Abstract:
Local-global principles in arithmetic geometry have been of great interest ever since the proof of the Hasse--Minkowski theorem, concerning quadric hypersurfaces, almost 100 years ago. A new and exciting chapter began with the introduction of the Brauer--Manin obstruction (BMO) in the 1970s, which can explain the failure of such principles. In 2007, Colliot-Th\'el\`ene and Xu developed a version of the BMO for integral points and studied it in the context of affine quadrics. In this talk, we will introduce semi-integral points and develop both local-global principles and the BMO in the semi-integral context. We will give general results for this obstruction before focusing on semi-integral points associated to quadrics, in keeping with historical precedent. This talk is based on joint work with Vladimir Mitankin and Masahiro Nakahara.