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Fungrim entry: a637cd

F={τ:τH  and  Re(τ)[12,12)  and  (τ>1  or  (τ=1  and  Re(τ)0))}\mathcal{F} = \left\{ \tau : \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(\tau) \in \left[-\frac{1}{2}, \frac{1}{2}\right) \;\mathbin{\operatorname{and}}\; \left(\left|\tau\right| > 1 \;\mathbin{\operatorname{or}}\; \left(\left|\tau\right| = 1 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(\tau) \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}(\tau) \in \left[-\frac{1}{2}, \frac{1}{2}\right) \;\mathbin{\operatorname{and}}\; \left(\left|\tau\right| > 1 \;\mathbin{\operatorname{or}}\; \left(\left|\tau\right| = 1 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(\tau) \le 0\right)\right) \right\}
Definitions:
Fungrim symbol Notation Short description
ModularGroupFundamentalDomainF\mathcal{F} Fundamental domain for action of the modular group
HHH\mathbb{H} Upper complex half-plane
ReRe(z)\operatorname{Re}(z) 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, Set(tau, For(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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2021-03-15 19:12:00.328586 UTC