Pro-p hom-form of the birational anabelian conjecture over sub-p-adic fields

Scott Corry, Florian Pop

Department of Mathematics, University of Pennsylvania, 209 S. 33rd St., Philadel-

phia, PA 19104, USA (e-mail: scorry@math.upenn.edu / pop@math.upenn.edu)

Received: date / Revised version: date

Abstract. We prove a Hom-form of the pro-p birational anabelian conjecture for function fields over sub-p-adic fields. Our starting point is the corresponding Theorem of Mochizuki in the case of transcendence degree 1.

1. Introduction

Let k be a fixed sub-p-adic field, i.e., a subfield of a function field over ℚ_{p}. Fix \overline{k}, an algebraic closure of k, and denote by G_{k} the absolute Galois group. Let F_{k} be the category of regular function fields K|k, and k-embeddings of function fields. Further, let G_{k} be the category of profinite groups, G, endowed with a surjective augmentation morphism, π_{G} : G → G_{k}, such that ker(π_{G}) is a pro-p group, and outer open G_{k}-homomorphisms. I.e., a morphism from G to H in G_{k} is of the form Inn_{G_{k}}(H) ∘ f, where f : G → H is an open homomorphism such that π_{G} = π_{H} ∘ f, and Inn_{G_{k}}(H) denotes the group of inner automorphisms of H which lie over G_{k}. Since G_{k} has trivial center, Inn_{G_{k}}(H) consists exactly of the inner automorphisms of H defined by an element of ker(π_{H}). Finally, we remark that there exists a naturally defined functor from F_{k} to G_{k}: for K|k from F_{k}, let K̃|K\overline{k} be a maximal pro-p extension. Then K̃|K is Galois, and Π_{K} := Gal(K̃|K) endowed with the projection π_{K} : Π_{K} → G_{k} is an object of G_{k}. Further, a morphism ı : K|k ↪ L|k in F_{k} extends uniquely to a \overline{k}-embedding \overline{ı} : K\overline{k} ↪ L\overline{k}, and \overline{ı} has prolongations ĩ: K̃ ↪ L̃. Each such prolongation ĩ: K̃ ↪ L̃ gives rise to an open G_{k}-homomorphism Φĩ: Π_{L} → Π_{K} defined by

Φĩ(g) = ĩ^{−1} ∘ g ∘ ĩ, g ∈ Π_{L} ,

and any two such prolongations are conjugate by an element from ker(π_{K}). Thus, sending each K|k from F_{k} to π_{K} : Π_{K} → G_{k} in G_{k} yields a well defined functor from F_{k} to G_{k}.

The purpose of this note is to prove the following Galois by pro-p Hom-form of the birational anabelian conjecture:

Theorem 1 The above functor from F_{k} to G_{k} is fully faithful, i.e., for regular function fields K|k and L|k, there is a canonical bijection

Hom_{k}(K, L) → Hom_{G_{k}}(Π_{L}, Π_{K}).

Equivalently, for fixed field extensions K̃|K and L̃|L as above, the map

ĩ ↦ Φĩ with Φĩ(g) = ĩ^{−1} ∘ g ∘ ĩ for g ∈ Π_{L} ,

is a bijection from the set of \overline{k}-embeddings ĩ: K̃ ↪ L̃ onto the set of all the open G_{k}-morphisms Π_{L} → Π_{K}.

Before embarking on the proof, the following comments are in order: first, if td(K|k) = 1, then Theorem 1 is a special case of Theorem 16.5 in the fundamental paper by Mochizuki [2]. Second, the above Theorem 1 implies the corresponding full profinite version, in which Π_{K} is replaced by the full absolute Galois group G_{K} of K. But naturally, the above Theorem 1 does not follow from the corresponding full profinite version. Finally, the full profinite version of Theorem 1 above was proved by Mochizuki in loc. cit., where it appears as Corollary 17.1. There he uses an inductive procedure on td(K|k), which is ill-suited to the pro-p situation, and hence he obtains only a profinite result. In our proof of Theorem 1 we use Mochizuki’s pro-p result for the transcendence degree one case, but instead of proceeding inductively on the transcendence degree, we will make use of the second author’s ideas as described in [3].

