Question about integral the form $\frac{dx}{y}$ on elliptic curve

Question

I've learn that for every elliptic curve $E$ we can define a map $\phi:E \rightarrow C$ by integral the form $\frac{dx}{y}$ from a fixed point $P$ to any $z \in E$ along some curve.Then we quotient the free abelian group generated by these two complex numbers: $\int_{A} \frac{dx}{y}$ and $ \int_{B} \frac{dx}{y}$ to make sure the map is well-defined,where $A,B$ are two generators of the fundamental group $\pi_{1}(E)=\mathbb{Z} \bigoplus \mathbb{Z}$.

The following are my questions 1)Why this two complex numbers are $\mathbb{R}-$linearly independent? 2)Why the map $\phi$ is injective?

I've heard that we can solve this two problems by using homology theory.Is there any elementary approach?

Thank you for your answer.

Answer 1

Assuming you work over $\mathbb C$, i.e. with Riemann surfaces, here are some possible complex-analytic approaches.

1. Riemann's bilinear relations.

Let $\mathcal R$ be a smooth Riemann surface of genus $g$ and $A_1,B_1,...,A_g,B_g$ a canonical basis of cycles. For any two closed differentials $\nu,\eta$ on $\mathcal R$ we have $$ \int_{\mathcal R}\nu\wedge\eta=\sum_{i=1}^g\left(\oint_{A_i}\nu\oint_{B_i}\eta-\oint_{A_i}\eta\oint_{B_i}\nu\right). $$ The proof is essentially Stokes' theorem.

In particular if $\mathcal R=E$ is a (smooth) elliptic curve, $A,B$ a canonical basis of cycles on $E$ then (denoting $\omega_A=\oint_A\frac {dx}y$ and $\omega_A=\oint_B\frac {dx}y$) $$ 0<\int_E\frac{dx\wedge d\overline{x}}{|y|^2}=Im(\omega_A\overline{\omega_B}) $$ implying that $\omega_A,\omega_B$ are $\mathbb R$-linearly independent.

2. Abel's Theorem.

If $\int_P^{z_1}\frac{dx}y=\int_P^{z_2}\frac{dx}y$ for some $z_{1,2}\in E$ (equality modulo periods $\omega_A,\omega_B$) then by Abel's theorem there is a meromorphic function on $E$ with exactly one pole at $z_2$ and one zero at $z_1$; this would establish a bi-holomorphism with the Riemann sphere, a contradiction.

Abel's theorem in general asserts that principal divisors are characterized by vanishing of the Abel map (period map). In the elliptic case you can directly prove the claim above by using elliptic theta functions; namely for any $c\not=0$ the function (wlog $\tau:=\omega_B/\omega_A$ is in the upper half-plane) $$ f(z)=c\frac{\vartheta(\int_{z_1}^{z}dx/y+\frac {1+\tau}2;\tau)}{\vartheta(\int_{z_2}^{z}dx/y+\frac {1+\tau}2;\tau)},\qquad\vartheta(z;\tau)=\sum_{n\in\mathbb Z}\exp(\pi i n^2\tau+2\pi in z), $$ would be well defined (by the quasi-periodicity properties of $\vartheta$) and would have only one zero (at $z_1$) and one pole (at $z_2$) and so would establish a bi-holomorphism between $E$ and the Riemann sphere, a contradiction.