$L$-functions of elliptic curves in ring class extensions of real quadratic fields via regularized theta liftings
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908587679285248 |
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| author | Van Order, Jeanine |
| author_facet | Van Order, Jeanine |
| contents | We derive new integral presentations for central derivative values of $L$-functions of elliptic curves defined over the rationals, basechanged to a real quadratic field $K$, twisted by ring class characters of $K$ in terms of sums along ``geodesics" corresponding to the class group of $K$ of automorphic Green's functions for certain Hirzebruch-Zagier-like arithmetic divisors on Hilbert modular surfaces. We also relate these sums to Birch-Swinnerton-Dyer constants and periods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_10277 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $L$-functions of elliptic curves in ring class extensions of real quadratic fields via regularized theta liftings Van Order, Jeanine Number Theory Primary 11F67, 11F27, 11F41, 11G40, Secondary 11F32, 11F46, 11G05, 11G18 We derive new integral presentations for central derivative values of $L$-functions of elliptic curves defined over the rationals, basechanged to a real quadratic field $K$, twisted by ring class characters of $K$ in terms of sums along ``geodesics" corresponding to the class group of $K$ of automorphic Green's functions for certain Hirzebruch-Zagier-like arithmetic divisors on Hilbert modular surfaces. We also relate these sums to Birch-Swinnerton-Dyer constants and periods. |
| title | $L$-functions of elliptic curves in ring class extensions of real quadratic fields via regularized theta liftings |
| topic | Number Theory Primary 11F67, 11F27, 11F41, 11G40, Secondary 11F32, 11F46, 11G05, 11G18 |
| url | https://arxiv.org/abs/2510.10277 |