Fungrim entry: 80279d

\operatorname{PSL}_2(\mathbb{Z}) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} : \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z}) \,\mathbin{\operatorname{and}}\, \left(c \gt 0 \,\mathbin{\operatorname{or}}\, \left(c = 0 \,\mathbin{\operatorname{and}}\, d \gt 0\right)\right) \right\}

TeX:

\operatorname{PSL}_2(\mathbb{Z}) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} : \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z}) \,\mathbin{\operatorname{and}}\, \left(c \gt 0 \,\mathbin{\operatorname{or}}\, \left(c = 0 \,\mathbin{\operatorname{and}}\, d \gt 0\right)\right) \right\}

Definitions:

Fungrim symbol	Notation	Short description
`PSL2Z`	$\operatorname{PSL}_2(\mathbb{Z})$	Modular group (canonical representatives)
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`Matrix2x2`	$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$	Two by two matrix
`SL2Z`	$\operatorname{SL}_2(\mathbb{Z})$	Modular group

Source code for this entry:

Entry(ID("80279d"),
    Formula(Equal(PSL2Z, SetBuilder(Matrix2x2(a, b, c, d), Tuple(a, b, c, d), And(Element(Matrix2x2(a, b, c, d), SL2Z), Or(Greater(c, 0), And(Equal(c, 0), Greater(d, 0))))))))

Topics using this entry

Modular transformations