K-groups of rings of algebraic integers and Iwasawa theory Miho AOKI Okayama University of Science 1
F/Q : finite abelian extension O F : the ring of algebraic integers of F K n (O F ) (n 0) H i (O F, Z p (n)) := Het í (Spec(O F[1/p]), Z p (n)) (i = 0, 1, 2, n Z) H i (G Σ (F ), Z p (n)) (:= lim H i (G Σ (F ), Z/p m Z(n))) Σ : all Archimedean primes & primes above p F Σ /F : the maximal algebraic Σ-ramified extension G Σ (F ) : =Gal(F Σ /F ) 2
p: odd prime number Conjecture (Quillen-Lichtenbaum) n 1 p-adic Chern class maps : K 2n+1 (O F ) Z Z p H 1 (O F, Z p (n + 1)) K 2n (O F ) Z Z p H 2 (O F, Z p (n + 1)) are isomorphisms. Remarks surjectivity & finite kernel [Soulé(1 n p 1), Dwyer and Friedlander (n 1))] K: totally imaginary field & p = 2 [Kahn] 3
Lemmas on H i (i = 1, 2) fr Zp H i (O F, Z p (n)) := H i (O F, Z p (n))/tor Zp H i (O F, Z p (n)) F := F (µ p ) G :=Gal(F /F ) w n (F ) :=max{n Gal(F (µ N )/F ) n = {1}} (n 0) w n (F ) = p w(p) n (F ) A m := p-part of the Σ-class groups of F m := F (µ p m+1) X := lim A m 4
Lemma (H 1 ) n 0 (1) tor Zp H 1 (O F, Z p (n)) Q p /Z p (n) G µ n w (p) n (F ) (2) fr Zp H 1 (O F, Z p (n)) Hom(G Σ (F ) ab Z p, Z p (n)) G Lemma (H 2 ) n 1 loc 2 0 X (n 1) G H 2 (O F, Z p (n)) p p µ 1 n w (p) 1 n (F p) µ 1 n w (p) 1 n (F ) 0 ( ) :=Hom(, Q p /Z p ) loc 2 : H 2 (G Σ (F ), Z p (n)) p p H2 (G Fp, Z p (n)) Keys of the Proofs Duality & Long exact sequence of Poitou-Tate. 5
K-groups and cyclotomic Z p -extensions F : totally real number field F m := F (µ p m+1), F := F (µ p ) :=Gal(F 0 /F ), Γ m :=Gal(F m /F 0 ), Γ :=Gal(F /F 0 ) G m :=Gal(F m /F ) Γ m G :=Gal(F /F ) Γ κ : G Z p cyclotomic character i.e. ζ τ = ζ κ(τ) for τ G, ζ µ p Ĝ Γ κ = ω κ 6
For even character χ Ĝm, L p (χ, s) s.t. L p (χ, 1 n) = L(χω n, 1 n) p p (1 χω n (p)n(p) n 1 )) for all integers n 1 0 Conjecture (L p ) L p (χ, 1 n) 0 for all even cheracters χ Ĝm and integers n 0. d i (n) := rank Zp H i (O Fm, Z p (n)) = corank Zp H i (O Fm, Q p /Z p (n)) Conjecture (d 2 )) d 2 (n) = 0 (n 1) Conjecture (d 1 ) d 1 (n) = [F m : Q]/2 (n 0, 1) [F m : Q]/2 + 1 (n = 0) 7
Remarks d 2 (1) = {p p p} 1, d 1 (1) = d 2 (1) + [F m : Q]/2 Finiteness of even-numbered K-groups d 2 (n) = 0 (n 2) Conjecture (L p ) + µ = 0 Conjecture (d 2 ) Conjecture (d 1 ) 8
Conjecture m 0, n 2 : integers. For all integers j, (K 2n 2 (Z[µ p m+1]) Z Z p ) ωj (K 2n 1 (Z[µ p m+1]) Z Z p ) ωj 0 if n \ j (mod 2) or n j (mod p 1), ( Λ/(f ω n j, g (m) 1 n )e ω j n+1 ) (n 1) otherwise. µ n D p,n,j if n j (mod 2), Hom Zp (Λ/(g (m) n )e ω n j, Z p (n)) otherwise. g (m) n = γ pm κ n (γ pm ) Λ := Z p [Γ] N p,n,j, D p,n,j : the power of p such that ψ Γ m L p (1 n, ω n j ψ) = N p,n,j D p,n,j Q p (N p,n,j, D p,n,j Z p, v p (N p,n,j ) = 0 or v p (D p,n,j ) = 0, N p,n,j p N p,n,j and D p,n,j p D p,n,j ) f ω n j Λ Z p [[T ]] such that f ω n j((1 + p) s 1) = L p (s, ω n j ) 9
