Linear dependence of quasi-periods over the rationals

Abstract

In this note we shall show that a lattice $\mathbb{Z}\omega _1+\mathbb{Z}\omega _2$ in $\mathbb{C}$ has $\mathbb{Q}$-linearly dependent quasi-periods if and only if $\omega _2/\omega _1$ is equivalent to a zero of the Eisenstein series $E_2$ under the action of $\mathrm{SL}_2(\mathbb{Z})$ on the upper half plane of $\mathbb{C}$.