nLab modular group

Idea

By the modular group one means alternatively

• the projective special linear group $PSL(2,\mathbb{Z})$ of integral Moebius transformations

  (cf. Serre 1973 §VII.1)

• the special linear group $SL(2,\mathbb{Z})$, being the mapping class group of the torus

  (cf. Stein 2007 Def. 1.1)

hence either the domain or codomain of the quotient map

$$SL_2(\mathbb{Z}) \longrightarrow SL_2(\mathbb{Z})/\{\pm 1\} \eqqcolon PSL_2(\mathbb{Z}) \,.$$

Related concepts

• modular form

References

• Morris Newman: The Classical Modular Group and Related Groups, chapter VIII in: Integral Matrices, Academic Press (1972)

• Jean-Pierre Serre, §VII.1: A Course in Arithmetic, Graduate Texts in Mathematics 7, Springer (1973) [doi:10.1007/978-1-4684-9884-4, pdf]

• William Stein, chapter 1 of: Modular Forms, a Computational Approach, Graduate Studies in Mathematics 79, AMS (2007) [doi:10.1090/gsm/079]

See also:

• Wikipedia: Modular group