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Automorphic factor

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Source updated July 18, 2026

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'.

01Definition

An automorphic factor of weight k is a function $\nu :\Gamma \times \mathbb {H} \to \mathbb {C}$ satisfying the four properties given below. Here, the notation $\mathbb {H}$ and $\mathbb {C}$ 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:

For a fixed $\gamma \in \Gamma$ , the function $\nu (\gamma ,z)$ is a holomorphic function of $z\in \mathbb {H}$ .
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.
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 $\delta$ .
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.

02Properties

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.

03Complex generalization

There exist non-holomorphic automorphic factors of the type

$\nu (\gamma ,z)=\upsilon (\gamma )(cz+d)^{\alpha }(c{\bar {z}}+d)^{\beta }$

where $\alpha ,\beta \in \mathbb {C}$ are arbitrary coweights. The condition $\nu (\gamma \delta ,z)=\nu (\gamma ,\delta z)\nu (\delta ,z)$ reduces to $\upsilon (\gamma \delta )=\upsilon (\gamma )\upsilon (\delta )$ if $\alpha -\beta \in \mathbb {Z}$ .

If $\Gamma =SL_{2}(\mathbb {Z} )$ is the modular group and $\Re \alpha ,\Re \beta \in [0,1)$ , then there exists a multiplier system such that

$\upsilon \left({\begin{smallmatrix}1&1\\0&1\end{smallmatrix}}\right)=e^{i{\frac {\pi }{6}}(\alpha -\beta )}\quad ,\quad \upsilon \left({\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}}\right)=e^{-i{\frac {\pi }{2}}(\alpha -\beta )}$

For $\eta (z)$ the Dedekind eta function, the modular form $f_{\alpha ,\beta }(z)=\eta (z)^{2\alpha }{\overline {\eta (z)^{2{\overline {\beta }}}}}$ is such that $f_{\alpha ,\beta }(\gamma (z))=\nu (\gamma ,z)f_{\alpha ,\beta }(z)$ for any $\gamma \in SL_{2}(\mathbb {Z} )$ .

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Sources and credits

This article is adapted from the Wikipedia article “Automorphic factor”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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