Legendre relation for elliptic curves

Question

Let $E$ be an elliptic curve over some subfield $k$ of $\mathbb{C}$, say given by an equation $$ y^2=4x^3+ax+b. $$ Then:

• $E(\mathbb{C})$ is a complex torus, so $H_1(E(\mathbb{C}), \mathbb{Q})$ is spanned by two cycles $\gamma_1$ and $\gamma_2$. Assume the basis $\{\gamma_1, \gamma_2\}$ is oriented.

• the algebraic de Rham cohomology $H^1_{dR}(E/k)$ is spanned by the differential forms $\frac{dx}{y}$ and $x \frac{dx}{y}$.

I've read several times that the following result is true

Legendre relation: The determinant of the period matrix with respect to these two basis is $2\pi i$.

Can someone indicate me how to prove this?

Is it really a result by Legendre? If so, where can I find it (in legendre's works)?

Answer 1

The standard proof (due, I believe, to Weierstrass) is as follows. The map $z\mapsto (\wp(z),\wp'(z))$ identifies $E(\mathbb{C})$ with the torus $\mathbb{C}/(\mathbb{Z}\omega _1+\mathbb{Z}\omega _2)$. Your differentials become $dz$ and $\wp(z)dz$, where $\wp$ is the Weierstrass $\wp$ function. Take for $\gamma _i$ the loop $t\mapsto a+ t\omega _i$ for some $a$. Then the periods of $dz$ are $\omega _1$ and $\omega _2$. Those of $\wp(z)dz$ are $\zeta (a+\omega _i)-\zeta (\omega _i)=\eta _i$, where $\zeta $ is the Weierstrass zeta function, a primitive of $-\wp$. The relation $\ \omega _1\eta _2-\omega _2\eta _1=2\pi i\ $ is obtained by integrating $\zeta $ along a fundamental parallelogram : see e.g. Chandrasekharan, Elliptic Functions, p. 50. The result (with a different language) appears indeed in Legendre' Traité des fonctions elliptiques, vol. I, pp. 60-61.