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

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 $t=0$).