automorphe

Automorphe, in mathematics, refers to automorphic forms, a class of highly symmetric functions that arise on quotients of Lie groups by discrete groups. They generalize classical modular forms and play a central role in number theory and representation theory. Broadly, an automorphic form is a function on a locally symmetric space that is invariant under a discrete group action and satisfies additional transformation, regularity, and growth conditions. The modern viewpoint often describes automorphic forms as functions on a group G that are invariant under a discrete subgroup Γ and transform in prescribed ways under a maximal compact subgroup K, with suitable behavior at infinity.

Classical examples include modular forms for the group SL2(Z) acting on the upper half-plane via Möbius transformations. A modular form of weight k is holomorphic on the upper half-plane and satisfies f((az+b)/(cz+d)) = (cz+d)^k f(z), with finiteness at cusps. Cusp forms are modular forms that vanish at cusps. More general automorphic forms can be real-analytic (Maass forms) or holomorphic, and may be considered on other groups G and discrete subgroups Γ, yielding a rich landscape of objects.

Automorphic forms connect to representation theory through automorphic representations, which encode forms as representations of adelic groups. They underpin major conjectures and the Langlands program, linking arithmetic, geometry, and analysis to L-functions and Galois representations.

Historically, automorphic forms originated with Poincaré’s work on automorphic functions and were extended by Hecke’s theory. The modern framework, developed in the 20th century, provides a unifying approach that continues to influence number theory, arithmetic geometry, and mathematical physics.