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
The function is called a multiplier system. Clearly,
,
while, if , then
which equals when k is an integer.
References[]
Robert Rankin, Modular Forms and Functions, (1977) Cambridge University Press ISBN0-521-21212-X. (Chapter 3 is entirely devoted to automorphic factors for the modular group.)
Categories:
Modular forms
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