Heights of CM cycles and derivatives of L-series

TL;DR

This paper generalizes the Gross-Zagier formula to higher weight modular forms using CM-cycle intersections on Kuga-Sato varieties, and applies it to bound Selmer and Tate-Shafarevich groups assuming non-vanishing L-function derivatives.

Contribution

It extends the Gross-Zagier formula to even weight modular forms and links it to bounds on Selmer and Tate-Shafarevich groups via Euler system methods.

Findings

Derived a Gross-Zagier type formula for higher weight modular forms.

Bounded Selmer and Tate-Shafarevich groups under non-vanishing L-function derivatives.

Connected arithmetic intersection theory with Iwasawa theory techniques.

Abstract

We extend the work of S. Zhang and Yuan-Zhang-Zhang to obtain a Gross-Zagier formula for modular forms of even weight in terms of an arithmetic intersection pairing of CM-cycles on Kuga-Sato varieties over Shimura curves. Combined with a result of the first author and de Vera-Piquero adapting Kolyvagin's method of Euler systems to this setting, we bound the associated Selmer and Tate-Shafarevich groups, assuming the non-vanishing of the derivative of the L-function at the central point.