The $m$-step solvable anabelian geometry of number fields

Given a number field $K$ and an integer $m\geq 0$, let $K_m$ denote the maximal $m$-step solvable Galois extension of $K$ and write $G_K^m$ for the maximal $m$-step solvable Galois group Gal$(K_m/K)$ of $K$. In this paper, we prove that the isomorphy type of $K$ is determined by the isomorphy type of $G_K^3$. Further, we prove that $K_m/K$ is determined functorially by $G_K^{m+3}$ (resp. $G_K^{m+4}$) for $m\geq 2$ (resp. $m \leq 1$). This is a substantial sharpening of a famous theorem of Neukirch and Uchida. A key step in our proof is the establishment of the so-called local theory, which in our context characterises group-theoretically the set of decomposition groups (at nonarchimedean primes) in $G_K^m$, starting from $G_K^{m+2}$.

field determines functorially the isomorphy type of the number field. An explicit description of the isomorphy type of the (prosolvable) Galois group of a number field seems to be out of reach for the time being. This prompts the natural question: Is it possible to prove any refinement of the Neukirch-Uchida theorem, whereby one replaces the full (prosolvable) Galois groups of number fields by some profinite quotients whose structure can be better approached/understood?
Class field theory provides a description of the maximal abelian quotient of the Galois group of a number field. The structure of the maximal m-step solvable quotient of the Galois group of a number field can be in principle approached via class field theory, say for small values of m as in (commutative) Iwasawa theory in case m = 2. In this paper we prove the following sharpening of the Neukirch-Uchida theorem. Write G m K and G m L for the maximal m-step solvable quotients of G K , G L and K m /K, L m /L for the corresponding subextensions of K/K and L/L, respectively (cf. Notations). Theorem 1 is not functorial in the sense that we do not know a priori how an isomorphism σ in the underlying statement relates to the isomorphism τ 3 . In this respect Theorem 1 is a sharpening of the weak version of the Neukirch-Uchida theorem originally proved in [Neukirch1]. The uniformity assertion in the Neukirch-Uchida theorem (existence and uniqueness of σ therein) was established by Uchida in [Uchida1], [Uchida2], using among others Neukirch's results. Uchida also removed the assumption originally imposed by Neukirch that at least one of K and L is Galois over the prime field Q.
Theorem 2 above is functorial. It implies in particular that the isomorphy type of K is functorially determined by the isomorphy type of G 4 K . As in the proof of the Neukirch-Uchida theorem, a key step in the proofs of Theorems 1 and 2 is the establishment of the so-called local theory, i.e., starting from an isomorphism of (quotients of) absolute Galois groups of number fields one establishes a one-to-one correspondence between the sets of their nonarchimedean primes and one between the corresponding decomposition groups. The latter is usually achieved via a purely group-theoretic characterisation of decomposition groups. (In the proof of the Neukirch-Uchida theorem decomposition groups are characterised group-theoretically using the method of Brauer groups.) In our context we prove the following (cf. Theorem 1.25 and Corollary 1.27 (i)).
The proof of our Theorem 1 follows then by applying the main theorem of loc. cit.. It would be interesting to investigate how precisely the conditions in the main theorem of loc. cit. relates to the Galois-theoretic information of K and how precisely this main theorem relates to our Theorem 1. Also, contrary to the proof of Theorem 1, our proof of Theorem 2 does not rely on the main (or any other) result in [Cornelissen-de Smit-Li-Marcolli-Smit]. In particular, Theorem 2(i) for m = 0 gives an alternative proof of Theorem 1.
• Our main Theorems 1 and 2 are bi-anabelian, i.e., with a reference to two, a priori distinct, number fields. A mono-anabelian version of Theorem 2 would be a version whereby one establishes a purely group-theoretic algorithm which starting solely from an isomorph of G n K , for suitable n ≥ 1, reconstructs an isomorph of the number field K. In [Hoshi] Hoshi establishes such an algorithm starting from an isomorph of G K . More generally, he also establishes such an algorithm starting from an isomorph of the maximal prosolvable quotient G sol K of G K , under the assumption that the maximal prosolvable extension K sol of K in K is Galois over the prime field Q. The algorithm relies crucially on this assumption, as well as the Neukirch-Uchida theorem itself. In fact, Hoshi's algorithm does not provide an alternative proof of the Neukirch-Uchida theorem but rather uses it in an essential way. In our context, one could adapt Hoshi's arguments to show, using Theorems 2 and 3, the following (details of proof may be considered in a subsequent work).
Let m ≥ 0 be an integer, and assume that K m is Galois over Q. Then there exists a purely group-theoretic algorithm which starting from G n K , for a suitable integer n ≥ m that can be made effective, reconstructs functorially the field K m together with the natural action of G m K on K m . • Finally, we mention that the authors prove an (a mono-anabelian) analogue of the main results of this paper for global function fields in positive characteristics (cf. [Saïdi-Tamagawa]).
Future perspectives. The Neukirch-Uchida theorem had several deep and substantial applications in anabelian geometry and has played a prominent role in the theory for more than 40 years. We hope that Theorem 2 will have a similar impact, for example in developing a new anabelian geometry, over finitely generated fields, stemming solely from the arithmetic of m-step solvable extensions of finitely generated fields and m-step solvable arithmetic fundamental groups, for small values of m.

Notations.
• Given a finite set H we write |H| for its cardinality.
• For a profinite group G let [G, G] be the closed subgroup of G which is (topologically) generated by the commutator subgroup of G. We write G ab def = G/[G, G] for the maximal abelian quotient of G.
