Arithmetic Hirzebruch-Zagier divisors and central derivative values of Rankin-Selberg $L$-functions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911204196220928 |
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| author | Van Order, Jeanine |
| author_facet | Van Order, Jeanine |
| contents | We give two distinct proofs of the Gross-Zagier formula in terms of sums of automorphic Green's functions realized as regularized theta lifts, including one involving arithmetic Hirzebruch-Zagier divisors on the Hilbert modular surface $X_0(N) \times X_0(N)$. We then describe applications to the refined conjecture of Birch and Swinnerton-Dyer. Through these calculations, we also describe known and conjectural relations of the central derivative values of Rankin-Selberg $L$-functions that appear to Fourier coefficients of certain half-integral weight forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_10303 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Arithmetic Hirzebruch-Zagier divisors and central derivative values of Rankin-Selberg $L$-functions Van Order, Jeanine Number Theory Primary 11F67, 11F27, 11F41, 11G40, Secondary 11F32, 11F46, 11G05, 11G18 We give two distinct proofs of the Gross-Zagier formula in terms of sums of automorphic Green's functions realized as regularized theta lifts, including one involving arithmetic Hirzebruch-Zagier divisors on the Hilbert modular surface $X_0(N) \times X_0(N)$. We then describe applications to the refined conjecture of Birch and Swinnerton-Dyer. Through these calculations, we also describe known and conjectural relations of the central derivative values of Rankin-Selberg $L$-functions that appear to Fourier coefficients of certain half-integral weight forms. |
| title | Arithmetic Hirzebruch-Zagier divisors and central derivative values of Rankin-Selberg $L$-functions |
| topic | Number Theory Primary 11F67, 11F27, 11F41, 11G40, Secondary 11F32, 11F46, 11G05, 11G18 |
| url | https://arxiv.org/abs/2510.10303 |