Local-global principles for torsors over arithmetic curves

Abstract

We consider local-global principles for torsors under linear algebraic groups, over function fields of curves over complete discretely valued fields. The obstruction to such a principle is a version of the Tate-Shafarevich group; and for groups with rational components, we compute it explicitly and show that it is finite. This yields necessary and sufficient conditions for local-global principles to hold. Our results rely on first obtaining a Mayer-Vietoris sequence for Galois cohomology and then showing that torsors can be patched. We also give new applications to quadratic forms and central simple algebras.

Lois de réciprocité supérieures et points rationnels

“…3.7] ont montré que si G est connexe et K-rationnel, alors X P (X , G) = 1. Dans [18,Cor. 5.9], ils montrent que pour tout groupe algébrique linéaire G sur K, X 0 (X , G) = ∪ P X P (X , G), où P parcourt tous les ensembles finis de points fermés de X contenant tous les points singuliers de X 0 .…”

Refinements to Patching and Applications to Field Invariants

“…Let Γ be a bipartite connected (multi-)graph, with vertex set V = V 1 ⊔ V 2 and edge set E. Suppose that we are given a Γ-field in the sense of [HHK14, Section 2.1.1]; i.e. a field F v for each v ∈ V and a field F e for each e ∈ E, together with an inclusion F v ֒→ F e whenever v is a vertex of e. These fields and inclusions define an inverse system of fields; and if the inverse limit is a field F then this is called a "factorization inverse system" over F ( [HHK15], Section 2), and the graph together with the associated fields is called a Γ/F -field ([HHK14], Section 2.1.1). In this situation, set F i = v∈V i F v for i = 1, 2 and set F 0 = e∈E F e .…”

“…Recall that for any field E and quadratic form q over E, H 1 (E, SO(q)) classifies quadratic forms of the same dimension and discriminant as q, with q corresponding to the distinguished element of the Galois cohomology set; see [KMRT98,29.29]. (In part (b) below we write µ 2 rather than Z/2Z as in [HHK15]; but these are equivalent since the characteristic is not two. )…”

“…Next, since char(k) = 2, there is a split cover ω : X ′ → X, say with function field extension F ′ /F and closed fiber X ′ , such that for each Q ′ ∈ S ′ := ω −1 (S) and each a ∈ F ′ Q ′ , there exist b ∈ F ′ and c ∈ F ′× Q ′ such that a = bc 2 . (See Corollaries 3.3(c) and 3.7 of [HHK13], the latter applied to the set of non-unibranched points of S.) By Proposition 5.1 of [HHK15], the set of isomorphism classes of fields F ′ Q ′ , for Q ′ ∈ X ′ , is the same as the set of isomorphism classes of fields F Q , for Q ∈ X. Also, since X is regular, the analogous assertion is true for the fields F v , by Proposition 7.6 of [HHK15].…”

“…To obtain our present results, we build on the patching framework that was used in our previous manuscripts [HH10], [HHK09], [HHK15], and [HHK13].…”