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'.

Definition

An automorphic factor of weight k refer to the upper half-plane and the complex plane 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:

1. is a holomorphic function.

2. for a fixed real number k.

3. is the fractional linear transform.

4. Here, I denotes the identity matrix.

Properties

Every automorphic factor may be written as

with

is called a multiplier system. Clearly,

1. ,

, then

when k is an integer.

Complex generalization

There exist non-holomorphic automorphic factors of the type

.

is the modular group, then there exists a multiplier system such that

the Dedekind eta function, the modular form.

References

• Robert Rankin, Modular Forms and Functions, (1977) Cambridge University Press . (Chapter 3 is entirely devoted to automorphic factors for the modular group.)

• . (For the complex generalization.)