Automorphic function
Encyclopedia
In mathematics, an automorphic function is a function on a space that is invariant under the action
of some group
, in other words a function on the quotient space
. Often the space is a complex manifold
and the group is a discrete group
.
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
of some group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
, in other words a function on the quotient space
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
. Often the space is a complex manifold
Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
and the group is a discrete group
Discrete group
In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one...
.