Artin–Tate theorem

The Artin–Tate theorem connects algebraic structures called number fields with their geometric and arithmetic properties. It states that certain complex relationships—specifically, the behavior of divisors, units, and Picard groups—can be understood using simpler, finite data. In essence, it provides a way to analyze the structure of algebraic varieties over number fields by breaking down their properties into more manageable, finite pieces, revealing deep links between geometric objects and number theory.