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
satisfying the four properties given below. Here, the notation
and
refer to the upper half-plane and the complex plane, respectively. The notation
is a subgroup of SL(2,R), such as, for example, a Fuchsian group. An element
is a 2×2 matrix
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
for a fixed real number k. - For all
and
, one has
Here,
is the fractional linear transform of
by
. - If
, then for all
and
, one has
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.)