Is there a standard example of two abelian varieties $A$, $B$ over some number field $k$ which are $k\_v$-isomorphic for every place $v$ of $k$ but not $k$-isomorphic ?



Everywhere locally isomorphic abelian varieties
(If you upvote this answer, please consider upvoting the answers by Felipe Voloch and David Speyer too, since this answer builds on their ideas.)


The smallest examples are in dimension $2$. Let $E$ be any elliptic curve over $\mathbf{Q}$ without complex multiplication, e.g., $X\_0(11)$. We will construct two twists of $E^2$ that are isomorphic over $\mathbf{Q}\_p$ for all $p \le \infty$ but not isomorphic over $\mathbf{Q}$.


Let $K:=\mathbf{Q}(\sqrt{-1},\sqrt{17})$. Let $G:=\operatorname{Gal}(K/\mathbf{Q}) = (\mathbf{Z}/2\mathbf{Z})^2$. Let $\alpha \colon G \to \operatorname{GL}\_2(\mathbf{Z}) = \operatorname{Aut}(E^2)$ be a homomorphism sending the two generators to the reflections in the coordinate axes of $\mathbf{Z}^2$, and let $A$ be the $K/\mathbf{Q}$-twist of $E^2$ given by $\alpha$. Define $\beta$ and $B$ similarly, but with the lines $y=x$ and $y=-x$ in place of the coordinate axes. The representations $\alpha$ and $\beta$ of $G$ on $\mathbf{Z}^2$ are not conjugate: only the former is such that the lattice vectors fixed by nontrivial elements of $G$ generate all of $\mathbf{Z}^2$. Thus $A$ and $B$ are not isomorphic over $\mathbf{Q}$.


On the other hand, every decomposition group $D\_p$ in $G$ is smaller than $G$ since $-1$ is a square in $\mathbf{Q}\_{17}$ and $17$ is a square in $\mathbf{Q}\_2$. Also, the restrictions of $\alpha$ and $\beta$ to any proper subgroup of $G$ are conjugate: any *single* line spanned by a primitive vector in $\mathbf{Z}^2$ can be mapped to any other by an element of $\operatorname{GL}\_2(\mathbf{Z})$. Thus $A$ and $B$ become isomorphic after base extension to $\mathbf{Q}\_p$ for any $p \le \infty$. $\square$


**Remark:** The abelian surfaces $A$ and $B$ constructed above are *isogenous* even over $\mathbf{Q}$, because the $\mathbf{Z}^2$ with one Galois action can be embedded into the $\mathbf{Z}^2$ with the other Galois action: rotate $45^\circ$ and dilate.


**Remark:** The nonexistence of examples in dimension $1$ follows from these two well-known facts:


1) Twists of an elliptic curve over a field $k$ of characteristic $0$ 
are classified by
$H^1(k,\mu\_n)=k^\times/k^{\times n}$ where $n$ is 2, 4, or 6. 


2) If $n<8$, the map $k^\times/k^{\times n} \to \prod\_v k\_v^\times/k\_v^{\times n}$ is injective. 


[**Edit:** This answer was edited to simplify the construction and to add those remarks at the end.]


