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 Andre-Oort conjecture is accordingly fundamental to the approach.

中文翻译：

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Heegner 点和复曲面周期的水平非消失

令 $F/\mathbb{Q}$ 是一个完全真实的域，而 $A$ 是 $F$ 上的模块化 $\GL_2$ 型阿贝尔变体。令 $K/F$ 为 CM 二次扩展。令 $\chi$ 是 $K$ 上的类群字符，使得 Rankin-Selberg 卷积 $L(s,A,\chi)$ 是根数为 $-1$ 的自对偶。我们表明类组字符 $\chi$ 的数量有界，使得 $L'(1, A, \chi) \neq 0$ 随 $K$ 判别式的绝对值而增加。我们还考虑了一种相当普遍的零阶情况。令 $\pi$ 是 $\GL_{2}(\BA_{F})$ 上的尖上同调自守表示。令 $\chi$ 是 $K$ 上的 Hecke 字符，使得 Rankin-Selberg 卷积 $L(s,\pi,\chi)$ 是根号为 $1$ 的自对偶。我们证明了具有固定 $\infty$ 类型和有界分支的 Hecke 字符 $\chi$ 的数量使得 $L(1/2, \pi, \chi) \neq 0$ 随$K$ 判别式的绝对值而增加。Gross-Zagier 公式和 Waldspurger 公式分别将问题与 Heegner 点的水平非消失和复曲面周期相关联。对于这两种情况，该策略是几何依赖于四元数 Shimura 品种的自产品上 CM 点的 Zariski 密度。安德烈-奥尔特猜想的最新结果 \cite{Ts, YZ, AGHP} 因此是该方法的基础。