Tite 素数巾導手実アーベル体の岩澤不変量 (Agebraic Number Theory and Reated Topics 2010) Author(s) 小松, 啓一 ; 福田, 隆 ; 森澤, 貴之 Citation 数理解析研究所講究録別冊 = RIMS Kokyuroku Bessa (2012), B32: 105-124 Issue Date 2012-07 URL http://hd.hande.net/2433/196246 Right Type Departmenta Buetin Paper Textversion pubisher Kyoto University
RIMS Kôkyûroku Bessatsu B32 (2012), 105 124 (Iwasawa invariants of rea abeian number fieds with prime power conductors) By (Keiichi Komatsu), (Takashi Fukuda), (Takayuki Morisawa) Abstract For each prime number ess than 10 4, we construct expicity an infinite famiy of number fieds for which both Iwasawa µ and λ invariants vanish. 1. k µ (k), λ (k), ν (k) k Z - k /k µ, λ, ν k /k n-th ayer k n -part e n e n = µ (k) n + λ (k)n + ν (k) (n >> 0) k µ (k) = λ (k) = 0 Greenberg (c.f. [6]) Received February 23, 2011. Revised September 13, 2011. 2000 Mathematics Subject Cassification(s): 2000 Mathematics Subject Cassification(s):11R30, 11R22, 11Y40 Key Words: Iwasawa invariants, Greenberg conjecture: Department of Mathematica Science, Schoo of Fundamenta Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan. e-mai: kkomatsu@waseda.jp Department of Mathematics, Coege of Industria Technoogy, Nihon University, 2-11-1 Shin-ei, Narashino, Chiba, Japan. e-mai: fukuda.takashi@nihon-u.ac.jp Department of Mathematica Science, Schoo of Fundamenta Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan. e-mai: da-vinci-0415@@moegi.waseda.jp c 2012 Research Institute for Mathematica Sciences, Kyoto University. A rights reserved.
106,, Probem 1.1. Probem 1.2. µ (k) = λ (k) = 0 k k µ (k) = λ (k) = 0 = 2 2 2 k/q k (cf. [9]) k µ 2 (k) = λ 2 (k) = ν 2 (k) = 0 k k/q k µ (k) = λ (k) = ν (k) = 0 [13] 2 µ 2 (k) = λ 2 (k) = 0 k µ 2 (k) = λ 2 (k) = 0 k Byeon [1] [11], Ono [12] k k µ (k) = λ (k) = ν (k) = 0 Ono, Byeon 1.1 k 1.1 p m 0 B p,m Q Z p - m-th ayer p = 2, 3 B p,m B 2,m = Q(2 cos 2π 2 m+2 ), B 2π 3,m = Q(2 cos 3 m+1 ). Ferrero-Washington [2] p µ (B p,m ) = 0 p = 2 1, 3 (mod 4) 2 c 1, 2 c 2 1 c p = 3 2 c 2 1 [ ] 1 2c + 2 og 2( 1) 2 if p = 2 (1.1) m p = [ 1 2c + 2 og 3( 1) + 1 ] 1 if p = 3 2 m p Theorem 1.3. p = 2, 3 p λ (B p,mp ) = 0 m 0 λ (B p,m ) = 0 λ (B p,mp ) = 0 Coroary 1.4. 10 4 m 0 λ (B 2,m ) = λ (B 3,m ) = 0.
