I have been trying to reconstruct some elliptic curves theory computationally and have gotten stuck on some period computations.

Specifically, let $$E_\lambda:\ y^2=x(x-1)(x-\lambda)$$ be the Legendre family of elliptic curves. The period integrals, (chosen to be) defined as $$\Omega=\Omega_\gamma(\lambda):=\int_\gamma \frac{dx}{\sqrt{x(x-1)(x-\lambda)}},$$ satisfy a second order differential equation, known as the Picard-Fuchs equation: $$ 4\lambda(1-\lambda)\frac{d^2\Omega}{d\lambda^2}+4(1-2\lambda)\frac{d\Omega}{d\lambda}-\Omega=0. $$

Question. Is there a basis, $\{\Omega_1(\lambda), \Omega_2(\lambda)\}$, of solutions around 0 for the Picard-Fuchs equation such that $$ E_\lambda(\mathbb{C})\cong \mathbb{C}/(\mathbb{Z}+\frac{\Omega_2(\lambda)}{\Omega_1(\lambda)}\mathbb{Z}) $$ for non-zero $\lambda$ (close to 0)?

Classic theory says that there is a unique holomorphic solution satisfying $\Omega(0)=1$, which is well known to be the hypergeometric function $$\Omega_1(\lambda)={}_2F_1(1/2, 1/2; 1; \lambda)=\sum_{n\ge0}\binom{-1/2}{n}^2\lambda^n.$$

Classic theory then tells us that there is a unique holomorphic function satisfying $\rho(0)=0$ and such that $$\Omega_2(\lambda)=C\left(\Omega_1(\lambda)\log{\lambda} + \rho(\lambda)\right)$$ is also a solution of the Picard-Fuchs equation, for all non-zero constants $C$, and that $\Omega_1, \Omega_2$ form a basis of solutions of the equation ($\Omega_2$ is a multi-valued map because of the appearance of $\log$). This $\rho$ is $F_1(1/2,1/2;\lambda)$, and is sometimes referred to as Fricke's hypergeometric function.

I have tried using these two solutions, $\Omega_1$ and $\Omega_2$, for different values of $C$, but I couldn't get the isomorphism in the question to be true. The most attractive value for $C$ is $\frac{1}{2\pi i}$, since then $$ \mathbb{Z}+\frac{\Omega_2(\lambda)}{\Omega_1(\lambda)}\mathbb{Z} = \mathbb{Z}+\frac{1}{2\pi i}\left(\log{\lambda}+\frac{\rho(\lambda)}{\Omega_1(\lambda)}\right) $$ and we can take a single value for the $\log$ term.

But for $\lambda=1/2$, we have: $$\frac{1}{2\pi i}\left(\log{\lambda}+\frac{\rho(\lambda)}{\Omega_1(\lambda)}\right) =0.058728i$$ which corresponds to $17.027i$ in the fundamental domain for $SL_2(\mathbb{Z})$, which doesn't sit well with the fact that $$E_{1/2}\cong\mathbb{C}/(\mathbb{Z}+i\mathbb{Z}).$$


I've given this another push and managed to find a basis that answers the question positively. But I don't get why it has to be this basis. Any comment on that would be much appreciated.

Set $C=1$ in the above explicit $\Omega_2$. Then set $$ \Omega'_1 := \pi\Omega_1\\ \Omega'_2 := (-4i\log{2})\Omega_1 - i\Omega_2. $$ Then we have $E_\lambda(\mathbb{C})\cong \mathbb{C}/(\mathbb{Z}+\frac{\Omega'_2(\lambda)}{\Omega'_1(\lambda)}\mathbb{Z}).$

In fact, these $\Omega'$ are: $$ \Omega'_1(\lambda)=\int_\lambda^1 \frac{dx}{\sqrt{x(x-1)(x-\lambda)}}\\ \Omega'_2(\lambda)=\int_1^\infty \frac{dx}{\sqrt{x(x-1)(x-\lambda)}}. $$

Other than seeing these two periods in a paper and trying them out, how does one get to them?


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