pt1 Rational points on varieties & the Brauer-Manin obstruction | Bianca Viray, Univ. of Washington

Lecture 1/3
In 1970, Manin introduced a powerful obstruction to the existence of rational points on varieties coming from the abelian reciprocity laws encoded in the Brauer group. This obstruction, now known as the Brauer-Manin obstruction, explains a large fraction of the failures of the local-to-global principle known to date. This mini-course will cover both the theoretical and computational aspects of the Brauer-Manin obstruction, with an emphasis on how the two inform each other. At the end of the mini-course, we will also review failures of the local-to-global principle that are not explained by the Brauer-Manin obstruction.

Prerequisites: Algebraic number theory, Brauer groups of number fields, including statements from class field theory. Familiarity with algebraic geometry, including Picard groups and étale cohomology.

Suggested reading: Gille, Philippe; Szamuely, Tamás Central simple algebras and Galois cohomology. Second edition of [MR2266528]. Cambridge Studies in Advanced Mathematics, 165. Cambridge University Press, Cambridge, 2017. xi+417 pp. ISBN: 978-1-316-60988-0; 978-1-107-15637-1 Skorobogatov, Alexei Torsors and rational points. Cambridge Tracts in Mathematics, 144. Cambridge University Press, Cambridge, 2001. viii+187 pp. ISBN: 0-521-80237-7