  Let $F/\mathbb{Q}$ be a totally real field and $A$ a modular $\GL_2$-type
abelian variety over $F$. Let $K/F$ be a CM quadratic extension. Let $\chi$ be
a class group character over $K$ such that the Rankin-Selberg convolution
$L(s,A,\chi)$ is self-dual with root number $-1$. We show that the number of
class group characters $\chi$ with bounded ramification such that $L'(1, A,
\chi) \neq 0$ increases with the absolute value of the discriminant of $K$.
  We also consider a rather general rank zero situation. Let $\pi$ be a
cuspidal cohomological automorphic representation over $\GL_{2}(\BA_{F})$. Let
$\chi$ be a Hecke character over $K$ such that the Rankin-Selberg convolution
$L(s,\pi,\chi)$ is self-dual with root number $1$. We show that the number of
Hecke characters $\chi$ with fixed $\infty$-type and bounded ramification such
that $L(1/2, \pi, \chi) \neq 0$ increases with the absolute value of the
discriminant of $K$.
  The Gross-Zagier formula and the Waldspurger formula relate the question to
horizontal non-vanishing of Heegner points and toric periods, respectively. For
both situations, the strategy is geometric relying on the Zariski density of CM
points on self-products of a quaternionic Shimura variety. The recent result
\cite{Ts, YZ, AGHP} on the Andr\'e-Oort conjecture is accordingly fundamental
to the approach.
