Horizontal non-vanishing of Heegner points and toric periods

Let $F\mathrm{/}\mathbb{Q}$ be a totally real field and $A$ a modular ${\text{GL}}_2$-type abelian variety over $F$. Let $K\mathrm{/}F$ 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 $L’(1, A, \chi) \neq 0$increaseswiththeabsolutevalueofthediscriminantof$K$. We also consider a rather general rank zero situation. Let $\pi$ be a cuspidal cohomological automorphic representation over ${\text{GL}}_2({\text{BA}}_F)$. 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 $\mathrm{\infty }$-type and bounded ramification such that $L(1\mathrm{/}2,\pi ,\chi )\mathrm{\ne }0$ 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.