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 p- p- -Leopoldt p- L- Kummer ζ(s p- L- ζ(s Drchlet L- p- Drchlet χ f χ Bernoull B n,χ Q[[T ]] (.2 f χ a= χ(at e at e f χt = B n,χ T n Drchlet L- L(s, χ Bernoull (.3 L( n, χ = B n,χ n, ( n Z χ f = ( = Bernoull B n = B n, p Q Q p- Q p Q p C p Drchlet
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(ζ p Q p q Drchlet χ = ω (.2 Q p [[T ]] Bernoull B n,ω Q p p- L- Kummer 5 p n 0 ( mod p n B,ω n B n n ( mod p Z p Bernoull (.4 B n (X = n ( n ( B X n, T ext e T = B n (X T n =0 B n,χ (.2 f χ F χ (.5 B n,χ = F n χ F χ a= χ(a B n ( a F χ Bernoull [Was] [AIK] Clausen-von Staudt n B n + p n p Z p n p 2. -Leopoldt p- L- p p = p C p p D C p p- 2
p- D f : D C p p- α D α D α s D α \{α} f(s = a n (s α n, Z, a 0, a n C p n= f α Laurent < 0 α a Laurent 0 f α p- α D p- f D p- [Gou] Prop.4.4.2 f(s = a n s n D α D ( n f(s = b m (s α m, b m = f (m (α = a n α n m m m=0 n=m C p C p [[s]] D -Leopoldt [KL] p- L- L p (s, χ ( -Leopoldt p- L- Drchlet χ D = {s C p s p < qp /(p ( > } p- χ p- L p (s, χ L p ( n, χ = ( χω n (pp n B n,χω (2. n, ( n Z n L p (s, D\{} p- s = p p- L- p- p- D = {s C p s p < qp /(p } (2.2 L p (s, χ = a { s + a n (s n, a n Q p, a = 0 : χ p : χ = χ s D χ = s D\{} (2.2 p- L- p- p- α D\{} D α = { s C p s α p < α p } D\{} s = α ( s α = α α 3 =0 ( s α α
Laurent L p (s, D\{} p- p- L- p- ( D C p p- f(s D f(s D C p p- f(s D α f(s α D α D f(s = (s α f(s, f(s = a n (s α n, 0, a 0 0 α f(s { α n } n D α α n α lm α n = α (α n α f(αn = f(α n = 0 f(α n = 0 n a 0 = f(α = lm f(αn = 0 f(s n D = {s C p s p < qp /(p } (2. p- χ χ( = (.2 n B n,χω n = 0 (2. χ p- L- L p (s, χ (2. (.3 n j ( mod p, 0 j < p (2.3 L p ( n, χ = ( χω j (pp n L( n, χω j, ( n Z p- L- L p (s, χ n n p j Drchlet L L(s, χω j p- Drchlet L Euler (2.4 L(s, χω j = N= χω j (N N s = l: ( χω j (ll s, Re(s > p N Euler χω j (pp n (2. (2.3 ζ(s s = L p (s, s = p Euler Kummer Euler χ Drchlet L s = (2.5 L(, χ = τ(χ f χ f χ a= χ (a log ζ a f χ, τ(χ = f χ a= χ(aζ a f χ L(, χ 0 τ(χ Gauss Drchlet L p- L 4
s = Euler (2.6 L p (, χ = ( χ(pp τ(χ f χ f χ a= χ (a log p ( ζ a f χ log p p- L p (, χ 0 L p (, χ 0 Leopoldt p- K p p p K p K p U p = { u K p u p = } U,p = { u K p u p < } U = U p U = U,p K E dagonal p p p p U U E = E U U E U E Z p K C r 2 r 2 Drchlet E Z-ran r + r 2 E Z p -ran Leopoldt Z p -ran E = r + r 2 δ = r + r 2 ( Z p -ran E 0 K Z p r 2 + + δ Leopoldt K Ax-Brumer Z p Z p K r = n = [K : Q] r 2 = 0 K R σ, σ 2,, σ n p K p K C K p C p U p = W U,p W p K p u U p u = w u w W u U,p K ε, ε 2,, ε n p- R p (K = det( log p ε σ j, j n K Drchlet X L p (, χ 0 ( χ X\{} R p (K 0 Z p -ran E = Z-ran E = n h(k d(k K K C p p- 2n h(kr p (K = d(k χ X\{} ( χ(pp L p (, χ p- Euler χ(pp K Leopoldt 5
