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Fungrim entry: 099301

λ ⁣(aτ+bcτ+d)={λ ⁣(τ),(a,b,c,d)(1,0,0,1)(mod2)1λ ⁣(τ),(a,b,c,d)(0,1,1,0)(mod2)1λ ⁣(τ),(a,b,c,d)(1,0,1,1)(mod2)11λ ⁣(τ),(a,b,c,d)(0,1,1,1)(mod2)λ ⁣(τ)1λ ⁣(τ),(a,b,c,d)(1,1,1,0)(mod2)λ ⁣(τ)λ ⁣(τ)1,(a,b,c,d)(1,1,0,1)(mod2)\lambda\!\left(\frac{a \tau + b}{c \tau + d}\right) = \begin{cases} \lambda\!\left(\tau\right), & \left(a, b, c, d\right) \equiv \left(1, 0, 0, 1\right) \pmod {2}\\1 - \lambda\!\left(\tau\right), & \left(a, b, c, d\right) \equiv \left(0, 1, 1, 0\right) \pmod {2}\\\frac{1}{\lambda\!\left(\tau\right)}, & \left(a, b, c, d\right) \equiv \left(1, 0, 1, 1\right) \pmod {2}\\\frac{1}{1 - \lambda\!\left(\tau\right)}, & \left(a, b, c, d\right) \equiv \left(0, 1, 1, 1\right) \pmod {2}\\\frac{\lambda\!\left(\tau\right) - 1}{\lambda\!\left(\tau\right)}, & \left(a, b, c, d\right) \equiv \left(1, 1, 1, 0\right) \pmod {2}\\\frac{\lambda\!\left(\tau\right)}{\lambda\!\left(\tau\right) - 1}, & \left(a, b, c, d\right) \equiv \left(1, 1, 0, 1\right) \pmod {2}\\ \end{cases}
Assumptions:τHand(abcd)SL2(Z)\tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})
TeX:
\lambda\!\left(\frac{a \tau + b}{c \tau + d}\right) = \begin{cases} \lambda\!\left(\tau\right), & \left(a, b, c, d\right) \equiv \left(1, 0, 0, 1\right) \pmod {2}\\1 - \lambda\!\left(\tau\right), & \left(a, b, c, d\right) \equiv \left(0, 1, 1, 0\right) \pmod {2}\\\frac{1}{\lambda\!\left(\tau\right)}, & \left(a, b, c, d\right) \equiv \left(1, 0, 1, 1\right) \pmod {2}\\\frac{1}{1 - \lambda\!\left(\tau\right)}, & \left(a, b, c, d\right) \equiv \left(0, 1, 1, 1\right) \pmod {2}\\\frac{\lambda\!\left(\tau\right) - 1}{\lambda\!\left(\tau\right)}, & \left(a, b, c, d\right) \equiv \left(1, 1, 1, 0\right) \pmod {2}\\\frac{\lambda\!\left(\tau\right)}{\lambda\!\left(\tau\right) - 1}, & \left(a, b, c, d\right) \equiv \left(1, 1, 0, 1\right) \pmod {2}\\ \end{cases}

\tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})
Definitions:
Fungrim symbol Notation Short description
ModularLambdaλ ⁣(τ)\lambda\!\left(\tau\right) Modular lambda function
HHH\mathbb{H} Upper complex half-plane
Matrix2x2(abcd)\begin{pmatrix} a & b \\ c & d \end{pmatrix} Two by two matrix
SL2ZSL2(Z)\operatorname{SL}_2(\mathbb{Z}) Modular group
Source code for this entry:
Entry(ID("099301"),
    Formula(Equal(ModularLambda(Div(Add(Mul(a, tau), b), Add(Mul(c, tau), d))), Cases(Tuple(ModularLambda(tau), CongruentMod(Tuple(a, b, c, d), Tuple(1, 0, 0, 1), 2)), Tuple(Sub(1, ModularLambda(tau)), CongruentMod(Tuple(a, b, c, d), Tuple(0, 1, 1, 0), 2)), Tuple(Div(1, ModularLambda(tau)), CongruentMod(Tuple(a, b, c, d), Tuple(1, 0, 1, 1), 2)), Tuple(Div(1, Sub(1, ModularLambda(tau))), CongruentMod(Tuple(a, b, c, d), Tuple(0, 1, 1, 1), 2)), Tuple(Div(Sub(ModularLambda(tau), 1), ModularLambda(tau)), CongruentMod(Tuple(a, b, c, d), Tuple(1, 1, 1, 0), 2)), Tuple(Div(ModularLambda(tau), Sub(ModularLambda(tau), 1)), CongruentMod(Tuple(a, b, c, d), Tuple(1, 1, 0, 1), 2))))),
    Variables(tau, a, b, c, d),
    Assumptions(And(Element(tau, HH), Element(Matrix2x2(a, b, c, d), SL2Z))))

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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 13:58:57.282983 UTC