On the defect in the generalized Grunwald--Wang problem

TL;DR

This paper investigates the limitations of the generalized Grunwald--Wang theorem, demonstrating that the expected finite obstructions in the special case do not always exist or have bounded order, especially over rational function fields.

Contribution

It shows that the obstruction group in the special case of the generalized Grunwald--Wang theorem can be infinite or unbounded, countering previous expectations.

Findings

Obstruction groups can be infinite in the special case.

Boundedness of the obstruction group does not hold universally.

Counterexamples are provided over rational function fields.

Abstract

The classical Grunwald--Wang theorem asserts that, unless we are in the so-called special case, local cyclic Galois extensions at finitely many completions of a number field can be approximated by a global cyclic extension. In the special case the obstruction is measured by a group of order 2. It has been known for a long time that the Grunwald--Wang theorem extends to a very general context of valued fields. Therefore it is natural to ask whether in the special case the obstruction is always measured by a finite group and if so, is the order of this group bounded independently of the number of places under consideration. We show that the answer to both questions is negative in general, already for rational function fields and discrete valuations coming from points of the affine line. This has some interesting links to the arithmetic of function fields over Q or Q_p.