L p (, χ 0 (2.6 p- L- [Was] 3. Drchlet χ f χ q lcm(f χ, q (3. lcm(f χ, q = dqp e, e 0, (p, d = d κ = + dq = dqp n ( n 0 n ( (3.2 Ker (Z/qp n Z (Z/qZ : a mod qp n a mod q p n κ mod qp n p a a = ω(a a Z p a ( mod q a mod qp n (3.2 (3.3 a κ n(a ( mod qp n, 0 n (a < p n Galos ( a mod a mod d, ω(a mod q, a mod qp n (Z/ Z (Z/dZ (Z/qZ }{{} (Z/dqZ κ mod qp n Gal(Q(ζ qn /Q Gal(Q(ζ dq /Q Gal(Q(ζ qn /Q(ζ dq = = Γ n σ a ( δ(a : ζ dq ζ a dq, γ n (a : ζ qn ζ κn(a (3.4 σ a = δ(aγ n (a Gal(Q(ζ qn /Q = Γ n χ ( n e (3.5 χ = θψ (Z/qn Z = Γ n, θ, ψ Γ n θ d qd Techmüller ω ψ Γ n qp e ( e 6
Q(ζ qp e Q p e Q e Z p Q /Q e-th layer χ θ Q(ζ qn /Q Stcelberger { } a ξ n = σa Q[Gal(Q(ζ qn /Q] (3.6 σ a Gal(Q(ζ qn /Q = 0<a<, (a,= a δ(a γ n (a Q[ Γ n ] { } x x [ x ] x { } x = x [ x ] ξ n Q p [ Γ n ] ωθ Q p [ ] (3.7 ε ωθ = ωθ (δ δ Q p [ ] Q p [ Γ n ] δ (3.8 ε ωθ ξ n = ξ n (θ ε ωθ ξ n (θ = a θω (aγ n (a Q q p [Γ n ] n 0<a<, (a, = Q p [Γ n ] ξ n (θ ε ωθ - (3.9 η n (θ = ( κγ n (κ ξ n (θ ( { } { } a aκ = κ θω (aγ n (aκ Q p [Γ n ] 0<a<, (a, = m n 0 φ m,n : Γ m Γ n (3.0 φ m,n : Q p [Γ m ] Q p [Γ n ], γ m (a γ n (a θ Q p (ζ fθ O θ ( η n (θ 2O[Γ n ] ( ξ n (θ 2O[Γ n ], θ ( φ m,n (η m (θ = η n (θ, φ m,n (ξ m (θ = ξ n (θ, ( m n 0 a 0 < a <, (a, = 7
{ a (3.9 C(a = κ } { } aκ Z p 2 ( p = 2 (3.9 η n (θ η n (θ = C(a θω (aγ n (aκ + C( a θω ( aγ n (( aκ a /2 = a /2 { } { } a qn a + =, ( a /2 C(a C( a θω (aγ n (aκ { } { } aκ (qn aκ + = C(a C( a = 2C(a 2Z p p = 2 ( γ n (a = γ n ( a θω ( a = θω (a a = a ω(a ξ n (θ ξ n (θ = a θ(aγ n (a a = ( a /2 = ( 2 = 2 a /2 a /2 a θ(aγ n (a + a /2 a θ(aγ n (a a θ(aγ n (a + a /2 a θ( aγ n ( a a /2 ω(a θ(aγ n (a θω (aγ n (a θ ( a θ(aγ n (a = a θ(aγ n (a a /2 = b mod Γ n b mod Γ n b mod Γ n ( a /2, a b ( mod ( a /2, a b ( mod a /2, a b ( mod b θ(aγ n (b mod O[Γ n ] θ(a b γ n (b b mod Γ n θ(a = θ(a = θ( δ = 0 2 2 a /2, a b ( mod a b ( mod δ a θ(aγ n (a 0 mod O[Γ n ] 2 a /2 a /2 a θ(aγ n (a 2O[Γ n ] p 2 ( p = 2 ( θω (aγ n (a = θω (aγ n (a a /2 = b mod Γ n b mod Γ n b mod Γ n ( a /2, a b ( mod ( a /2, a b ( mod a /2, a b ( mod 8 θω (a θ(a γ n (b γ n (b mod 2O[Γ n ]
0 mod 2O[Γ n ] p = 2 ( m n 0 φ m,n (ξ m (θ = q m = q m = q m = q m 0<a<q m, (a,q m = 0<b<, (b,= 0<b<, (b, = 0<b<, (b, = = ξ n (θ pm n 2 a θω (aγ n (a (b + θω (bγ n (b 0 <p m n ( θω (b (b + γ n (b 0 <p m n θω (b q ( m p m n b + γ n (b 2 θω (bγ n (b 0<b<, (b,= θω (bγ n (b = θω ( bγ n ( b = b b b θω (bγ n (b 0 φ m,n (ξ m (θ = ξ n (θ φ m,n ( κγ m (κ = κγ n (κ φ m,n (η m (θ = η n (θ ( (3.9 φ m,n Z p Galos Γ = Gal(Q(ζ dp /Q(ζ dq = lm Γ n κz p = + pz p Z p γ = lm γ n (κ : ζ qn ζq κ n ( n 0 O O[[Γ]] = lm O[Γ n ] O Λ = O[[T ]] lm ξ n (θ f(t, θ, θ lm η n (θ g(t, θ κγ κ( + T γ + T (3. O[[Γ]] Λ O[Γ n ] Λ/( + T pn γ n (κ + T mod ( + T pn η n (θ g(t, θ mod ( + T pn 9
θ Λ (3.2 f(t, θ = g(t, θ κ( + T θ = (3.2 f(t, Stcelberger f(t, θ (3.0 (3.9 ( { } { } a aκ g(t, θ κ θω (a( + T n(a (3.3 0<a<, (a, = mod ( + T pn 4. p- L- p- (4. ( n+ X n log p ( + X = n n= n ( n+ /n p = p vp(n n, n log p ( + X + X = κ = + dq (4.2 log p κ p = log p ( + dq p dq p = q p- (4.3 exp(x = X n n p p log n log p < v p( < n p, p n p p log n log p < < p n p p p /(p ( < [Was] [Gou] p pz p = 2 2 n p (4.4 q p n p + p 2 < p (4.2 D = {s C p s p < qp /(p } (4.5 s log p κ p = s p log p κ p < qp /(p q < p /(p 0
D p- (4.6 κ s = exp( s log p κ = ( log p κ n s n s κ s κ s κ s = ( + n=2 n 2 n = (p + j, ( log p κ n s ( n s log p κ j p p = 2 n = + j = (4.4 p s D p (log p κ n s n p = log p κ n p s n p p < p q n+ p n p = p j p p < p j p n 2 (4.7 (log p κ n s n p (log p κ n < s log p κ p s n p < ( log s = D κ s p κ n Z p ( log p κ n log p κ p p q n=2 (4.8 κ s + q s Z p [[s]] κ s p = s log p κ p < p /(p ( s D ψ p (4.9 ζ ψ = ψ(κ = χ(κ Q p (ζ ψ π = ζ ψ O ψ (4.6 ζ ψ κ s πo ψ [[s]] s (4.0 ζ ψ κ s = c n (s n, c n πo ψ ( n 0 D
p- L- χ = θψ θ ψ θ f(t, θ (4. L p (s, χ = f( ζ ψ κ s, θ. ζ ψ κ s step p- θ f(t, θ = s D (4.8 =0 z T n Λ = O[[T ]] ζ ψ κ s p = ζ ψ ( κ s + ( ζ ψ p < ( ζ ψ κ s p 0 ( f( ζ ψ κ s, θ D ( f( ζ ψ κ s, θ = z c n (s n t +t 2 + +t =n = =0 =0 ( z t +t 2 + +t =n c t c t2 c t (s n c t c t2 c t p π p 0, ( a n = f( ζ ψ κ s, θ = ( z =0 t +t 2 + +t =n c t c t2 c t a n (s n Q p [[s ]] θ = g( ζ ψ κ s, Q p [[s ]] D g(t, f(t, = κ ( + T f( ζ ψ κ s, = ζ ψκ s g( ζ ψ κ s, ζ ψ κ s κ D s ζ ψ κ s κ ψ ζ ψ D ζ ψ κ s κ = 0 κ pn (s = ( n s = κ ζ ψ p = ( ζ ψ + dq p = ζ ψ p ζ p p = p /(p (4.8 s D ζ ψ(κ s p = κs p < κ ζ ψ κ ζ ψ p ζ ψ κ s κ = κ ζ ψ ( ζψ (κ s = κ ζ ψ κ ζ ψ 2 ( ζψ (κ s =0 κ ζ ψ
(4.8 ζ ψ(κ s q κ ζ ψ π O ψ[[s]] s ζ ψ (κ s = c n (s n, c n q κ ζ ψ π O ψ ( n 0 θ s D ζ ψ κ s κ s D f( ζ ψ κ s, Q p [[s ]] ψ = χ = D\{} ζ κ s κ = κ(s ( κ s = (κ log p κ s ( (log p κ n + (s s n κ s κ = κ s ζ = s D (4.7 (log p κ n (s n p < n=2 ζ κ s κ = (κ log p κ s ( =0 n=2 n=2 (log p κ n (s n (4.7 n 2 (log p κ n < p D\{} ζ κ s κ s Q p[[s ]] f( κ s, D\{} s Laurent s = χ f( ζ ψ κ s, θ f( ζ ψ κ s, θ = n= a n (s n a n Q p, χ a = 0 D p- χ p- step 2 n 0 < a <, (a, = { } a { } a aκ κa = ( + dqa a a 2 a 2 = κ κ n(a a κa κ n(a+ ( mod n (a = n (a + θω (a = θω (a (3.3 g(t, θ a 2 θω (a ( + T n(a mod ( + T pn 0<a<, (a, = n T = ζ ψ κ m ( m N ( + T pn = (ζ ψ κ m pn = (( + dq pn m 0 ( mod T =ζψ κ m 3
