Fungrim entry: a637cd

\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
`ModularGroupFundamentalDomain`	$\mathcal{F}$	Fundamental domain for action of the modular group
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`HH`	$\mathbb{H}$	Upper complex half-plane
`Re`	$\operatorname{Re}\!\left(z\right)$	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, 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"))

Topics using this entry

Modular transformations