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? 



Can the simplicity of abelian varieities be implied by the reduction
Here is a slightly different proof. We have the following facts:



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> (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 $C$. 
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> (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.
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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. 


