Geometric picture of invariant differential of an elliptic curve - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T16:26:07Z http://mathoverflow.net/feeds/question/82597 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/82597/geometric-picture-of-invariant-differential-of-an-elliptic-curve Geometric picture of invariant differential of an elliptic curve Tom Copeland 2011-12-04T04:12:15Z 2012-02-24T09:57:10Z <p><strong>What is the geometric meaning of $\omega=dx/(2y+a_1x+a_3)$ for an elliptic curve?</strong> </p> <p>This question is an adjunct to MO <a href="http://mathoverflow.net/questions/52241/formal-group-laws-and-l-series" rel="nofollow">Q1</a> on formal laws and L-series, which motivated <a href="http://mathoverflow.net/questions/81332/formal-group-laws-l-series-and-flow-equations" rel="nofollow">Q2</a>. Q1 (Silverman) and <a href="http://www.math.mcgill.ca/darmon/pub/Articles/Research/36.NSF-CBMS/chapter.pdf" rel="nofollow">Darmon</a> (pg. 6) state: </p> <p>The invariant holomorphic differential form (Neron differential) attached to an <a href="http://en.wikipedia.org/wiki/Elliptic_curve" rel="nofollow">elliptic curve</a> is </p> <p>$\omega=dx/(2y+a_1x+a_3)$. </p> <p>(Ancilliary question: Relation to <a href="http://en.wikipedia.org/wiki/Weierstrass_elliptic_function" rel="nofollow">Weierstrass's elliptic functions</a>?)</p> <p>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?"</p> <p>E.g., J. Hoffman in <a href="https://www.math.lsu.edu/~hoffman/papers/elmod.pdf" rel="nofollow">Topics in Elliptic Curves and Modular Forms</a> gives for the Jacobi quartic family of elliptic curves</p> <p>$\omega=dx/(1+2\kappa x^{2}+x^{4})^{1/2}=\sum_{n=0}^{\infty}L_{n}(\kappa)x^{2n}dx$ </p> <p>with $L_{n}(\kappa)$ the Legendre polynomials. </p> http://mathoverflow.net/questions/82597/geometric-picture-of-invariant-differential-of-an-elliptic-curve/82612#82612 Answer by Tom Copeland for Geometric picture of invariant differential of an elliptic curve Tom Copeland 2011-12-04T11:18:47Z 2012-02-24T09:57:10Z <p>A paper by <a href="http://www.kryakin.com/files/Invent_mat_%282_8%29/23/23_01.pdf" rel="nofollow">John Tate</a> (pg. 1 and 2) gives a clear derivation of the diff. form:</p> <p>Reparametrize the elliptic curve </p> <p>$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4 x+a_6$ </p> <p>with $p(z)=x+(a_1^2+4a_2)/12$ and $p^{'}(z)=2y+a_1x+a_3$ to obtain</p> <p>$(p^{'})^2=4p^3-g_2p-g_3$, defining the Weierstrass elliptic fct., and </p> <p>$\omega=dp(z)/p^{'}(z)=dz=dx/(2y+a_1x+a_3)$. </p> <p>Per Dan's comment, a coordinate transformation of $x=u^2x^{'}+r$ and $y=u^3y^{'}+su^2x^{'}+t$<br> leaves $\omega^'=u\omega$.</p> <p>Given $\sigma=p(z)$ and the inverse $z=p^{-1}(\sigma)$, </p> <p>$dz=(p^{-1}(\sigma))^{'}d\sigma=(p^{-1}(\sigma))^{'}p^{'}(z)dz$, so</p> <p>$(p^{-1}(\sigma))^{'}=1/p^{'}(z)$ and $dz=d\sigma/p^{'}(z)=\omega$.</p> <p>The amplitwist interpretation of differentiation and inversion presented by Tristan Needham in his book <a href="http://usf.usfca.edu/vca//PDF/vca-toc.pdf" rel="nofollow">Visual Complex Analysis</a> provides a geometric interpretation of these differential relations.</p> <p>Consider as an analogy $P(\theta)=sin(\theta), P^{'}(\theta)=cos(\theta), and P^2+(P^{'})^2=1$.</p>