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 Some consequences of Leopoldt’s conjecture (II) July 2, 2020 3 / 18
Set-up • p : prime • F/E : Galois extension of number fields K ab S ( p ) ∆ := Gal( F/E ) • K : Galois extension of both F and E such X that Γ := Gal( K/F ) ∼ = Z r p for some r ≥ 1 K G := Gal( K/E ) . Γ ∼ = Z r p • Λ := Z p [[Γ]] ∼ = Z p [[ T 1 , . . . , T r ]] F Ω := Z p [[ G ]] G ∆ • S : a finite set of places of E , S ⊇ S p ∪ S ∞ E K S : the maximal extension of K unramified outside of S X := G ab K,S ( p ) = H 1 ( G K,S , Z p ) Some consequences of Leopoldt’s conjecture (II) July 2, 2020 4 / 18
Set-up Let p ∈ S f • K p := Z p [[ G / G p ]] , an Ω -module (in general G p ⋪ G ) K := � p ∈ S f K p K 0 := ker( aug : K → Z p ) • D p := ( G K p ) ab ( p ) I p := inertia subgroup of D p D p := Ω ˆ ⊗ Z p [[ G p ]] D p I p := Ω ˆ ⊗ Z p [[ G p ]] I p Some consequences of Leopoldt’s conjecture (II) July 2, 2020 5 / 18
Set-up Let p ∈ S f • K p := Z p [[ G / G p ]] , an Ω -module (in general G p ⋪ G ) K := � p ∈ S f K p K 0 := ker( aug : K → Z p ) • D p := ( G K p ) ab ( p ) I p := inertia subgroup of D p D p := Ω ˆ ⊗ Z p [[ G p ]] D p I p := Ω ˆ ⊗ Z p [[ G p ]] I p For a locally compact Λ -(or Ω -)module, we abbreviated E i Λ ( M ) := Ext i Λ ( M, Λ) and set M ∗ := E 0 Λ ( M ) = Hom Λ ( M, Λ) . Some consequences of Leopoldt’s conjecture (II) July 2, 2020 5 / 18
Useful lemmas 2: Useful lemmas Some consequences of Leopoldt’s conjecture (II) July 2, 2020 6 / 18
Useful lemmas We saw: Lemma (4.2.2) Let p ∈ S f and suppose Γ p � = 0 . Consider • ε p = 0 if K p contains the unram. Z p -extension of E p , 1 otherwise. • ε ′ p = ε p δ r p , 1 , where r p = rank Z p If ε ′ p = 1 , assume that K ⊇ µ p ∞ . Then the following diagram K ε p 0 I p D p 0 p ι ε ′ I ∗∗ D ∗∗ 0 K p 0 p p p commutes and has exact rows. If ε ′ p = 1 , then ι = Id . Some consequences of Leopoldt’s conjecture (II) July 2, 2020 7 / 18
Useful lemmas Relation E j Λ ( K ) ↔ E j Λ ( K 0 ) ? Lemma (4.2.4) a For 1 ≤ j < r − 1 , we have E j Λ ( K ) ∼ = E j Λ ( K 0 ) For j > r , we have E j Λ ( K ) = E j Λ ( K 0 ) = 0 . b If r � = r p for all p ∈ S f , then Λ ( K ) ∼ • E r = E r Λ ( K 0 ) = 0 • The sequence 0 → E r − 1 ( K ) → E r − 1 ( K 0 ) → Z p → 0 is exact Λ Λ c If r = r p for some p ∈ S f , then ( K ) ∼ • E r − 1 = E r − 1 ( K 0 ) Λ Λ • The sequence 0 → Z p → E r Λ ( K ) → E r Λ ( K 0 ) → 0 is exact Some consequences of Leopoldt’s conjecture (II) July 2, 2020 8 / 18