2. Proof of Theorem 1

The proof of Theorem 1 will have two parts:

i) Given an open G_{k}-homomorphism Φ : Π_{L} → Π_{K}, there exists a \overline{k}-embedding ĩ_{Φ} : K̃ ↪ L̃ which defines Φ as indicated above, i.e., such that Φ = Φĩ_{Φ}.

ii) The map ĩ ↦ Φĩ is injective.

We begin by recalling some basic facts from Kummer Theory. First, for every K|k from F_{k}, let 𝕋_{p,K} = lim_{⟵}_{e} μ_{p^{e},K} be the p-adic Tate

module of 𝔾_{m,K}. Then via the canonical projection π_{K} : Π_{K} → G_{k}, we can/will identify 𝕋_{p,K} with 𝕋_{p,k}, and denote it simply by 𝕋_{p}. Second, let K̂ denote the p-adic completion of K^{×}, and ȷ_{K} : K^{×} → K̂ the completion morphism. We note that ker(ȷ_{K}) is the maximal p^{∞}- divisible subgroup of K^{×}, hence by the structure of K|k it follows that ker(ȷ_{K}) = μ′ is the group of roots of unity in k of order prime to p. Further, K^{×}/k^{×} is a free abelian group, hence it injects into its p-adic completion (K^{×}/k^{×}) which itself equals K̂/k̂. Next, recall that Kummer Theory yields a canonical ismorphism of p-adically complete groups δ_{K} : K̂ → H^{1}(Π_{K}, 𝕋_{p}). Therefore we will make the identification K̂ = H^{1}(Π_{K}, 𝕋_{p}), if this does not lead to confusion. By the functoriality of Kummer Theory, the surjective projection π_{K} : Π_{K} → G_{k} gives rise to an embedding H^{1}(π_{K}) : k̂ ↪ K̂ which is nothing but the p-adic completion of the structural morphism k ↪ K. Furthermore, if K|k and L|k are objects from F_{k}, and Φ : Π_{L} → Π_{K} is an open G_{k}-morphism, then by functoriality we get an embedding of p-adically complete groups

H^{1}(Φ) : K̂ = H^{1}(Π_{K}, 𝕋_{p}) → H^{1}(Π_{L}, 𝕋_{p}) = L̂

which identifies k̂ ⊂ K̂ with k̂ ⊂ L̂. Finally note that if Φ = Φĩ is defined by a morphism ı : K|k ↪ L|k from F_{k}, then H^{1}(Φĩ) = î is nothing but the p-adic completion of the k-embedding ı : K|k ↪ L|k, and therefore one has:

(†)  H^{1}(Φĩ) ∘ ȷ_{K}(x) = ȷ_{L} ∘ ı(x), x ∈ K^{×}.

Proof of i): Claim 1. H^{1}(Φ) ∘ ȷ_{K}(K^{×}) ⊆ ȷ_{L}(L^{×}). Proof of Claim 1 : Consider t ∈ K^{×} arbitrary. First, if t ∈ k^{×}, then H^{1}(Φ) identifies ȷ_{K}(t) with ȷ_{L}(t) by the discussion above. Second, let t ∈ K^{×}\k^{×}. Then the inclusion k(t) ⊆ K is a morphism in F_{k}, hence gives rise canonically to an open G_{k}-morphism Φ_{t} : Π_{K} → Π_{k(t)}. But then Φ_{t} ∘ Φ : Π_{L} → Π_{k(t)} is an open G_{k}-homomorphism too. Since td(k(t)|k) = 1, it follows by Theorem 16.5 from [2] that Φ_{t} ∘ Φ is defined by a k-embedding ı_{t} : k(t) → L, i.e., of the form Φ_{t} ∘ Φ = Φĩ_{t}. Hence by the assertion (†) above, H^{1}(Φ_{t} ∘ Φ) : k̂(t) → L̂ is exactly the p-adic completion of ı_{t} : k(t) → L, and we get:

H^{1}(Φ_{t} ∘ Φ) ∘ ȷ_{k(t)}(x) = ȷ_{L} ∘ ı_{t}(x) ∈ ȷ_{L}(L^{×}), x ∈ k(t)^{×}.

