Let $A$ and $B$ be isogenous abelian varieties defined over a field $k$.
Suppose $A(L)\cong B(L)$ for all finite extensions $L$ of $k$. Does this imply that $A\cong B$?
It would be different if we do not suppose $A$ and $B$ isogenous?
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4$\begingroup$ What happens when $k$ equals $\mathbb{C}$? $\endgroup$– Jason StarrCommented Jul 20, 2018 at 12:54
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2$\begingroup$ Not with complete local fields e.g. Tate curves over a p-adic field with same valuations of j invariants $\endgroup$– David LampertCommented Jul 20, 2018 at 13:25
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$\begingroup$ $k$ is itself a finite extension of $k$ $\endgroup$– YCorCommented Jul 20, 2018 at 14:39
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4$\begingroup$ Perhaps you want to ask: Let $f: A \to B$ be a $k$-isogeny. If $f$ induces an isomorphism on $L$-valued points for all $L/k$ finite, is $f$ an isomorphism? $\endgroup$– user19475Commented Jul 20, 2018 at 16:28
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$\begingroup$ No, over a finite field $k$, there are elliptic curves $E,E'$ with isomorphic group of points over every finite extension of $k$, but $E$ is not isomorphic to $E'$ (it has to be isogenous by Tate theorem though). See the appendix in the 2001 paper "Group Structure of Elliptic Curves over Finite Fields" by Christian Wittmann. I don't know if this addresses the above (very good) comment. $\endgroup$– WatsonCommented May 21, 2021 at 16:09
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