a g( ζ ψ κ m, θ a a 2 ω (a θ(a ψ(a a m a a 2 θω (a ζ n(a ψ (κ n(a m a 2 χω m (a a m ( mod (κa m = ( a + a 2 m a m + ma m a 2 ( mod qn 2 qn 2 g( ζ ψ κ m, θ m ( a χω m (a (κa m a m a ( χω m (κ κ m a a χω m (a ma m a 2 χω m (a a m ( ζ ψ κm a χω m (κa (κa m a χω m (a a m χω m (a a m ( mod q 2 n g( ζ ψ κ m, θ ( ζ ψ κm χω m (a a m m a ( mod m m 0 ( n (4.2 f( ζ ψ κ m, θ = g( ζ ψκ m, θ ζ ψ κm = m lm n 0<a<, (a,= χω m (a a m (.5 (4.3 B m,χω m = a= χω m (a q m n B m ( a (.4 B m,χω m = a= ( χω m (a a m m 2 am + qn( 2 q n χω m (a a m m 2 a= a= χω m (a a m ( mod a= χω m (a a m q n a= χω m ( a ( a m a= χω m (a a m ( mod 4
0 n ( χω m (pp m B m,χω m = ( χω m (pp m lm n = lm n = lm n = lm n a= q n a= a= χω m (a a m χω m (a a m χω m (pp m lm n χω m (a a m lm 0<a<, (a,= n χω m (a a m (4.2 (4.4 q n a= q n a= χω m (pa (pa m f( ζ ψ κ m, θ = ( χω m (pp m B m,χω m m χω m (a a m χ = d = = qp n ζ ψ = a (a = s = m m = (p p M (4.2 lm (s f( s κs, = lm (p M pm f( κ (p pm, = lm M lm n = lm n qp n = lm n qp n qp n 0<a<qp n, (a,p= 0<a<qp n, (a,p= 0<a<qp n, (a,p= = lm n (p qp n qp n (a a (p pm lm M a (p pm = p f( κ s, s = p p- f( ζ ψ κ s, θ p- L- (2.2 5. Kummer p- L- Kummer p m n 0 ( mod p m n ( mod (p p a m n a (4.8 κ s + p s Z p [[s]] ( κ m ( κ n = p(m n ( p- 0 ( mod p a 5
f(t, ω m p- ( p m B m = L p ( m, ω m = f( κ m, ω m m f( κ n, ω n = L p ( n, ω n = ( p n B n ( mod p a n Z p Kummer n 0 ( mod p n κ s + p s Z p [[s]] κ n 0 ( mod p f(t, ω n p- B,ω n = ( ω n (p B,ω n = L p (0, ω n = f(0, ω n f( κ n, ω n = L p ( n, ω n = ( p n B n, n B n ( mod p n Z p 6. Drchlet L- Bernoull p- L- Stcelberger Galos annhlator Stcelberger p- L- (2.6 p- L- Stcelberger Herbrand-Rbet p- L- Drchlet L- p- p- L- p- p 0 < p Techmüller χ = θ = ω ωθ = ω f ω = p p- Z p Q(ζ p /Q(ζ p Galos = Gal(Q(ζ p /Q Γ = Gal(Q(ζ p /Q(ζ p Γ n = Gal(Q(ζ p n+/q(ζ p κ = + p Γ γ : ζ p n+ ζ κ p ( n 0 n+ Λ = Z p [ ][[Γ]] ε = ω (δ δ Z p [ ] p Λ δ 6
Λ ω - Λ = Z p [[T ]] γ ε γ + T (6. ε Λ = Zp [[Γ]]ε Z ( p [[Γ]] Z ( p [[T ]] = Λ lm ξ n (ω ε lm ξ n (ω f(t, ω ( = Z p δ ω (δ- Z ( p Z p ( ω Stcelberger ξ n ε - ξ n (ω f(t, ω f(t, ω p- L- (6.2 L p (s, ω = f( κ s, ω Z p Q(ζ p /Q(ζ p n-th layer Q(ζ p n+ p-sylow A n ω - A ( n = ε A n X = lm A n = p 2 =0 X ( X ( = ε X = lm A ( n (6. torson Λ- f(t, ω torson Λ- X ( p- Mazur-Wles < < p (6.3 char Λ X ( = f(t, ω Λ char Λ X ( Λ- (6.4 0 ( fnte X ( Λ/g ( Λ ( fnte 0 g ( p dstngushed char Λ X ( = g ( Λ 7
