3
$\begingroup$

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)?

$\endgroup$

1 Answer 1

3
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ Would that "different language" be French? $\endgroup$ Oct 30, 2013 at 22:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.