( ) 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

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1 ( ) 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 ( ) : Tate Weil 6 7 Z p - 2 [Iw1] 2 21 K k k 1 k K K 1

2 k(t) K K k = k K ( ) v k K ( ) v κ(v) k [κ(v) : k] deg(v) K D(K) D(K) := Zv v: f K v ord v (f) Z div : K D(K) Cl(K) P = v n vv D(K) deg(p ) := v n v deg(v) deg(p ) = 0 0 D 0 (K)( D(K)) div(k ) D 0 (K) Cl 0 (K) := D 0 (K)/div(K ) 0 0 Cl 0 (K) Cl(K) Z Cl 0 (K) K : P D(K) l(p ) := dim k {f K div(f) + P 0} {0} ( v n vv 0 v n v 0 ) P D(K) deg(p ) l(p ) + 1 K K L L k L k L k L K L L L L Cl(L) Cl(L ) Cl 0 (L) Cl 0 (L ) L/K Cl(L) Gal(L/K) N L/K : Cl(L) Cl(K) N L/K : Cl 0 (L) Cl 0 (K) L/K L K L k L/K L Cl(L) := lim Cl(M) Cl 0 (L) := lim Cl 0 (M) M M M L K 2

3 K k k Kk K k = k Kk = K k k Kk k Kk /K k Cl 0 (Kk ) Cl 0 (Kk ) = lim Cl 0 (Kk ) k k k 22 p k K K ZQ p /Z p V (K) V (K) := {f (1/p n ) K Q p /Z p v ord v (f) p n } f (1/p n ) V (K) v (ord v(f)/p n )v V (K) Cl 0 (K)[p ] k Z Q p /Z p (21) 0 k Z Q p /Z p V (K) Cl 0 (K)[p ] 0 k p k 1 p : µ p k K p- K urp (22) V (K) Gal(K urp /K) µ p (f (1/p n ) σ) σ( pn f)/ p n f ( p ) (23) Gal(K urp /K) = HomZ p (V (K) µ p ) k Z Q p /Z p = 0 (21) V (K) = Cl 0 (K)[p ] (23) (24) Gal(K urp /K) = HomZ p (Cl 0 (K)[p ] µ p ) K 0 K K/K 0 (22) Gal(K/K 0 )- (23) (24) Gal(K/K 0 )- Gal(K urp /K) Gal(K/K 0 ) 3 [We2] [SU] 3

4 31 F K F 31 (cf [SU] 692) Cl 0 (K) K Re(s) > 1 ζ K (s) := (1 κ(v) s ) 1 v: κ(v) v 32 (Hasse-Weil cf [We2] Chap VII 7 Theorem 4) K g q = F (i) 2g P K (T ) Z[T ] ζ K (s) = P K (q s ) (1 q s )(1 q 1 s ) P K (T ) ζ K (s) (ii) : ( ) 1 P K (T ) = q g T 2g P K qt (iii) P K (0) = 1 P K (1) = Cl 0 (K) 32 7 K K ur KF K 33 (cf [SU] 812) ρ K : Cl(K) Gal(K ur /K) v Frob v (v Frobenius ) dense Cl 0 (K) : ρ K0 : Cl 0 (K) = Gal(K ur /KF) L/K L (cf [SU]) L/K N L/K : Cl 0 (L) Cl 0 (K) Gal(L ur /LF) Gal(K ur /KF) ρ K0 ρ L0 4

5 4 F K KF( = K F F) F l K 1 l : (41) KF = l-nk(µ n ) Gal(KF/K) = Gal(F/F) Gal(F/F) Frobenius Gal(KF/K) Frob p F KF p- KF urp Y (K) := Gal(KF urp /KF) Gal(KF/K) Y (K) 41 (Weil) (i) g K Z p Y (K) = Z 2g p (ii) P K (T ) K ( 32 (i)) P K (T ) = det(1 FrobT : Y (K) Z p Q p ) k := Q(ζ p ) k /k Z p - k = k(µ p ) ((41) ) k urp k p- X(k) := Gal(k urp /k ) X(k) Y (K) 41 X(k) X(k) (Gal(k/Q) 1 ) X(k) Γ = Gal(k /k) Z p [[Γ]]- Λ := Z p [[T ]] Γ γ Z p [[Γ]] = Λ γ T + 1 X(k) Λ- Gal(k/Q) χ 1 f(t χ) Λ p L L p (s χ) κ : Γ Z p f(κ(γ) s 1 χ) = L p (s χ) (f(t χ) γ ) Weierstrass f(t χ) Q χ (T ) Q(T ) := Q χ (T ) Z p [T ] χ 1: 5

