日本数学会・２０１１年度年会（早稲田大学）・総合講演

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1 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 ( ) 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 p σ : O K O K σ K K σ 2. K, k j : X X k smooth X \ X =: Z = r i=1 Z i X O K smooth affine p scheme X Z X Z = r i=1 Z i Z i (1 i r) X {t i =0} {t i } r i=1 X {t i} d i=1 Z = r i=1 Z i, X p Z K = r i=1 Z i,k, X K λ [0, 1) Γ U λ := {x X K t i (x) λ for 1 i r} U λ X K j λ (U 0 = X K.) U λ F F X K j F j F := lim j ρ, (F Uρ ) λ ρ (1) E := (E, ) (X, X) E X K j O X K : E E j O XK j Ω 1 Leibniz rule, X K U λ ( : ( : ), ( : ), ( : )) 2010 Mathematics Subject Classification: 12H25, 14F35 overconvergent isocrystal, p-adic differential equation

2 Ẽ Leibniz rule, : Ẽ Ẽ U λ Ω1 Uλ (E, ) =j (Ẽ, ) η (0, 1), ρ [λ, 1), e Γ(U ρ, Ẽ), 1 (2.1.1) i! i (e) η i 0(i N d, i ) ( = d j=1 jdt i i =(i 1,..., i d ) i = d j=1 i j j, Γ(U ρ, Ẽ) norm) (2) E := (E, ) (X,Z) E O X K : E E OX Ω 1 (log Z K X K ) Leibniz rule, K E (2.1.1) ( Ẽ E U λ U ) λ (X, X) Isoc (X, X), (X,Z) Isoc log (X,Z) X = X (X, X) X Isoc(X) (X, X) (X,Z) X, Z (X, X)/O K (X, X) k j : X X. 2.1 j : Isoc log (X,Z) Isoc (X, X) 2.2. (1) (X, X) (X,Z) (2) E (X, X) E( ) Christol-Mebkhout, André, Kedlaya, Mebkhout K k λ [0, 1) Γ [λ, 1) 1 p A 1 K[λ, 1) p A 1 K [λ, 1) t A K[λ, 1) := Γ(A 1 K [λ, 1), O A 1 K [λ,1) ) Robba R R := lim A λ 1 K[λ, 1) R A K [λ, 1) A K [0, 1).

3 3.1. (1) (E, ) R(resp. A K [λ, 1)) E R(resp. A K [λ, 1)) : E Edt Leibniz rule (2) (E, ) A K [0, 1) E A K [0, 1) : E Edlog t Leibniz rule p ( ) (1) R A K [0, 1) (2) R (E, ) A 1 K [0, 1) E( ) A K [0, 1)(K K ) (E, ) R A K [λ, 1) ( (E, ).) ρ [λ, 1) K(t) ρ K(t) ρ-gauss norm ( n a nt n ρ = max n a n ρ n norm ρ ) A K [λ, 1) K(t) ρ E ρ := E AK [λ,1) K(t) ρ ρ : E ρ E ρ dt = E ρ E ρ n ρ G n (n N) (E, ) ρ intrinsic generic IR(E,ρ) IR(E,ρ) := min(1,ρ 1 lim G n /n! 1/n ρ ) n ( norm norm max.) lim ρ 1 IR(E,ρ) =1 (E, ) solvable (E, ) solvable 1 ρ IR(E,ρ) =ρ b b Q 0 b (E, ) slope solvability ( -Trihan[23].) slope slope ([5], [22] ) 3.3 (slope, Christol-Mebkhout). (E, ) R solvable (E, ) = s Q 0 (E s, s ) (E s, s ) slope s. slope = 0 3.2(1) 3.4. (1) a Z p p Liouville x s s 1, a s a, x s s 1, a s + a p 1 (2) Σ Z p /Z (NLD) Σ Z p a, b a b p Liouville

