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Fungrim entry: a637cd

F={τ:τHandRe ⁣(τ)[12,12)and(τ>1or(τ=1andRe ⁣(τ)0))}\mathcal{F} = \left\{ \tau : \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(\tau\right) \in \left[-\frac{1}{2}, \frac{1}{2}\right) \,\mathbin{\operatorname{and}}\, \left(\left|\tau\right| \gt 1 \,\mathbin{\operatorname{or}}\, \left(\left|\tau\right| = 1 \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(\tau\right) \le 0\right)\right) \right\}
The choice to include the left or right boundary is arbitrary; the present definition follows Cohen and simplifies the treatment of reduced binary quadratic forms.
References:
  • H. Cohen, A Course in Computational Algebraic Number Theory, Springer, 1993
TeX:
\mathcal{F} = \left\{ \tau : \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(\tau\right) \in \left[-\frac{1}{2}, \frac{1}{2}\right) \,\mathbin{\operatorname{and}}\, \left(\left|\tau\right| \gt 1 \,\mathbin{\operatorname{or}}\, \left(\left|\tau\right| = 1 \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(\tau\right) \le 0\right)\right) \right\}
Definitions:
Fungrim symbol Notation Short description
ModularGroupFundamentalDomainF\mathcal{F} Fundamental domain for action of the modular group
SetBuilder{f ⁣(x):P ⁣(x)}\left\{ f\!\left(x\right) : P\!\left(x\right) \right\} Set comprehension
HHH\mathbb{H} Upper complex half-plane
ReRe ⁣(z)\operatorname{Re}\!\left(z\right) Real part
ClosedOpenInterval[a,b)\left[a, b\right) Closed-open interval
Absz\left|z\right| Absolute value
Source code for this entry:
Entry(ID("a637cd"),
    Formula(Equal(ModularGroupFundamentalDomain, SetBuilder(tau, tau, And(Element(tau, HH), Element(Re(tau), ClosedOpenInterval(Neg(Div(1, 2)), Div(1, 2))), Or(Greater(Abs(tau), 1), And(Equal(Abs(tau), 1), LessEqual(Re(tau), 0))))))),
    Description("The choice to include the left or right boundary is arbitrary; the present definition follows Cohen and simplifies the treatment of reduced binary quadratic forms."),
    References("H. Cohen, A Course in Computational Algebraic Number Theory, Springer, 1993"))

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2019-06-18 07:49:59.356594 UTC