107 Remark. m 0 λ 2 (B 2,m ) = λ 3 (B 3,m ) = 0 Remark. p = 2 3, 5 (mod 8) B 2,m [7] B 2,m (m 0) λ (B 2,m ) = 0 p = 3 2, 4, 5, 7 (mod 9) B 3,m [7] B 3,m (m 0) λ (B 3,m ) = 0 2. 1, 1 λ (B p,m ) = 0 m = G(B p,m /Q) ψ : m Q idempotent e ψ Z [ m ] λ (B p,m ) e ψ = 1 m σ m Tr(ψ(σ))σ 1 λ (B p,m ) = ψ λ,ψ (B p,m ) Tr Q (ψ( m )) Q trace ψ m Q λ (B p,mp ) λ,ψ (B p,m ) Lemma 2.1. 1 m m p m m p m Q ψ λ,ψ (B p,m ) = 0 λ (B p,mp ) = 0 (, ψ) Bernoui λ,ψ (B p,m ) = 0 Lemma 2.2. B 1,ω 1 ψ = 1 λ,ψ (B p,m ) = 0 B 1,ω 1 ψ < 10 4 B 1,ω 1 ψ 1 (, ψ) p = 2 7 p = 3 4 (, ψ) (, ψ) ψ p m B 1,ω 1 ψ 1 ψ ( ) p m 1 ζ k = exp(2π 1/k) p = 2 ζ 2 n+2 ζ 5 2 n+2 p = 3 ζ 3 n+1 ζ 4 3 n+1 m σ m = σ g 2 Q 1 p m η m η m g 1 p m (mod )
108,, m ψ m ψ m (σ) = η m m = ψ m B 1,ω 1 ψ 1 ψ = ψ k m P ψ(t ) ψ [8, Coroary 2] (, ψ) p m 1 [8] (C1) (H Pi,n) = (H i,n ) n = 2 λ,ψ (B p,m ) = 0 Tabe 1. p = 2 ψ case P ψ (T ) mod 2 31 ψ 1 (C) T + 186 1429969 193 ψ 25 6 (A) T + 33389 5521195777 257 ψ 97 7 (A) T + 12593 52145949697 521 ψ 3 (A) T + 204753 18101857409 641 ψ 17 7 (A) T + 223068 1213630714369 3617 ψ 23 5 (A) T + 11965036 60569710224641 4513 ψ 17 5 (A) T + 15930890 235307606264321 Tabe 2. p = 3 ψ case P ψ (T ) mod 2 73 ψ 1 (C) T + 2263 56018449 109 ψ 14 3 (A) T + 2289 1888152283 487 ψ 61 4 (C) T + 39934 280668166291 1621 ψ 55 4 (A) T + 2207802 16560570765169 3. Bernoui, p 4 if p = 2 q = p if p > 2 Q ω mod Teichmüer ψ mod qp m p m p a Z
109 ω 1 ψ(a) = ω 1 (a)ψ(a) Bernoui B 1,ω 1 ψ B 1,ω 1 ψ = 1 qp m qp m aω 1 (a)ψ(a) Q a=1 - p j (3.1) { i + j mod qp m 0 i < qp m } = { i 0 i < qp m } qp m B 1,ω 1 ψ = 1 = 0 i<qp m 0 j< = 0 j< + 1 ω 1 (j) 0 j< 0 j< ω 1 (j) (i + j)ω 1 (i + j)ψ(i + j) 0 i<qp m iψ(i + j) jω 1 (j) 0 i<qp m iψ(i + j) 0 i<qp m ψ(i + j) B 1,ω 1 ψ - B 1,ω 1 ψ = 1 (3.1) mod 0 if j = 0 ω 1 (j) 1 j (mod ) if 1 < j < p 2 3 p = 2 2 s 1 if 1 (mod 4) 2 s + 1 if 3 (mod 4) p = 3 s if 1 (mod 4) c = s + 1 if 3 (mod 4) 2 s 2 1, c = s (1.1) m p
