Let $E$ be an elliptic curve over $\overline{\mathbb{Q}}$ defined by a Weierstrass equation
$$
y^2=4x^3+g_2x+g_3.
$$ Then $H^1_{dR}(E/\overline{\mathbb{Q}})$ is spanned by the classes of the differential forms $\omega_1=\frac{dx}{y}$ and $\omega_2=x\frac{dx}{y}$ and "the" period matrix of $E$ consists of the four numbers
$$
\int_{\gamma_i} \omega_j
$$ where $\gamma_1, \gamma_2$ are generators of $H_1(E(\mathbb{C}), \mathbb{Q})$.
Theorem. The degree of transcendence of the field spanned by these numbers is at least $2$.
I am trying to understand why this implies that $\int_\gamma \omega_1$ and $\int_\gamma \omega_2$ are algebraically independent whenever $E$ has complex multiplication. This will follow if one shows that the ratio between periods attached to the same differential form is algebraic.
For $\omega_1$ I can see what's going on: by uniformization,
$$
E(\mathbb{C}) \simeq \mathbb{C} / \mathbb{Z} \oplus \mathbb{Z} \tau
$$ for some $\tau$ in the upper half plane. Complex multiplication forces $\tau$ to be imaginary quadratic. The differential form $\omega_1$ corresponds to $dz$ on the complex torus and the periods become $\int_0^1 dz=1$ and $\int_0^\tau dz=\tau$, so the quotient is $\tau$ which is algebraic.
What about the other differential form? I know that $\omega_2=x \frac{dx}{y}$ corresponds to $\mathcal{P}(z)dz$ where $\mathcal{P}$ is the Weierstrass function attached to the lattice. Then the periods $\int_{\gamma_i} \omega_2$ are minus the periods of $\zeta$, the primitive of $-\mathcal{P}$. Why their quotient is algebraic in this situation?
Best Answer
For the elliptic curve $E$ in the original post, we have two periods $\lambda_i = \int_{\gamma_i} \frac{dx}{y}$, $i=1,2$, and quasi-periods $\eta_i = \int_{\gamma_i} \frac{x\,dx}{y}$, $i=1,2$.
Theorem (Schneider 1936): Assume that $E$ is defined over $\overline{\mathbb{Q}}$. I.e., $g_2$, $g_3 \in \overline{\mathbb{Q}}$.
Schneider's results were improved over the years, starting with work of Baker, Coates, and Masser, to results on $\overline{\mathbb{Q}}$-linear combinations of periods, quasi-periods, and elliptic logarithms. Parts 2 and 3 of the following theorem should settle the original question.
Theorem (Masser 1975): Again assume that $E$ is defined over $\overline{\mathbb{Q}}$. Let $V$ be the $\overline{\mathbb{Q}}$-linear span of the six numbers $1$, $\pi$, $\lambda_1$, $\lambda_2$, $\eta_1$, and $\eta_2$.
A couple of further comments: