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,…,G_r) attached to semi-abelian varieties G₁,…,G_r over a field k is isomorphic to Ext_{$¥mathcal{M}$k}^r($¥mathbb{Z}$,G₁[-1] ⊗ … ⊗ G_r[-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 G_i are Jacobians of smooth curves over a perfect field and we take $¥mathcal{M}$_k as Voevodsky's category of motives DM_-^eff(k).