Automorphic factor explained

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$$\backslash nu\; :\; \backslash Gamma\; \backslash times\; \backslash mathbb\; \backslash to\; \backslash Complex$$satisfying the four properties given below. Here, the notation

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 matrixwith 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 H .

1. For all

z\in H and \gamma\in\Gamma , one has $$\backslash vert\backslash nu(\backslash gamma,z)\backslash vert\; =\; \backslash vert\; cz\; +\; d\backslash vert^k$$ for a fixed real number k.

1. For all

z\in H and \gamma,\delta \in \Gamma , one has $$\backslash nu(\backslash gamma\backslash delta,\; z)\; =\; \backslash nu(\backslash gamma,\backslash delta\; z)\backslash nu(\backslash delta,z)$$ Here, \delta z is the fractional linear transform of z by \delta .

1. If

- I\in\Gamma , then for all z\in H and \gamma \in \Gamma , one has $$\backslash nu(-\backslash gamma,z)\; =\; \backslash nu(\backslash gamma,z)$$ 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} 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.)