reduction of elliptic curves - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T10:15:35Z http://mathoverflow.net/feeds/question/106785 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106785/reduction-of-elliptic-curves reduction of elliptic curves kiseki 2012-09-10T07:31:22Z 2012-09-10T15:22:46Z <p>Let \$X\$ be an elliptic curve over a complete local field.</p> <p>The definition of semi-abelian reduction is: "the Neron model of \$X\$ is a semi-abelian scheme". On the other hand, the definition of semi-stable reduction is: "the minimal regular model of \$X\$ is semi-stable."</p> <p>For elliptic curves, are the two definitions equivalent? How to prove it?</p> http://mathoverflow.net/questions/106785/reduction-of-elliptic-curves/106787#106787 Answer by Qing Liu for reduction of elliptic curves Qing Liu 2012-09-10T08:01:28Z 2012-09-10T08:01:28Z <p>The Néron model of \$X\$ is the smooth locus of the minimal regular model of \$X\$ (see Bosch-Lütkebohmert-Raynaud: Néron models, §1.5). The equivalence is then clear using the classification of Kodaira-Néron of the types of reduction of the minimal regular model. </p> <p>Note that if \$g(X)\ge 2\$, then it is also true that \$X\$ has semi-stable reduction if and only if its Jacobian has semi-abelian reduction (Deligne-Mumford, based on Raynaud's description of Néron models of Jacobians). </p> <p>If \$g(X)=1\$ but \$X\$ doesn't have a rational point, then the statement is no longer true. But one can show that the Jacobian of \$X\$ has semi-abelian reduction if and only if the type of the reduction of \$X\$ is a multiple of \$I_n\$, \$n\ge 0\$ (use the fact that \$X\$ covers its Jacobian. The additive reduction case is treated in a paper of Lorenzini, Raynaud and myself in 2004). </p> <p>You don't need the completeness hypothesis on the base DVR.</p>