• Given a profinite group G, and a prime number l, we write G (l) for the maximal pro-l quotient of G, and G (l ′ ) for the maximal prime-to-l quotient of G. • Let G be a profinite group and consider the derived series and refer to it as 4 the (maximal) i-step solvable quotient of G (thus G 1 = G ab , G 2 is the maximal metabelian quotient of G, ...). By definition, ← −i≥0 G i and refer to it as the (maximal) prosolvable quotient of G.
• Given a profinite group G we write Sub(G) for the set of closed subgroups of G, and C(G) for the centre of G. For H ∈ Sub(G), we write N G (H) for the normaliser of H in G. • Given a profinite group G and a prime number l, we write G l an l-Sylow subgroup of G, which is defined up to conjugation. • Let G be a profinite group, H ⊂ G a closed subgroup, and l a prime number. We . Accordingly, commensurability (resp. l-commensurability) relation is (resp. is not in general) an equivalence relation on Sub(G). • Given an abelian profinite group A, we write A tor for the closure in A of the torsion subgroup A tor of A, and set A / tor def = A/A tor . Given a profinite group G, we set G ab / tor def = (G ab ) / tor . • Given a field K, we write K an algebraic closure of K, K sep for the maximal separable extension of K contained in K, and G K for the absolute Galois group Gal(K sep /K) of K. • Given a field K and an integer m ≥ 0, we write K m /K for the maximal m-step solvable subextension of K sep /K, which corresponds to the quotient • Given a field K, and H ⊂ Aut(K) a group of automorphisms of K, we write K H ⊂ K for the subfield of K which is fixed under the action of H. • A number field is a finite field extension of the field of rational numbers Q. For an (a possibly infinite) algebraic extension F of Q, we write Primes F (resp. Primes na F ) for the set of primes (resp. nonarchimedean primes) of F . We often identify Primes na Q with the set of prime numbers. For Q ⊂ F ⊂ F ′ ⊂ Q and p ∈ Primes na F ′ , we write p F ∈ Primes na F for the image of p in Primes na F . Further, for a set of primes S ⊂ Primes F , we write S(F ′ ) for the set of primes of F ′ above the primes in S: • Given an algebraic extension K of Q and a nonarchimedean prime p of K, we write κ(p) for the residue field at p. Further, when K is a number field, we write K p for the completion of K at p, and, in general, we write K p for the 5 • Given a number field K and a nonarchimedean prime p ∈ Primes na K above a prime p ∈ Primes Q , K p /Q p is a finite extension. We write d p , e p , f p , and N (p) for the local degree [K p : Q p ], the ramification index of K p /Q p , the residual degree [κ(p) : F p ], and the norm |κ(p)|, respectively, where O K is the ring of integers of K. (Thus, d p = e p f p and N (p) = p f p .) • Let K be a number field, and p a prime number which splits as ( which is a monotone non-decreasing finite sequence of positive integers. For each monotone non-decreasing finite sequence A of positive integers, write P K (A) ⊂ Primes na Q for the set of prime numbers with splitting type A in K. Two number fields K 1 , K 2 are called arithmetically equivalent if P K 1 (A) = P K 2 (A) for every such sequence A.
• Let l be a prime number and setl def = l (resp.l def = 4) for l = 2 (resp. l = 2). Then the multiplicative group Z × l is canonically decomposed into the direct product • Given a prime number l, a profinite group G, and a character χ : G → Z × l , we write χ : G → 1 +lZ l for the character defined by χ(g) = χ(g) (g ∈ G). = {m ∈ M | rm = 0 for some non-zero-divisor r ∈ R}. Given a ∈ R we write (a) = a R ⊂ R for the principal ideal of R generated by a. An Rsubmodule N of M is called R-cofinite if the quotient M/N is a finitely generated R-module. §1. The local theory. In this section we establish the local theory necessary to prove Theorems 1 and 2. We use the notations in the Introduction.

Structure of local Galois groups.
Let K be a number field, p ∈ Primes na K a nonarchimedean prime above a prime p ∈ Primes Q ,p a prime of K above p, and Dp ⊂ G K the decomposition group at p. Thus, Dp ∼ ← Gal(Kp/K p ) is isomorphic to the absolute Galois group of K p (cf. [Neukirch-Schmidt-Wingberg], (8.1.5) Proposition). We write Dp ։ D tamẽ p ։ D ur p for the maximal tame and unramified quotients of Dp, respectively (cf. loc. cit., discussion before (7.5.2) Proposition), and set Ip (i) The surjective map D m p ։ Dp m is an isomorphism. In particular, the natural surjective maps Gal(Kp/K p ) ab ։ D 1 p ։ Dp 1 are isomorphisms. (ii) If m ≥ 1, then p is the unique prime number l such that log l |D is the maximal normal pro-p subgroup of Dp m . (vii) For each 2 ≤ j ≤ m (resp. 0 ≤ j ≤ m − 1), the kernel of the projection Dp m ։ Dp j is pro-p (resp. infinite).
(viii) If m ≥ 2 and l is a prime number, then the inflation maps H 2 (D tamẽ Proof. (i) This follows, by induction on m ≥ 0, from the fact that the natural map D 1 p → G 1 K is injective (cf. [Gras], III, 4.5 Theorem) and applying this to finite extensions of K corresponding to various open subgroups of G m−1 K .
(ii)(iii)(iv) These assertions follow immediately from local class field theory.
