Overview of Moret-bailly’s Theorem on Global Points

Let K be a global field and let XK be a geometrically integral and separated K-scheme of finite type (a “K-variety”). Let Σ be a finite non-empty set of places of K, and suppose we are given local points xv ∈ XK(Kv) for each v ∈ Σ. It is natural to ask if we may be able to find x ∈ XK(K) that is arbitrarily close to xv for the natural topology on XK(Kv) (arising from the topology on Kv) for all v ∈ Σ. That is, is the image of the map XK(K) → ∏ v∈ΣXK(Kv) dense? In general the answer is no because XK(K) may be empty. But there is a reasonable weakening of the question for which one does not see immediate counterexamples: if K ′/K is a finite extension in which each place of Σ is totally split (i.e., K ′ ⊗K Kv is a product of copies of Kv, so K ′/K is separable) then for each place v′ on K ′ over a place v ∈ Σ we get a map XK(K ′) → XK(K ′ v′) = XK(Kv), and so we can ask if there exists some such K ′/K and a point x′ ∈ XK(K ′) whose image in XK(Kv) for each place v′ on K ′ over v ∈ Σ is in a neighborhood of xv that we specify in advance. This weaker kind of global approximation question turns out to have an affirmative answer when the xv’s lie in the smooth locus X K , and this is a consequence of a stronger result that is the main theorem of Moret-Bailly in [8]. We now formulate this main theorem, and then we show how it provides an affirmative answer to the preceeding question (as well as a key result used by Taylor in [10]). In §2ff. below we shall discuss Moret-Bailly’s proof. The setup for Moret-Bailly’s theorem goes as follows. We fix a global field K as above and set B = SpecR for R a ring of S-integers of K (with S a finite non-empty set of places of K that contains all of the archimedean places). We also fix a non-empty finite set of places Σ of K that is a proper subset of S, which is to say that Σ is non-empty and avoids the places coming from B but does not exhaust all places away from B. (In particular, S needs to have at least two elements, so B = Z is not permitted.) This latter condition is called incompleteness of Σ (with respect to B) in [8]. We also give ourselves the following geometric data: a separated surjective map f : X → B of finite type with irreducible X and geometrically irreducible generic fiber XK , as well as finite Galois extensions Lv/Kv for each v ∈ Σ and Gal(Lv/Kv)-stable non-empty open subsets Ωv ⊆ X K (Lv) (in particular X K is non-empty, so XK is generically K-smooth). For example, if XK is K-smooth then we can take Lv = Kv and Ωv = XK(Kv) for all v ∈ Σ if these latter sets are non-empty. (In all interesting examples we have Lv = Kv, but for technical reasons related to preliminary reduction steps in the main proof we have to allow the generality indicated above.) The main theorem of [8] is: