automorphic

Automorphic describes objects in mathematics that are defined by or exhibit symmetry under a group action, typically involving a reductive group acting on a homogeneous space. The term is most often encountered in the theory of automorphic forms, where a function is automorphic if it satisfies a specified invariance or transformation rule with respect to a discrete subgroup and a corresponding automorphy factor.

In the classical setting, automorphic forms are functions on the upper half-plane that transform in a prescribed way under a discrete group such as SL2(Z). Modular forms are a well-known subset. More generally, automorphic forms can be holomorphic or non-holomorphic and may be defined for various groups, including GL(n) and other reductive groups, over number fields. They encode arithmetic and analytic information and often satisfy growth and symmetry conditions appropriate to the space on which they live.

A modern, representation-theoretic approach frames automorphic forms as automorphic representations. These are irreducible constituents of spaces of automorphic forms on G(A), where G is a reductive group defined over a number field F and A denotes the adeles of F. This adelic framework unifies many classical constructions and connects to harmonic analysis on quotient spaces.

Associated L-functions are attached to automorphic forms and representations, generalizing Dirichlet and modular L-functions. They play a central role in number theory, including conjectural links to Galois representations as envisioned in the Langlands program. Automorphic theory thus sits at the intersection of analysis, algebraic geometry, and arithmetic, shaping a broad and active area of research.