CSUN Algebra, Number Theory, and Discrete Mathematics Seminar

The classical Hasse-Minkowski theorem states that if two quadratic forms over $\mathbf{Q}$ become isomorphic over $\mathbf{R}$ and over ${\mathbf{Q}}_p$ for all prime numbers $p$, then they are isomorphic over $\mathbf{Q}$. This is one of the first examples of the so-called local-global principle, or Hasse principle. Just after the Hasse-Minkowski theorem, Hasse obtained another important result, the so-called Hasse cyclic norm principle: in cyclic extensions of number fields, local norms are global norms. Many local-global results were proved since then, and this theme became a central one in number theory. It is well-known that the Hasse principle does not always hold, and most of the counter-examples are explained by the Brauer-Manin obstruction. The aim of this talk is to present some old and new results concerning the Hasse principle and the Brauer-Manim obstruction. The main examples of the talk concern norm equations, in particular recent results joint with Tingyu Lee and Parimala.