6 42 ( ) (i) X(k) µ- 0 (Ferrero-Washington) Λ- Z p - X(k) = Z λ k p (ii) X(k) Λ char Λ (X(k) )( Z p [T ]) char Λ (X(k) ) = Q(T ) 42 (i) 41 (i) X(k) λ- λ k 2 2g Z p - µ (ii) : 43 (cf [Wa] p292) Λ- M Γ γ T + 1( Λ) M µ- 0 char Λ (M) = det((t + 1) γ : M Z p Q p ) ( T M ) (ii) Q(T ) = det((t + 1) γ : X(k) Z p Q p ) 41 (ii) p L Z p - (1) KF/K (2) Gal(KF/K) = Gal(F/F) = Ẑ ( Λ ) (3) Frob Gal(KF/K) (4) P K (T ) p 1 n Q(µ n ) 42 ([Ho] [Ku] ) Q p

7 51 [DK] [AG] Chap V VII k ( ): {k } {k projective smooth } C/k K(C) K {K } { generic point } projective smooth C/k projective smooth K C K g C M/k C M := C Spec(k) Spec(M) M K(C M ) K k M Weil C/k Jac(C) k g M/k functorial Jac(C)(M) = Cl 0 (K(C M )) k /k (51) Jac(C)(k ) = Cl 0 (Kk ) (Kk = K k k ) k /k Gal(k /k) Gal(Kk /K) Gal(Kk /K) = Gal(k /k) (51) p k k Jac(C)[p n ] := Jac(C)(k)[p n ] Jac(C)[p n ] = (Z/p n ) 2g Jac(C)[p ] = (Q p /Z p ) 2g Weil Gal(k/k)- (52) Jac(C)[p n ] Jac(C)[p n ] µ p n (Weil pairing) p n p n+1 p (53) Jac(C)[p ] T p (Jac(C)) µ p n T p (Jac(C)) Jac(C) (p-)tate T p (Jac(C)) := lim n Jac(C)[p n ] ( p ) T p (Jac(C)) = Z 2g p 7

8 (53) Gal(k/k)- (54) T p (Jac(C)) = Hom(Jac(C)[p ] µ p ) (24) (51) (54) 51 Kk p- Kk urp Gal(Kk/K) = Gal(k/k)- Gal(Kk urp /Kk) = T p (Jac(C)) 52 Weil F K K C F p V p := T p (Jac(C)) Z p Q p V p Gal(F/F) Weil (Weil cf [We1]) P K (T ) 32 (i) P K (T ) = det(1 FrobT : V p ) 53 ζ K (s) = exp( n C(F(n)) q ns ) n ( F = q F(n) [F(n) : F] = n ) : C(F(n)) = 1 + q n Tr(Frob n : V p ) (Frob n Gal(F/F(n)) ) 41 : 51 Gal(KF/K) = Gal(F/F)- Gal(KF urp /KF) = T p (Jac(C)) 41 (i) T p (Jac(C)) = Z 2g p (ii) Weil pairing 32 (ii) (52) p T p (Jac(C)) T p (Jac(C)) lim n µ p n < >: V p V p Q p (1) 8

9 Q p (1) := (lim µ p n) n Z p Q p Frob q 1 x y V p < Frobx Froby >= q < x y > det(1 FrobT : V p ) = (qt ) 2g det(frob : V p ) 1 det(1 Frob(qT ) 1 : V p ) < > (55) det(frob : V p ) = q g p L p L 6 Z p - 61 Riemann-Hurwitz K L k g(k) g(l) 61 (Riemann-Hurwitz cf [Iw1] p 311 [DK] 814) L/K tame 2g(L) 2 = [L : K](2g(K) 2) + (e(v ) 1) e(v ) L/K v v :L 4 k = Q(ζ p ) Z p - λ - (X(k) λ- λ k ) ( 2 ) k CM λ - CM λ - Riemann-Hurwitz ( [Ki] [Iw5] p- ) 62 p- K k p k K K p- ( ) 62 (Grothendieck cf [NSW] Theorem 1012) Gal( K/K) 2g x 1 x 2 x 2g [x 1 x 2 ][x 3 x 4 ] [x 2g 1 x 2g ] = 1 pro-p 9

10 Z p - p- [Wi] [NSW] Chap X 7 Z p - [Iw4] 12 [Wa] Z p - F p F F F 1 p µ p F(µ p ) Gal(F(µ p )/F) Z p F(µ p )/F F Gal(F /F) = Z p F F p- F /F n F n K Z p - K K K := KF K F Γ := Gal(K /K) Γ = Gal(F /F) = Z p K /K n K n K n = KF n K n F n 72 p-part A(K) := Cl 0 (K)[p ] K p- K urp F /F p- K K urp KF K urp = K A(K) = Gal(K urp /K ) L/K A(L) Gal(L urp /L ) N L/K A(K) Gal(K urp /K ) A n := A(K n ) X(K) := lim A n n 10

