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arXiv · 2408.04375

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

David T. -B. G. Lilienfeldt·Ari Shnidman
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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 $t=0$).

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David T. -B. G. Lilienfeldt, Ari Shnidman. 2024-08-08. Derivatives of Rankin-Selberg $L$-functions and heights of generalized Heegner cycles. https://arxiv.org/abs/2408.04375

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