RIMS Kôkyûroku Bessatsu B32 (2012), (Iwasawa invariants of rea abeian number fieds with prime power conductors) By (Keiichi Komatsu), (Takashi

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Bessa (2012), B32: 105-124 Issue Date 2012-07 URL http://hd.hande.

1 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: Issue Date URL Right Type Departmenta Buetin Paper Textversion pubisher Kyoto University

2 RIMS Kôkyûroku Bessatsu B32 (2012), (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, Revised September 13, 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, Okubo, Shinjuku, Tokyo , Japan. e-mai: kkomatsu@waseda.jp Department of Mathematics, Coege of Industria Technoogy, Nihon University, 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, Okubo, Shinjuku, Tokyo , Japan. e-mai: da-vinci-0415@@moegi.waseda.jp c 2012 Research Institute for Mathematica Sciences, Kyoto University. A rights reserved.

3 106,, Probem 1.1. Probem 1.2. µ (k) = λ (k) = 0 k k µ (k) = λ (k) = 0 = 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 m 0 λ (B 2,m ) = λ (B 3,m ) = 0.

4 107 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 ) = , 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 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 )

5 108,, 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 ψ 25 6 (A) T ψ 97 7 (A) T ψ 3 (A) T ψ 17 7 (A) T ψ 23 5 (A) T ψ 17 5 (A) T Tabe 2. p = 3 ψ case P ψ (T ) mod 2 73 ψ 1 (C) T ψ 14 3 (A) T ψ 61 4 (C) T ψ 55 4 (A) T Bernoui, p 4 if p = 2 q = p if p > 2 Q ω mod Teichmüer ψ mod qp m p m p a Z

6 109 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

7 110,, 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

8 111 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 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

9 112,, 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 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 (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

10 113 = 8191 c = 14 1 m 26 (3.1) m = = 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 = m 2c 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

11 114,, 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 )

12 115 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 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 )

13 116,, () 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()

14 117 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 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 )

15 118,, 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 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

16 119 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 )

17 120,, 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 (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 }

18 121 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 y j = ( ) ai (1 z xxi γ xj σ ) (0 j p m 1) n 1 x H 4 p 5 TC C

19 122,, ( 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 m > m p λ (B p,m ) λ (B p,mp ) = λ,m,ψ. m p <m m ψ ψ m Q

20 123 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 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 B 1,ω 1 ψ = 1 λ,m,ψ = 0 Coroary 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), [2] B. Ferrero and L. Washington, The Iwasawa invariant µ p vanishes for abeian number fieds, Ann. Math. 109 (1979), [3] T. Fukuda and K. Komatsu, Weber, 8, [4] T. Fukuda, K. Komatsu and T. Morisawa, On λ-invariants of Z -extensions over rea abeian number fieds of prime power conductors, preprint, [5] T. Fukuda and H. Taya,,, 12 (2002), [6] R. Greenberg, On the Iwasawa invariants of totay rea number fieds. Amer. J. Math. 98(1976),

21 124,, [7] K. Horie, Certain primary components of the idea cass group of the Z p -extension over the rationas, Tohoku Math. J. 59 (2007), [8] H. Ichimura and H. Sumida, On the Iwasawa Invariants of certain rea abeian fieds II, Inter. J. Math. 7 (1996), [9] K. Iwasawa, A note on cass numbers of agebraic number fieds, Abh. Math. Sem. Univ. Hamburg 20 (1956), [10] T. Morisawa, A Cass Number Probem in the Cycotomic Z 3 -extension of Q, Tokyo J. Math. 32 (2009), [11] J. Nakagawa and K. Horie, Eiptic curves with no rationa points, Proc. Amer. Math. Soc. 104 (1988), [12] K. Ono, Indivisibiity of cass numbers of rea quadratic fieds, Compositio Math. 119 (1999), [13] M. Ozaki and H. Taya, On the Iwasawa λ 2 -invariants of certain famiies of rea quadratic fieds, Manuscripta Math. 94 (1997), no. 4, [14] L. C. Washington, Introduction to Cycotomic Fieds, 2nd edition, Graduate Texts in Math., 83, Springer-Verag, New York, Heideberg, Berin, 1997.

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