(v) This follows from the fact that D ur p ≃ Z is abelian. (vi) This follows from (i), the fact that D tamẽ p is metabelian (cf. [Neukirch-Schmidt-Wingberg], (7.5.2) Proposition), and Ker(Dp ։ D tamẽ p ) is the maximal normal pro-p subgroup of Dp (cf. loc. cit., (7.5.7) Corollary (i)). (vii) Let 2 ≤ j ≤ m. Then, by (vi), there is a factorisation Dp m ։ Dp j ։ D tamẽ p , and Ker(Dp m ։ Dp j ) is a subgroup of the pro-p group Ker(Dp m ։ D tamẽ p ). Thus, Ker(Dp m ։ Dp j ) is also pro-p. Next, let 0 ≤ j ≤ m − 1. Then Ker(Dp m ։ Dp j ) contains Ker(Dp m ։ Dp where the first equality follows from (i). So, it suffices to prove that Dp[i] ab is infinite for i ≥ 1. Let Dp ⊃ Ip ։ I tamẽ p (≃Ẑ (p ′ ) ) be the inertia and the tame inertia groups, and I wild p = Ker(Ip ։ I tamẽ p ) the wild inertia group. By (i) and local class field theory, Im(Ip → G 1 K ) is a direct product of a finitely generated pro-p abelian group and a prime-to-p finite cyclic group. By (v) (resp. (vi)), for i ≥ 1 (resp. i ≥ 2), Dp where the horizontal maps are cup products. By local Tate duality, the lower horizontal map gives a perfect pairing, hence is surjective (as H 1 (Dp, F p (1)) = K × p /(K × p ) p = 0). Thus, it suffices to show that the natural inflation maps where the horizontal and vertical maps are inflation and restriction maps, respectively. Here, the vertical maps are isomorphisms since |∆p| is prime to p (as it divides p − 1). The lower horizontal map is also an isomorphism since N ab is torsion free for i = 1, . . . , m − 1, H must be trivial, as desired.
1.2. Separatedness in G m K . Let K be a number field, p, p ′ ∈ Primes na K , and D p , D p ′ ⊂ G K the decomposition groups at p, p ′ , respectively. Then it is well-known (cf. [Neukirch-Schmidt-Wingberg], (12.1.3) Corollary) that the following separatedness property holds: This does not hold as it is, if K and G K are replaced by K m and G m K , respectively (cf. Propositions 1.3 and 1.13). However, we show certain weaker separatedness properties hold in the latter situation.
In this case, one has Proposition 1.3. Let m ≥ 1 be an integer, K a number field, p, p ′ ∈ Primes na K m , D p , D p ′ ⊂ G m K the decomposition groups at p, p ′ , respectively, and p, p ′ the images of p, p ′ in Primes na K m−1 , respectively. Then the following are equivalent.
are infinite as follows from Proposition 1.1(vii). The implication (i) =⇒ (ii) for m = 1 follows from [Gras], III, 4.16.7 Corollary. Thus, we prove the implication (i) =⇒ (ii) for m ≥ 2. For this, assume that (i) holds, and set one obtains p M ′ = p ′ M ′ by applying the implication (i) =⇒ (ii) for m = 1 (which we have already established) to the number field M ′ . Thus, one has p M = p ′ M , a fortiori. As M/K is an arbitrary finite subextension of K m−1 /K, this shows that p = p ′ , as desired.
Lemma 1.4. Let G be a profinite group and H a closed subgroup of G. Let l be a prime number. Consider the following conditions (i)-(v).
The main result in this subsection is the following.
Proposition 1.5. Let K be a number field, and K/K a (an infinite) Galois extension such that Q ab ⊂ K. Then the natural surjective map: Primes na K ab ։ Dec( K ab /K) is bijective.
Corollary 1.6. Let K be a number field and m ≥ 2 an integer. Then the natural surjective map Primes na Proof of Corollary 1.6. Apply Proposition 1.5 to K = K m−1 .
Corollary 1.7. With the notations and assumptions in Proposition 1.5, the centraliser of Gal( K ab /K) in Aut( K ab ) is trivial, and Gal( K ab /K) is centre free. In particular, for m ≥ 2, the centraliser of G m K in Aut(K m ) is trivial, and G m K is centre free.
Proof of Corollary 1.7. By definition, the natural action (by conjugacy) on Dec( K ab /K) of the centraliser of Gal( K ab /K) in Aut( K ab ) is trivial, hence, by Proposition 1.5, that on Primes na K ab is trivial. The first assertion follows from this, together with Lemma 1.8 below (applied to E = F = K). The second assertion follows from the first. The third and the fourth assertions follow from the first and the second, applied to K = K m−1 .
Lemma 1.8. Let E, F be algebraic extensions of Q. Then the natural map Proof. Let σ 1 , σ 2 ∈ Isom (fields) (E, F ) and assume that they induce the same bijec- By assumption, σ induces the identity on Primes na F , hence σ ′ 0 and σ ′ 00 induces the identity on Primes na F ′ 0 and Primes na F ′ 00 , respectively. Now, by Chebotarev's density theorem, there exists p ∈ Primes na F 00 which splits completely in F ′ 00 /F 00 . Then Gal(F ′ 00 /F 00 ) = σ ′ 00 acts regularly (i.e. transitively and freely) on the set of primes in Primes na F ′ 00 above p. As this action is also trivial, one must have F ′ 00 = F 00 , hence Proposition 1.5 follows immediately from the implication (ii) =⇒ (i) in the following Proposition 1.9. Proposition 1.9. With the notations and assumptions in Proposition 1.5, let follow immediately, while the implication (vi) =⇒ (vii) follows from [Gras], III, (4.6.17) Corollary (applied to various finite subextensions of K/K), as in the proof of Proposition 1.3, (i) =⇒ (ii). So, we may concentrate on proving the implication (iv) =⇒ (i). We start with some lemmas.
