On a branched Zp-cover of Q-homology 3-spheres

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1 Zp 拡大と分岐 Zp 被覆 GL1 表現の変形理論としての岩澤理論 SL2 表現の変形理論 On a branched Zp -cover of Q-homology 3-spheres 植木 潤 九州大学大学院数理学府 D2 December 23, 2014 植木 潤 九州大学大学院数理学府 D2 On a branched Zp -cover of Q-homology 3-spheres

2 Z p Z p Arithmetic topology p Z p := lim n Z/p n Z p Z p = { n a n p n a n Z} Z Z p Z Z p R[T ] R[[T ]] n mod p n : Z p Z/p n Z

3 Z p Arithmetic topology K S 3 X = S 3 \ K τ : π 1 (X ) Z Z-cover X X n Z Z/nZ τ τ n Ker τ n < π 1 (X ) n X n X K (Fox ) K n M n M Γ := Gal(X X ) = Deck(X X ) = t, Λ := Z[Γ] Λ H 1 (X ) H 1 (X ) Λ = K,1 (t) H 1 (M n ) K,1 (t)

4 Arithmetic topology Z p Arithmetic topology F /k QHS 3 N M primes S = {p 1,..., p s } L = K 1... K s π 1 (Spec O k \ S) π 1 (M \ L) (Artin ) Hurwicz 1 = Cl(k) = Gal(kab ur 1(M) ab = H 1 (M) = Gal(Mab unbr M) #Cl(k) < #H 1 (M) < ( M QHS 3 ) iff Alexander-Fox Z p k /k Z X X Alexander -Mazur π ab

5 Z p Arithmetic topology 1 Z p Arithmetic topology 2 Z p Z p Λ Γ X 3 4 GL 1 Z p GL 1 5 Appendix: 2014 April

6 Z p Z p Z p Λ Γ X k k /k Z p def Gal(k /k) = Z p. k /k Z p (in C) p n k n /k k = k 1 k 2... k n... k = n k n. Q 1 p Q(ζ p ) Z p p Z p Z p n 0 Z p k /k n0

7 Z p Z p Z p Λ Γ X Cl(k) p-part Cl(k) (p) Theorem ( ) k /k Z p p n k n λ, µ, ν n 0 n > n 0 #Cl(k n ) (p) = p λn+µpn +ν. λ, µ, ν [Iwasawa 1959] On Γ-extensions of algebraic number fields

8 Z p Z p Λ Γ X Z p Z p -cover p n A n Definition L M QHS 3 p n {h n : M n M} L M QHS 3 Z p -cover n h n h n+1 subcover h n,n+1 : M n+1 M n h n+1 = h n h n,n+1 Remark. X = M \ L {Z p -cover br. over L} 1:1 { τ : π 1 (X ) Z p } up to isom.

9 Z p Z p Z p Λ Γ X L M QHS 3 X := M \ L τ : π 1 (X ) Z p τ : π 1 (X ) Z p 1:1 Z p -cover br. over L (up to isom) τ : π 1 (X ) Z τ mod p n : π 1 (X ) Z/p n Z τ : π 1 (X ) Z p L Z p -cover Z-cover L = K 1 K 2 S 3 µ i H 1 (X ) K i p 1 mod 4 1 Z p τ : µ 1 1, µ 2 1 Z-cover Z p -cover

10 Z p Z p Z p Λ Γ X Theorem ( U. ) L QHS 3 Z p -cover {h n : M n M} λ, µ, ν n 0 n > n 0 #H 1 (M n ) (p) = p λn+µpn +ν. λ, µ, ν [Hillman, Matei, Morishita 2006] for K S 3 [Kadokami, Mizusawa 2008] for L M, Z-cover

11 Λ 1 Z p Z p Λ Γ X Z p Γ = t Λ := Z p [[ t ]] = Z p [[T ]]; t 1 + T Lemma (Washington, Chapter 13) E Λ (1) E Λ iff E/(p, T ) (2) E Λ E Λ r i Λ/(f e i i ) j Λ/(p m j ), r, e i, m k Z, f i Z p [T ], f i Weierstrass E E def hom f : E E such that Ker(f ), Coker(f ) <

12 Λ 2 Z p Z p Λ Γ X Lemma ( ( ) 2.45, 2.50) E Λ (1) E/(t pn 1) char Λ (E) iff (t pn 1) (2) (1) λ, µ, ν n 0 #(E/(t pn 1)) = p λn+µpn +ν, n > n 0 (3) t pn 1 1 f (t 1) #E/( tpn 1 f (t 1) )