By functoriality we have H^{1}(Φ_{t} ∘ Φ) = H^{1}(Φ) ∘ H^{1}(Φ_{t}), and H^{1}(Φ_{t}) is the p-adic completion of the inclusion k(t) ⊆ K, hence H^{1}(Φ_{t}) ∘ ȷ_{k(t)}(x) = ȷ_{K}(x) for x ∈ k(t)^{×}. Combining these equalities, we finally get

(†′)  H^{1}(Φ) ∘ ȷ_{K}(x) = ȷ_{L} ∘ ı_{t}(x) ∈ ȷ_{L}(L^{×}), x ∈ k(t)^{×}.

Since t was arbitrary, this concludes the proof of Claim 1. □

Next let us identify K^{×}/μ′ and L^{×}/μ′ with their images in K̂, respective L̂, via the p-adic completion homomorphisms ȷ_{K}, respectively ȷ_{L}. Then by Claim 1 above, H^{1}(Φ) maps K^{×}/μ′ into L^{×}/μ′, and identifies k^{×}/μ′ ⊂ K^{×}/μ′ with k^{×}/μ′ ⊂ L^{×}/μ′. Further, H^{1}(Φ) is nothing but the p-adic completion of its restriction to K^{×}/μ′. Modding out by k^{×}/μ′ thus yields an embedding of free abelian groups α : K^{×}/k^{×} ↪ L^{×}/k^{×} canonically defined by H^{1}(Φ). Thinking now of (K, +) and (L, +) as infinite dimensional k-vector spaces, P(K) := K^{×}/k^{×} and P(L) := L^{×}/k^{×} are their projectivizations. Then α : P(K) ↪ P(L) is an inclusion which respects the multiplicative structures.

Claim 2. The map α : P(K) ↪ P(L) preserves collineations. Proof of Claim 2 : A line in P(K) is the image of a two-dimensional k-subspace of K, say 𝔩_{t_{1}t_{2}} := kt_{1} +kt_{2}, where t_{1}, t_{2} ∈ K are k-linearly independent. Note that 𝔩_{t_{1}t_{2}} = t_{1} · 𝔩_{t}, where t = t_{2}/t_{1} and 𝔩_{t} := k + kt. Since multiplication by t_{1}/k^{×} is a line-preserving automorphism of K^{×}/k^{×} = P(K), α is multiplicative, and multiplication by α(t_{1}/k^{×}) is a line-preserving automorphism of P(L), it suffices to show that α maps the lines 𝔩_{t}, t ∈ K\k, to lines in P(L). In order to do this, remark that by relation (†′) above we have:

H^{1}(Φ) ∘ ȷ_{K}(kt + k)^{×} = ȷ_{L} ∘ ı_{t}(kt + k)^{×} = ȷ_{L}(kı_{t}(t) + k)^{×}.

Thus 𝔩_{t} ⊂ P(K) is mapped bijectively onto 𝔩_{ı_{t}(t)} ⊂ P(L). □

Let L_{0}^{×} ⊆ L^{×} denote the preimage of α(P(K)) in L, and let us set L_{0} := L_{0}^{×} ∪ {0_{L}} ⊆ L. Since α is collineation preserving, it follows that L_{0} is a field containing k, and α : P(K) → P(L_{0}) is a collineation preserving bijection. By the Fundamental Theorem of projective geometry, see Artin [1], we conclude that α is induced by an isomorphism of k-vector spaces α_{K} : (K, +) → (L_{0}, +), which is unique up to k-semilinear homotheties. As in [3], it follows that setting ı_{K} := α(1_{K})^{−1} · α_{K}, the resulting map ı_{K} : K → L is actually a k-isomorphism of fields, whose projectivization equals α. In particular, the p-adic completion of ı_{K} equals H^{1}(Φ), and ı_{K} is the unique embedding of fields K ↪ L with this property.

