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

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1 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) Gal(F /F ) W K W K K Wel σ W K n(σ) Z φ(σ) = Fr n(σ) q W + K = { σ W K n(σ) 0 } X K Hc (X K, Q l ) ([Hu3], [Hu5]) Gal(K/K) W K X K K 0 σ W + K ) ([M1, Theorem 1.1]) σ Hc (X K, Q l ) σ α m ι: Q l C ι(α) = q m/2 ) ([M2, Theorem 1.1]) σ 2 dm X =0 ( 1) Tr ( σ ; H c (X K, Q l )) l ) Wel ) l [Oc, Proposton 2.1, Theorem 2.4] K [Oc] de Jong alteraton ([dj]) ), ) 1

2 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 A Spec B O K X X K X X F X X X X 1 l ) X K 1.1 K 1.1 X K x X x U X O K U U = (U ) rg rg Raynaud U A K f 1,..., f n A T 1,..., T n det( f / T j ) A T 1,..., T n /(f 1,..., f n ) 0 A V g 1,..., g n A T 1,..., T n g f + V 2 2

3 A T 1,..., T n /(f 1,..., f n ) = A T 1,..., T n /(g 1,..., g n ) det( g / T j ) A T 1,..., T n /(g 1,..., g n ) [Hu3, Proposton 1.7.1] K Sp K T 1,..., T n /(f 1,..., f m ) (m n) A = K T m+1,..., T n 1.2 g 1,..., g m K T 1,..., T n A T 1,..., T n /(f 1,..., f m ) = A T 1,..., T n /(g 1,..., g m ) det( g / T j ) A T 1,..., T n /(g 1,..., g m ) g 1,..., g m O K k U = Spec O K [T 1,..., T n, T n+1 ]/(g 1,..., g m, T n+1 π k ) = det( g / T j ) K X U 1,..., U n U 1 U k 1.3 X U U O K X X 3 X rg X Spf O K U Y X U Y U rg = U ([BL, Lemma 4.4]) X = n =1 U Čech E s,t 1 = 1 1 < < s m Hc t ( ) (U1 U s ) K, Q l = H s+t c (X K, Q l ) E 1 l X 3

4 RψQ l 1.4 ([Hu3, Theorem 5.7.6]) X O K Gal(K/K) Hc ( (X ) rg, Q ) = K l H c (X F, RψQ l ) 1.5 H c (X F, RψQ l ) X O K Hc (X K, Q l ) 2 X 0 0 X O K Delgne ) 1.6 X O K σ W + K ( 1) Tr ( σ ; Hc (X F, RψQ l )) l 1.2 de Jong alteraton 1.6 X O K X O K Zarsk Spec O K [T 0,..., T n ]/(T 0 T r π) π K 0 r n X O K d Γ X OK X Γ X OK X pr 1 X d σ W + K ( 1) Tr ( Γ σ ; H c (X F, RψQ l )) l 4

5 1.8 Γ X OK X pr X p Γ RΓ c (X F, Rψ X Q l ) ( ) RΓ c (Γ F, p 1 Rψ X Q l ) RΓ c (Γ F, Rψ Γ Q l ) = RΓ c (X F, Rp 2! Rψ Γ Q l ) RΓ c (X F, Rψ X Rp 2! Q l ) RΓ c (X F, Rψ X Q l ) H c ( ) p Γ = X 1.6 ) ) σ X O K 1.7 ([Sa, Theorem 0.1]) 1.7 [Sa, 3] X [dj, Theorem 6.5] L K O L Y alteraton f : Y X L K K K K X X OK O K L K [Sa, Lemma 1.11] L K Galos L E τ Gal(L/K) Y τ = Y OL σ O L f τ : Y τ X f (Y K ) L = τ Gal(L/K) Y τ L, ( ) H c YF, RψQ = l H ( c Y τ E, RψQ ) l τ Gal(L/K) Γ K X K X K f K f K Γ K Fulton-MacPherson refned Gysn map [Ful, 6, 8] Y K Y K Γ K L (Y K Y K ) L = τ,τ Gal(L/K) Y L τ Y τ L Γ L (τ, τ ) Y τ Y τ Γ τ,τ σ W + K ( 1) Tr ( Γ σ ; Hc (X F, RψQ l )) = 1 deg f τ Gal(L/K) ( 1) Tr ( Γ στ,τ σ ; H c (Y τ E, RψQ l )) 5

