Finding mappings between j-invariants for Calabi-Yau threefolds - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T18:20:15Z http://mathoverflow.net/feeds/question/104835 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/104835/finding-mappings-between-j-invariants-for-calabi-yau-threefolds Finding mappings between j-invariants for Calabi-Yau threefolds Jimeree 2012-08-16T13:56:25Z 2012-08-20T08:27:40Z <p>Suppose we have an elliptic curve of the form:</p> <p>$y^{2}=x^{3}-f\left(z_{1},z_{2}\right)x-g\left(z_{1},z_{2}\right)$</p> <p>This describes a Calabi-Yau threefold as an elliptic fibration over $\mathbb{P}^{1}\times\mathbb{P}^{1}$, where $\left(z_{1},z_{2}\right)$ are the coordinates of the $\mathbb{P}^{1}\times\mathbb{P}^{1}$; $f$ is a homogenous polynomial of degree 8 in each of the $z_i$; and $g$ is a homogenous polynomial of degree 12 in each of the $z_i$. The $j$-invariant should be:</p> <p>$j=\frac{4f^{3}}{4f^{3}-27g^{2}}$</p> <p>Now suppose that this $j$-invariant is also expressed in terms of $t\left(z_{1},z_{2}\right)$. Specifically, I am looking at the Index 36 $j$-invariants on page 5 here: <a href="http://mysite.science.uottawa.ca/asebbar/publi/mcse.pdf" rel="nofollow">http://mysite.science.uottawa.ca/asebbar/publi/mcse.pdf</a>.</p> <p>I want to find a substitution of the form:</p> <p>$t=\frac{P\left(z_{1},z_{2}\right)}{Q\left(z_{1},z_{2}\right)}$</p> <p>Where $P$ and $Q$ are polynomials of some appropriate degree, such that we reproduce the form the $j\left(f,g\right)$ above. Is there any general procedure for doing this? Or is it just frustrating guesswork?</p> <p>Many thanks!</p>