13 Z p Γ X Z p Z p Λ Γ X k n p k n X n := Gal( k n /k n ) = CFT Cl(k n ) (p) k n /k k := k n k Γ := Gal(k /k) = t = Z p X = lim X n = Gal( k /k ) X Λ Lemma Z p k /k (1) k /k Cl(k n ) (p) = X /(t p n 1) (2) k /k s Y s.t., (X : Y) < Cl(k n ) (p) = X /( t pn 1 t 1 )Y

14 Zp 拡大と分岐 Zp 被覆 GL1 表現の変形理論としての岩澤理論 SL2 表現の変形理論 Zp 拡大の岩澤類数公式 分岐 Zp 被覆と岩澤型公式 Λ 加群の補題 Γ と X Zp 拡大の Γ と X の図 植木 潤 九州大学大学院数理学府 D2 On a branched Zp -cover of Q-homology 3-spheres

15 Z p Γ X Z p Z p Λ Γ X n p M n M n X n := Gal( M n /M n ) = Hur H 1 (M n ) (p) M n M M n Γ = t := lim Gal(M n M) = Z p X := lim X n X Λ Lemma L Z p -cover (1) L H 1 (M n ) (p) = X /(t p n 1)X (2) L (X : Y) < Y H 1 (M n ) (p) = X /( t pn 1 t 1 )Y Remark.

16 Zp 拡大と分岐 Zp 被覆 GL1 表現の変形理論としての岩澤理論 SL2 表現の変形理論 Zp 拡大の岩澤類数公式 分岐 Zp 被覆と岩澤型公式 Λ 加群の補題 Γ と X 分岐 Zp 被覆の Γ と X の図 植木 潤 九州大学大学院数理学府 D2 On a branched Zp -cover of Q-homology 3-spheres

17 Z p 1 Z p Z p Λ Γ X Λ G n,n := Gal( M N M n ), G n := lim N G n,n (t pn 1)X = [G n, G n ] G n /X = Γ n Γ n = t pn X [G n, G n ] (t pn 1)x = t pn x t pn x 1 = [ t pn, x], (x X ) (t pn 1)X compact Hasudorff (1) I n < G n < G 1 I n X = 1, G n = In X H 1 (M n ) (p) = Gal( Mn /M n ) = G n /I n [G n, G n ] = I n X /I n (t pn 1)X = X /(t pn 1)X

18 Z p 2 Z p Z p Λ Γ X (2) L = K 1... K s K i I i < G 1 Γ = Z p Z p t i I i t Y = (t 1)X, t 2 t1 1,..., t st1 1 < G 1 (X : Y) < H 1 (M n ) (p) = X /( t pn 1 t 1 )Y X /(t pn 1) = Gn ab X n t pn i < G 1 H 1 (M n ) (p) = Xn = (X /(t p n 1)X )/( t pn 1 = X /( (t pn 1)X, t pn 1 tj n,..., tpn s,..., tpn s X ) X ) = X / (t pn 1)X, t pn 2 t pn 1,..., ts pn t pn 1. = (t j t 1 1 )(t 1 t j t 1 1 t 1 1 )(t1 2 t j t 1 1 t 2 1 )...(t n 1 1 t j t 1 1 t (n 1) 1 )t1 n t X (x X tx = x t = t 1 xt 1 1 ) t pn j t pn 1 = (t j t 1 1 ) 1+t+...+tp n 1 = ( tpn 1 ) (1 t)x, t t 1 2t 1 1,..., t s t 1 1 = ( tp n 1 t 1 )Y.

19 Z p Z p Λ Γ X TLNcover QHS 3 Z p -cover Remark. (1) [K i ] = 0 in H 1 (M) iff K i (2) L = K i [K i ] = 0 τ : µ i 1 Z p -cover TLN Alexander TLN L (t) iff M n QHS 3.

20 Theorem (periodic knot, Gordon 1972) K S 3 Z-cover M S 3 r M r S 3 H 1 (M r ) = H 1 (M r+m ), for all r m 1 (t)/ 2 (t) t m 1 i (t) i-th elementary ideal (= (i + 1)-th Fitting ideal) #H 1 (M r ) < Q-HS 3 Z p -cover 1 (t)/ 2 (t) iff iff λ = µ = 0.

