The Neukirch-Uchida theorem with restricted ramification Ryoji - PowerPoint PPT Presentation

The Neukirch-Uchida theorem with restricted ramification Ryoji Shimizu RIMS, Kyoto University This presentation is based on my paper with the same title, whose preprint will be uploaded in a few weeks. 1 / 35

Introduction Let K be a number field and S a set of primes of K . We write K S / K for the maximal extension of K unramified outside S and G K , S for its Galois group. The goal of this talk is to prove the following generalization of the Neukirch-Uchida theorem under as few assumptions as possible: “For i = 1 , 2, let K i be a number field and S i a set of primes of K i . If G K 1 , S 1 and G K 2 , S 2 are isomorphic, then K 1 and K 2 are isomorphic.” For this, as in the proof of the Neukirch-Uchida theorem, we first characterize group-theoretically the decomposition groups in G K , S , and then obtain an isomorphism of fields using them. 2 / 35

Notations def • G ( L / K ) = Gal( L / K ) : the Galois group of a Galois extension L / K • K : a separable closure of a field K def • G K = G ( K / K ) • K : a number field (i.e. a finite extension of the field of rational numbers Q ) • P = P K : the set of primes of K • P ∞ = P K , ∞ : the set of archimedean primes of K • P l = P K , l : the set of primes of K above a prime number l • S : a subset of P K def • S f = S \ P K , ∞ • S ( L ) : the set of primes of L above the primes in S for an algebraic extension L / K For convenience, we consider that an algebraic extension L / K is ramified at a complex prime of L if it is above a real prime of K . 3 / 35

Previous works The Neukirch-Uchida theorem (Uchida, 1976). Let K 1 and K 2 be number fields. If G K 1 ≃ G K 2 , then K 1 ≃ K 2 . This is in the case that S i = P K i for i=1,2. 4 / 35

Previous works Theorem (Ivanov, 2017). For i = 1 , 2, let K i be a number field and S i a set of primes of K i . Assume G K 1 , S 1 ≃ G K 2 , S 2 and that the following conditions hold: (a) K i / Q is Galois for i = 1 , 2 and K 1 is totally imaginary. (b) There exist two odd prime numbers p such that P K 1 , p ⊂ S 1 . (c) There exists an odd prime number p such that P K 2 , p ⊂ S 2 and S i is sharply p -stable for i = 1 , 2. (d) For i = 1 , 2, S i is 2-stable and is sharply p -stable for almost all p . Then K 1 ≃ K 2 . Let K be a number field and S a set of primes of K . We say that S is stable if there are a subset S 0 ⊂ S and an ǫ ∈ R > 0 such that for any finite subextension K S / L / K , S 0 ( L ) has Dirichlet density δ ( S 0 ( L )) > ǫ . 5 / 35

One of main results Theorem 4.2. For i = 1 , 2, let K i be a number field and S i a set of primes of K i with P K i , ∞ ⊂ S i . Assume G K 1 , S 1 ≃ G K 2 , S 2 and that the following conditions hold: (a) K i / Q is Galois for i = 1 , 2 and K 1 is totally imaginary. (b) There exist two different prime numbers p such that for i = 1 , 2, P K i , p ⊂ S i . (c) For one i , there exists a totally real subfield K i , 0 ⊂ K i and a set of nonarchimedean primes T i , 0 of K i , 0 such that δ ( T i , 0 ( K i )) � = 0. a (d) For the other i , δ ( S i ) � = 0. Then K 1 ≃ K 2 . a Let K be a number field and S a set of primes of K . We say that δ ( S ) ̸ = 0 if S has positive Dirichlet density or does not have Dirichlet density. 6 / 35

Previous works Theorem (Ivanov, 2013). Let K be a number field and P ∞ ⊂ S a finite set of primes of K . Assume that there exist two different prime numbers p such that P p ⊂ S , and write l for one of them. Assume ( G K , S , l ) are given. Then the data of the l -adic cyclotomic character of an open subgroup of G K , S is equivalent to the data of the decomposition groups in G K , S at primes in S f ( K S ). In the proof, the injectivity of H 2 ( G K , S , µ l ∞ ) → ⊕ H 2 ( D p , µ l ∞ ) p ∈ S plays an impotant role. Even if S is not finite, we can obtain the “bi-anabelian” version of this result. In order to use this, in § 1 we recover the l -adic cyclotomic character of an open subgroup of G K , S . 7 / 35

Contents Recovering the l -adic cyclotomic character 1 Local correspondence and recovering the local invariants 2 The existence of an isomorphism of fields 3 Main results 4 8 / 35

§ 1. Recovering the l -adic cyclotomic character (1/7) Let K be a number field, and fix a prime number l . def • Σ = Σ K = { l , ∞} ( K ) = P l ∪ P ∞ • K ∞ / K : a Z l -extension • Γ = G ( K ∞ / K ) • K ∞ , 0 / K : the cyclotomic Z l -extension def • Γ 0 = Γ K , 0 = G ( K ∞ , 0 / K ) Note that K ∞ / K is unramified outside Σ. • γ p : the Frobenius element in Γ at p ∈ P K \ Σ • Γ p = � γ p � : the decomposition group in Γ at p ∈ P K \ Σ • S : a set of primes of K In § 1, we assume that Σ ⊂ S . Then µ l ∞ ⊂ K S , and we write χ ( l ) = χ ( l ) K for the ∗ . l -adic cyclotomic character G K , S → Aut( µ l ∞ ) = Z l 9 / 35

