Derivatives of Rankin-Selberg $L$-functions and heights of generalized Heegner cycles
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909281648902144 |
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| author | Lilienfeldt, David T. -B. G. Shnidman, Ari |
| author_facet | Lilienfeldt, David T. -B. G. Shnidman, Ari |
| contents | Let $f$ be a newform of weight $2k$ and let $χ$ 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 $L(f,χ,s)$ 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 $t=0$). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_04375 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Derivatives of Rankin-Selberg $L$-functions and heights of generalized Heegner cycles Lilienfeldt, David T. -B. G. Shnidman, Ari Number Theory Algebraic Geometry 11G40, 11G15, 14C25 Let $f$ be a newform of weight $2k$ and let $χ$ 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 $L(f,χ,s)$ 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 $t=0$). |
| title | Derivatives of Rankin-Selberg $L$-functions and heights of generalized Heegner cycles |
| topic | Number Theory Algebraic Geometry 11G40, 11G15, 14C25 |
| url | https://arxiv.org/abs/2408.04375 |