The Modular Group and Its Subgroups

Xueli Wang () and Dingyi Pei ()
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Xueli Wang: South China Normal University, Department of Mathematics
Dingyi Pei: Guangzhou University, Institute of Mathematics and Information Science

Chapter Chapter 3 in Modular Forms with Integral and Half-Integral Weights, 2012, pp 45-64 from Springer

Abstract: Abstract Let $$ SL_2 (\mathbb{R}) = \left\{ {\left. {\left( {\begin{array}{*{20}c} a & b \\ c & d \\ \end{array} } \right)} \right|a,b,c,d \in \mathbb{R},ad - bc = 1} \right\}. $$ For any $$ \sigma = \left( {\begin{array}{*{20}c} a & b \\ c & d \\ \end{array} } \right) \in SL_2 (\mathbb{R}) $$ define a transformation on the whole complex plane as follows $$ \sigma (z) = \frac{{az + b}} {{cz + d}}. $$ It is easy to prove $$ \operatorname{Im} \left( {\sigma (z)} \right) = \frac{{\operatorname{Im} \left( z \right)}} {{\left| {cz + d} \right|^2 }}. $$ .

Keywords: Equivalence Class; Conjugate Classis; Discrete Subgroup; Modular Group; Fuchsian Group (search for similar items in EconPapers)
Date: 2012
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DOI: 10.1007/978-3-642-29302-3_3

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