Algebraic curve mapping to elliptic curve - how to check whether this is possible? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T23:02:01Z http://mathoverflow.net/feeds/question/105965 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105965/algebraic-curve-mapping-to-elliptic-curve-how-to-check-whether-this-is-possible Algebraic curve mapping to elliptic curve - how to check whether this is possible? Peter Mueller 2012-08-30T16:21:15Z 2012-08-30T16:21:15Z <p><em>Question:</em> Let \$C\$ be an algebraic curve over some field (like the rationals) given by a plane projective model (possibly with singularities). Is there an easy way to see if this curve has a non-trivial rational map to an elliptic curve?</p> <p>Criteria involving the Jacobian (something like \$J(C)\$ has a subgroup of codimension \$1\$) wouldn't help, as the curves I am interested in have a big genus (\$>10\$) and high degrees in both variables, so it is very unlikely that anything about their Jacobians can be computed.</p> <p><em>Background:</em> This question arose from an attempt to study rational points (over \$\mathbb Q\$) on certain curves which possibly are coverings of lower genus curves. In particular, if one could compute a covering map to an elliptic curve, one could efficiently look for rationals points.</p>