Motivic interpretation of Milnor K-groups attached to Jacobian varieties

Abstract

In the paper [Som90], Somekawa conjectures that his Milnor K-group K(k,G1,…,Gr) attached to semi-abelian varieties G1,…,Gr over a field k is isomorphic to Ext$¥mathcal{M}$kr($¥mathbb{Z}$,G1[-1] ⊗ … ⊗ Gr[-1]) where $¥mathcal{M}$k is a certain category of motives over k. The purpose of this note is to prove this conjecture, when the varieties Gi are Jacobians of smooth curves over a perfect field and we take $¥mathcal{M}$k as Voevodsky's category of motives DM-eff(k).