Shimura's reciprocity law

Shimura's reciprocity law is a profound result in the theory of numbers, particularly in the context of modular forms and the Langlands program. It generalizes classical reciprocity laws, such as those established by Gauss and later by Artin, to a broader setting involving Shimura varieties and abelian varieties. In essence, Shimura’s reciprocity law connects the arithmetic properties of abelian varieties defined over number fields to the values of certain automorphic forms.