Useful lemmas Lemma (4.2.4) a For 1 ≤ j < r − 1 , we have E j Λ ( K ) ∼ = E j Λ ( K 0 ) For j > r , we have E j Λ ( K ) = E j Λ ( K 0 ) = 0 . b If r � = r p for all p ∈ S f , then • E r Λ ( K ) = E r Λ ( K 0 ) = 0 • The sequence 0 → E r − 1 ( K ) → E r − 1 ( K 0 ) → Z p → 0 is exact Λ Λ c If r = r p for some p ∈ S f , then • E r − 1 ( K ) ∼ = E r − 1 ( K 0 ) Λ Λ • The sequence 0 → Z p → E r Λ ( K ) → E r Λ ( K 0 ) → 0 is exact Idea: 1 Long Ext sequence of 0 → K 0 → K → Z p → 0 Λ ( Z p ) = Z δ r,j 2 E j (A.13) p 3 E j p ) δ r p ,j (4.1.13) p ∈ S f ( K ι Λ ( K ) = � Some consequences of Leopoldt’s conjecture (II) July 2, 2020 9 / 18
Eigenspaces 3: Eigenspaces Some consequences of Leopoldt’s conjecture (II) July 2, 2020 10 / 18
Eigenspaces Simplification of the setting: • ∆ abelian • G = Γ × ∆ , therefore Ω = Λ[∆] • F ∋ ζ p Some consequences of Leopoldt’s conjecture (II) July 2, 2020 11 / 18
Eigenspaces Simplification of the setting: • ∆ abelian • G = Γ × ∆ , therefore Ω = Λ[∆] • F ∋ ζ p ∗ . Let We are interested in characters ψ : ∆ → Q p • O ψ := Z p [ ψ ] . We have a surjection Z p [∆] ։ O ψ . • Λ ψ := O ψ [[Γ]] = Λ[ ψ ] • For a Z p [∆] -module M , let M ψ = M ⊗ Z p [∆] O ψ Note that Ω ψ = Λ[∆] ⊗ Z p [∆] O ψ ∼ = Λ ψ (as compact O ψ -algebras). Some consequences of Leopoldt’s conjecture (II) July 2, 2020 11 / 18
Eigenspaces Remark: letting Q := Frac(Λ) , we have (Wedderburn) Q [∆] ∼ � = Q ( ψ ) ψ ∈ Irr(∆) / ∼ Some consequences of Leopoldt’s conjecture (II) July 2, 2020 12 / 18
Eigenspaces Remark: letting Q := Frac(Λ) , we have (Wedderburn) Q [∆] ∼ � = Q ( ψ ) ψ ∈ Irr(∆) / ∼ If p ∤ | ∆ | , the decomposition is finer: Ω = Λ[∆] ∼ � = Λ[ ψ ] , ψ ∈ Irr(∆) / ∼ hence Ω -modules really decompose as M = Ω ⊗ Ω M ∼ � M ψ . = ψ ∈ Irr(∆) / ∼ Some consequences of Leopoldt’s conjecture (II) July 2, 2020 12 / 18
Eigenspaces Remark: letting Q := Frac(Λ) , we have (Wedderburn) Q [∆] ∼ � = Q ( ψ ) ψ ∈ Irr(∆) / ∼ If p ∤ | ∆ | , the decomposition is finer: Ω = Λ[∆] ∼ � = Λ[ ψ ] , ψ ∈ Irr(∆) / ∼ hence Ω -modules really decompose as M = Ω ⊗ Ω M ∼ � M ψ . = ψ ∈ Irr(∆) / ∼ In general, however, we only know Ω ֒ → � ψ Λ[ ψ ] (maximal order). Some consequences of Leopoldt’s conjecture (II) July 2, 2020 12 / 18
Eigenspaces Lemma (4.3.1) a If weak Leopoldt holds for K , then rank Λ ψ X ψ = r 2 ( E ) + r ψ 1 ( E ) , where r ψ 1 ( E ) = no. of real places of E at which ψ is odd (i.e. ψ | ∆ p � = 1 ). b Let p ∈ S f . If Γ p � = 0 or ψ | ∆ p � = 1 , then rank Λ ψ D ψ p = [ E p : Q p ] . Some consequences of Leopoldt’s conjecture (II) July 2, 2020 13 / 18
Eigenspaces Lemma (4.3.1) a If weak Leopoldt holds for K , then rank Λ ψ X ψ = r 2 ( E ) + r ψ 1 ( E ) , where r ψ 1 ( E ) = no. of real places of E at which ψ is odd (i.e. ψ | ∆ p � = 1 ). b Let p ∈ S f . If Γ p � = 0 or ψ | ∆ p � = 1 , then rank Λ ψ D ψ p = [ E p : Q p ] . General proof : local and global class field theory. It uses Nekovář’s Euler-Poincaré characteristic formulas: 2 ( − 1) j − 1 rank Λ ψ H j ( G E, Σ , B ψ ) ∨ = � � rank Λ ψ (Ω ψ (1)) G Ev j =0 v ∈ S ∞ and 2 ( − 1) j rank Λ ψ H j ( G E p , B ψ ) ∨ = [ E p : Q p ] . � j =0 Some consequences of Leopoldt’s conjecture (II) July 2, 2020 13 / 18
