Some consequences of Leopoldts conjecture (II) Antonio Mejas Gil - - PowerPoint PPT Presentation

Some consequences of Leopoldt’s conjecture (II)

Antonio Mejías Gil July 2, 2020

Outline

1 Set-up 2 Useful lemmas 3 Eigenspaces

Set-up

1: Set-up

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Set-up

• p: prime
• F/E: Galois extension of number fields

∆ := Gal(F/E)

• K: Galois extension of both F and E such

that Γ := Gal(K/F) ∼ = Zr

p for some r ≥ 1

G := Gal(K/E).

• Λ := Zp[[Γ]] ∼

= Zp[[T1, . . . , Tr]] Ω := Zp[[G]]

• S: a finite set of places of E, S ⊇ Sp ∪ S∞

KS: the maximal extension of K unramified

• utside of S

X := Gab

K,S(p) = H1(GK,S, Zp)

Kab

S (p)

K F E

X Γ ∼ = Zr

p

∆ G

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Set-up

Let p ∈ Sf

• Kp := Zp[[G/Gp]], an Ω-module (in general Gp ⋪ G)

K :=

p∈Sf Kp

K0 := ker(aug: K → Zp)

• Dp := (GKp)ab(p)

Ip := inertia subgroup of Dp Dp := Ω ˆ ⊗Zp[[Gp]] Dp Ip := Ω ˆ ⊗Zp[[Gp]] Ip

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Set-up

Let p ∈ Sf

• Kp := Zp[[G/Gp]], an Ω-module (in general Gp ⋪ G)

K :=

p∈Sf Kp

K0 := ker(aug: K → Zp)

• Dp := (GKp)ab(p)

Ip := inertia subgroup of Dp Dp := Ω ˆ ⊗Zp[[Gp]] Dp Ip := Ω ˆ ⊗Zp[[Gp]] Ip For a locally compact Λ-(or Ω-)module, we abbreviated Ei

Λ(M) := Exti Λ(M, Λ)

and set M ∗ := E0

Λ(M) = HomΛ(M, Λ).

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Useful lemmas

2: Useful lemmas

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Useful lemmas

We saw: Lemma (4.2.2) Let p ∈ Sf and suppose Γp = 0. Consider

• εp = 0 if Kp contains the unram. Zp-extension of Ep, 1 otherwise.
• ε′

p = εpδrp,1, where rp = rank Zp

If ε′

p = 1, assume that K ⊇ µp∞. Then the following diagram

Ip Dp Kεp

p

I∗∗

p

D∗∗

p

K

ε′

p

p ι

commutes and has exact rows. If ε′

p = 1, then ι = Id.

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Useful lemmas

Relation Ej

Λ(K) ↔ Ej Λ(K0)?

Lemma (4.2.4)

a For 1 ≤ j < r − 1, we have Ej Λ(K) ∼

= Ej

Λ(K0)

For j > r, we have Ej

Λ(K) = Ej Λ(K0) = 0. b If r = rp for all p ∈ Sf, then

• Er

Λ(K) ∼

= Er

Λ(K0) = 0

• The sequence 0 → Er−1

Λ

(K) → Er−1

Λ

(K0) → Zp → 0 is exact

c If r = rp for some p ∈ Sf, then

• Er−1

Λ

(K) ∼ = Er−1

Λ

(K0)

• The sequence 0 → Zp → Er

Λ(K) → Er Λ(K0) → 0 is exact

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Useful lemmas

Lemma (4.2.4)

a For 1 ≤ j < r − 1, we have Ej

Λ(K) ∼

= Ej

Λ(K0)

For j > r, we have Ej

Λ(K) = Ej Λ(K0) = 0.

b If r = rp for all p ∈ Sf, then

• Er

Λ(K) = Er Λ(K0) = 0

• The sequence 0 → Er−1

Λ

(K) → Er−1

Λ

(K0) → Zp → 0 is exact

c If r = rp for some p ∈ Sf, then

• Er−1

Λ

(K) ∼ = Er−1

Λ

(K0)