Now let K′|K be a finite Galois extension contained in K̃, and k′ := K′ ∩ \overline{k}. Then Π_{K′} is an open normal subgroup of Π_{K}, and Φ^{−1}(Π_{K′}) = Π_{L′} for some finite Galois extension L′|L contained in L̃. Since Φ is a G_{k}-homomorphism, it follows that L′ ∩ \overline{k} = k′, and K′|k′ and L′|k′ are regular function fields over k′ which is a sub-p-adic field. The restriction Φ′ of Φ to Π_{L′} yields an open G_{k′}-homomorphism Φ′ : Π_{L′} → Π_{K′}. Mutatis mutandis, we obtain from Φ′ a k′-embedding ı_{K′} : K′ ↪ L′ such that its p-adic completion is H^{1}(Φ′). The compatibility relation res_{Π_{L}}^{Π_{L′}} ∘ H^{1}(Φ′) = H^{1}(Φ) ∘ res_{Π_{K}}^{Π_{K′}} translates into the fact that ı_{K} is the restriction of ı_{K′} to K. Note that extensions of the form K′|K above exhaust K̃|K, so taking limits we obtain a \overline{k}-embedding ĩ_{Φ} : K̃ ↪ L̃ such that (ĩ_{Φ})|_{K′} = ı_{K′} for all K′|K. Claim 3. Φĩ_{Φ} = Φ. Proof of Claim 3 : Indeed, for K′|K and the corresponding L′|L as above, Φ yields a surjection of Galois groups

Ψ′ : Gal(L′|L) → Gal(K′|K) .

Note that H^{1}(Π_{K′}, 𝕋_{p}) is a Gal(K′|K) module, and correspondingly for L′|L. Moreover, for all σ ∈ Gal(L′|L) we have the following commutative diagram:

H^{1}(Π_{K′}, 𝕋_{p}) ⟶^{H^{1}(Φ′)} H^{1}(Π_{L′}, 𝕋_{p})
Ψ′(σ)↓  ↓^{σ}
H^{1}(Π_{K′}, 𝕋_{p}) ⟶^{H^{1}(Φ′)} H^{1}(Π_{L′}, 𝕋_{p}).

which via the Kummer Theory isomorphisms translates into:

ı_{K′} ∘ Ψ′(σ) = σ ∘ ı_{K′}, σ ∈ Gal(K′|K).

Hence taking limits over all the K′|K, we get as required:

ĩ_{Φ} ∘ Φ(g) = g ∘ ĩ_{Φ}, g ∈ Π_{L} . □

Proof of ii): Suppose we are given a \overline{k}-embedding ĩ: K̃ ↪ L̃, and let Φĩ : Π_{L} → Π_{K} be the corresponding open G_{k}-homomorphism. For every finite Galois sub-extension L′|L of L̃|L and the corresponding K′ := ĩ^{−1}(L′), the restriction of Φĩ to Π_{L′} defines an open G_{k′}- homomorphism Φ_{ĩ,L′} : Π_{L′} → Π_{K′}, where k′ := \overline{k} ∩ L′. By the discussions above, the Kummer morphism H^{1}(Φ_{ĩ,L′}) is the p-adic completion of ı_{K′} := ĩ|_{K′}, and in particular, H^{1}(Φ_{ĩ,L′}) dermines ı_{K′} uniquely. By taking limits, it follows that ĩ is uniquely determined by the family of Kummer morphisms H^{1}(Φ_{ĩ,L′}) with L′|L as above. Assertion ii) is thus proven. □ This concludes the proof of Theorem 1.

References

1. E. Artin, Geometric algebra, Interscience Publishers, Inc., New York (1957).

2. S. Mochizuki, The local pro-p anabelian geometry of curves, Invent. Math. 138 (1999), pp. 319-423.

3. F. Pop, The birational anabelian conjecture – revisited, unpublished manuscript, available at www.math.upenn.edu/∼pop.