Let F/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 χ be
a class group character over K such that the Rankin-Selberg convolution
L(s,A,χ) is self-dual with root number −1. We show that the number of
class group characters χ with bounded ramification such that L′(1,A,χ)=0 increases with the absolute value of the discriminant of K.
We also consider a rather general rank zero situation. Let π be a
cuspidal cohomological automorphic representation over \GL_{2}(\BA_{F}). Let
χ be a Hecke character over K such that the Rankin-Selberg convolution
L(s,π,χ) is self-dual with root number 1. We show that the number of
Hecke characters χ with fixed ∞-type and bounded ramification such
that L(1/2,π,χ)=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.Comment: Adv. Math., to appear. arXiv admin note: text overlap with
arXiv:1712.0214