Lemma 1.10. Let F be a number field, F ′ /F a finite abelian extension, q, q ′ ∈ Primes na F ′ , with q = q ′ . Then there exists a subextension F ′′ /F of F ′ /F such that F ′′ /F is cyclic of prime power order; is a cyclic subgroup of prime power order. As σ = 1, there exists at least one i such that the image of σ under the projection G/D ։ C i is nontrivial. Now the subextention F ′′ /F of F ′ /F corresponding to the quotient G ։ G/D ։ C i satisfies the conditions. Lemma 1.11. Let F be a number field, and q ∈ Primes na F . For each integer m > 0, write ψ F,q,m for the natural map Then, for each pair of positive integers m, n, the natural map Ker(ψ F,q,mn ) → Ker(ψ F,q,m ) induced by the natural projection Proof of Lemma 1.11. Consider the following commutative diagram in which the two rows are exact. (Here, the first horizontal arrows are induced by the m-th power maps and the second horizontal arrows are the natural projections.) Observe that the vertical arrows are surjections. Indeed, for each integer k > 0, (F × q ) k is open in F × q , and the image of the natural inclusion F × ֒→ F × q is dense. Now, the assertion follows from diagram-chasing.
Proof of Proposition 1.9, (iv) =⇒ (i). Assume that (iv) holds. (Then (vii) holds, a fortiori, i.e. p K = p ′ K .) Suppose that (i) does not hold, i.e. p = p ′ . Then there exists a finite subextension L/K of K ab /K such that p L = p ′ L . Set M def = L ∩ K. Then M/K is a finite subextension of K/K and L/M is a finite extension. As M ⊂ K, one has p M = p ′ M . Note that we may replace L (and M , correspondingly) by any finite extension of L contained in K ab .
Step 1. Reduction: We may assume that L/M is abelian. Indeed, if we replace L by the Galois closure L 1 of L/M (and M by M 1 def = L 1 ∩ K), this holds.
Step 2. Reduction: We may assume that L/M is cyclic of order l r for some prime number l and some integer r ≥ 0 and that p M splits completely in L/M . Indeed, this follows from Lemma 1.10, applied to L/M .
Step 3. Reduction: We may assume that the conditions of Step 2 and the condition that ζ l r ∈ M hold. Indeed, if we replace L by L(ζ l r ) (and M by L(ζ l r ) ∩ K), this holds with r replaced by r ′ below. First, one has M (ζ l r ) ⊂ L(ζ l r )∩M Q ab ⊂ L(ζ l r ) ∩ K. Next, consider the commutative diagram of fields: As L/M is cyclic of order l r and p M splits completely in L/M , we see that L(ζ l r )/L(ζ l r ) ∩ K is cyclic of order l r ′ with r ′ ≤ r and p L(ζ l r )∩ K splits completely in L(ζ l r )/L(ζ l r ) ∩ K. (Observe Gal(L(ζ l r )/(L(ζ l r ) ∩ K)) ⊂ Gal(L(ζ l r )/M (ζ l r )) ֒→ Gal(L/M ).) As ζ l r ′ ∈ M (ζ l r ) ⊂ L(ζ l r ) ∩ K, we are done.
Step 3 is the final reduction step and we will not replace L (or M ) any more.
Step 4. r > 0. Indeed, otherwise, L = M , which contradicts p L = p ′ L and p M = p ′ M .
Step 5. L/M is linearly disjoint from M (ζ l ∞ )/M . Indeed, one has Step 6. The l-adic cyclotomic character and l-commensurability. Let χ (l) cycl : factors through Gal( K/K), hence, a fortiori, through Gal( K ab /K). By abuse, write χ cycl . Since the number of l-power roots of unity in each finite extension of M p is finite (as M p is a finite extension of Q p , where p is the characteristic of κ(p)), χ Step 7 As This, together with the fact that ζ l r+r 0 ∈ M ′ , implies L ′′ /M ′ is cyclic of order l r+r 0 . Further, as g ∈ M × , the extension L ′′ /M is Galois. By the choice of g, the image of Step 8. End of proof. Consider the following commutative diagram in which the horizontal isomorphisms arise from Kummer theory, the vertical arrows are induced by the natural inclusion M ֒→ M ′ , and the equality follows from the fact that ζ l r+r 0 ∈ M ′ . Here, the image of g mod ( i.e., τ ∈ Gal(L ′′ /M ′ ) l r = Gal(L ′′ /L ′ ). This implies p ′ L ′ = τ p L ′ = p L ′ , a contradiction.
In Proposition 1.9, the implication (v) =⇒ (i) fails in general. (For example, it fails in the case K = K m−1 treated in Corollary 1.6.) More precisely, we show Proposition 1.13 below. Let P 0 be a subset of Primes na Q . Definition 1.12. (i) We say that a profinite group is P 0 -perfect, if it admits no nontrivial pro-l abelian quotient for any l ∈ P 0 . (ii) We say that a field F ⊂ Q is P 0 -perfect, if Gal(Q/F ) is P 0 -perfect. Proposition 1.13. Assume P 0 Primes na Q . Then, with the assumptions in Proposition 1.5, the following are equivalent.
