Gan--Gross--Prasad cycles and derivatives of $p$-adic $L$-functions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914366538907648 |
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| author | Disegni, Daniel Zhang, Wei |
| author_facet | Disegni, Daniel Zhang, Wei |
| contents | We study the p-adic analogue of the arithmetic Gan-Gross-Prasad (GGP) conjectures for unitary groups. Let $Π$ be a conjugate-selfdual cuspidal automorphic representation of GL_{n} x GL_{n+1} over a CM field, which is algebraic of minimal regular weight at infinity. We first show the rationality of twists of the ratio of L-values of $Π$ appearing in the GGP conjectures. Then, when $Π$ is p-ordinary at a prime p, we construct a cyclotomic p-adic L-function $L_p(M_Π)$ interpolating those twists. Finally, under some local assumptions, we prove a precise formula relating the first derivative of $L_p(M_Π)$ to the p-adic heights of Selmer classes arising from arithmetic diagonal cycles on unitary Shimura varieties. We deduce applications to the p-adic Beilinson-Bloch-Kato conjecture for the motive attached to $Π$. All proofs are based on some relative-trace formulas in p-adic coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_08401 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gan--Gross--Prasad cycles and derivatives of $p$-adic $L$-functions Disegni, Daniel Zhang, Wei Number Theory 11G40, 11F67, 11G18, 11F33, 11F72 We study the p-adic analogue of the arithmetic Gan-Gross-Prasad (GGP) conjectures for unitary groups. Let $Π$ be a conjugate-selfdual cuspidal automorphic representation of GL_{n} x GL_{n+1} over a CM field, which is algebraic of minimal regular weight at infinity. We first show the rationality of twists of the ratio of L-values of $Π$ appearing in the GGP conjectures. Then, when $Π$ is p-ordinary at a prime p, we construct a cyclotomic p-adic L-function $L_p(M_Π)$ interpolating those twists. Finally, under some local assumptions, we prove a precise formula relating the first derivative of $L_p(M_Π)$ to the p-adic heights of Selmer classes arising from arithmetic diagonal cycles on unitary Shimura varieties. We deduce applications to the p-adic Beilinson-Bloch-Kato conjecture for the motive attached to $Π$. All proofs are based on some relative-trace formulas in p-adic coefficients. |
| title | Gan--Gross--Prasad cycles and derivatives of $p$-adic $L$-functions |
| topic | Number Theory 11G40, 11F67, 11G18, 11F33, 11F72 |
| url | https://arxiv.org/abs/2410.08401 |