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Fundamental groups, reciprocity laws, and Diophantine equations



Abstract
In the late 19th century, Kurt Hensel introduced non-Archimedean completions of algebraic number fields, which was used soon afterwards by Minkowski and Hasse to study Diophantine equations via the so- called local-to-global principles. During the first world war, Takagi proved the existence theorem for class fields, leading up to his lecture at the ICM in Strasbourg, and laid the foundation for Artin’s reciprocity law of class field theory. In this lecture, we will remark briefly on subsequent developments, concluding with a description of recent attempts to refine Hasse principles using arithmetic fundamental groups and non-abelian reciprocity laws.