Automorphic factor

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 is a function

satisfying the four properties given below. Here, the notation and refer to the upper half-plane and the complex plane, respectively. The notation Γ is a subgroup of SL(2,R), such as, for example, a Fuchsian group. An element is a 2x2 matrix

with a, b, c, d real numbers, satisfying ad−bc=1.

An automorphic factor must satisfy:

1. For a fixed , the function ν(γ,z) is a holomorphic function of .

2. For all and , one has

for a fixed real number k.

3. For all and , one has

ν(γδ,z) = ν(γ,δz)ν(δ,z)

Here, δz is the fractional linear transform of z by δ.

4.If , then for all and , one has

ν( − γ,z) = ν(γ,z)

Here, I denotes the identity matrix.

Properties

Every automorphic factor may be written as

ν(γ,z) = υ(γ)(cz + d)^k

with

The function is called a multiplier system. Clearly,

υ(I) = 1,

while, if , then

υ( − I) = e ^{− iπk}

References

• Robert Rankin, Modular Forms and Functions, (1977) Cambridge University Press ISBN 0-521-21212-X. (Chapter 3 is entirely devoted to automorphic factors for the modular group.)