14 Proof. The implication (ii) =⇒ (i) just follows from the proof of Proposition 1.9, (iv) =⇒ (i). (The assumption P 0 Primes na Q is used to ensure (vii) there, via (vi) there.) More precisely, the only new input is the observation that, under the assumption that (ii) holds, l in Step 2 there automatically belongs to Primes na Q \ P 0 . Indeed, otherwise, i.e., if l ∈ P 0 , one must have L ⊂ K, since L/M is cyclic of lpower order, M ⊂ K, and K is P 0 -perfect.
To prove the implication (i) =⇒ (ii), suppose that K is not P 0 -perfect. Then there exists l 0 ∈ P 0 such that K admits a finite cyclic extension L of degree l 0 , which then descends to a finite cyclic extension L/M of degree l 0 over some field M with K ⊂ M ⊂ K and [M : K] < ∞. (Thus, in particular, L = L K.) By Chebotarev's density theorem, p M splits completely in L/M for some p ∈ P K ab . We fix such p.
Next, as D p ( K ab /M ) is a quotient of G M p , it is prosolvable. So, there exists an l ′ 0 -Hall subgroup D of D p ( K ab /M ), that is, D is pro-prime-to-l 0 and (D p ( K ab /M ) : D) is a (possibly infinite) power of l 0 . Now, consider the exact sequence of profinite abelian groups which induces an exact sequence of pro-l 0 abelian groups where () l 0 refers to the l 0 -Sylow subgroups. As Gal( L/ K) = Gal(L K/ K) ∼ → Gal(L/M ), the Galois group Gal( K ab /M ) acts (by conjugation) trivially on Gal( L/ K), hence, a fortiori, so does D ⊂ D p ( K ab /M ) ⊂ Gal( K ab /M ). But as D is a proprime-to-l 0 group, the sequence obtained by taking the D-fixed parts 1 → (Gal( K ab / L) l 0 ) D → (Gal( K ab / K) l 0 ) D → Gal( L/ K) D = Gal( L/ K) → 1 remains exact. Thus, a generator of Gal( L/ K) ≃ Z/l 0 Z lifts to an element τ of (Gal( K ab / K) l 0 ) D ⊂ Gal( K ab / K) D . As the image of τ in Gal(L/M ) is nontrivial and p M splits completely in L/M , it follows that τ ∈ D p ( K ab /M ), or, equivalently, On the other hand, as D fixes (i.e. commutes with) τ , one has τ Dτ −1 = D. Note that D (resp. τ Dτ −1 = D) is an l ′ 0 -Hall subgroup of D p ( K ab /M ) (resp. τ D p ( K ab /M )τ −1 = D p ′ ( K ab /M )). Namely, D is an l ′ 0 -Hall subgroup of both D p ( K ab /M ) and D p ′ ( K ab /M ), which implies that D p ( K ab /M ) and D p ′ ( K ab /M ) are l-commensurable for every l ∈ Primes na Q \ {l 0 }, hence, in particular, for every l ∈ Primes na Q \ P 0 . This contradicts (i), as p ′ = p.

Correspondence of decomposition groups.
Finally, we establish local theory in our context. More precisely, starting from an isomorphism of the maximal (m + 2)-step solvable Galois groups of number fields, we establish a one-to-one correspondence between the sets of their nonarchimedean primes and a one-to-one correspondence between the corresponding decomposition groups (cf. Corollary 1.27). This is achieved via a purely group-theoretic characterisation of decomposition groups (cf. Theorem 1.25).
As in the proof of the Neukirch-Uchida theorem, our proof is based on the following local-global principle.
Proposition 1.14. Let K be a number field, and l a prime number. For a prime p ∈ Primes K , let D p ⊂ G K be a decomposition group at p (D p is only defined up to conjugation). Then the following hold. (i) If p ∈ Primes na K , then H 2 (D p , F l (1)) ∼ → F l . (ii) If l is odd or K is totally imaginary, then there exists a natural injective homomorphism where the direct sum is over all nonarchimedean primes p ∈ Primes na K . Proof. Assertions (i) and (ii) are well-known (cf. [Neukirch-Schmidt-Wingberg], (8.1.5) Proposition, (7.1.8) Theorem (ii), (9.1.10) Corollary, and the fact that H 2 (D p , F l (1)) = 0 if p is archimedean and either l = 2 or K is totally imaginary).
Corollary 1.15. Let L be an algebraic extension of Q, and l a prime number. For a prime p ∈ Primes L , let D p ⊂ G L be a decomposition group at p (D p is only defined up to conjugation). Then the following hold.
where the product is over all nonarchimedean primes p ∈ Primes na L . Proof. The first half of assertion (i) and assertion (ii) are reduced to assertion (i) and assertion (ii) of Proposition 1.14, respectively, by taking the inductive limits. The second half of assertion (i) also follows from assertion (i) of Proposition 1.14, and the fact that for open subgroups D ′ ⊂ D ⊂ D p ( ∼ ← G Q p ), the restriction map from H 2 (D, F l (1)) = F l to H 2 (D ′ , F l (1)) = F l is the (D : D ′ )-multiplication.
The following lemma will be of important use later.
Lemma 1.16. Let G be a profinite group, m ≥ 0 an integer and N a finite discrete G m -module. For a closed subgroup F ⊂ G m , write F (resp. F 1 ) for the inverse image of F in G (resp. in G m+1 ). Then the natural map H 2 (F, N ) Proof. We have the following commutative diagram where the horizontal sequences are the exact sequences arising from the Hochschild-Serre spectral sequences, the vertical maps are inflation maps, and G[m + 1, m] act trivially on N . The middle vertical map is an isomorphism since it is the identity. Now, our assertion follows by an easy diagram chasing.