Brumer conjecture and Coates-Sinnott conjecture F : totally real number field K/F : finite abelian extension G :=Gal(K/F ) S : finite set of primes of F containing all Archimedean primes & primes which ramify in K Definition (Partial zeta function) For σ G, ζ F,S (σ, s) = Na s (Re(s) > 1). (a,k/f )=σ a is prime to S Theorem [Klingen and Siegel] For all integers n 0, ζ F,S (σ, n) Q. Definition (Higer Stickelberger elements) For an integer n 0, θ S (n) = σ G ζ F,S (σ, n)σ 1 Q[G]. 10
Theorem (Integrality) [Iwasawa, Coates, Sinnott, Cassou-Noguès, Deligne and Ribet] For all integers n 0, ann Z[G] (µ n+1 w n+1 (K) ) θ S(n) Z[G]. Lemma n 0 ann Z[G] (µ n+1 w n+1 (K) ) = Na n+1 (a, K/F ) integral ideals a prime to S and w n+1 (K) Z[G]. (Na n+1 (a, K/F ))θ S (n) = σ G δ n+1(σ, a)σ 1, where δ n+1 (σ, a) = Na n+1 ζ F,S (σ, n) ζ F,S (σ(a, K/F ), n). Then Theorem (Integrality) δ n+1 (σ, a) Z. 11
Theorem (Congruence) [Iwasawa, Coates, Sinnott, Doligne and Ribet] Assume S contains all primes above p. δ n+1 (σ, a) (Nb σ a) n δ 1 (σ, a) mod w n (K)Z p where b σ is an integral ideal of F s.t. (b σ, K/F ) = σ. Theorem [Coates] Theorem (Integrality) + Theorem (Congruence) p-adic L-function exsists. 12
Conjecture (Brumer) ann Z[G] (µ K ) θ S (0) ann Z[G] (Cl K ). Conjecture (Brumer-p)) p : prime number. ann Zp [G](µ K Z Z p ) θ S (0) ann Zp [G](Cl K Z Z p ). Remarks Conjecture (Brumer) F = Q [Stickelberger]. Conjecture (Brumer-p) CM fields K such that p [K : F ] [Wiles] Conjecture (Brumer-p) CM fields K under some assumptions including a condition on the µ-invariant. [Greither] 13
Conjecture (Coates-Sinnott) n 1 : integer (1) ann Z[G] (tor Z (K 2n+1 (O K ))) θ S (n) Z[G] (2) ann Z[G] (tor Z (K 2n+1 (O K ))) θ S (n) ann Z[G] (K 2n (O K )) Conjecuture (Coates-Sinnott-p) p: prime number, n 1: integer (1) ann Zp [G](tor Zp (K 2n+1 (O K ) Z Z p )) θ S (n) Z p [G] (2) ann Zp [G](tor Zp (K 2n+1 (O K ) Z Z p )) θ S (n) ann Zp [G](K 2n (O K ) Z Z p ) Remarks Conjecture (Quillen-Lichtenbaum) + Lemma (H 1 ) + Theorem (Integrality) Conjecture (Coates-Sinnott-p) (1) 14
In cases F = Q and finite abelian extensions K/Q For n = 1 and odd prime numbers p [Coates and Sinnott] For integers n and odd prime numbers p (under Conjecture(Q-L)) [Nguyen Quang Do] For all results below, assume Conjecture(Q-L) In cases finite abelian extensions K/F of totally real fields both F and K For even integers n and odd prime numbers p under some assumption on the µ-invariant [Nguyen Quang Do] In cases finite abelian extensions K/F of totally real fields F and CM fields K For odd prime numbers p under some assumptions [Burns and Greither] For odd integers n and odd prime numbers p v p such that p n[k : F ] w(p) n (K v ) [Banaszak] w n (p) (K) 15
