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 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 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>