etale cover of a hyperelliptic curve - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T19:58:06Z http://mathoverflow.net/feeds/question/44639 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/44639/etale-cover-of-a-hyperelliptic-curve etale cover of a hyperelliptic curve Michael Zhang 2010-11-03T03:16:01Z 2010-11-03T15:38:35Z <p>Since every hyperelliptic curve \$C\$ of genus \$g\$ can be written as \$y^2=f(x)\$, is there an way to write an equation of a new curve \$C'\$ which has an etale cover over \$C\$ just using \$x\$ and \$y\$?</p> http://mathoverflow.net/questions/44639/etale-cover-of-a-hyperelliptic-curve/44689#44689 Answer by Daniel Loughran for etale cover of a hyperelliptic curve Daniel Loughran 2010-11-03T15:38:35Z 2010-11-03T15:38:35Z <p>Since nobody has answered this yet, here is my attempt, however Im not sure if it is exactly what you want. Hopefully it will at least lead you to a solution of your problem if you have not seen this kind of stuff already.</p> <p>If your curve \$C\$ is defined over a field \$k\$ in which \$f(x)\$ splits as \$f(x)=f_1(x)f_2(x)\$, then we can define \$C'\$ via: \$\$C': y_1^2 = f_1(x) , y_2^2 = f(x).\$\$</p> <p>Then the natural map \$(x,y_1,y_2) \mapsto (x,y_1y_2)\$ extends to an étale map on the compactifications.</p> <p>However your question asks for \$C'\$ to "be written in terms of \$x\$ and \$y\$". If by this you mean present an affine patch of \$C'\$ which is a subset of \$\mathbb{A}^2\$, then \$C'\$ is at least birational to such an expression by the primitive element theorem and this shouldn't be too hard to write down - however it might be singular.</p> <p>I hope this helps.</p>