Theorem p : odd prime number Assume: I. Conjecture (Quillen-Lichtenbaum). II. S contains all primes above p. III. Conjecture (Brumer-p) for K m /F (K m = K(µ p m+1)) with sufficiently large m. v p w(p) n (K v ) IV. w (p) n (K) = 1. Then, Conjecture (Coates-Sinnott-p) is true. Remarks on the condition IV In particular, the condition IV holds in the following cases p does not ramify in K and (p 1) n. p does not split in K. The condition yields that the localization map : H 2 ét(spec(o K [1/p]), Z p (n + 1)) v p H 2 (K v, Z p (n + 1)) is a zero map. 16
Outline of the proof Proposition Assume condition II. For all integers m, n 0 and integral ideals a which are coprime to S, (Na n+1 (a, K m /F )) ζ F,S (σ, n)σ 1 σ G m (Na n+1 (a, K m /F )) σ G m ζ F,S (σ, 0)κ m (σ) n σ 1 (mod p m+1 Z p ). (Proof: Use Theorem (Congruence)) By Condition I and IV, K 2n (O K ) Z Z p H 2 (O K, Z p (n + 1)) X (n) Gal(K /K) X /IX (n) where I = σ κ n (σ) σ Gal(K /K). Coose m 0 satisfying the following conditions: X /IX p m+1 All ramified primes in K /K m are totally ramified. ν m,e (X /IX ) = 0 (ν m,e = 1+γ pe +γ 2pe + +γ pm p e ) 17
Put ζ = ζ p m+1. For a ζ n (X /IX )(n) = (X /IX ) µ n p m+1, (Na n+1 (a, K/F ))θ S (n) (a ζ n ) = (Na n+1 (a, K m /F )) σ G m ζ F,S (σ, n)σ 1 (a ζ n ) = (Na n+1 (a, K m /F )) σ G m ζ F,S (σ, 0)κ m (σ) n σ 1 (a ζ n ) = {Na n (Na (a, K m /F )) σ G m ζ F,S (σ, 0)σ 1 a} ζ n = 0. 18
(1) p8 Conjecture (L p ) + µ = 0 Conjecture (d 2 ) Conjecture (d 1 ), µ = 0 (,, µ).. (2) p5 Lemma (H 1 )(H 2 ) [S] + ) (3) p7 Conjecture (L p ) [KNF] p687 Conjecture (D k ) + µ = 0 Conjecture (L p ). (4) p7 Conjecture (d 2 ) [KNF] p683 p683 Conjecture (C m ) [CL] p521 Conjecture 3.1 (5) p8 Conjecture(d 2 ) Conjecture(d 1 ) ( [S] p192 Satz 6) (6) p9 Conjecture ( [A1]) (7) p12 Theorem [Coates] ( [C] p319 Theorem 4.4) (8) Brumer conjecture and Coates-Sinnott conjecture ( [A2]) [A1] M. Aoki, On some exact sequences in the theory of cyclotomic fields, Journal of Algebra, 2008, 320, 4156-4177. [A2] M.Aoki, A note on the Coates-Sinnott conjecture, to appear in Bull. Lond. Math. Soc. [C] J. Coates, p-adic L-functions and Iwasawa s theory, Algebraic Number Fields : L-functions and Galois properties (ed. A. Fröhlich), Academic Press, London (1977) 269-353. [CL] J. Coates and S. Lichtenbaum, On l-adic zeta functions, Ann. of Math. (2), 98 (1973)498-550. [KNF] M. Kolster, T. Nguyen Quang Do and V. Fleckinger, Twisted S-units, p-adic class number formulas, and the Lichtenbaum conjectures, Duke Math. J. 84 (1996), 679-717. [S] P. Schneider, Über gewisse Galoiscohomologiegruppen, Math. Z. 168 (1979) 181-205. 19