Good reduction of isogenous abelian varieties over finitely generated fields - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T08:56:07Z http://mathoverflow.net/feeds/question/81884 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/81884/good-reduction-of-isogenous-abelian-varieties-over-finitely-generated-fields Good reduction of isogenous abelian varieties over finitely generated fields Martin Orr 2011-11-25T13:58:54Z 2011-11-25T13:58:54Z <p>Let $K$ be a finitely generated field over $\mathbb{Q}$. Let $A$ and $B$ be abelian varieties over a field $K$, isogenous over some finite extension $L$ of $K$. I want to ask if they have the same points of good reduction.</p> <p>More precisely, suppose that $S$ is a normal affine variety over $\mathbb{Q}$ with function field $K$, such that there is an abelian scheme $\mathcal{A}$ over $S$ with generic fibre $A$.</p> <p>Does there exist an abelian scheme $\mathcal{B}$ over $S$ with generic fibre isomorphic to $B$, and such that the isogeny $A_L \to B_L$ extends to an isogeny $\mathcal{A} \times_S T \to \mathcal{B} \times_S T$ where $T$ is the normalisation of $S$ in $L$?</p> <p>I believe that this is true if I suppose that $L=K$, as then we can extend the kernel of the isogeny to a finite subgroup scheme of $\mathcal{A}$ and form the quotient.</p> <p>I ask this because the analogous statement is true if we replace $S$ by the spectrum of the ring of integers of a number field, by the Néron-Ogg-Shafarevich criterion. But I am not sure if this extends to $\dim S > 1$.</p>