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Number theory Automorphic forms Object

Modular group

Builds on Special linear groupMöbius transformationUpper half-planeQuotient group

Description

The modular group is the special linear group SL2(ℤ)={(abcd):a,b,c,d∈ℤ, ad−bc=1}\mathrm{SL}_2(\Z) = \set{\begin{pmatrix} a & b \\ c & d \end{pmatrix} : a, b, c, d \in \Z, \ ad - bc = 1} under matrix multiplication. It acts on the upper half-plane ℍ\mathbb{H} by Möbius transformations τ↦(aτ+b)/(cτ+d)\tau \mapsto (a\tau + b)/(c\tau + d). The matrices II and −I-I both act as the identity map of ℍ\mathbb{H}, so many authors call the quotient group PSL2(ℤ)=SL2(ℤ)/{±I}\mathrm{PSL}_2(\Z) = \mathrm{SL}_2(\Z) / \set{\pm I} the modular group. The group is generated by S=(0−110),T=(1101),S = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, \qquad T = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, which act by Sτ=−1/τS\tau = -1/\tau and Tτ=τ+1T\tau = \tau + 1.

Remarks

The generators satisfy S2=−IS^2 = -I and (ST)3=−I(ST)^3 = -I. Check: ST=(0−111)ST = \begin{pmatrix} 0 & -1 \\ 1 & 1 \end{pmatrix}, (ST)2=(−1−110)(ST)^2 = \begin{pmatrix} -1 & -1 \\ 1 & 0 \end{pmatrix}, and (ST)3=(−100−1)(ST)^3 = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}. The group PSL2(ℤ)\mathrm{PSL}_2(\Z) is the free product of a cyclic group of order 22 and a cyclic group of order 33. The set {τ∈ℍ:∣τ∣≥1, ∣Re⁡τ∣≤1/2}\set{\tau \in \mathbb{H} : \abs{\tau} \ge 1, \ \abs{\operatorname{Re} \tau} \le 1/2} is a fundamental domain for its action on ℍ\mathbb{H}. The modular forms are the functions on ℍ\mathbb{H} with a prescribed behaviour under this group.

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  1. Complex numbers object
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Further reading

  1. Web https://en.wikipedia.org/wiki/Modular_group
About this page
TagVEKV1EM77B
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Kindobject
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MSC11F06, 20H05
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UsesSpecial linear group, Möbius transformation, Upper half-plane, Quotient group
Same as
Changed2 November 2026

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