L-values of elliptic curves twisted by cubic characters
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913199498985472 |
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| author | Angdinata, David Kurniadi |
| author_facet | Angdinata, David Kurniadi |
| contents | Given a rational elliptic curve $ E $ of analytic rank zero, its L-function can be twisted by an even primitive Dirichlet character $ χ$ of order $ q $, and in many cases its associated central algebraic L-value $ \mathcal{L}(E, χ) $ is known to be integral. This paper derives some arithmetic consequences from a congruence between $ \mathcal{L}(E, 1) $ and $ \mathcal{L}(E, χ) $ arising from this integrality, with an emphasis on cubic characters $ χ$. These include $ q $-adic valuations of the denominator of $ \mathcal{L}(E, 1) $, determination of $ \mathcal{L}(E, χ) $ in terms of Birch--Swinnerton-Dyer invariants, and asymptotic densities of $ \mathcal{L}(E, χ) $ modulo $ q $ by varying $ χ$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_09927 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | L-values of elliptic curves twisted by cubic characters Angdinata, David Kurniadi Number Theory 11G05, 11G40 Given a rational elliptic curve $ E $ of analytic rank zero, its L-function can be twisted by an even primitive Dirichlet character $ χ$ of order $ q $, and in many cases its associated central algebraic L-value $ \mathcal{L}(E, χ) $ is known to be integral. This paper derives some arithmetic consequences from a congruence between $ \mathcal{L}(E, 1) $ and $ \mathcal{L}(E, χ) $ arising from this integrality, with an emphasis on cubic characters $ χ$. These include $ q $-adic valuations of the denominator of $ \mathcal{L}(E, 1) $, determination of $ \mathcal{L}(E, χ) $ in terms of Birch--Swinnerton-Dyer invariants, and asymptotic densities of $ \mathcal{L}(E, χ) $ modulo $ q $ by varying $ χ$. |
| title | L-values of elliptic curves twisted by cubic characters |
| topic | Number Theory 11G05, 11G40 |
| url | https://arxiv.org/abs/2401.09927 |