Automorphic factor

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'.

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 \nu : \Gamma \times \mathbb{H} \to \Complex satisfying the four properties given below. Here, the notation \mathbb{H} and \Complex 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. ==Properties== 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. \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}) . ==References== Robert Rankin, Modular Forms and Functions, (1977) Cambridge University Press . (Chapter 3 is entirely devoted to automorphic factors for the modular group.) . (For the complex generalization.)

Text derived from https://en.wikipedia.org/w/index.php?oldid=1364776814 under CC BY-SA 4.0; modifications have been made.