The following result is the main step towards establishing the desired local theory in our context. Proposition 1.17. Let K be a number field, and l a prime number. Let F ⊂ G m K be a closed subgroup. (We use the notations in Lemma 1.16 for G = G K and F .) Set L def = (K m ) F , and assume that either l is odd or L is totally imaginary. For each nonarchimedean primep ∈ Primes na L , write Fp ⊂ F = G L for a decomposition group atp (defined up to conjugation in F ). Then there exists a natural injective homomorphism where the product is over all nonarchimedean primesp ∈ Primes na L . Proof. We have natural maps where the first map is an inflation map and the second map is a product of restriction maps. The first map is injective by Lemma 1.16 and the second map is injective by Corollary 1.15 (ii). Thus, the composite is also injective, as desired.
Definition 1.18. Let l be a prime number. Given a profinite group F , we say that F is an l-decomposition like group if there exists an exact sequence 1 → F 1 → F → F 2 → 1 where F 1 , F 2 are free pro-l of rank 1 (i.e., isomorphic to Z l ).
Lemma 1.19. Let F be an l-decomposition like group and D a closed subgroup of F . Consider the following conditions: Proof. The implications (i) =⇒ (ii), (i) =⇒ (iv), (vi) =⇒ (vii) are trivial. We prove (v) =⇒ (vi). As D is an l-decomposition like group, there exists an exact sequence 1 → D 1 → D → D 2 → 1 with D 1 , D 2 ≃ Z l . Then by the Hochschild-Serre spectral sequence and the fact that cd l (Z l ) = 1 and H 1 (Z l , F l ) ≃ F l , one has H 2 (D, F l ) ∼ → H 1 (D 2 , H 1 (D 1 , F l )) ≃ H 1 (D 2 , F l ) ≃ F l , as desired. (For the second isomorphism, observe that the isomorphism H 1 (D 1 , F l ) ≃ F l is automatically D 2equivariant, since any homomorphism from the pro-l group D 2 to F × l is trivial.) In particular, by applying (v) =⇒ (vi) to D = F , one has H 2 (F, F l ) ≃ F l . Thus, the implication (ii) =⇒ (iii) follows.
Next, as F is an l-decomposition like group, there exists an exact sequence H 1 (D 1 , F l ))) is nontrivial, one must have D 1 = F 1 and D 2 = F 2 , hence D = F . This shows (iii) =⇒ (i). Next, if D is open in F , then D 1 and D 2 must be open in F 1 and F 2 , respectively. As any open subgroup of Z l is isomorphic to Z l , this shows (iv) =⇒ (v). Further, as H 2 (D, F l ) H 1 (D 1 , F l )), (vii) implies D 1 = 1 and D 2 = 1. As any nontrivial closed subgroup of Z l is open, this shows (vii) =⇒ (iv). Finally, as D is an extension of D 2 ֒→ F 2 ≃ Z l by D 1 ֒→ F 1 ≃ Z l , one has cd l (D) ≤ 2. Thus, (vii) ⇐⇒ (viii) follows (cf. [Neukirch-Schmidt-Wingberg], (3.3.2) Proposition). This completes the proof.
Definition 1.20. Let m ≥ 2 be an integer, F ⊂ G m K a closed subgroup, and l a prime number. Then we say that F satisfies condition (⋆ l ) if the following two conditions hold. (⋆ l )(a) F is an l-decomposition like group (cf. Definition 1.18). (⋆ l )(b) With the notations in Lemma 1.16, H 2 (F, F l ) = 0.
Remark 1.21. We use the notations in Definition 1.20. (i) Assume that F satisfies condition (⋆ l )(a). Then the F -module F l (1) is isomorphic to the trivial module F l , as any homomorphism from the pro-l group F to F × l is trivial. This, together with Lemma 1.19, (i) =⇒ (vi), shows that condition (⋆ l )(b) is equivalent to saying that H 2 (F, F l (1)) ≃ F l . (ii) Condition (⋆ l ) is group-theoretic in the following sense: One can detect purely group-theoretically whether or not a closed subgroup F ⊂ G m K satisfies (⋆ l ), if we start from (the isomorphy type of) G m+1 K . Proposition 1.22. Let m ≥ 2 be an integer, F ⊂ G m K a closed subgroup, and l a prime number. Then F satisfies condition (⋆ l ) if and only if F is an open subgroup of an l-Sylow subgroup of the decomposition group D p ⊂ G m K at some p ∈ Primes na K m with residue characteristic = l. Further, then the image p ∈ Primes na K m−1 of p is uniquely determined by F .
