Topic summary
Automorphic forms
Extracted from the Wikipedia article Automorphic form.
Definition
A more straightforward but technically advanced definition using class field theory, constructs automorphic forms and their correspondent functions as embeddings of Galois groups to their underlying global field extensions. In this formulation, automorphic forms are certain finite invariants, mapping from the idele class group under the Artin reciprocity law. Herein, the analytical structure of its L-function allows for generalizations with various algebro-geometric properties; and the resultant Langlands program. To oversimplify, automorphic forms in this general perspective, are analytic functionals quantifying the invariance of number fields in an abstract sense, therefore indicating the 'primitivity' of their fundamental structure. It is a mathematical tool for analyzing the invariant constructs of many numerical structure.