Fungrim entry: e46697

$\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathbb{H}} E_{2 k}\!\left(\tau\right) = \left\{ \gamma \circ \tau : \tau \in \mathop{\operatorname{zeros}\,}\limits_{z \in \mathcal{F}} E_{2 k}\!\left(z\right) \,\mathbin{\operatorname{and}}\, \gamma \in \operatorname{PSL}_2(\mathbb{Z}) \right\}$
Assumptions:$k \in \mathbb{Z}_{\ge 2}$
TeX:
\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathbb{H}} E_{2 k}\!\left(\tau\right) = \left\{ \gamma \circ \tau : \tau \in \mathop{\operatorname{zeros}\,}\limits_{z \in \mathcal{F}} E_{2 k}\!\left(z\right) \,\mathbin{\operatorname{and}}\, \gamma \in \operatorname{PSL}_2(\mathbb{Z}) \right\}

k \in \mathbb{Z}_{\ge 2}
Definitions:
Fungrim symbol Notation Short description
Zeros$\mathop{\operatorname{zeros}\,}\limits_{P\left(x\right)} f\!\left(x\right)$ Zeros (roots) of function
EisensteinE$E_{k}\!\left(\tau\right)$ Normalized Eisenstein series
HH$\mathbb{H}$ Upper complex half-plane
SetBuilder$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$ Set comprehension
ModularGroupAction$\gamma \circ \tau$ Action of modular group
ModularGroupFundamentalDomain$\mathcal{F}$ Fundamental domain for action of the modular group
PSL2Z$\operatorname{PSL}_2(\mathbb{Z})$ Modular group (canonical representatives)
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("e46697"),
Formula(Equal(Zeros(EisensteinE(Mul(2, k), tau), tau, Element(tau, HH)), SetBuilder(ModularGroupAction(gamma, tau), Tuple(gamma, tau), And(Element(tau, Zeros(EisensteinE(Mul(2, k), z), z, Element(z, ModularGroupFundamentalDomain))), Element(gamma, PSL2Z))))),
Variables(k),
Assumptions(And(Element(k, ZZGreaterEqual(2)))))

Topics using this entry

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

2019-08-19 14:38:23.809000 UTC