = (, b), ( n+, b n+ ) = ( n + b n, n b n ). { n }, {b n } lim n n = lim b n n b M(, b) Theorem (C.F. Guss 799 ) Mple M(,
$\ell$進層のSwan導手とunit-root overconvergent $F$-isocrystalの特性サイクルについて (Algebraic Number Theory and Related Topics 2007)
\cdots RIMS Kôkyûroku Bessatsu B12 (2009), 51 56 l 進層の Swan 導手と unit root overconvergent F isocrystal の特性サイクルについて By 阿部知行 (Tomoyuki Abe) * Abstract この要約ではまず P. Berthelot による数論的 \mathscr{d} 加群の理論を用いて unit
G 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
25 7 18 1 1 1.1 v.s............................. 1 1.1.1.................................. 1 1.1.2................................. 1 1.1.3.................................. 3 1.2................... 3
平成 19 年度 ( 第 29 回 ) 数学入門公開講座テキスト ( 京都大学数理解析研究所, 平成 19 ~8 年月 72 月日開催 30 日 ) R = T, Fermat Wiles, Taylor-Wiles R = T.,,.,. 1. Fermat Fermat,. Fermat, 17
R = T, Fermat Wiles, Taylor-Wiles R = T.,,.,. 1. Fermat Fermat,. Fermat, 17, 400.. Descartes ( ) Corneille ( ), Milton ( ), Velázquez ( ), Rembrandt van Rijn ( ),,,. Fermat, Fermat, Fermat, 1995 Wiles
chap9.dvi
9 AR (i) (ii) MA (iii) (iv) (v) 9.1 2 1 AR 1 9.1.1 S S y j = (α i + β i j) D ij + η j, η j = ρ S η j S + ε j (j =1,,T) (1) i=1 {ε j } i.i.d(,σ 2 ) η j (j ) D ij j i S 1 S =1 D ij =1 S>1 S =4 (1) y j =
( ) (, ) arxiv: hgm OpenXM search. d n A = (a ij ). A i a i Z d, Z d. i a ij > 0. β N 0 A = N 0 a N 0 a n Z A (β; p) = Au=β,u N n 0 A
( ) (, ) arxiv: 1510.02269 hgm OpenXM search. d n A = (a ij ). A i a i Z d, Z d. i a ij > 0. β N 0 A = N 0 a 1 + + N 0 a n Z A (β; p) = Au=β,u N n 0 A-. u! = n i=1 u i!, p u = n i=1 pu i i. Z = Z A Au
(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t
![(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t](/thumbs/67/56612994.jpg "(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t")
6 6.1 6.1 (1 Z ( X = e Z, Y = Im Z ( Z = X + iy, i = 1 (2 Z E[ e Z ] < E[ Im Z ] < Z E[Z] = E[e Z] + ie[im Z] 6.2 Z E[Z] E[ Z ] : E[ Z ] < e Z Z, Im Z Z E[Z] α = E[Z], Z = Z Z 1 {Z } E[Z] = α = α [ α ]
.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
meiji_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