• The sequence 0 → Zp → Er

Λ(K) → Er Λ(K0) → 0 is exact

Idea:

1 Long Ext sequence of 0 → K0 → K → Zp → 0 2 Ej Λ(Zp) = Zδr,j p

(A.13)

3 Ej Λ(K) = p∈Sf (Kι p)δrp,j (4.1.13)

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Eigenspaces

3: Eigenspaces

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Eigenspaces

Simplification of the setting:

• ∆ abelian
• G = Γ × ∆, therefore Ω = Λ[∆]
• F ∋ ζp

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Eigenspaces

Simplification of the setting:

• ∆ abelian
• G = Γ × ∆, therefore Ω = Λ[∆]
• F ∋ ζp

We are interested in characters ψ: ∆ → Qp

∗. Let

• Oψ := Zp[ψ]. We have a surjection Zp[∆] ։ Oψ.
• Λψ := Oψ[[Γ]] = Λ[ψ]
• For a Zp[∆]-module M, let M ψ = M ⊗Zp[∆] Oψ

Note that Ωψ = Λ[∆] ⊗Zp[∆] Oψ ∼ = Λψ (as compact Oψ-algebras).

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Eigenspaces

Remark: letting Q := Frac(Λ), we have (Wedderburn) Q[∆] ∼ =

• ψ∈Irr(∆)/∼

Q(ψ)

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Eigenspaces

Remark: letting Q := Frac(Λ), we have (Wedderburn) Q[∆] ∼ =

• ψ∈Irr(∆)/∼

Q(ψ) If p ∤ |∆|, the decomposition is finer: Ω = Λ[∆] ∼ =

• ψ∈Irr(∆)/∼

Λ[ψ], hence Ω-modules really decompose as M = Ω ⊗Ω M ∼ =

• ψ∈Irr(∆)/∼

M ψ.

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Eigenspaces

Remark: letting Q := Frac(Λ), we have (Wedderburn) Q[∆] ∼ =

• ψ∈Irr(∆)/∼

Q(ψ) If p ∤ |∆|, the decomposition is finer: Ω = Λ[∆] ∼ =

• ψ∈Irr(∆)/∼

Λ[ψ], hence Ω-modules really decompose as M = Ω ⊗Ω M ∼ =

• ψ∈Irr(∆)/∼

M ψ. In general, however, we only know Ω ֒ →

ψ Λ[ψ] (maximal order).

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Eigenspaces

Lemma (4.3.1)

a If weak Leopoldt holds for K, then rankΛψ Xψ = r2(E) + rψ 1 (E),

where rψ

1 (E) = no. of real places of E at which ψ is odd (i.e.

ψ|∆p = 1).

b Let p ∈ Sf. If Γp = 0 or ψ|∆p = 1, then rankΛψ Dψ p = [Ep : Qp].

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Eigenspaces

Lemma (4.3.1)

a If weak Leopoldt holds for K, then rankΛψ Xψ = r2(E) + rψ 1 (E),

where rψ

1 (E) = no. of real places of E at which ψ is odd (i.e.

ψ|∆p = 1).

b Let p ∈ Sf. If Γp = 0 or ψ|∆p = 1, then rankΛψ Dψ p = [Ep : Qp].

General proof: local and global class field theory. It uses Nekovář’s Euler-Poincaré characteristic formulas:

2

• j=0

(−1)j−1 rankΛψ Hj(GE,Σ, Bψ)∨ =

• v∈S∞

rankΛψ(Ωψ(1))GEv and

2

• j=0

(−1)j rankΛψ Hj(GEp, Bψ)∨ = [Ep : Qp].

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Eigenspaces

(Very) simplified proof: r = 1, F = E a) WeakLeopoldt(K) ⇒ rankΛ X = r2(F) Let:

• Fn := KΓpn
• Mn := maximal abelian pro-p extension of

Fn unramified outside S. One has F∞ ⊆ M0 ⊆ M1 ⊆ · · · ⊆ M∞ = (KS)ab(p).