Proof. First, we prove the 'only if' part of the first assertion. So, assume that F is an open subgroup of an l-Sylow subgroup of the decomposition group D p ⊂ G m K at some p ∈ Primes na K m with residue characteristic p = l. By Proposition 1.1(vi), F is isomorphically mapped onto an open subgroup F of an l-Sylow subgroup of D tamẽ p (wherep is any element of Primes K above p) by the surjection D p ։ D tamẽ p . Note that there exists an exact sequence with D ur p ≃ Z and I tamẽ p ≃ Z (p ′ ) . This implies that F ( ∼ → F ) satisfies (⋆ l )(a). Again by Proposition 1.1(vi), there exists a closed subgroup F ′ of Dp which is isomorphically mapped onto F by the surjection Dp ։ D p . The isomorphism F ′ ∼ → F factors as F ′ ֒→ F ։ F , where F ′ ֒→ F is the natural inclusion and F ։ F is the natural surjection induced by the surjection In particular, the inflation map H 2 (F, F l ) → H 2 ( F , F l ) is injective, which implies H 2 (F, F l ) ∼ → H 2 (F, F l ). Now, by the fact that F satisfies condition (⋆ l )(a), together with Lemma 1.19, (i) =⇒ (vi), one has H 2 (F, F l ) ( Next, we prove the 'if' part of the first assertion. So, assume that F satisfies condition (⋆ l ). Let L def = (K m ) F . We claim that L is totally imaginary. Indeed, otherwise, one has an embedding L ֒→ R, which extends to an embedding K m ֒→ C. To these embeddings, a homomorphism Gal(C/R) → G m K is associated. As , this homomorphism is injective. This is absurd, since F is torsion free as it satisfies (⋆ l )(a). Together with this, Proposition 1.17 and Remark 1.21(i) imply that we have an injective homomorphism where the product is over all nonarchimedean primesp ∈ Primes na L . This map is nontrivial since F satisfies condition (⋆ l )(b). Thus, there existsp ∈ Primes L such that the map H 2 (F, F l ) → H 2 ( Fp, F l ) is nontrivial. In particular, the map The former nontriviality, together with Lemma 1.19, (iii) =⇒ (i), implies that F = Fp. The latter nontriviality, together with Corollary 1.15(i), implies that Fp is l-open in a decomposition subgroup of G Q at p, where p is the residue characteristic ofp. In particular, F = Fp is l-open in a decomposition subgroup D p of G m K at p (where p ∈ Primes na K stands for the image ofp), as desired. Next, take a finite abelian extension K ′ of K p with [K ′ : Q p ] > 1, which corresponds to an open subgroup H of D p containing D p [1] (cf. Proposition 1.1(i)). As F ∩ H is l-open in H, the natural map (F ∩ H) ab ⊗ Z l Q l → H ab ⊗ Z Q l must be surjective. Here, by Lemma 1.19, (iv) =⇒ (v), F ∩ H is an l-decomposition like group, hence (toplogically) generated by 2 elements and dim Q l ((F ∩ H) ab ⊗ Z l Q l ) ≤ 2. On the other hand, by local class field theory, together with the fact that m ≥ 2, one has dim(H ab ⊗ Z Q l ) = 1 (resp. [K ′ : Q p ] + 1 > 2) if l = p (resp. l = p). Therefore, one must have l = p, as desired.
Finally, the second assertion follows immediately from Proposition 1.3. This completes the proof.
Let m ≥ 2 be an integer. For a prime number l, write Proposition 1.24. We use the above notations. Let m ≥ 2 be an integer, and l 1 = l 2 prime numbers. Then one has in Sub(G m−1 K ). In particular, the subset Dec(K m−1 /K) ⊂ Sub(G m−1 K ) can be recovered group-theoretically from G m+1 K .
Proof. The first assertion follows from the various definitions and Proposition 1.23. The second assertion follows from the first and the fact that for each prime number l, the G m−1 K -set D m,l , hence the subset St(D m,l ) ⊂ Sub(G m−1 K ), can be recovered group-theoretically from G m+1 K (cf. Remark 1.21(ii)).
We have a natural surjective map Primes na Theorem 1.25. Let m ≥ 2 (resp. m = 1) be an integer. The G m K -set Dec(K m /K) (resp. the set Dec(K 1 /K)), hence the G m K -set Primes na K m (resp. the set Primes na K ), can be recovered group-theoretically from G m+2 K .
For an integer m ≥ 1, let χ cycl : G m K (։ G 1 K ) → Z × be the cyclotomic character. Theorem 1.26. Let m ≥ 3 be an integer. Then χ cycl can be recovered grouptheoretically from G m K . Proof. We may assume m = 3, and show that for each prime number l, the l-part of the cyclotomic character χ (l) cycl : G 2 K → Z × l can be recovered group-theoretically from G 3 K . We claim that the following group-theoretic characterisation of χ (l) cycl holds: Let χ : G 2 K → Z × l be a character. Then, χ = χ (l) cycl if and only if for every F ∈ D 2,l , every h ∈ F ∩ G K [2, 1] and every g ∈ N G 2 K (F ∩ G K [2, 1]), one has ghg −1 = h χ(g) . (Recall that the definition of the set D 2,l (⊂ Sub(G 2 K )) involves G 3 K .) To prove this claim, we first determine the normaliser N G 2 K (F ∩ G K [2, 1]) for each F ∈ D 2,l . By Proposition 1.22, F is an open subgroup of an l-Sylow subgroup of the decomposition group D p ⊂ G 2 K at some p ∈ Primes na K 2 with residue characteristic = l, and the image p ∈ Primes na K 1 of p is uniquely determined by F . Then one has N G 2 for the inertia and the tame inertia groups. By Proposition 1.
where (D p ∩G K [2, 1]) l stands for the l-Sylow subgroup of the profinite abelian group D p ∩ G K [2, 1], and the last isomorphism follows from Proposition 1.1(vi). As F is an open subgroup of an l-Sylow subgroup of the decomposition group D where p 0 is the image of p in Primes na K , hence, in particular, (by local class field theory) the l-Sylow group of D p is isomorphic to the direct product of a finite cyclic group of l-power order (that is the l-Sylow subgroup of κ(p 0 ) × ) and Z l . It follows from this that the inclusion (D p ∩ G K [2, 1]) (l) ֒→ I (l) 1] is nontrivial. Therefore, by Proposition 1.3, (i) =⇒ (ii), one has p = gp = π(g)p, which implies π(g) ∈ D p . Thus, ]. Now, we prove the 'only if' part of the claim. As the inclusion F ∩G K [2, 1] ֒→ I (l) p is compatible with the actions of D p (by conjugation), one has ghg −1 = h χ (l) cycl (g) for every h ∈ F ∩ G K [2, 1] and every g ∈ D p . On the other hand, as G K [2, 1] is abelian and χ (l) , as desired. Finally, we prove the 'if' part of the claim. For this, it suffices to show that G 2 K is (topologically) generated by N G 2 K (F ∩ G K [2, 1]) (F ∈ D 2,l ). As in the preceding arguments, one has F ⊂ D p for some p ∈ Primes K 2 and N G 2 K (F ∩ G K [2, 1]) = D p · G K [2, 1]. So, it suffices to prove that G 1 K is generated by D p , where p is the image of p in Primes K 1 . This follows from Chebotarev's density theorem, together with (the surjectivity in) Proposition 1.23(i), as desired.