6 1.7 X = Y τ 1.11 L K Galos σ W + K X O L d Γ X σ OL X Γ X σ OL X pr 1 X σ d ( 1) Tr ( Γ σ ; H c (X F, RψQ l )) l 1.3 X 1.12 X O K d D 1,..., D m X F {1,..., m} I D I = I D p D (p) = I {1,...,m},#I=p+1 D I E s,t 1 = max(0, s) H t 2 c ( (s+2) D, Q l ( ) ) = Hc s+t (X F, RψQ l ) ( ) F RψQ l I K RψQ l [Sa, 2.1, 2.2] 1.13 ( ) ([RZ], [Sa]) E 1 RψQ l [Sa, Proposton 2.20] ( ) 1.14 ([M2, Theorem 5.5.3]) X, D 1.12 X d D Γ X O K X d Γ X OK X pr 1 X p D (p) D (p) d p 6

7 Γ (p) Γ (p) D (p) D (p) pr 1 D (p) E s,t 1 = ( max(0, s) Ht 2 (s+2) c D, Q l ( ) ) F Γ (s+2) F H s+t c (X F, RψQ l ) E s,t 1 = ( max(0, s) Ht 2 (s+2) c D, Q l ( ) ) Hc s+t (X, RψQ F l ). Γ l [Sa] 3 Chow Γ (p) refned Gysn map Chern X O K X OK X X [M2] [M2] 1.14 E 1 σ E 1 σ σ Gal(K/K) Gal(F /F ) σ Fr X : X X q Frobenus σ = (Fr X )n(σ) Γ σ ( ) E X F d Γ X X d Γ X X pr 1 X ( 1) Tr ( Γ ; Hc (X F, Q l )) l Γ (n) = (Fr n X 1) (Γ) [Fuj, Proposton 5.3.4, Proposton 5.4.1] [Var] N n N ( 1) Tr ( Γ (Fr X )n ; Hc (X F, Q l )) = deg( X Γ (n) ) 7

8 X Fr n X (Γ) n F l Γ Fr X Γ l Fr X p 0 ( 1) Tr ( Γ ; Hc (X F, Q l )) Z[1/p] 1.16 ( 1) Tr ( Γ ; Hc (X F, Q l )) 1.7 Bloch-Esnault [BE, Proposton 3.4] X alteraton X K 2 l K 0 ) X 2.1 Huber Huber 2.1 ([Hu4, Theorem 2.1]) X, Y K f : X Y char K = 0, dm Y 1 X Z/l n F R f! F Y Z/l n [Hu4, Defnton 1.1] Z/l n Z/l n K p Remann dm Y 2 ([Hu4, Example 2.2]) 2.2 ([Hu4, Corollary 2.7]) char K = 0 X K Y X U = X \ Y ε K U(ε) U X Y ε [Hu4, 2.6] Q(ε) U U(ε) 8

9 l ε l > 0 0 < ε ε l ) ( ) Hc( U(ε)K, Q l H c UK, Q l Q l Z/l Z/l Z/l n Y Y 1 f f f : X A 1 0 A 1 Y 2.1 F = R f! Z/l ε > 0 B(ε) \ {0} F Hc (D(ε) K \ {0}, F ) = 0 B(ε) ε D(ε) ε 2.3 char K = 0 X K Y X U = X \ Y Hc (U K, Q l ) H c (X K, Q l ) H c (Y K, Q l ) H+1 c (U K, Q l ) Z/l n [Hu3, Remark v)] H c (U K, Z/ln ) H c (X K, Z/ln ) H c (Y K, Z/ln ) H +1 c (U K, Z/l n ) 2.1 H c (X K, Z/ln ), H c (Y K, Z/ln ) Z/l n Hc (U K, Z/ln ) Z/l n { Hc (U K, Z/ln ) } { n, H c (X K, Z/l n ) } n, { Hc (Y K, Z/ln ) } Mttag-Leffler n lm n Hc (U K, Z/ln ) lm H n c (X K, Z/ln ) lm H n c (Y K, Z/ln ) lm n H +1 c (U K, Z/l n ) [Hu5, Theorem 3.3] Hc (U K, Z l ) = lm H n c (U K, Z/ln ) 2.2 [Hu5, Theorem 3.1] X, Y Hc (U K, Z l ) H c (X K, Z l ) H c (Y K, Z l ) H+1 c (U K, Z l ) Q l 9