< < p Weerstrass (6.3 dstngushed P alg P ana (6.5 char Λ X ( = p µalg f(t, ω Λ = p µana P alg Λ, p µalg P alg = P ana Λ, λ ana = degp ana g (, λ alg = degp alg λ alg λ ana µ alg µ ana p- λ-, µ- [Was] < < p λ alg = λ ana µ alg = µ ana Ferrero-Washngton = µ ana = 0 µ alg CM X ( Λ [Was] Z p - (6.6 X ( Z λalg p Λ/f(T, ω Λ Z λana p Q p V alg = X ( Q p Q p [T ]/( g (, dm Q p V alg = λ alg V ana = Λ/f(T, ω Λ Q p Q p [T ]/(P ana, dmq p V ana = λ ana T T : x T x p V alg (6.7 p-det( T V alg = X ( /T X ( = A ( 0 = p v p(a ( 0 char( T V alg = g ( = P alg f(0, ω = L p (0, ω = B,ω V ana p p-det( T V ana = Λ/( T, f(t, ω = Z p /f(0, ω Z p (6.8 = Z p /( B,ω Z p = p v p(b,ω char( T V ana = P ana 8
( h (Q(ζ p = 2 p (6.9 2 B,ω <p, f(0, ω 0 T f(t, ω (6.7 (6.8 V alg V ana (6.0 p-det( T V alg = p v p(a ( 0 char( T V alg = P alg p-det( T V ana = p v p(b,ω char( T V ana = P ana Stcelberger p c (c σ c ξ n Z[Γ n ] (c σ c ξ n A n = 0 σ c = δ(cγ n (c < p (6. (c ω (cγ n (c ξ n (ω A ( n = ε ( (c σ c ξ n A n = 0 c p lm (c ω (cγ n (c = c ω (c( + T (c Λ, c ω (cγ n (c Z p [Γ n ] (6.2 f(t, ω X ( = lm ξ n (ω A ( n = 0 ( (6.4 f(t, ω f(t, ω Λ/g ( Λ Λ Λ/g ( Λ Λ (6.3 ( f(t, ω Λ/g ( Λ ( f(t, ω ( g ( P ana P alg = 0 g ( (6.0 V alg V ana < < p (.5 B,ω = p p a= a ω (a p p mod Z p 9
0 = 0 (6. = c = κ = + p pb,ω A ( 0 = 0 (6.9 A ( p-det( T V alg = p v p(a ( 0 = p vp(b,ω = p-det( T V ana ( deg char( T ( ( ( V alg = deg P alg = deg P ana = deg char( T V ana Q p (6.0 ω p- Herbrand-Rbet < < p A ( 0 = 0 p B p (6.4 v p (A ( 0 = 0 v p (B,ω = 0 Mazur-Wles A < < p 2 ( g (, g ( 2 = ( P ana P alg deg P alg = ( deg P alg = P ana P ana A X ( Λ P alg 20
[AIK],, 200. [Gou] F. Q. Gouvêa, p-adc Numbers 2nd ed. Unverstext, Sprnger, 997. [Iwa] K. Iwasawa, On p-adc L-functons Ann. of Math., 89 (969, 98 205 ( II 62 69. [Iwa2] K. Iwasawa, Lectures on p-adc L-functons Ann. Math. Studes #74, Prnceton Unversty Press, 972. [KL] T. Kubota and H. Leopoldt, Ene p-adsche Theore der Zetawerte ( Tel I: Enfuhrung der p-adschen Drchletschen L-Funtonen, J. Rene. Angew. Math., 23 (964, 328 339. [Kum] E. E. Kummer, Über ene allgemene Egenschaft der ratonalen Entwclungscoeffcenten ener bestmmten Gattung analytscher Functonen, J. Rene. Angew. Math., 4 (85, 368 372 ( I 358 362. [KKS],,, 998. [Was] L. C. Washngton Introducton to cyclotomc felds 2nd ed. Graduate Texts n Math. vol.83, Sprnger-Verlag New Yor, 997. Yasush Mzusawa 69-8555 3-4- e-mal : mzusawa@aane.waseda.jp 2