Fungrim entry: a637cd

$\mathcal{F} = \left\{ \tau : \tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \operatorname{Re}(\tau) \in \left[-\frac{1}{2}, \frac{1}{2}\right) \,\mathbin{\operatorname{and}}\, \left(\left|\tau\right| > 1 \,\mathbin{\operatorname{or}}\, \left(\left|\tau\right| = 1 \,\mathbin{\operatorname{and}}\, \operatorname{Re}(\tau) \le 0\right)\right) \right\}$
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
ModularGroupFundamentalDomain$\mathcal{F}$ Fundamental domain for action of the modular group
HH$\mathbb{H}$ Upper complex half-plane
Re$\operatorname{Re}(z)$ Real part
ClosedOpenInterval$\left[a, b\right)$ Closed-open interval
Abs$\left|z\right|$ 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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2019-10-05 13:11:19.856591 UTC