21 Fitting ideal. Z p 1 (t)/ 2 (t) iff λ = µ = 0.

22 Fitting ideal. Z p 1 (t)/ 2 (t) iff λ = µ = 0.. Yes. Λ X j Λ/(f j ) s.t., f j f j+1 i (t) = j i f j (i + 1)-th Fitting ideal f 1 = 1 (t)/ 2 (t) iff X 0 iff λ = µ = 0. : Z/2Z Z/4Z Z/3Z = Z/2Z Z/12Z by CRT

23 Z p Z p GL 1 Z[[T ]] u = ±1 + T f (T ) p Z p = lim Z/p n Z Z p = {x + py x 0 mod p} u 1 mod p u 1 + pz p (1 + pz p ) = Z p ; 1 + p 1 non canonical Z p (1 + pz p )

24 Z p GL 1 R Z p m R R/m R = Fp ρ : π 1 (X ) GL 1 (F p ) ρ : π 1 (X ) GL 1 (R) such that ρ mod m R = ρ ρ R 1 Z p Z p = (1 + pzp ) GL 1 (F p ) R = Z p

25 Z p GL 1 Z p R s.t., R/m R = F p ρ : π 1 (X ) GL 1 (F p ) ρ ρ univ : π 1 (X ) R univ ρ ρ : π 1 (X ) GL 1 (R) φ : R univ R φ ρ univ = ρ GL 1 (F p ) R univ = Λ = Z p [[T ]] : ρ univ : π 1 (X ) Λ = Z p [[T ]]; µ 1 + T.

26 GL 1 (F p ) Z p GL 1 ρ R Defo( ρ, R) R Defo( ρ, R) 1 + m R ; ρ ρ(1), Hom(Z p [[T ]], R) 1 + m R ; φ 1 + φ(t ) R Defo( ρ, R) R Hom(Z p [[T ]], R) R 1 + m R R Defo( ρ, R) Z p [[T ]] Defo( ρ, R) Hom(Z p [[T ]], R); ρ (φ ρ : f (T ) f (ρ(1) 1)) ρ univ : H 1 (X ) GL 1 (Z p [[T ]]); µ 1 + T.

27 Z p GL 1 X = S 3 \ K ρ univ : π 1 (X ) Λ GL 1 (F p ) Alexander ( ) Λ = Z p [[Γ]] X = lim H 1 (X p n, Z p ) = H 1 (X, Λ) = H 1 (π 1 (X ), Λ) = H 1 (ρ univ ). H 1 (ρ univ ) = Λ/( K (t)).

28 Appendix: 2014 April [Morishita Takakura Terashima U.] (on arxiv) = =

29 Appendix: 2014 April [Morishita Takakura Terashima U.] (on arxiv) = = Thurston Dehn Mazur R=T

30 Zp 拡大と分岐 Zp 被覆 GL1 表現の変形理論としての岩澤理論 SL2 表現の変形理論 SL2 表現の変形理論の展望 Appendix: 2014 April Thank you for listening! 植木 潤 九州大学大学院数理学府 D2 On a branched Zp -cover of Q-homology 3-spheres

31 Zp 拡大と分岐 Zp 被覆 GL1 表現の変形理論としての岩澤理論 SL2 表現の変形理論 植木 潤 九州大学大学院数理学府 D2 SL2 表現の変形理論の展望 Appendix: 2014 April On a branched Zp -cover of Q-homology 3-spheres

32 Zp 拡大と分岐 Zp 被覆 GL1 表現の変形理論としての岩澤理論 SL2 表現の変形理論 植木 潤 九州大学大学院数理学府 D2 SL2 表現の変形理論の展望 Appendix: 2014 April On a branched Zp -cover of Q-homology 3-spheres

33 Zp 拡大と分岐 Zp 被覆 GL1 表現の変形理論としての岩澤理論 SL2 表現の変形理論 植木 潤 九州大学大学院数理学府 D2 SL2 表現の変形理論の展望 Appendix: 2014 April On a branched Zp -cover of Q-homology 3-spheres

34 Zp 拡大と分岐 Zp 被覆 GL1 表現の変形理論としての岩澤理論 SL2 表現の変形理論 植木 潤 九州大学大学院数理学府 D2 SL2 表現の変形理論の展望 Appendix: 2014 April On a branched Zp -cover of Q-homology 3-spheres

35 Zp 拡大と分岐 Zp 被覆 GL1 表現の変形理論としての岩澤理論 SL2 表現の変形理論 植木 潤 九州大学大学院数理学府 D2 SL2 表現の変形理論の展望 Appendix: 2014 April On a branched Zp -cover of Q-homology 3-spheres

36 Zp 拡大と分岐 Zp 被覆 GL1 表現の変形理論としての岩澤理論 SL2 表現の変形理論 植木 潤 九州大学大学院数理学府 D2 SL2 表現の変形理論の展望 Appendix: 2014 April On a branched Zp -cover of Q-homology 3-spheres

37 Zp 拡大と分岐 Zp 被覆 GL1 表現の変形理論としての岩澤理論 SL2 表現の変形理論 植木 潤 九州大学大学院数理学府 D2 SL2 表現の変形理論の展望 Appendix: 2014 April On a branched Zp -cover of Q-homology 3-spheres

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