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Local-global principles for norms

André MacEdo, University of Reading
Tuesday, March 2, 2021 - 10:00am to 11:00am
Abstract: Given an extension L/K of number fields, we say that the Hasse norm principle (HNP) holds if every non-zero element of K which is a norm everywhere locally is in fact a global norm from L. If L/K is cyclic, the original Hasse norm theorem states that the HNP holds. More generally, there is a cohomological description (due to Tate) of the obstruction to the HNP for Galois extensions. 
In this talk, I will introduce this principle and present work developing explicit methods to study it for non-Galois extensions. As a key application, I will describe how these methods can be used to characterize the HNP for extensions whose normal closure has Galois group A_n or S_n. I will additionally discuss some recent generalizations of these methods to study the Hasse principle and weak approximation for products of norms as well as consequences in the statistics of these local-global principles.
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