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# Fungrim entry: 21839d

$\lambda\!\left(\frac{a \tau + b}{c \tau + d}\right) = \lambda(\tau)$
Assumptions:$\tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z}) \,\mathbin{\operatorname{and}}\, \begin{pmatrix} a & b \\ c & d \end{pmatrix} \equiv \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \pmod {2}$
TeX:
\lambda\!\left(\frac{a \tau + b}{c \tau + d}\right) = \lambda(\tau)

\tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z}) \,\mathbin{\operatorname{and}}\, \begin{pmatrix} a & b \\ c & d \end{pmatrix} \equiv \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \pmod {2}
Definitions:
Fungrim symbol Notation Short description
ModularLambda$\lambda(\tau)$ Modular lambda function
HH$\mathbb{H}$ Upper complex half-plane
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("21839d"),
Formula(Equal(ModularLambda(Div(Add(Mul(a, tau), b), Add(Mul(c, tau), d))), ModularLambda(tau))),
Variables(tau, a, b, c, d),
Assumptions(And(Element(tau, HH), Element(Matrix2x2(a, b, c, d), SL2Z), CongruentMod(Matrix2x2(a, b, c, d), 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.

2019-12-30 15:00:46.909060 UTC