TITLE: A local-to-global principle for isogeny
QUESTION [10 upvotes]: If two elliptic curves over $\mathbb{Q}$ are $\mathbb{Q}_p$-isogneous for almost all primes $p$, then they are $\mathbb{Q}$-isogenous.  
This follows from the fact that they have the same number of $\mathbb{F}_p$-points for almost all $p$, hence their $L$-functions have the same local factors at all these $p$, therefore a combination of "multiplicity one" and Faltings' isogeny theorem implies that they are $\mathbb{Q}$-isogenous.  Correct me if I'm wrong.
Here $\mathbb{Q}$ can be replaced by any number field $k$.
Question :  Does the same argument work for any two abelian varieties $A$, $B$ over $k$ ?  It should, since $H^i$ is $\wedge^i H^1$ for $i>0$.
If so, this explains why Poonen's abelian surfaces $A,B$  (everywhere locally isomorphic but not isomorphic) are $\mathbb{Q}$-isogenous.

REPLY [7 votes]: A propos Dimitrov's answer, one should also mention two beautiful papers of Masser and Wustholz. 
MR1037140 Masser, D. W.(1-MI); Wüstholz, G.(CH-ETHZ); Estimating isogenies on elliptic curves. Invent. Math. 100 (1990), no. 1, 1–24.
MR1217345 Masser, David(1-MI); Wüstholz, Gisbert(CH-ETHZ); Isogeny estimates for abelian varieties, and finiteness theorems. Ann. of Math. (2) 137 (1993), no. 3, 459–472. 
Given a number field $K$ and abelian varieies $A/K$ and $A'/K$ that are isogenous over $K$, they give an explicit value $M(K,A,A')$ such that there is a $K$-isogeny $A\to A'$ of degree less then $M$. (The first article treats the case of elliptic curves.) Such estimates have many applications, including a proof of the hardest implication of the theorem stated in Keller's answer (by an argument appearing in Tate's original paper). As in the articles by the Chudnovskys, the tools used in the proof are those of Diophantine approxmation and transcendence theory.