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`	$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$	Zeros (roots) of function
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`HH`	$\mathbb{H}$	Upper complex half-plane
`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), ForElement(tau, HH)), Set(ModularGroupAction(gamma, tau), For(Tuple(gamma, tau)), And(Element(tau, Zeros(EisensteinE(Mul(2, k), z), ForElement(z, ModularGroupFundamentalDomain))), Element(gamma, PSL2Z))))),
    Variables(k),
    Assumptions(And(Element(k, ZZGreaterEqual(2)))))

Topics using this entry

Eisenstein series