11 X(K) Γ Z p - Z p [[Γ]] Γ γ (71) Z p [[Γ]] Λ γ T + 1 X(K) Λ := Z p [[T ]]- ( 4 ) 71 Λ- (72) X(K) = Gal(K urp /K ) (Γ Gal(K urp /K ) ) K /K 72 ω n (T ) := (1 + T ) pn 1 Λ A n = X(K)/ωn X(K) A n X(K) Λ- λ µ ν- λ K µ K ν K n A n = p λ Kn+µ K p n +ν K Proof (72) ω n X(K) M n M n /K n K urp K n K urp n K urp K urp n M n M n /K n K urp n M n K urp n = M n X(K)/ω n X(K) = Gal(K urp n /K ) = A n K urp 73 Z p - X(K) λ K µ K X(K) = Z λ K p λ K 2g (g K ) µ K = 0 X(K) Λ- Frob Γ Γ = Gal(F /F) Frobenius (71) Γ Frob 1 (Hyp): Cl 0 (K)[p] = (Z/p) 2g 74 K (Hyp) X(K) Γ Frob 1 Λ- char Λ (X(K)) = q g P K (T + 1) P K (T ) 32 (i) 75 C K Cl 0 (K)[p] = Jac(C)(F)[p] (Hyp) Jac(C)(F)[p] = Jac(C)[p] Weil-pairing F µ p K F /F KF KF (Hyp) (F Jac(C)[p] F(Jac(C)[p]) ) 11

12 74 C K K (Hyp) 76 (Hyp) Jac(C)(F )[p ] = Jac(C)[p ] Proof Gal(F(Jac[p ])/F) GL 2g (Z p ) GL 2g (Z/p) pro-p F p- F F(Jac[p ]) (Hyp) F µ p F µ p (24) (51) (54) (72) 76 (73) X(K) = T p (Jac(C)) 76 Jac(C)[p ] Gal(F/F) Gal(F /F) (73) Gal(F /F)- Γ Frob 1 X(K)( = T p (Jac(C))) µ 0 char(x(k)) = det((t + 1) Frob 1 : T p (Jac(C)) Z p Q p ) ( 43) det((t + 1) Frob 1 ) = det( Frob 1 ) det(1 Frob(T + 1)) 52 (55) 77 Jac(C)(F )[p ] = Jac(C)[p ] (Hyp) 74 K F := F(Jac(C)[p ]) p G Gal(F /F) = Z p G F Z p F Gal(F /F ) = Z p Gal(F /F ) = G X(K) = X(KF ) G G p X(KF ) G X(KF ) = Z 2g p 2g Z p - 73 X(K) char Λ (X(K)) Z p [T ] q g P K (T + 1) K = KF q = F 79 A := Cl 0 (K )[p ] A = lim A n n (Hyp) A = (Qp /Z p ) 2g (24) (72) : lim n A n = Hom(lim A n µ p ) n 12

13 References [AG] Cornell G and Silverman J H eds: Arithmetic Geometry Springer-Verlag New York (1986) [DK] : (1998) [Ho] Horie K: CM fields with all roots of unity Compositio Math 74 (1990) 1 14 [Iw1] : (1973) [Iw2] : 15 (1963) 65 67; Collected Papers [40] [Iw3] Iwasawa K: Analogies between number fields and function fields Some Recent Advances in the Basic Sciences Vol 2 (1969) pp ; Collected Papers [47] [Iw4] Iwasawa K: On Z l -extensions of algebraic number fields Ann of Math (2) 98 (1973) ; Collected Papers [52] [Iw5] Iwasawa K: Riemann-Hurwitz formula and p-adic Galois representations for number fields Tohoku Math J (2) 33 (1981) ; Collected Papers [58] [Iw6] : (1993) [Ki] Kida Y: l-extensions of CM-fields and cyclotomic invariants J Number Theory 12 (1980) [Ku] Kurihara M: On the ideal class groups of the maximal real subfields of number fields with all roots of unity J European Math Soc 1 (1999) [NSW] Neukirch J Schmidt A Wingberg K: Cohomology of number fields Grundlehren der Mathematischen Wissenschaften 323 (2000) [SU] : (1998) [Wa] Washington LC: Introduction to cyclotomic fields 2nd edition GTM 83 Springer-Verlag New York (1997) [We1] Weil A: Courbes algébriques et variétés Abéliennes Hermann Paris (1971) [We2] Weil A: Basic Number Theory 3rd edition Grundlehren der Mathematischen Wissenschaften 144 Springer-Verlag New York-Berlin (1974) [Wi] Wingberg K: On the maximal unramified p-extension of an algebraic number field J Reine Angew Math 440 (1993) yhachi@mathgakushuinacjp 13

[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,