110,, Theorem 3.1. ψ m m p + 1 mod qp m p m B 1,ω 1 ψ = 1. 1 m m p B 1,ω 1 ψ = 1 1 m 2c 2 2c 1 m m p 4. p = 2, 1 m 2c 2 B 1,ω 1 ψ mod 2 m+2 2 m ψ 2 m 1 Q - η m Q 1 2 m c m m 2 m Q - d m [Q (η m ) : Q ] c m d m = 2 m 1 1 if 1 (mod 4), 1 m s 2 m s if 1 (mod 4), s + 1 m d m = 1 if 3 (mod 4), m = 1 2 if 3 (mod 4), 2 m s 2 m s if 3 (mod 4), s + 1 m 2 m 1 if 1 (mod 4), 1 m s 2 s 1 if 1 (mod 4), s + 1 m c m = 1 if 3 (mod 4), m = 1 2 m 2 if 3 (mod 4), 2 m s 2 s 1 if 3 (mod 4), s + 1 m ζ 2 n+2 ζ 5 2 n+2 m = G(B 2,m /Q) σ m ψ m ψ m (σ) = η m m = { ψ k m 0 k < 2 m } m 2 m
111 ψ = ψm k { 1 k < 2 m k : odd } if 1 (mod 4), 1 m s { 1 k < 2 s k : odd } if 1 (mod 4), s + 1 m X m = { 1 } if 3 (mod 4), m = 1 { 1 k < 2 m 1 k : odd } if 3 (mod 4), 2 m s { 1 k < 2 s 1, 2 s < k < 2 s + 2 s 1 k : odd } if 3 (mod 4), s + 1 m X m = c m { ψm k k X m } m 2 m Q - B 1,ω 1 ψm k (3.1) O(2m+2 ) (4.1) B 1,ω 1 ψ k m (k X m) m s+1 O(2 s 1 2 m+2 ) s (eg. = 8191 s = 13) B 1,ω 1 ψ m (3.1) (4.2) B 1,ω 1 ψ m = 2 m 1 a i η i m 2 B 1,ω 1 ψ = m 1 2 m 1 a m k i ηm ki = b i ηm i (4.1) O(2 m+2 + 2 s 1 2 m ) ψ = ψ m (3.1) ψ m (± 5 i mod 2 m+2 ) = η i m { ψ m (j) 0 j < 2 m+2 } (j ψ m (j) = 0) (4.2) B 1,ω 1 ψ k m B(X) = 2 m 1 b i X i η m X η m if 1 (mod 4), 1 m s X 2m s η s if 1 (mod 4), s + 1 m F (X) = X + 1 if 3 (mod 4), m = 1 X 2 a m X + 1 if 3 (mod 4), 2 m s X 2m s a s+1 X 2m s 1 1 if 3 (mod 4), s + 1 m
112,, B(X) = F (X)G(X) + R(X), degr < degb B 1,ω 1 ψ k m = R(η m) (4.3) B 1,ω 1 ψ k m d m 1 = c i η i m (k X k ) mod 1 O(2 m ) (4.3) O(2 m+2 + 2 s 1 2 m + 2 s 1 2 m ) = O(2 m (4 + 2 s )) η m (1 m s) g 2 η m g 1 2 m (mod ) 1 a m = Tr Q (η m )/Q (η m ) (2 m s + 1) [3] Lemma 4.1. a 2 = 0 a m (3 m s + 1) a m = 2 + a m 1 (3 m s) a s+1 = 2 + a s Q mod F Lemma 4.2. 3 (mod 4) a F a F ( a ) = 1 = a = ± a +1 4 η m Q (η m ) Z B 1,ω 1 ψ k (4.3) Lemma 4.3. B 1,ω 1 ψ k m d m 1 = c i η i m (c i Z ) B 1,ω 1 ψ k m 0 (mod ) c i 0 (mod ) for a 0 i d m 1. 1 2 2
113 = 8191 c = 14 1 m 26 (3.1) m = 26 2 28 8191 = 2198754820096 2.1 10 12 TC C 2 5. p = 2, 2c 1 m m 2 p.387]) B 1,ω 1 ψ Sinnott-Washington (c.f. [14, 1 h(t ) = ω 1 (1 + 2 c i)t i Z [T ] h(t ) ψ m Lemma 5.1. m 2c 1 Q 1 2 m+2 c η m+2 c h(η m+2 c ) 0 (mod ) mod 2 m+2 2 m ψ B 1,ω 1 ψ 0 (mod ) h(η m+2 c ) O() m + 2 c s + 1 [Q (η m+2 c ) : Q ] = 2 m+2 c s h(η m+2 c ) O(2 m+2 c s ) k X m+2 c h(η k m+2 c) O(2 s 1 )O(2 m+2 c s ) = O(2 m+1 c ) = 8191, c = 14, m = m 2 = 32 2 m+1 c = 2 19 = 524288 1 m 2c 2 3 6. p = 3, 1 m 2c 2 B 1,ω 1 ψ mod 3 m+1 3 m ψ 2 3 m 1 Q - η m Q 1 3 m c m m 3 m Q - d m [Q (η m ) : Q ] c m d m = 2 3 m 1 2 TC C TC C 3 C TC