(iii) Let p ∈ Primes na K and q def = φ(p) ∈ Primes na L . Then d p = d q , e p = e q , f p = f q , and N (p) = N (q). In particular, K and L are arithmetically equivalent (cf. Notations).
(iv) Let p ∈ Primes na K m (resp. p ∈ Primes na K ) and q def = φ m (p) ∈ Primes na L m (resp. q def = φ(p) ∈ Primes na L ). Let D p , D q be the decomposition subgroups of G m K , G m L , respectively, corresponding to p, q, respectively, I p , I q the inertia subgroups of D p , D q , respectively, and Frob p ∈ D p /I p , Frob q ∈ D q /I q the Frobenius elements. Then the isomorphism D p ∼ → D q induced by τ m (cf. (i)) restricts to an isomorphism I p and N (p ′ ) = N (q ′ ). In particular, K ′ and L ′ are arithmetically equivalent.
(v) Write p (resp. p) for the image of p ′ in Primes na K (resp. Primes na Q ) and set q = φ(p) ∈ Primes na L . Takep ∈ Primes na K m above p ′ and setq def = φ m (p) ∈ Primes na L m . 23 Then q and p (resp.q) are below (resp. is above) q ′ (cf. (i) and (ii)). By (i) and (iv), one has τ m : Dp ∼ → Dq and τ m : where the first and the second formulae follow from (iii), the third formula follows form the first and the second, and the last formula follows from the third. This finishes the proof of Corollary 1.27. §2. Proof of Theorem 1.
For a number field K let I K be the multiplicative monoid freely generated by the elements of Primes na K (cf. [Cornelissen-de Smit-Li-Marcolli-Smit]), endowed with the norm function defined by p → N (p) (p ∈ Primes na K ). Let K, L be number fields and τ : G 3 Then p (resp. q) is unramified in the extension K ′ /K (resp. L ′ /L) if and only the image of I p (resp. I q ) in Gal(K ′ /K) (resp. Gal(L ′ /L)) is trivial. In particular, p is unramified in K ′ /K if and only if q is unramified in L ′ /L, and the isomorphism Gal(K ′ /K) ∼ → Gal(L ′ /L) induced by τ 1 : G 1 K ∼ → G 1 L maps the image of Frob p to that of Frob q (cf. Corollary 1.27 (iv)). Thus, by the main theorem of [Cornelissen-de Smit-Li-Marcolli-Smit], there exists a field isomorphism σ : K ∼ → L (cf. loc. cit. Theorem 3.1, the equivalence (ii) ⇐⇒ (iv)). This finishes the proof of Theorem 1. §3. Proof of Theorem 2.
Let m ≥ 0 be an integer, K, L number fields, and τ m+3 : G m+3 K ∼ → G m+3 L an isomorphism of profinite groups. We show the existence of a field isomorphism σ m : K m ∼ → L m such that τ m (g) = σ m gσ −1 m for every g ∈ G m K , where τ m : G m K ∼ → G m L is the isomorphism induced by τ m+3 . We follow Uchida's method in [Uchida2]. Let K ′ /K be a finite Galois subextension of K m /K with Galois group H, corresponding to a normal open subgroup U ⊂ G m K , and V def = τ m (U ) corresponding to a finite Galois subextension L ′ /L of L m /L with Galois group J. The isomorphism τ m induces naturally an isomorphism τ : H ∼ → J. Let T (K ′ ) be the (finite) set of field isomorphisms σ : K ′ ∼ → L ′ such that τ (h) = σhσ −1 for every h ∈ H. It is easy to see that {T (K ′ )} K ′ /K forms a projective system, and the projective limit lim ← − K ′ /K T (K ′ ) consists of isomorphisms σ m : K m ∼ → L m satisfying the condition in Theorem 2. Further, if T (K ′ ) = ∅ for any K ′ as above, then the projective limit lim ← − K ′ /K T (K ′ ) over all such finite sets T (K ′ ) would be nonempty.

24
Now, the same proof as in [Uchida2] shows that T (K ′ ) = ∅. More precisely, the proof in loc. cit. applies as it is by noting the following. First, one needs to know that K ′ and L ′ above are arithmetically equivalent, which in our case follows from Corollary 1.27(v). Second, one needs to know that certain finite abelian extensions of K ′ and L ′ introduced (and denoted by m j=0 M 1,j and m j=0 M 2,j ) in loc. cit. are arithmetically equivalent. Since these extensions are contained in K m+1 and L m+1 , respectively, and correspond to each other via τ m+1 : G m+1 K ∼ → G m+1 L , this follows again from Corollary 1.27(v) (applied to m + 1 and m + 3 = (m + 1) + 2 instead of m and m + 2). The rest of the proof that T (K ′ ) = ∅ is the same as in loc. cit. This finishes the proof of (i).