Can the simplicity of abelian varieities be implied by the reduction A is an abelian variety over number field K, with simple good reduction at a finite field $\kappa$, can we deduce that $A$ itself is simple? 
 A: Here is a slightly different proof. We have the following facts:
 

(1) If $B, C$ are abelian varieties over $K$, then the Néron model of $B\times C$ is the product of the Néron models (this is a simple onsequence of the universal property of Néron models). In particular, if $B\times C$ has good reduction, then $B, C$ also have good reduction, and the reduction of $B\times C$ is the product of the reductions of $B$ and $C$. 



(2) If $A$ and $B$ are isogeneous abelian varieties over $K$ and if $A$ has good reduction, then $B$ has good reduction and any isogeny $A\to B$ induces an isogeny on the reductions.


Proof of (2): if $A\to B$ is an isogeny of degree $n$, then there exists an isogeny $B\to A$ such that the composition $A\to B\to A$ is the multiplication-by-$n$ map $[n]_A: A\to A$ on $A$. On the reductions we have the maps $A_k \to B_k^0 \to A_k$ whose composition is $[n]_{A_k}$. Consider a positive integer $\ell$ prime to $n$ and to $\mathrm{char}(k)$. As the restriction of $[n]_{A_k}$ to $A_k[\ell]$ is an isomorphism, $A_k[\ell]\to B^0_k[\ell]$ is injective, so $B^0_k[\ell]$ contains $(\mathbb Z/\ell \mathbb Z)^{2\dim B}$. Therefore $B_k^0$ is an abelian variety and $B$ has good reduction. Finally $A_k\to B_k=B_k^0$ is an isogeny because its kernel is contained in $A_k[n]$. 
This is a special case of Néron-Ogg-Shafarevich criterion (Serre-Tate: Good reduction of abelian varieties, §1).  
Now to answer the original quesiton, if $A$ was not simple, then it is isogeneous to a product of abelian varieties $B\times C$. Then $B\times C$, $B, C$ have good reduction by (2) and (1), and $A_k$ is isogeneous to $B_k\times C_k$. So $A_k$ would not be simple.  
