Automorphic Form

In mathematics, an automorphic form is a function on a Lie group G that satisfies a specific invariance condition with respect to a discrete subgroup Γ, together with growth and smoothness conditions. The classical prototype is the modular form on the upper half-plane, invariant under the action of the modular group SL(2, Z). Automorphic forms generalize this construction to arbitrary reductive groups over algebraic number fields and their adele rings. An automorphic form on G(A_K) / G(K) — where A_K is the Adele Ring — is a function that is invariant under the discrete subgroup G(K), transforms according to a representation of a maximal compact subgroup, and satisfies a moderate growth condition. The space of automorphic forms decomposes into irreducible representations called automorphic representations, and these are the objects that the Langlands Program proposes to match with Galois representations.

The automorphic form is not a generalization of the modular form; it is the modular form seen without the distortion of working over the complex numbers alone. The upper half-plane is a beautiful but parochial picture. The adele quotient is the universe.