Derivatives of Rankin-Selberg $L$-functions and heights of generalized Heegner cycles

TL;DR

This paper establishes a deep connection between the derivatives of Rankin-Selberg L-functions and the heights of generalized Heegner cycles, extending classical formulas to higher weights and more general settings.

Contribution

It generalizes the Gross-Zagier and Zhang formulas by relating derivatives of Rankin-Selberg L-functions to heights of generalized Heegner cycles for higher weights and characters.

Findings

Central derivatives of L-functions equal heights of cycles up to explicit constants

Extension of Gross-Zagier formula to higher weights

Extension of Zhang's higher weight formula

Abstract

Let $f$ be a newform of weight $2k$ and let $\chi$ be an unramified imaginary quadratic Hecke character of infinity type $(2t, 0)$, for some integer $0 < t \leq k-1$. We show that the central derivative of the Rankin-Selberg $L$-function $L(f,\chi,s)$ is, up to an explicit positive constant, equal to the Beilinson-Bloch height of a generalized Heegner cycle. This generalizes the Gross-Zagier formula (the case $k = 1$) and Zhang's higher weight formula (the case $t=0$).