114,, 1 if 1 (mod 3), 1 m s 3 m s if 1 (mod 3), s + 1 m d m = 2 if 2 (mod 3), 1 m s 2 3 m s if 2 (mod 3), s + 1 m 2 3 m 1 if 1 (mod 3), 1 m s 2 3 s 1 if 1 (mod 3), s + 1 m c m = 3 m 1 if 2 (mod 3), 1 m s 3 s 1 if 2 (mod 3), s + 1 m ζ 3 m+1 ζ3 4 m+1 m = G(B 3,m /Q) σ m ψ m ψ m (σ) = η m m = { ψm k 0 k < 3 m } m 3 m ψ = ψm k { 1 k < 3 m 3 k } if 1 (mod 3), 1 m s { 1 k < 3 s 3 k } if 1 (mod 3), s + 1 m X m = { 1 k < 3m 1 3 k } if 2 (mod 3), 1 m s 2 { 1 k < 3s 1 3 k } if 2 (mod 3), s + 1 m 2 X m = c m { ψm k k X m } m 3 m Q - p = 2 ψ m ψ m (± 4 i mod 3 m+1 ) = ηm i (j 3 ψ m (j) = 0) η m X η m if 1 (mod 3), 1 m s X 3m s η s if 1 (mod 4), s + 1 m X 2 a m X + 1 if 2 (mod 3), 1 m s X 2 3m s a s X 3m s + 1 if 2 (mod 3), s + 1 m B 1,ω 1 ψ k m d m 1 = c i ηm i (k X k )
115 η m (1 m s) g 2 η m g 1 3 m (mod ) a m = Tr Q (η m )/Q (η m ) (1 m s + 1) [10] Lemma 6.1. a 1 = 1 X 3 3X a m 1 = 0 (2 m s) ( ) a m p = 3 η m Q (η m ) Z 4.3 7. p = 3, 2c 1 m m 3 Sinnott-Washington (c.f. [14, p.387]) 1 h(t ) = ω 1 (1 + 3 c i)t i Z [T ] Lemma 7.1. m 2c 1 Q 1 3 m+1 c η m+1 c h(η m+1 c ) 0 (mod ) mod 3 m+1 3 m ψ B 1,ω 1 ψ 0 (mod ) 8. ψ m = G(B p,m /Q) p m ψ - L- L (s, ψ) L (s, ψ) = g ψ ((1 + qp m ) 1 s 1) g ψ (T ) Z [[T ]] () g ψ (T ) distinguished P ψ (T ) Z [T ] (8.1) g ψ (T ) = u ψ (T )P ψ (T )
116,, () u ψ (T ) Z [[T ]] u ψ (0) 0 (mod ) P ψ (T ) P ψ (T ) Stickeberger 1 ξ n = 2qp m n+1 qp m n+1 a=1 ( ) 1 aω 1 B,n /Q (a)ψ(a) Z [Γ n ] a Γ n = G(B,n /Q) = G(B p,m B,n /B p,m ) ( ) B,n /Q a Frobenius (a, p) 1 ω 1 (a)ψ(a) = 0 ξ n 2qp m ξ n = 1 n+1 = 0 j< n+1 = 0 j< n+1 (j,)=1 0 i<qp m + 1 n+1 (i n+1 + j)ω 1 (i n+1 + j)ψ(i n+1 + j) 0 j< n+1 ( ) 1 iω 1 (j)ψ(i n+1 B,n /Q + j) j 0 i<qp m ( ) 1 jω 1 B,n /Q (j) ψ(i n+1 + j) j 0 j< n+1 0 i<qp m ( ) 1 ω 1 B,n /Q (j) iψ(i n+1 + j) j 0 i<qp m g 2 n 0 ( 1) n 2qp m ξ n = j=0 (Z/ n+1 Z) = g + n+1 Z ( ) j qp ω 1 (g j ) B,n /Q m 1 g iψ(i n+1 + (g j mod n+1 )). ( ) 1 B,n /Q i n+1 + j γ = ( ) B,n /Q 1 + qp m = ( B,n /Q g ) rn g r n 1 + qp m (mod n+1 ) r n r 0 O()
