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arXiv · 2504.15767

Is there a Birch and Swinnerton-Dyer conjecture for Dedekind zeta functions?

Christopher Deninger
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Abstract

A Birch and Swinnerton-Dyer conjecture for number fields $K / \mathbb{Q}$ would assert that $dim V_K = ord_{s = 1/2} \zeta_K (s)$ 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 $\zeta_K (s)$ 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.

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Christopher Deninger. 2025-04-22. Is there a Birch and Swinnerton-Dyer conjecture for Dedekind zeta functions?. https://arxiv.org/abs/2504.15767

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