TITLE: Is an abelian variety with a Galois invariant, rank one submodule of its Tate module, CM?
QUESTION [12 upvotes]: Let $A$ be an absolutely simple abelian variety over a number field $K$. Assume that, for some prime $p$, the Tate module $T_p A$ has a submodule of rank one, invariant under the absolute Galois group of $K$. Does it follow that $A$ is has CM?
For elliptic curves, I guess this follows from Serre's open image theorem. That's all I know. I would be surprised if there was a counterexample as it would be a way of constructing abelian extensions of $K$ using non-CM abelian varieties, which would be surprising.

REPLY [13 votes]: Yes.  This follows from the main result of the following paper of Zarhin.

MR0885780 (88h:14046)
Zarhin, Yu. G.
Endomorphisms and torsion of abelian varieties.
Duke Math. J. 54 (1987), no. 1, 131–145. 

His result, specialized to the $K$-simple case, is the following (fantastic) theorem.
Let 
$A$ be a $K$-simple abelian variety defined over a number field $K$.  The following are equivalent:
(i) $A(K^{\operatorname{ab}})[\operatorname{tors}]$ is infinite.
(ii) $A$ is of CM-type over $K$.  
Your hypotheses imply that there is infinite torsion over the abelian extension cut out by 
the action of Galois on the one-dimensional subspace (the Galois group is contained in $\mathbb{Z}_{p}^{\times}$), so Zarhin's theorem applies.