### Ranks of elliptic curves in families of quadratic twists.

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The main result of this paper implies that if an abelian variety over a field $F$ has a maximal isotropic subgroup of $n$-torsion points all of which are defined over $F$, and $n\ge 5$, then the abelian variety has semistable reduction away from $n$. This result can be viewed as an extension of Raynaud’s theorem that if an abelian variety and all its $n$-torsion points are defined over a field $F$ and $n\ge 3$, then the abelian variety has semistable reduction away from $n$. We also give information about the Néron models...

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