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
![{\displaystyle \nu :\Gamma \times \mathbb {H} \to \mathbb {C} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/24f8871e7832c87b8cc102671721e7cc27622e23)
satisfying the four properties given below. Here, the notation
![{\displaystyle \mathbb {H} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e050965453c42bcc6bd544546703c836bdafeac9)
and
![{\displaystyle \mathbb {C} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7)
refer to the upper half-plane and the complex plane, respectively. The notation
![{\displaystyle \Gamma }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19)
is a subgroup of SL(2,R), such as, for example, a Fuchsian group. An element
![{\displaystyle \gamma \in \Gamma }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4dfac36fd2ffa28cf37de8b15068ce0079b4aca)
is a 2×2 matrix
![{\displaystyle \gamma ={\begin{bmatrix}a&b\\c&d\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1ed537d2e00bd2082f4a3f60c62f6c389ccceac)
with
a,
b,
c,
d real numbers, satisfying
ad−
bc=1.
An automorphic factor must satisfy:
- For a fixed
, the function
is a holomorphic function of
. - For all
and
, one has ![{\displaystyle \vert \nu (\gamma ,z)\vert =\vert cz+d\vert ^{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c06fa86ba6c8633aa3b9df3147e8e71a5b040776)
for a fixed real number k. - For all
and
, one has ![{\displaystyle \nu (\gamma \delta ,z)=\nu (\gamma ,\delta z)\nu (\delta ,z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20e252016bab91d9cf210fe1d167049b212efbd2)
Here,
is the fractional linear transform of
by
. - If
, then for all
and
, one has ![{\displaystyle \nu (-\gamma ,z)=\nu (\gamma ,z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a355f755e44c87f63fbf863af09d9513fd025e1e)
Here, I denotes the identity matrix.
Properties
Every automorphic factor may be written as
![{\displaystyle \nu (\gamma ,z)=\upsilon (\gamma )(cz+d)^{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/485d5161a3bccd89266809af572843b51cc9e7fd)
with
![{\displaystyle \vert \upsilon (\gamma )\vert =1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f80150eea1cfc9d8ba58aa99f870874572c666a7)
The function
is called a multiplier system. Clearly,
,
while, if
, then
![{\displaystyle \upsilon (-I)=e^{-i\pi k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a3d24883dc2550262c7e35cfa120fdf288e0792)
which equals
when k is an integer.
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.)