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Fungrim entry: b23575

H={γτ:τFλandγSL2(Z)andγ(1001)(mod2)}\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
HHH\mathbb{H} Upper complex half-plane
SetBuilder{f ⁣(x):P ⁣(x)}\left\{ f\!\left(x\right) : P\!\left(x\right) \right\} Set comprehension
ModularGroupActionγτ\gamma \circ \tau Action of modular group
ModularLambdaFundamentalDomainFλ\mathcal{F}_{\lambda} Fundamental domain of the modular lambda function
SL2ZSL2(Z)\operatorname{SL}_2(\mathbb{Z}) Modular group
Matrix2x2(abcd)\begin{pmatrix} a & b \\ c & d \end{pmatrix} Two by two matrix
Source code for this entry:
Entry(ID("b23575"),
    Formula(Equal(HH, SetBuilder(ModularGroupAction(gamma, tau), Tuple(tau, gamma), And(Element(tau, ModularLambdaFundamentalDomain), Element(gamma, SL2Z), CongruentMod(gamma, Matrix2x2(1, 0, 0, 1), 2))))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-09-15 11:00:55.020619 UTC