Geometric picture of invariant differential of an elliptic curve

2011-12-04T04:12:15Z

What is the geometric meaning of $\omega=dx/(2y+a_1x+a_3)$ for an elliptic curve?

This question is an adjunct to MO Q1 on formal laws and L-series, which motivated Q2. Q1 (Silverman) and Darmon (pg. 6) state:

The invariant holomorphic differential form (Neron differential) attached to an elliptic curve is

$\omega=dx/(2y+a_1x+a_3)$.

(Ancilliary question: Relation to Weierstrass's elliptic functions?)

I'd like to broaden the question as a community wiki to ask, "What are some interesting manifestations of this one-form in various families of elliptic curves?"

E.g., J. Hoffman in Topics in Elliptic Curves and Modular Forms gives for the Jacobi quartic family of elliptic curves

$\omega=dx/(1+2\kappa x^{2}+x^{4})^{1/2}=\sum_{n=0}^{\infty}L_{n}(\kappa)x^{2n}dx$

with $L_{n}(\kappa)$ the Legendre polynomials.

Answer by Tom Copeland for Geometric picture of invariant differential of an elliptic curve

2011-12-04T11:18:47Z

A paper by John Tate (pg. 1 and 2) gives a clear derivation of the diff. form:

Reparametrize the elliptic curve

$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4 x+a_6$

with $p(z)=x+(a_1^2+4a_2)/12$ and $p^{'}(z)=2y+a_1x+a_3$ to obtain

$(p^{'})^2=4p^3-g_2p-g_3$, defining the Weierstrass elliptic fct., and

$\omega=dp(z)/p^{'}(z)=dz=dx/(2y+a_1x+a_3)$.

Per Dan's comment, a coordinate transformation of $x=u^2x^{'}+r$ and $y=u^3y^{'}+su^2x^{'}+t$
leaves $\omega^'=u\omega$.

Given $\sigma=p(z)$ and the inverse $z=p^{-1}(\sigma)$,

$dz=(p^{-1}(\sigma))^{'}d\sigma=(p^{-1}(\sigma))^{'}p^{'}(z)dz$, so

$(p^{-1}(\sigma))^{'}=1/p^{'}(z)$ and $dz=d\sigma/p^{'}(z)=\omega$.

The amplitwist interpretation of differentiation and inversion presented by Tristan Needham in his book Visual Complex Analysis provides a geometric interpretation of these differential relations.

Consider as an analogy $P(\theta)=sin(\theta), P^{'}(\theta)=cos(\theta), and P^2+(P^{'})^2=1$.

Geometric picture of invariant differential of an elliptic curve - MathOverflow

Geometric picture of invariant differential of an elliptic curve

Answer by Tom Copeland for Geometric picture of invariant differential of an elliptic curve