Fungrim entry: b23575

\mathbb{H} = \left\{ \gamma \circ \tau : \tau \in \mathcal{F}_{\lambda} \;\mathbin{\operatorname{and}}\; \gamma \in \operatorname{SL}_2(\mathbb{Z}) \;\mathbin{\operatorname{and}}\; \gamma \equiv \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \pmod {2} \right\}

TeX:

\mathbb{H} = \left\{ \gamma \circ \tau : \tau \in \mathcal{F}_{\lambda} \;\mathbin{\operatorname{and}}\; \gamma \in \operatorname{SL}_2(\mathbb{Z}) \;\mathbin{\operatorname{and}}\; \gamma \equiv \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \pmod {2} \right\}

Definitions:

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

Source code for this entry:

Entry(ID("b23575"),
    Formula(Equal(HH, Set(ModularGroupAction(gamma, tau), For(Tuple(tau, gamma)), And(Element(tau, ModularLambdaFundamentalDomain), Element(gamma, SL2Z), CongruentMod(gamma, Matrix2x2(1, 0, 0, 1), 2))))))

Topics using this entry

Modular lambda function