Moving the origin of an elliptic curve - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T06:02:39Z http://mathoverflow.net/feeds/question/76551 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/76551/moving-the-origin-of-an-elliptic-curve Moving the origin of an elliptic curve Prachi 2011-09-27T19:55:11Z 2011-09-27T20:07:18Z <p>Suppose one has an elliptic curve \$E = (C,O)\$ over a field \$k\$ where \$C\$ is a non-singular genus one curve over \$k\$ and \$O\$ is a \$k\$-rational point on \$C\$. By moving \$O\$ on \$C\$ one gets a family of elliptic curves. Is this family trivial in the sense that all the curves are mutually isomorphic? If not (as I suspect), what are the isomorphism classes of elliptic curves one gets this way and how are they related to the original curve \$E\$? </p> <p>Special cases of \$k = \mathbb{C}\$ and \$\mathbb{Q}\$ would be specially helpful to know about, for \$\mathbb{C}\$ in terms of lattices, and for \$\mathbb{Q}\$ in terms of the associated modular forms after Wiles et al. </p> <p>(Depending on the answer) should CM curves be distinguished?</p> <p>Thanks!</p>