117 Lemma 8.1. r i+1 r i (mod ( 1) i ) (i 0) r i+1 { r i + k( 1) i 0 k 1 } Proof. g r i+1 1 + qp m (mod i+2 ) g r i (mod i+1 ) g r i+1 r i 1 (mod i+1 ) r i+1 r i 0 (mod ϕ( n+1 )) r n O( n+1 ) O((n + 1)) xr n 1 (mod n ) (8.2) ( 1) n 2qp m ξ n = j=0 ω 1 (g j )(γ 1 ) xj qp m 1 iψ(i n+1 + (g j mod n+1 )). ψ ψ = ψ k m ψ m ψ m (± 5 i mod 2 m+2 ) = η i m if p = 2 ψ m (± 4 i mod 3 m+1 ) = η i m if p = 3 (8.2) P ψ (T ) 2qp m (8.1) u ψ (T ) B 1,ω 1 ψ 1 degp ψ = 1 (8.2) mod n P ψ (T ) mod n η m Z η m g 1 p m (mod )
118,, 1 p m η m ) (g 1 n 1 p m (mod n ) ω (8.2) (8.3) ξ n n 1 ω(a) a n 1 (mod n ) a i (γ 1 ) i (mod n ) (a i Z) (8.3) n 1 ( 1 + T ) i g ψ (T ) a i 1 + qp m (mod n ) b i (1 + T ) i (mod n ) n 1 g ψ (T ) T min(n + 1, n 1) = n + 1 2 g ψ (T ) 0 g ψ (T ) (1 + T )g ψ (T ) + b i (i = n 1,..., 0) 1 + T n + 2 [5, 5.3] n+1 g ψ (T ) c i T i (mod (T n+2, n )) P ψ (T ) T + α (mod n ) α 0 (mod ), α 0 (mod 2 ) 9. (1 + T ) n 1 if ψ() 1 W (T ) = (1 + T ) n 1 T if ψ() = 1
119 Y (T )P ψ (T ) a (mod W (T )) (9.1) W (T ) = Y (T )P ψ (T ) + a Y (T ) P ψ (T ) mod n (9.1) mod n W ( α) 0 (mod n ) W (T ) (T + α)y (T ) (mod n ) Y (T ) mod n Y (T ) T γ 1 Y (T ) 1+T Y (T ) = Y 1 (1+T ) Y 1 (T ) W (T 1) (T 1 + α)y 1 (T ) (mod n ) Lemma 9.1. b 0 = 1, b i+1 = (1 α)b i (i 0) {b i } k 0 (k 1 T k 1 = (T 1 + α) b k 1 i T i) + b k 1 Proof. k = 0 k k + 1 T k+1 1 = T (T k 1) + T 1 (k 1 = (T 1 + α) b k 1 i T i+1) + (b k 1)T + T 1 ( k = (T 1 + α) b k i T i) + b k (T 1 + α) + (1 α)b k 1 i=1 ( k = (T 1 + α) b k i T i) + b k+1 1 k = n b n 1 (mod n )
120,, Lemma 9.2. b 0 = 0, b i+1 = (1 α)b i + 1 (i 0) {b i } k 0 T k (k 2 1 T 1 = (T 1 + α) b k 1 i T i) + b k Proof. k = 0 k k + 1 T k+1 1 T 1 = T (T k 1) + T 1 T 1 = T T k 1 T 1 + 1 (k 2 = (T 1 + α) b k 1 i T i+1) + b k T + 1 (k 1 = (T 1 + α) b k i T i) + b k (T 1 + α) + (1 α)b k + 1 i=1 (k 1 = (T 1 + α) b k i T i) + b k+1 k = n b n 0 (mod n ) ψ = ψm k idempotent e ψ = 1 ψ(σ 1 )σ Z [ m ] m σ m e ψ Y 1 (γ) c n = N Q(ζf )/B p,m B,n (1 ζ f ), f = qp m n+1 g 2 x 1, x 2 Z ζ f = ζ n+1ζ qp m x 1 g n (mod n+1 ), x 1 1 (mod qp m ) x 2 1 (mod n+1 ), x 2 1 (mod qp m ) H = { x i 1x j 2 mod f 0 i <, 0 j 1 }