4 a Z p a p Liouville a = n 1 pf(n) f(1) = 1,f(n +1)=p f(n) a p Liouville Σ ( Σ.) 3.5. Σ Z p /Z R (E, ) Σ (resp. Σ ) (E, ) (R,d+ αdlog t)(α α mod Z Σ ) (resp. ) Σ =0 (resp. ) Christol-Mebkhout, Dwork slope = 0 R μ (E, ) p exponent Exp(E) (Z p /Z) μ / e ([4], [11], [7]) e (Exp(E) p 1 monodromy 1 log 2πi.) Christol-Mebkhout([4], [7], [22] ) 3.2(1) ( ) ( Ext ) 3.6 (p Fuchs Christol-Mebkhout). μ N Σ Z p /Z (NLD) Σ := Im(Σ μ (Z p /Z) μ (Z p /Z) μ / e ) (E, ) μ R slope = 0 Exp(E) Σ (E, ) Σ (E, ) A K [0, 1) 3.2(2) K k σ ϕ : R R ϕ(t) =t p R (E, ) Frobenius h N R ϕ h,r (E, ) (E, ) k((t)) k R ( K K )Robba R(k) André[1], Mebkhout[24], Kedlaya[15] 3.2(2) ([22] ) 3.7 (p monodromy André, Mebkhout, Kedlaya). (E, ) Frobenius R k((t)) k (E, ) R(k) ( A K [0, 1).) 3.7 ([8] ) André R(k) K K Frobenius MCF(R(k) K K) ( k ) ( 3.3 ) MCF(R K K) pro 3.6 Exp(E) =0 Crew[10], [33] Mebkhout (E, ) k K K 1 R(k) K K ( (E, ) (E, ).) Exp(E) =0 Crew, Kedlaya R Frobenius ( )

5 Mebkhout p non-liouville R k R(k) Σ ( Σ={p non-liouville }/Z). 4. K, k j : X X k 3.7 (X, X) E Frobenius 2.2(2) p q := p f k F q. F :(X, X) (X, X) Frobenius (q ) (σ f ) : Spf O K Spf O K E F F E F 4.1. (X, X) F (X, X) E Ψ:F E E (E, Ψ) (X, X) F F -Isoc (X, X) F 2.2(2) [27] ([18] ) 4.2 ( F ). E (X, X) F alteration (proper surjective generically etale ) f : (X, X ) (X, X) X smooth, X \ X X f E (X,Z ) ( Z := X \ X ) E (X, X) F 4.2 de Jong[12] dim X =1 3.7 Kedlaya[14] ( -Trihan[23].) 4.3. dim X =1 4.2 f f X finite etale [27] 4.2 F rigid cohomology F rigid cohomology Kedlaya[16] 4.2 : unit-root ( 5.6 ) F [34] Kedlaya([18], [19], [20], [21]) 4.2

6 4.4 (Kedlaya) Exp(E) = (1) j : X X k smooth X \ X =: Z = r i=1 Z i Z i Z i,k X K Z i,k A 1 K [0, 1) X K L i := (Frac Γ(Z i,k, O Zi,K )) ( sup norm ) R i L i Robba (X, X) E := (E, ) Z i,k generic point R i (E i, i ) monodromy, monodromy 4.5. (X, X) E := (E, ) monodromy (resp. monodromy), 1 i r (E i, i ) (resp. ) ( (E i, i ) slope = 0 Exp(E i )=0.) exponent j : X X,Z = r i=1 Z i, X, Z = r i=1 Z i 2.1 (X,Z) E := (E, ) Z i,k residue res i End OZi,K (E Zi,K ) 0 P i (x) Z p [x] P i (res i )=0 res i Z p [x] (E, ) Z i,k exponent 4.6. Σ= r i=1 Σ i Z r p (X,Z) E := (E, ) residue 1 i r (E, ) Z i,k exponent 0 Kedlaya[18] (X, X) ( )monodromy ( ) 4.7. X, X,Z ( ) ( ) E Isoc log (X,Z) (4.7.1) j = E Isoc (X, X) :. residue monodromy (4.7.2) j : Isoc(X) = ( ) E Isoc (X, X). monodromy (4.7.2)( ) 4.8. X X X smooth Y X 2 Isoc (X, X) Isoc (X \Y,X)