10 2.4 l X K j : U X Hc (U K, Q l ) = H (X K, j! Q l ) X = P 1, U = Ω 1 Drnfeld P 1 K [Hu5, Example 2.7] Hc (U K, Z/ln ) = H (X K, j! Z/l n ) 2.2 l l dm X X X Y dm Y < dm X U = X \ Y 2.3 U l l, l q 2 ε 0 = mn(ε l, ε l ), U = U(ε 0 ) Hc ( U K, Q ) ( ) ( l H c UK, Q l, H c U K, Q ) ( ) l H c UK, Q l U K σ W + K ( 1) Tr ( σ ; H c (U K, Q l )) ( 1) Tr ( σ ; H c (U K, Q l )) U l 3 ) ) ) E 1 l E 1 ) ([M1, Lemma 4.3]) l 1.1 X 1.3 alteraton l alteraton [M1] 10

11 1.11 Galos 1.14 Wel E 1 Frobenus Wel ([De, Corollare 3.3.3, Corollare 3.3.4]) char K = 0 X l [BE] S. Bloch, H. Esnault, Künneth projectors for open varetes, preprnt, math.ag/ , [BL] S. Bosch, W. Lütkebohmert, Formal and rgd geometry I. Rgd spaces., Math. Ann. 295 (1993), no. 2, [Ca] H. Carayol, Nonabelan Lubn-Tate theory, Automorphc forms, Shmura varetes, and L-functons, Vol. II (Ann Arbor, MI, 1988), 15 39, Perspect. Math., 11, Academc Press, Boston, MA, [dj] A. J. de Jong, Smoothness, sem-stablty and alteratons, Inst. Hautes Études Sc. Publ. Math., No. 83 (1996), [De] P. Delgne, La conjecture de Wel II, Inst. Hautes Études Sc. Publ. Math. No. 52 (1980), [Fuj] K. Fujwara, Rgd geometry, Lefschetz-Verder trace formula and Delgne s conjecture, Invent. Math. 127 (1997), no. 3, [Ful] W. Fulton, Intersecton theory, Second edton. Ergebnsse der Mathematk und hrer Grenzgebete. 3. Folge. A Seres of Modern Surveys n Mathematcs, 2, Sprnger- Verlag, Berln, [Hu1] R. Huber, Contnuous valuatons, Math. Z. 212 (1993), no. 3, [Hu2] R. Huber, A generalzaton of formal schemes and rgd analytc varetes, Math. Z. 217 (1994), no. 4, [Hu3] R. Huber, Étale cohomology of rgd analytc varetes and adc spaces, Aspects of Mathematcs, E30. Fredr. Veweg & Sohn, Braunschweg, [Hu4] R. Huber, A fnteness result for the compactly supported cohomology of rgd analytc varetes, J. Algebrac Geom. 7 (1998), no. 2,

12 [Hu5] R. Huber, A comparson theorem for l-adc cohomology, Composto Math. 112 (1998), no. 2, [M1] Y. Meda, On the acton of the Wel group on the l-adc cohomology of rgd spaces over local felds, preprnt, math.nt/ , [M2] Y. Meda, On l-ndependence for the étale cohomology of rgd spaces over local felds, preprnt, math.ag/ , [Oc] T. Ocha, l-ndependence of the trace of monodromy, Math. Ann. 315 (1999), no. 2, [RZ] M. Rapoport, Th. Znk, Über de lokale Zetafunkton von Shmuravaretäten. Monodromefltraton und verschwndende Zyklen n unglecher Charakterstk, Invent. Math. 68 (1982), no. 1, [Sa] T. Sato, Weght spectral sequences and ndependence of l, J. Inst. Math. Jusseu 2 (2003), no. 4, [Var] Y. Varshavsky, Lefschetz-Verder trace formula and a generalzaton of a theorem of Fujwara, preprnt, math.ag/ ,

[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 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,