Let $A$ be an abelian variety over a field $k$, and let $N$ be a finite subgroup of $A$. Suppose that $N$ is also defined over $k$, or at least that all Galois automorphisms fixing $k$ leave $N$ invariant. Then I've heard it stated that there is an abelian variety $B / k$ and an isogeny $\phi: A \rightarrow B$ with kernel $N$, such that $\phi$ is defined over $k$. I can't seem to find this stated anywhere (except in Silverman, where it is stated for elliptic curves as an exercise) and am not sure how to prove it. Does anyone know how to prove this and/or where to cite it in the literature?

Take the 2-minute tour
×

MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Zariski-locallyon the target to form the quotient $B$ of this torsor by $N$ using justaffineconsiderations. This quotient is then seen to be what we sought. A merit of this argument (over the one in Mumford's book) is that it works with abelian schemes overanybase scheme; no projectivity hypotheses. – user30379 Feb 15 '13 at 3:19