Horizontal non-vanishing of Heegner points and toric periods

TL;DR

This paper proves that the number of characters with non-vanishing L-values related to Heegner points and toric periods increases with the discriminant of the quadratic extension, using geometric methods and recent advances in the André-Oort conjecture.

Contribution

It establishes new non-vanishing results for L-functions associated with Heegner points and toric periods, leveraging geometric techniques and recent progress on the André-Oort conjecture.

Findings

Number of class group characters with non-zero derivatives increases with discriminant.

Number of Hecke characters with non-zero central L-values increases with discriminant.

Uses Zariski density of CM points and recent André-Oort results.

Abstract

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.