Fungrim home page

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}(\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"))

Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-10-05 13:11:19.856591 UTC