Horizontal non-vanishing of Heegner points and toric periods

Let F/Q be a totally real field and A a modular 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 ₂(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 {Ts, YZ, AGHP} on the André-Oort conjecture is accordingly fundamental to the approach.