121 c n = x H(1 ζ x f ) 4 x γ 1 + qp m (mod n+1 ), x γ 1 (mod qp m ) x σ 1 (mod n+1 ), x σ 5 (mod 2 m+2 ) if p = 2 x σ 1 (mod n+1 ), x σ 4 (mod 3 m+1 ) if p = 3 x γ, x σ Z Y 1 (γ) e ψ n 1 p m 1 j=0 a i γ i (mod n ) b i σ i (mod n ) 1 (mod qp m n+1 ) g z Z Lemma 9.3. ( ( p m 1( n 1 j=0 λ,ψ (B p,m ) = 0 z g 1 n g 1 p m (mod f) x H ) ) bj ) 1 ai n (1 z xxi γ xj σ ) 1 (mod ) 9.3 O(qp m n+1 ) O(qp m n+1 ) mod n 9.3 mod p = 2, m = 5, = 4513, n = 2 Xeon 2GHz 3 9.3 25 5 9.3 60 y j = ( ) ai (1 z xxi γ xj σ ) (0 j p m 1) n 1 x H 4 p 5 TC C
122,, ( y j ) ( p m 1 j=0 y b j j ) 1 n 1 (mod ) 10. p p m, n 1 G(B p,m B, /B, ) G(B p,m /Q) m B p,m Z - n-th ayer B p,m B,n -part A m,n ψ : m Q idempotent e ψ = 1 m σ m Tr(ψ(σ))σ 1 Z [ m ] A m,n A m,n ψ-part A m,n,ψ = ε ψ A m,n Tr Q (ψ( m )) Q trace A m,n A m,n = ψ A m,n,ψ ψ m Q n λ,m,ψ 0, ν,m,ψ A m,n,ψ = λ,m,ψ n + ν,m,ψ (n >> 0) λ,m = λ (B p,m ) (10.1) λ,m = ψ λ,m,ψ ψ m Q ψ Kerψ B p,m ψ m A m,n,ψ = Am,n,ψ (10.1) λ,m = (10.2) 1 m m ψ λ,m,ψ ψ m Q (10.2) Lemma 10.1. m > m p λ (B p,m ) λ (B p,mp ) = λ,m,ψ. m p <m m ψ ψ m Q
123 ψ m ω mod Teichmüer ψ = ψ 1 ω λ,m,ψ λ,m,ψ (10.3) λ,m,ψ λ,m,ψ λ,m,ψ Bernoui B 1,ω 1 ψ Lemma 10.2. B 1,ω 1 ψ = 1 λ,m,ψ = 0 Proof. B 1,ω 1 ψ 0 (mod ) ξ 0 0 (mod ) Mazur-Wies ξ 0 0 (mod ) λ,m,ψ = 0 (10.3) 10.1 Coroary 10.3. B 1,ω 1 ψ = 1 λ,m,ψ = 0 Coroary 10.4. m > m p = λ (B p,m ) = λ (B p,mp ). 1.3 Remark. Tabe 1,2 case (A) λ,ψ (B p,m ) = 0 (cf. [8, Remark 4]) 11 3 References [1] D. Byeon, Indivisibiity of cass numbers and Iwasawa λ-invariants of rea quadratic fieds, Compositio Math. 126 (2001), 249 256. [2] B. Ferrero and L. Washington, The Iwasawa invariant µ p vanishes for abeian number fieds, Ann. Math. 109 (1979), 377 395. [3] T. Fukuda and K. Komatsu, Weber, 8, http://tnt.math.metro-u.ac.jp/ac/2007/proceedings/ [4] T. Fukuda, K. Komatsu and T. Morisawa, On λ-invariants of Z -extensions over rea abeian number fieds of prime power conductors, preprint, 2010. [5] T. Fukuda and H. Taya,,, 12 (2002), 293 306. [6] R. Greenberg, On the Iwasawa invariants of totay rea number fieds. Amer. J. Math. 98(1976), 263 284.
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