The idea of an analogy between number fields and Riemann surfaces goes back to Richard Dedekind and Heinrich M. Weber in the nineteenth century.
The terminology may be due to Weil, who wrote his Basic Number Theory (1967) in part to work out the parallelism.
The development of Arakelov theory and its exploitation by Gerd Faltings in his proof of the Mordell conjecture is a dramatic example.
Artin's reciprocity law implies a description of the abelianization of the absolute Galois group of a global field K that is based on the Hasse local–global principle.
[4][5] Let L⁄K be a Galois extension of global fields and CL stand for the idèle class group of L. The maps θv for different places v of K can be assembled into a single global symbol map by multiplying the local components of an idèle class.