Let $X$ be an elliptic curve over a complete local field.


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."


For elliptic curves, are the two definitions equivalent? How to prove it?



reduction of elliptic curves
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. 


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). 


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). 


You don't need the completeness hypothesis on the base DVR.


