Week 04.05.2026 – 10.05.2026

We describe two local-global questions generalizing some classical ones considered by Hasse and Cassels, respectively. Along with explaining the relationship between these two problems, we depict some modified Tate-Shafarevich groups whose vanishings imply an affirmative answer to both of them. We also give an overview of the most recent results achieved for the two problems.

Abstract: The inverse Galois problem has many refinements. In this talk, we focus on the existence of abelian Galois extensions with restricted ramification. The Gras--Munnier formula provides a quick test for whether such extensions exist. We discuss how the Gras--Munnier formula corresponds to a Brauer--Manin obstruction to strong approximation on the classifying stack BG. This is joint work with Daniel Loughran.