Fungrim entry: 127a52

\begin{pmatrix} a & b \\ c & d \end{pmatrix} \circ \tau = \frac{a \tau + b}{c \tau + d}

Assumptions:

\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z}) \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}

TeX:

\begin{pmatrix} a & b \\ c & d \end{pmatrix} \circ \tau = \frac{a \tau + b}{c \tau + d}

\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z}) \,\mathbin{\operatorname{and}}\, \tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`ModularGroupAction`	$\gamma \circ \tau$	Action of modular group
`Matrix2x2`	$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$	Two by two matrix
`SL2Z`	$\operatorname{SL}_2(\mathbb{Z})$	Modular group
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("127a52"),
    Formula(Equal(ModularGroupAction(Matrix2x2(a, b, c, d), tau), Div(Add(Mul(a, tau), b), Add(Mul(c, tau), d)))),
    Assumptions(And(Element(Matrix2x2(a, b, c, d), SL2Z), Element(tau, HH))))

Topics using this entry

Modular transformations