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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.15767 |
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| _version_ | 1866908928860749824 |
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| author | Deninger, Christopher |
| author_facet | Deninger, Christopher |
| contents | A Birch and Swinnerton-Dyer conjecture for number fields $K / \mathbb{Q}$ would assert that $dim V_K = ord_{s = 1/2} ζ_K (s)$ for some vector space functorially attached to $K$. Presently there is no natural candidate for the $V_K$'s. However, assuming $V_K$ is of a cohomological nature and assuming a conjecture of Serre on the vanishing order of $ζ_K (s)$ at $s = 1/2$ we show that such functors $K \mapsto V_K$ (with natural extra structures) exist and are all isomorphic. Their common automorphism group is $2$-torsion and abelian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_15767 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Is there a Birch and Swinnerton-Dyer conjecture for Dedekind zeta functions? Deninger, Christopher Number Theory A Birch and Swinnerton-Dyer conjecture for number fields $K / \mathbb{Q}$ would assert that $dim V_K = ord_{s = 1/2} ζ_K (s)$ for some vector space functorially attached to $K$. Presently there is no natural candidate for the $V_K$'s. However, assuming $V_K$ is of a cohomological nature and assuming a conjecture of Serre on the vanishing order of $ζ_K (s)$ at $s = 1/2$ we show that such functors $K \mapsto V_K$ (with natural extra structures) exist and are all isomorphic. Their common automorphism group is $2$-torsion and abelian. |
| title | Is there a Birch and Swinnerton-Dyer conjecture for Dedekind zeta functions? |
| topic | Number Theory |
| url | https://arxiv.org/abs/2504.15767 |