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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2310.20678 |
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| _version_ | 1866912710888783872 |
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| author | Kriz, Daniel Nordentoft, Asbjørn Christian |
| author_facet | Kriz, Daniel Nordentoft, Asbjørn Christian |
| contents | We define new objects called 'horizontal $p$-adic $L$-functions' associated to $L$-values of twists of elliptic curves over $\mathbb{Q}$ by characters of $p$-power order and conductor prime to $p$. We study the fundamental properties of these objects and obtain applications to non-vanishing of finite order twists of central $L$-values, making progress toward conjectures of Goldfeld and David--Fearnley--Kisilevsky. For general elliptic curves $E$ over $\mathbb{Q}$ we obtain strong quantitative lower bounds on the number of non-vanishing central $L$-values of twists by Dirichlet characters of fixed order $d\equiv 2 \mod 4$ greater than two. We also obtain non-vanishing results for general $d$, including $d = 2$, under mild assumptions. In particular, for elliptic curves with $E[2](\mathbb{Q}) = 0$ we improve on the previously best known lower bounds on the number of non-vanishing $L$-values of quadratic twists due to Ono. Finally, we obtain results on simultaneous non-vanishing of twists of an arbitrary number of elliptic curves with applications to Diophantine stability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_20678 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Horizontal $p$-adic $L$-functions Kriz, Daniel Nordentoft, Asbjørn Christian Number Theory We define new objects called 'horizontal $p$-adic $L$-functions' associated to $L$-values of twists of elliptic curves over $\mathbb{Q}$ by characters of $p$-power order and conductor prime to $p$. We study the fundamental properties of these objects and obtain applications to non-vanishing of finite order twists of central $L$-values, making progress toward conjectures of Goldfeld and David--Fearnley--Kisilevsky. For general elliptic curves $E$ over $\mathbb{Q}$ we obtain strong quantitative lower bounds on the number of non-vanishing central $L$-values of twists by Dirichlet characters of fixed order $d\equiv 2 \mod 4$ greater than two. We also obtain non-vanishing results for general $d$, including $d = 2$, under mild assumptions. In particular, for elliptic curves with $E[2](\mathbb{Q}) = 0$ we improve on the previously best known lower bounds on the number of non-vanishing $L$-values of quadratic twists due to Ono. Finally, we obtain results on simultaneous non-vanishing of twists of an arbitrary number of elliptic curves with applications to Diophantine stability. |
| title | Horizontal $p$-adic $L$-functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2310.20678 |