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?
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$\begingroup$ Isn't this in Weil? $\endgroup$– Felipe VolochFeb 14, 2013 at 3:15
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$\begingroup$ Do you mean, in his "varietes abeliennes" book? $\endgroup$– Jeff YeltonFeb 14, 2013 at 3:30
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6$\begingroup$ This is in Mumford's book on Abelian Varieties (for a finite subgroup scheme). Mumford assumes that the ground field is algebraically closed, but the proof doesn't need this. Alternatively, if you are willing to assume that the ground field is perfect, then the statement follows from the algebraically closed case + descent. $\endgroup$– anonFeb 14, 2013 at 4:30
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$\begingroup$ Choose $n > 0$ killing $N$, so $N \subset A[n]$. By fppf descent theory, the finite fppf map $[n]:A \rightarrow A$ identifies the target as the quotient $A/A[n]$. So the source is an fppf $A[n]$-torsor over the target. Using the torsor viewpoint (and so more descent theory), we can work Zariski-locally on the target to form the quotient $B$ of this torsor by $N$ using just affine considerations. 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 over any base scheme; no projectivity hypotheses. $\endgroup$– user30379Feb 15, 2013 at 3:19
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