Reciprocity obstruction to strong approximation over p-adic function fields

TL;DR

This paper constructs specific rational varieties over p-adic function fields and demonstrates that a reciprocity obstruction fully explains the failure of strong approximation for these varieties.

Contribution

It introduces a reciprocity obstruction that accounts for the failure of strong approximation and shows it is the only obstruction for certain classes of varieties over p-adic function fields.

Findings

Strong approximation fails for constructed varieties.

Reciprocity obstruction accounts for this failure.

Obstruction is the only one for classifying varieties of tori.

Abstract

Over function fields of p-adic curves, we construct stably rational varieties in the form of homogeneous spaces of SL_n with semisimple simply connected stabilizers and we show that strong approximation away from a non-empty set of places fails for such varieties. The construction combines the Lichtenbaum duality and the degree 3 cohomological invariants of the stabilizers. We then establish a reciprocity obstruction which accounts for this failure of strong approximation. We show that this reciprocity obstruction to strong approximation is the only one for counterexamples we constructed, and also for classifying varieties of tori. We also show that this reciprocity obstruction to strong approximation is compatible with known results for tori. At the end, we explain how a similar point of view shows that the reciprocity obstruction to weak approximation is the only one for classifying varieties of tori over p-adic function fields.