7 valuation theory E X k(x) v 4.2 [18] 4.7 (4.7.1) 1 [19] 1 = k = k [20] v Q r ( 3.7 [17] ) [21] r 4.4 p [30] f :(X, X) (Y,Y ) X Y proper (X, X) F Y U (U, Y ) F 4.4 Caro- [2] 4.4 k smooth overholonomic F -D D Grothendieck 6 de Jong Kedlaya 4.4 p Exp(E) =0 Exp(E) =0 j : X X k smooth X \ X =: Z = r i=1 Z i Z i Robba R i (X, X) E := (E, ) Z i,k generic point R i (E i, i ) 4.5 Σ monodromy, Σ monodromy Σ= r i=1 Σ i Z r p /Zr Σ i (NLD) Σ (NLD) 5.1. Σ= r i=1 Σ i Z r p/z r (NLD) (X, X) (E, ) Σ monodromy (resp. Σ monodromy), 1 i r (E i, i ) Σ i (resp. Σ i ) (Σ i (E i, i ) slope= 0 Exp(E i ) Σ i ( 3.6 ).) exponent 5.2. Σ= r i=1 Σ i Z r p (X,Z) (E, ) exponent Σ 1 i r (E, ) Z i,k exponent Σ i residue 1 i r (E, ) Z i,k residue

8 (X, X) Σ monodromy ([28], [32]) (1) ( ) 5.3. X, X,Z Σ = r i=1 Σ i Z r p/z r (NLD) τ = r i=1 τ i : Z r p/z r Z r p Zr p Z r p/z r section ( ) ( ) E Isoc log (X,Z) (5.3.1) j = E Isoc (X, X) :. exponent τ(σ) Σ monodromy E Isoc log (X,Z) ( ) E Isoc (5.3.2) j : exponent τ(σ) = (X, X). Σ monodromy residue Frobenius 4.2 X log (5.3.1) di Proietto[26]. Σ monodromy [29] ( 2.2(1).) 5.4. k X, X,Z Σ = r i=1 Σ i Z r p /Zr (NLD) (X, X) E := (E, ) (1) E Σ monodromy. (2) C X Z 1 E (C,C) ( C := X C) (C, C) Σ i monodromy ( i C Z Z i ). E =(E, ) Robba R i (E i, i ) C X C Z Robba (E C, C ) (E i, i ) slope p exponent C Z Z i C X (E C, C ) slope p exponent slope p exponent ( 4.8) 4.8 (X, X) X X ( ) [31] Frobenius X 5.5. X X smooth X \ X Y Y X 2 F -Isoc (X, X) F -Isoc (X \ Y,X \ Y )

9 X = A d k,x = Ga m,k Ad a k (0 a d), 5.5 X X 5.5 X \ X =: Z = r i=1 Z i π 1 (X) p V monodromy Z i v i k(x) k(x) vi V (X, X) F (E, Ψ) X x x x (E, Ψ) σ f Newton polygon x 0 (E, Ψ) unit-root [31] 5.6. X X ( ) ( ) π1 (X) K σ (X, X) unit-root (5.6.1). p monodromy F 1 [33] [34] , 5.6 (X,Z) adjusted parabolic (X,Z) (E α ) α Z r (p) := (E α, α ) α Z r (p) e i (1 i r) N r (E α+ei ) α = (Eα (Z i )) α ( E α (Z i ) (E α, α ) (O X K (Z i,k ), d).) n transition maps E [nα]/n E α (Adjusted) α =(α i ) i Z r (p) E α exponent r i=1 ([ α i, α i +1) Z (p) ). (X,Z) adjusted parabolic F adjusted parabolic (E α ) α ind-object Ψ : lim F E = α lim E α ((E α ) α, Ψ) (X,Z) adjusted parabolic F E := ((E α ) α, Ψ) 0 X x E (σ f ) Newton polygon endpoint x μ(e) E generically semistable (gss) U X 0 E E U μ(e ) μ(e) [32] 5.7. X, X,Z π1 t(x) Z i v i (1 i r) X tame ( ) (X,Z) adjusted parabolic π t 1 (X) = (5.7.1) K σ p F. generically semistable, μ = 0

