Algebraic independence of the exponential and Weierstrass $\wp$-functions

Abstract

Let $\Omega$ be a lattice in $\mathbb{C}$ with algebraic invariants and complex multiplication, let $\mathcal{E}$ be the elliptic curve associated with $\Omega$, and let $\wp$ be the Weierstrass function relative to $\Omega$. Set $k:=\mathrm{End}(\mathcal{E}){\otimes }_{\mathbb{Z}}\mathbb{Q}.$ We prove that if $t_1,\dots ,t_s$ are $\mathbb{Q}$-linearly independent algebraic numbers and $p_1,\dots ,p_n$ are $k$-linearly independent algebraic numbers, then the $s+n$ numbers \[ \mathrm{e}^{t_1},\dots,\mathrm{e}^{t_s}, \wp(p_1),\dots,\wp(p_n) \] are algebraically independent over $\overline{\mathbb{Q}}$. The proof uses the Tannakian description of the Lie algebra of the unipotent radical of the $1$-motive associated with these points.