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,

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1 affinoids analytic reduction Raynaud visualization Zariski-Riemann K. Hensel p A. Ostrowski Q 1930 W. Schöbe 1940 M. Krasner 1

2 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, d) X totally disconnected 2 X

3 3 Q R Q R Q Harvard J. Tate Tate Inventiones Grauert-Remmert 1966 Tate Weierstrass affinoid 1969 Gerritzen-Grauert affinoid R. Kiehl 1967 A B 1972 M. Raynaud 2.2. Tate. Tate Tate Tate curve 3 E : y 2 + xy x 3 + b 2 x + b 3 = 0 ى) = b 3 + b b 2b 3 432b b3 2 0), C E m Z C /Z E(C) an C w q m w x(w) = (1 q m w) 2 2 q m (1 q m ) 2, y(w) = m Z m 1 (q m w) 2 (1 q m w) 3 + q m (1 q m ) 2. q b 2, b 3 n 3 q n b 2 = b 2 (q) = 5 1 q n = 5q(1 + 9q + 28q2 + ), n 1 b 3 = b 3 (q) = n 1 (7n 5 + 5n 3 )q n 12(1 q n ) m 1 = q(1 + 23q + 154q 2 + ).

4 4 Tate C C p C p Q p p- Tate w ( x(w) = (1 w) 2 + q m w (1 q m w) 2 + q m w 1 (1 q m w 1 ) 2 2 q m ) (1 q m ) 2 y(w) = m 1 w 2 (1 w) 3 + m 1 ( q 2m w 2 (1 q m w) 3 q m w 1 (1 q m w 1 ) 3 + q m ) (1 q m ) 2 r 1 w r 2, w q m ε, w 1 q m ε (m Z) R(r 1, r 2, ε) Figure 2. R(r 1, r 2, ε) Tate x = x(w) y = y(w) C w = q m m Z Runge C D D C q Z π: (G m,k ) cl E cl cl Tate q Z C p C K = C p K X X an X cl X an = X

5 5 π underlying sets π π: (G m,k ) an E an Tate Krasner Tate Tate rigidify A 1 k = Spec k[t] ى = {z C z < 1} D 1 K = {z K z 1}. 3. Tate Hilbert 3.1. k k a (X a) Spec k[x] k Spec k[x] k[x]

6 6 K K K V = {a K a 1}, m V = {a K a < 1} K V 1 0 a m V a a- V 0 a m V a a- 1 K = V[ 1 a ] K V[X] a- Zariski V[X] Zar V[X] 1 + av[x] 3.2. D 1 K a (X a) Spm V[X]Zar V K = Spm V[X] Zar [ 1 a ], K Spm X an = X Runge V[X] a- Zariski a- V X = lim n 0 V[X]/a n V[X]. K X = V X [ 1 a ] K X = { n=0 a n X n an K, a n 0 }. K X Tate Tate K X 1 D 1 K K X V[X] Zar [ 1 a ] D 1 K = Spm V[X]Zar [ 1 a ] = Spm K X K X

7 D 1 K = (Spm K X,, O) O U O(U) = U Runge Tate Krasner O O Tate Tate D 1 K K X m V X V X m V X V X a m V V X /m V X k[x] m V X k[x] red: Spm K X m (m V X mod m V X ) Spec k[x] Tate K D 1 K P1 (K) z 1 P 1 (K) = P 1 (V) (x : y) (x : y) P 1 (k) x = (x mod m V ) D 1 K

8 8 1 1 D 1 K A1 k = Spec k[x] 8 red 0 Figure 3. D 1 K = Spm K X D 1 K Zariski K X V V X 8 8 red red 0 0 Figure 4. Grothendieck Grothendieck

9 Tate acyclicity. D = D 1 K K X Grothendieck τ D Zariski red Zariski red O D Spf V X admissible blow-up a Tate 3.3 (Tate acyclicity). O 4. D = D 1 K = (Spm K X, τ D, O D ) Tate Tate Grothendieck τ D Gerritzen-Grauert Raynaud 1972 Tate 4.1. Raynaud. K V V m V 0 a m V a K V 1 a V K = Frac(V) = V[ 1 a ] Zariski

10 10 Raynaud 4.1 (Raynaud 1972). rig X X rig { } { } coherent V-formal / coherent rigid schemes of finite type admissible spaces over K blow-ups coherent quasi-compact quasi-separated Raynaud Tate-Raynaud (C) Tate-Raynaud (H) R. Huber adic (Z) Zariski-Riemann (B) Berkovich (Z) (B) Grothendieck Berkovich Berkovich Top ( (C), (H), (Z) ) Top ( (B) ) Berkovich. Berkovich Tate-Raynaud affinoid Tate K X 1, ldots, X n A Spm A Berkovich A M (A) Spm A

11 11 P 1 K Berkovich P 1 tree Tate-Raynaud P 1 K Tate-Raynaud P 1 K Figure 5. Berkovich Berkovich Berkovich Grothendieck Berkovich Berkovich visualization Huber adic Zariski-Riemann

12 12 sober Zariski coherent X X := lim coherent X cofinal Zariski X Raynaud Zariski Raynaud s viewpoint of rigid geometry Zariski s viewpoint of birational geometry Geometry of models Rigid analytic geometry Geometry of formal schemes + Zariski-Riemann space Figure address:

[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,

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