10 5.6 π t 1 (X) p (Z (p)/z) r monodromy unit-root F 5.3 τ F adjusted parabolic F unit-root residue gss μ =0 Mehta-Seshadri[25] C unitary polystable parabolic vector bundle Weng[35] [1] Y. André, Filtrations de type Hasse-Arf et monodromie p-adique, Invent. Math. 148(2002), no. 2, [2] D. Caro and N. Tsuzuki, Overholonomicity of overconvergent F -isocrystals over smooth varieties, arxiv: v1. [3] G. Christol and Z. Mebkhout, Sur le théorème de l indice des équations différentielles p-adiques I, Ann. Inst. Fourier (Grenoble) 43(1993), no. 5, [4] G. Christol and Z. Mebkhout, Sur le théorème de l indice des équations différentielles p-adiques II, Ann. of Math. (2) 146(1997), no. 2, [5] G. Christol and Z. Mebkhout, Sur le théorème de l indice des équations différentielles p-adiques III, Ann. of Math. (2) 151(2000), no. 2, [6] G. Christol and Z. Mebkhout, Sur le théorème de l indice des équations différentielles p-adiques IV, Invent. Math. 143(2001), no. 3, [7] G. Christol and Z. Mebkhout, Équations différentielles p-adiques et coefficients p-adiques sur les courbes, Astérisque No. 279(2002), [8] P. Colmez, Les conjectures de monodromie p-adiques, Séminaire Bourbaki, Astérisque No. 290(2003), [9] R. Crew, Specialization of crystalline cohomology, Duke Math. J. 53(1986), no. 3, 749?757. [10] R. Crew, F -isocrystals and p-adic representations, Proc. Sympos. Pure Math. 46, Part 2 (1987), [11] B. Dwork, On exponents of p-adic differential modules, J. Reine Angew. Math. 484(1997), [12] A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. No. 83(1996), [13] N. M. Katz, Slope filtration of F -crystals, Astérisque, 63(1979), [14] K. S. Kedlaya, Semistable reduction for overconvergent F -isocrystals on a curve, Math. Res. Lett. 10(2003), no. 2-3, [15] K. S. Kedlaya, A p-adic local monodromy theorem, Ann. of Math. (2) 160(2004), no. 1, [16] K. S. Kedlaya, Finiteness of rigid cohomology with coefficients, Duke Math. J. 134(2006), no. 1, [17] K. S. Kedlaya, The p-adic local monodromy theorem for fake annuli, Rend. Semin. Mat. Univ. Padova 118(2007), [18] K. S. Kedlaya, Semistable reduction for overconvergent F -isocrystals I: Unipotence and

11 logarithmic extensions, Compos. Math. 143(2007), no. 5, [19] K. S. Kedlaya, Semistable reduction for overconvergent F -isocrystals II: A valuationtheoretic approach, Compos. Math. 144(2008), no. 3, 657?672. [20] K. S. Kedlaya, Semistable reduction for overconvergent F -isocrystals III: Local semistable reduction at monomial valuations, Compos. Math. 145(2009), no. 1, [21] K. S. Kedlaya, Semistable reduction for overconvergent F-isocrystals IV: Local semistable reduction at nonmonomial valuations, arxiv: v4, to appear in Compos. Math. [22] K. S. Kedlaya, p-adic Differential Equations, Cambridge Studies in Advanced Mathematics 125, Cambridge Univ. Press, [23] S. Matsuda and F. Trihan, Image directe supérieure et unipotence, J. Reine Angew. Math. 569(2004), [24] Z. Mebkhout, Analogue p-adique du théor `me de Turrittin et le théorème de la monodromie p-adique. Invent. Math. 148(2002), no. 2, [25] V. B. Mehta and C. S. Seshadri, Moduli of vector bundles on curves with parabolic structures, Math. Ann. 248(1980), no. 3, [26] V. di Proietto, On p-adic differential equations on semistable varieties, arxiv: v2. [27] A. Shiho, Crystalline fundamental groups II Log convergent cohomology and rigid cohomology, J. Math. Sci. Univ. Tokyo 9 (2002), no. 1, [28] A. Shiho, On logarithmic extension of overconvergent isocrystals, Math. Ann. 348(2010), no. 2, [29] A. Shiho, Cut-by-curves criterion for the log extendability of overconvergent isocrystals, arxiv: v1, to appear in Math. Z. [30] A. Shiho, Relative log convergent cohomology and relative rigid cohomology III, arxiv: v1. [31] A. Shiho, Purity for overconvergence, arxiv: v1. [32] A. Shiho, Parabolic log convergent isocrystals, arxiv: v1 (revision in preparation). [33] N. Tsuzuki, Finite local monodromy of overconvergent unit-root F -isocrystals on a curve. Amer. J. Math. 120(1998), no. 6, [34] N. Tsuzuki, Morphisms of F -isocrystals and the finite monodromy theorem for unit-root F -isocrystals. Duke Math. J. 111(2002), no. 3, [35] L. Weng, Stability and arithmetic, Adv. Study in Pure Math. 58(2010),

[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,