• Xn := Gal(Mn/Fn), so X = lim

← −n Xn

M∞ Mn K Fn F

X Xn Γpn Γ

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Eigenspaces

(Very) simplified proof: r = 1, F = E a) WeakLeopoldt(K) ⇒ rankΛ X = r2(F) Let:

• Fn := KΓpn
• Mn := maximal abelian pro-p extension of

Fn unramified outside S. One has F∞ ⊆ M0 ⊆ M1 ⊆ · · · ⊆ M∞ = (KS)ab(p).

• Xn := Gal(Mn/Fn), so X = lim

← −n Xn Then XΓpn = Gal(Mn/K) and therefore 0 → XΓpn → Xn → Γpn → 0 (1) is exact.

M∞ Mn K Fn F

X Xn Γpn Γ

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Eigenspaces

Recall that r2(Fn) + 1 + dp(Fn) = no. of indep. Zp-extensions of Fn = rankZp Xn

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Eigenspaces

Recall that r2(Fn) + 1 + dp(Fn) = no. of indep. Zp-extensions of Fn = rankZp Xn

• On the left, r2(Fn) = pnr2(F) (Zp-extensions are unramified at

archimedean places)

• On the right, the sequence (1) 0 → XΓpn → Xn → Γpn → 0 shows

that rankZp XΓpn = rankZp Xn − 1. Therefore, rankZp XΓpn = pnr2(F) + dp(Fn).

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Eigenspaces

Recall that r2(Fn) + 1 + dp(Fn) = no. of indep. Zp-extensions of Fn = rankZp Xn

• On the left, r2(Fn) = pnr2(F) (Zp-extensions are unramified at

archimedean places)

• On the right, the sequence (1) 0 → XΓpn → Xn → Γpn → 0 shows

that rankZp XΓpn = rankZp Xn − 1. Therefore, rankZp XΓpn = pnr2(F) + dp(Fn). General fact: for all n large enough, rankZp XΓpn = pn rankΛ X + c for some c independent of n. Hence rankΛ X = r2(F) by Weak Leopoldt.

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Eigenspaces

b) Γp = 0 ⇒ rankΛ Dp = [Fp : Qp] Let Λp := Zp[[Γp]] (Iwasawa algebra). Note that rankΛ Dp = rankΛ Λ ⊗Λp Dp = rankΛp Dp. Let K

ab p (p) (resp. F ab p (p)) be the maximal abelian

pro-p extension of Kp (resp. Fp). In particular, F

ab p (p) ⊆ K ab p (p).

K

ab p (p)

F

ab p (p)

Kp Fp

Dp Γp

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Eigenspaces

b) Γp = 0 ⇒ rankΛ Dp = [Fp : Qp] Let Λp := Zp[[Γp]] (Iwasawa algebra). Note that rankΛ Dp = rankΛ Λ ⊗Λp Dp = rankΛp Dp. Let K

ab p (p) (resp. F ab p (p)) be the maximal abelian

pro-p extension of Kp (resp. Fp). In particular, F

ab p (p) ⊆ K ab p (p).

Similarly to before, (Dp)Γp = Gal(F

ab p (p)/Kp), so

0 → (Dp)Γp → Gal(Fp

ab(p)/Fp) → Γp → 0

is exact.

K

ab p (p)

F

ab p (p)

Kp Fp

Dp Γp

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Eigenspaces

Hence rankZp(Dp)Γp = rankZp Gal(Fp

ab(p)/Fp) − rankZp Γp

But rankZp Gal(Fp

ab(p)/Fp) = no. of indep. Zp-extensions of Fp

= [Fp : Qp] + 1 (local CFT)

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Eigenspaces

Hence rankZp(Dp)Γp = rankZp Gal(Fp

ab(p)/Fp) − rankZp Γp

But rankZp Gal(Fp

ab(p)/Fp) = no. of indep. Zp-extensions of Fp

= [Fp : Qp] + 1 (local CFT) General fact: rankΛp Dp = rankZp(Dp)Γp − rankZp(Dp)Γp =[Fp : Qp] − 0

• Some consequences of Leopoldt’s conjecture (II)

July 2, 2020 17 / 18

Thank you for your attention