Neron models of elliptic curves with level N structure? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T14:48:20Z http://mathoverflow.net/feeds/question/51252 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/51252/neron-models-of-elliptic-curves-with-level-n-structure Neron models of elliptic curves with level N structure? Hugo Chapdelaine 2011-01-05T22:08:19Z 2011-01-05T22:34:18Z <p>In the Deligne-Rapoport paper entitled "Les schemas de modules de courbes elliptiques" the following is written (I translated in english):</p> <p>Let $E$ be an elliptic curve with $\Gamma(N)$-level structure defined over $\mathbb{C}((T))$. Let $E'$ be the minimal model of $E$ over $\mathbb{C}[[T]]$. It may happen that $E'$ has bad reduction (i.e. when one reduces modulo $T$). So let $A$ be the neron model of $E'$ over the d.v.r. $\mathbb{C}[[T]]$. Then it seems that the special fiber of $A$ (i.e. when $T=0$) is always isomorphic to $\mathbb{C}^{\times}\times\mathbb{Z}/kN$ for some suitable integer $k$.</p> <p><strong>Q: How come this $N$ shows up in the special fiber of $A$ ?</strong></p> <p>This is a little bit strange since in the definition of a Neron model no such $N$ appears. </p> http://mathoverflow.net/questions/51252/neron-models-of-elliptic-curves-with-level-n-structure/51255#51255 Answer by Joe Silverman for Neron models of elliptic curves with level N structure? Joe Silverman 2011-01-05T22:34:18Z 2011-01-05T22:34:18Z <p>Roughly speaking, the N-torsion defined over the base injects into the Neron model (in characteristic 0), so the special fiber of the Neron model needs to have a subgroup isomorphic to Z/NZ x Z/NZ, since by assumption there is full level N structure. The special fiber (since there's bad reduction) has the form C^* x F, where F is a finite cycle group. So F contains an N-torsion element, which means F = Z/kN.</p>