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 $dim{V}_K=or{d}_{s=1/2}{\zeta }_K(s)$ 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.

Final version. To appear in Journal of Number Theory