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

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.