A proof of $p$-adic Gross--Zagier theorem via BDP formula
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908966541328384 |
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| author | Büyükboduk, Kâzım Neamti, Peter |
| author_facet | Büyükboduk, Kâzım Neamti, Peter |
| contents | This paper provides a new proof of the $p$-adic Gross--Zagier formula for the $p$-adic $L$-function associated with the base change of a normalised cuspidal eigen-newform $f$ of weight $k \geq 2$ (and families of such) to an imaginary quadratic field $K$. Our results encompass both the classical $p$-ordinary cases and non-ordinary scenarios, including new cases where $k > 2$ and $\mathrm{ord}_p(a_p(f)) > 0$. Unlike the traditional approach of comparing geometric and analytic kernels, we employ a ``wall-crossing'' strategy centred on the BDP formula and the theory of Beilinson--Flach elements. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_13854 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A proof of $p$-adic Gross--Zagier theorem via BDP formula Büyükboduk, Kâzım Neamti, Peter Number Theory Primary 11G40, Secondary 11G18, 11F67, 11R23, 11F33 This paper provides a new proof of the $p$-adic Gross--Zagier formula for the $p$-adic $L$-function associated with the base change of a normalised cuspidal eigen-newform $f$ of weight $k \geq 2$ (and families of such) to an imaginary quadratic field $K$. Our results encompass both the classical $p$-ordinary cases and non-ordinary scenarios, including new cases where $k > 2$ and $\mathrm{ord}_p(a_p(f)) > 0$. Unlike the traditional approach of comparing geometric and analytic kernels, we employ a ``wall-crossing'' strategy centred on the BDP formula and the theory of Beilinson--Flach elements. |
| title | A proof of $p$-adic Gross--Zagier theorem via BDP formula |
| topic | Number Theory Primary 11G40, Secondary 11G18, 11F67, 11R23, 11F33 |
| url | https://arxiv.org/abs/2604.13854 |