An analytic viewpoint on the Hasse principle

TL;DR

This paper introduces a geometric approach to analyze the Hasse principle over function fields of curves, verifying it for specific families of projective homogeneous spaces and extending known cases.

Contribution

It presents a novel geometric method for studying the Hasse principle on Berkovich analytic curves, expanding its verification to new classes of algebraic varieties.

Findings

Hasse principle verified for certain projective homogeneous spaces

Extended the principle to quadratic forms and homogeneous varieties over unitary groups

Provided a geometric framework for future research on local-global principles

Abstract

Working on Berkovich analytic curves, we propose a geometric approach to the study of the Hasse principle over function fields of curves defined over a complete discretely valued field. Using it, we show the Hasse principle to be verified for certain families of projective homogeneous spaces. As a consequence, we also obtain said principle in the already known cases of quadratic forms and homogeneous varieties over unitary groups.