# Automorphic factor

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In [mathematics](/source/mathematics), an '''automorphic factor''' is a certain type of [analytic function](/source/analytic_function),  defined on [subgroup](/source/subgroup)s of [SL(2,R)](/source/SL(2%2CR)), appearing in the theory of [modular form](/source/modular_form)s. The general case, for general groups, is reviewed in the article '[factor of automorphy](/source/factor_of_automorphy)'.

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

An automorphic factor must satisfy:
# For a fixed <math>\gamma\in\Gamma</math>, the function <math>\nu(\gamma,z)</math> is a [holomorphic function](/source/holomorphic_function) of <math>z\in\mathbb{H}</math>.
# For all <math>z\in\mathbb{H}</math> and <math>\gamma\in\Gamma</math>, one has <math display="block">\vert\nu(\gamma,z)\vert = \vert cz + d\vert^k</math> for a fixed real number ''k''.
# For all <math>z\in\mathbb{H}</math> and <math>\gamma,\delta \in \Gamma</math>, one has <math display="block">\nu(\gamma\delta, z) = \nu(\gamma,\delta z)\nu(\delta,z)</math> Here, <math>\delta z</math> is the [fractional linear transform](/source/fractional_linear_transform) of <math>z</math> by <math>\delta</math>.
# If <math>-I\in\Gamma</math>, then for all <math>z\in\mathbb{H}</math> and <math>\gamma \in \Gamma</math>, one has <math display="block">\nu(-\gamma,z) = \nu(\gamma,z)</math> Here, ''I'' denotes the [identity matrix](/source/identity_matrix).

==Properties==
Every automorphic factor may be written as 

:<math>\nu(\gamma, z)=\upsilon(\gamma) (cz+d)^k</math>

with 
:<math>\vert\upsilon(\gamma)\vert = 1</math>

The function <math>\upsilon:\Gamma\to S^1</math> is called a '''multiplier system'''.  Clearly,

:<math>\upsilon(I)=1</math>, 
while, if <math>-I\in\Gamma</math>, then

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

== Complex generalization ==
There exist non-holomorphic automorphic factors of the type 

:<math>\nu(\gamma, z)=\upsilon(\gamma) (cz+d)^\alpha (c\bar z+d)^\beta</math>

where <math>\alpha,\beta\in\mathbb{C}</math> are arbitrary coweights. The condition <math display="block">\nu(\gamma\delta, z) = \nu(\gamma,\delta z)\nu(\delta,z)</math> reduces to <math>\upsilon(\gamma\delta)=\upsilon(\gamma)\upsilon(\delta)</math> if <math>\alpha-\beta\in\mathbb{Z}</math>. 

If <math>\Gamma=SL_2(\mathbb{Z})</math> is the [modular group](/source/modular_group) and <math>\Re\alpha,\Re\beta\in [0,1)</math>, then there exists a multiplier system such that 
:<math>\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)} 
</math>
For <math>\eta(z)</math> the [Dedekind eta function](/source/Dedekind_eta_function), the [modular form](/source/modular_form) <math>f_{\alpha,\beta}(z)=\eta(z)^{2\alpha}\overline{\eta(z)^{2\overline{\beta}}} </math> is such that <math>f_{\alpha,\beta}(\gamma(z)) = \nu(\gamma,z)f_{\alpha,\beta}(z)</math> for any <math>\gamma\in SL_2(\mathbb{Z})</math>.

==References==
* [Robert Rankin](/source/Robert_Alexander_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.)''
* {{citation |last=Pasles | first = Paul C. | journal = Acta Arithmetica | title = Multiplier systems | volume = 108 | number = 3 | pages = 235-243 | year = 2003 | issn = 0065-1036 | url = https://www.pasles.com/}}. ''(For the complex generalization.)''

Category:Modular forms

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Adapted from the Wikipedia article [Automorphic factor](https://en.wikipedia.org/wiki/Automorphic_factor) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Automorphic_factor?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.