Eigenspaces (Very) simplified proof : r = 1 , F = E a) WeakLeopoldt( K ) ⇒ rank Λ X = r 2 ( F ) Let: • F n := K Γ pn M ∞ • M n := maximal abelian pro- p extension of X F n unramified outside S . One has M n F ∞ ⊆ M 0 ⊆ M 1 ⊆ · · · ⊆ M ∞ = ( K S ) ab ( p ) . K X n Γ pn • X n := Gal( M n /F n ) , so X = lim − n X n ← F n Γ F Some consequences of Leopoldt’s conjecture (II) July 2, 2020 14 / 18
Eigenspaces (Very) simplified proof : r = 1 , F = E a) WeakLeopoldt( K ) ⇒ rank Λ X = r 2 ( F ) Let: • F n := K Γ pn M ∞ • M n := maximal abelian pro- p extension of X F n unramified outside S . One has M n F ∞ ⊆ M 0 ⊆ M 1 ⊆ · · · ⊆ M ∞ = ( K S ) ab ( p ) . K X n Γ pn • X n := Gal( M n /F n ) , so X = lim − n X n ← F n Γ Then X Γ pn = Gal( M n /K ) and therefore 0 → X Γ pn → X n → Γ p n → 0 F (1) is exact. Some consequences of Leopoldt’s conjecture (II) July 2, 2020 14 / 18
Eigenspaces Recall that r 2 ( F n ) + 1 + d p ( F n ) = no. of indep. Z p -extensions of F n = rank Z p X n Some consequences of Leopoldt’s conjecture (II) July 2, 2020 15 / 18
Eigenspaces Recall that r 2 ( F n ) + 1 + d p ( F n ) = no. of indep. Z p -extensions of F n = rank Z p X n • On the left, r 2 ( F n ) = p n r 2 ( F ) ( Z p -extensions are unramified at archimedean places) • On the right, the sequence (1) 0 → X Γ pn → X n → Γ p n → 0 shows that rank Z p X Γ pn = rank Z p X n − 1 . Therefore, rank Z p X Γ pn = p n r 2 ( F ) + d p ( F n ) . Some consequences of Leopoldt’s conjecture (II) July 2, 2020 15 / 18
Eigenspaces Recall that r 2 ( F n ) + 1 + d p ( F n ) = no. of indep. Z p -extensions of F n = rank Z p X n • On the left, r 2 ( F n ) = p n r 2 ( F ) ( Z p -extensions are unramified at archimedean places) • On the right, the sequence (1) 0 → X Γ pn → X n → Γ p n → 0 shows that rank Z p X Γ pn = rank Z p X n − 1 . Therefore, rank Z p X Γ pn = p n r 2 ( F ) + d p ( F n ) . General fact: for all n large enough, rank Z p X Γ pn = p n rank Λ X + c for some c independent of n . Hence rank Λ X = r 2 ( F ) by Weak Leopoldt. Some consequences of Leopoldt’s conjecture (II) July 2, 2020 15 / 18
Eigenspaces b) Γ p � = 0 ⇒ rank Λ D p = [ F p : Q p ] Let Λ p := Z p [[Γ p ]] (Iwasawa algebra). Note that rank Λ D p = rank Λ Λ ⊗ Λ p D p = rank Λ p D p . ab p ( p ) K ab ab Let K p ( p ) (resp. F p ( p ) ) be the maximal abelian pro- p extension of K p (resp. F p ). In particular, D p ab ab ab F p ( p ) F p ( p ) ⊆ K p ( p ) . K p Γ p F p Some consequences of Leopoldt’s conjecture (II) July 2, 2020 16 / 18
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