The elliptico--toric conjecture

Let $\mathcal{E}$ be an elliptic curve with period lattice $\Omega$, invariants $g_2,g_3$, and endomorphism field $k$. Let $s,n\geq0$, let $\mathrm{e}^{t_1},\dots,\mathrm{e}^{t_s}$ be points of $\mathbb{G}_m(\mathbb{C})$, and let $P_1,\dots,P_n$ be points of $\mathcal{E}(\mathbb{C})$, with $P_i=[\wp(p_i):\wp'(p_i):1]$. Let $K$ be the field defined in the source, containing the exponential and elliptic-function data. Define $\langle t_\ell\rangle_\ell$ as the $\mathbb{Q}$-span of the classes of the $t_\ell$ in $\mathbb{C}/2\pi\mathrm{i}\mathbb{Q}$, and $\langle p_i\rangle_i$ as the $k$-span of the classes of the $p_i$ in $\mathbb{C}/(\Omega\otimes_\mathbb{Z}\mathbb{Q})$. Elliptico--toric conjecture.

$$\operatorname{tran.deg} K(\omega_1,\omega_2,\eta_1,\eta_2) \geq \dim_{\mathbb{Q}}\langle t_\ell\rangle_\ell+\frac{4}{[k:\mathbb{Q}]}+2\dim_k\langle p_i\rangle_i.$$

This is attributed in the source to Bertrand and is equivalent there to a Grothendieck--André generalized period conjecture for a suitable 1-motive. It remains open.