Automorphic factor

Definition

An automorphic factor of weight k is a function \nu : \Gamma \times \mathbb{H} \to \Complex satisfying the four properties given below. Here, the notation \mathbb{H} and \Complex refer to the upper half-plane and the complex plane, respectively. The notation \Gamma is a subgroup of SL(2,R), such as, for example, a Fuchsian group. An element \gamma \in \Gamma is a 2×2 matrix \gamma = \begin{bmatrix}a&b \c & d\end{bmatrix} with a, b, c, d real numbers, satisfying ad−bc=1.

An automorphic factor must satisfy:

1. For a fixed \gamma\in\Gamma, the function \nu(\gamma,z) is a holomorphic function of z\in\mathbb{H}.
2. For all z\in\mathbb{H} and \gamma\in\Gamma, one has \vert\nu(\gamma,z)\vert = \vert cz + d\vert^k for a fixed real number k.
3. For all z\in\mathbb{H} and \gamma,\delta \in \Gamma, one has \nu(\gamma\delta, z) = \nu(\gamma,\delta z)\nu(\delta,z) Here, \delta z is the fractional linear transform of z by \delta.
4. If -I\in\Gamma, then for all z\in\mathbb{H} and \gamma \in \Gamma, one has \nu(-\gamma,z) = \nu(\gamma,z) Here, I denotes the identity matrix.

Properties

Every automorphic factor may be written as

:\nu(\gamma, z)=\upsilon(\gamma) (cz+d)^k

with :\vert\upsilon(\gamma)\vert = 1

The function \upsilon:\Gamma\to S^1 is called a multiplier system. Clearly,

:\upsilon(I)=1, while, if -I\in\Gamma, then

:\upsilon(-I)=e^{-i\pi k} which equals (-1)^k when k is an integer.

References

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