�� �� § 1. Recovering the l -adic cyclotomic character (2/7) { 4 if l = 2, def We set ˜ l = We have the following commutative diagram: l if l � = 2. χ ( l ) � ∗ (1 + ˜ ∗ ) tor G K , S l Z l ) × ( Z l Z l pr 1 w � 1 + ˜ Γ 0 l Z l We write w = w K : Γ 0 → 1 + ˜ l Z l for the bottom homomorphism. w → 1 + ˜ Note that χ ( l ) | G K ( µ ˜ l Z l ) | G K ( µ ˜ l )) = ( G K , S ↠ Γ 0 l )) . l ) , S ( K ( µ ˜ l ) , S ( K ( µ ˜ The goal of this section is the following. Theorem 1.7. Assume that δ ( S ) � = 0. Then the surjection G K , S ↠ Γ 0 and the character w : Γ 0 → 1 + ˜ l Z l are characterized group-theoretically from G K , S (and l ). We will see the sketch of the proof of Theorem 1.7. 10 / 35

§ 1. Recovering the l -adic cyclotomic character (3/7) • Λ = Λ Γ def − n Z l [Γ / Γ l n ] : the complete group ring of Γ = Z l [[Γ]] = lim ← def = (Ker( G K , S ↠ Γ) ( l ) ) ab • X S = X Γ S Note that X S is constructed group-theoretically from G K , S ↠ Γ by its very definition, and X S has a natural structure of Λ-module. • ( S \ Σ) fd def = { p ∈ S \ Σ | p is finitely decomposed in K ∞ / K } • ( S \ Σ) cd def = { p ∈ S \ Σ | p is completely decomposed in K ∞ / K } Note that S \ Σ = ( S \ Σ) fd ⨿ ( S \ Σ) cd . For p ∈ ( S \ Σ) fd with µ l ⊂ K p , the local l -adic cyclotomic character ∗ factors as G K p ↠ Γ p → Z l ∗ because G K p → Aut( µ l ∞ ) = Z l ∗ for the second Γ p = G ( K p ( µ l ∞ ) / K p ), where we write χ ( l ) p : Γ p → Z l l ⊂ K p and Γ = Γ 0 , we have w | Γ p = χ ( l ) homomorphism. Further, when µ ˜ p . 11 / 35

§ 1. Recovering the l -adic cyclotomic character (4/7) We have the following structure theorem for the Λ-module X S . Lemma 1.1. Assume that the weak Leopoldt conjecture holds for K ∞ / K . Then there exists an exact sequence of Λ-modules ∏ 0 → J p → X S → X Σ → 0 , p ∈ S \ Σ where X Σ is a finitely generated Λ-module and Λ / � γ p − χ ( l )  µ l ⊂ K p and p ∈ ( S \ Σ) fd , p ( γ p ) � ,   J p = Λ / l t p , µ l ⊂ K p and p ∈ ( S \ Σ) cd ,  0 , µ l �⊂ K p ,  where l t p = # µ ( K p )[ l ∞ ]. We set J = J Γ def = ∏ J p ⊂ X S . p ∈ S \ Σ 12 / 35

§ 1. Recovering the l -adic cyclotomic character (5/7) Lemma 1.2. The weak Leopoldt conjecture is true for K ∞ / K if and only if H 2 ( G ( K S / K ∞ ) , Q l / Z l ) = 0. Further, the weak Leopoldt conjecture is true for K ∞ , 0 / K . Note that H 2 ( G ( K S / K ∞ ) , Q l / Z l ) can be reconstructed group-theoretically from G K , S ↠ Γ since G ( K S / K ∞ ) = Ker( G K , S ↠ Γ) and Q l / Z l is a trivial G ( K S / K ∞ )-module. Lemma 1.3. Assume that µ l ⊂ K . Then #( S \ Σ) cd < ∞ if and only if X S [ l ∞ ] is a finitely generated Λ-module. Further, ( S \ Σ) cd = ∅ for K ∞ , 0 / K . Note that X S [ l ∞ ] also can be reconstructed group-theoretically from G K , S ↠ Γ. 13 / 35

§ 1. Recovering the l -adic cyclotomic character (6/7) Definition 1.4. Let M ⊂ X S be a Λ-submodule whose quotient X S / M is a finitely generated Λ-module. We set l Z l )) prim and x ∈ M \ { 0 } For ( γ, α ) ∈ (Γ × (1 + ˜ � � � def A Γ ρ : Γ → 1 + ˜ � = l Z l , M � with γ − α ∈ Ann Λ ( x ), ρ ( γ ) = α � l Z l )) prim def where (Γ × (1 + ˜ = (Γ × (1 + ˜ l Z l )) \ (Γ × (1 + ˜ l Z l )) l . Note that this set is constructed from M and Γ. Proposition 1.5. Assume that µ ˜ l ⊂ K , Γ = Γ 0 and # S = ∞ . Let M ⊂ J be a Λ-submodule whose quotient J / M is a finitely generated Λ-module. Then A Γ 0 M = { w } . Proposition 1.6. Assume that µ ˜ l ⊂ K , Γ � = Γ 0 , δ ( S ) � = 0, the weak Leopoldt conjecture is true for K ∞ / K and #( S \ Σ) cd < ∞ . Let M ⊂ X S be a Λ-submodule whose quotient X S / M is a finitely generated Λ-module. Then A Γ M = ∅ . 14 / 35