Local-global principles for homogeneous spaces over some geometric global fields

Abstract: Local-global principles are a classical type of problem in Number Theory, both over number fields and over global fields in positive characteristic. Concerning the local-global principle for the existence of rational points, there is a classic obstruction known as the Brauer-Manin obstruction, which is conjectured to explain all failures of this principle for homogeneous spaces of connected linear groups. In the last few years, there has been an increasing interest in fields of a more ``geometric" nature that are amenable to local-global principles as well. These include, for instance, function fields of curves over discretely valued fields, by analogy with the positive characteristic case. It is in this context that I will present recent work with Diego Izquierdo on local-global principles for homogeneous spaces with connected stabilizers. We will see that, although some of the known results for number fields have direct analogs (that can be obtained in the same way), the particularities of these new fields bring up new counterexamples that cannot be explained by the Brauer-Manin obstruction, contrary to the number field case.

number theoryrepresentation theory

Audience: researchers in the topic

Number Theory and Representations in Valparaiso