In mathematics, an automorphic factor is a certain type of analytic function, defined on subgroups of SL(2,R), appearing in the theory of modular forms.
The general case, for general groups, is reviewed in the article 'factor of automorphy'.
An automorphic factor of weight k is a function
satisfying the four properties given below.
refer to the upper half-plane and the complex plane, respectively.
is a subgroup of SL(2,R), such as, for example, a Fuchsian group.
with a, b, c, d real numbers, satisfying ad−bc=1.
An automorphic factor must satisfy: Every automorphic factor may be written as with The function
is called a multiplier system.