Horizontal non-vanishing of Heegner points and toric periods
Let F be a totally real number field and A a modular GL₂-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 GL₂ (AF). 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 , ,  on the André–Oort conjecture is accordingly fundamental to the approach.
Additional Information© 2019 Elsevier Inc. Received 26 September 2019, Accepted 9 October 2019, Available online 27 December 2019. We are grateful to Haruzo Hida for encouragement and suggestions. The topic was initiated by the joint work of the first-named author with Haruzo Hida regarding horizontal non-vanishing (). We thank Li Cai for his assistance and Hae-Sang Sun for stimulating conversations regarding horizontal non-vanishing. We thank Nicolas Templier for helpful comments and suggestions. We also thank Farrell Brumley, Henri Darmon, Najmuddin Fakhruddin, Dimitar Jetchev, Mahesh Kakde, Chandrashekhar Khare, Philippe Michel, C.-S. Rajan, Jacques Tilouine, Vinayak Vatsal, Xinyi Yuan, Shou-Wu Zhang and Wei Zhang for instructive conversations about the topic. The first named author is grateful to MCM Beijing for the continual warm hospitality. The article was conceived in Beijing during his first MCM visit. Finally, we are indebted to the referee.
Submitted - 1712.01465.pdf