Local-global principles for zero-cycles on homogeneous spaces over arithmetic function fields

Abstract

We study the existence of zero-cycles of degree one on varieties that are defined over a function field of a curve over a complete discretely valued field. We show that localglobal principles hold for such zero-cycles provided that local-global principles hold for the existence of rational points over extensions of the function field. This assertion is analogous to a known result concerning varieties over number fields. Many of our results are shown to hold more generally in the henselian case.

Local-global Galois theory of arithmetic function fields

“…As an application of our descent results, we obtain a more explicit version of a local-global principle that appeared in [CHHKPS17]. That result concerned the index of a variety over F , and related it to its index over the fields F P and F U .…”

“…As in [CHHKPS17], given a collection Ω of overfields of F and an F -scheme Z, we say that pZ, Ωq satisfies a local-global principle for rational points if the following holds: ZpF q ‰ ∅ if and only if ZpLq ‰ ∅ for all L P Ω. In particular, we will consider the collection of overfields Ω X ,P consisting of the overfields F P , F U for P and U as in Notation 2.8.…”

“…(See the paragraph preceding Theorem 2.6 for the terminology.) In [CHHKPS17], we considered a related question: in the situation as above, let U be a nonempty connected affine open subset of X, and let F U be the fraction field of the ring p R U of formal functions along U. (See Notation 2.2.)…”

“…In Section 2.1 we provide a positive answer if charpkq " 0 and X is unibranched at P (Theorem 2.6), but show that there are always counterexamples if X is not unibranched at P (Remark 2.7(b)). In Section 2.2 we combine Theorem 2.6 and a result from [CHHKPS17] to obtain an explicit local-global principle for zero-cycles on varietes over F under the characteristic zero hypothesis (Corollary 2.10). We show in Section 2.3 that if charpkq " p ą 0, then there are always degree p separable extensions of F P that do not descend to extensions of F (Proposition 2.15).…”

Patching over Analytic Fibers and the Local-Global Principle

“…Field patching, introduced by Harbater and Hartmann in [17], and extended by the aforementioned authors and Krashen in [18], has recently seen numerous applications and is the crucial ingredient in an ongoing series of papers (see e.g. [18], [19], [21], [20], [7]).…”