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In this paper we describe the category of motives for an elliptic curve in the sense of Voevodsky as a derived category of dg modules over a commutative differential graded algebra in the category of representations over some reductive group.
Using Dold--Puppe category approach to the duality in topology, we prove general duality theorem for the category of motives. As one of the applications of this general result we obtain, in particular, a generalization of Friedlander--Voevodskys dual
After introducing the Ogus realization of 1-motives we prove that it is a fully faithful functor. More precisely, following a framework introduced by Ogus, considering an enriched structure on the de Rham realization of 1-motives over a number field,
Let k be a number field, and let S be a finite set of k-rational points of P^1. We relate the Deligne-Goncharov contruction of the motivic fundamental group of X:=P^1_k- S to the Tannaka group scheme of the category of mixed Tate motives over X.
We construct nontrivial L-equivalence between curves of genus one and degree five, and between elliptic surfaces of multisection index five. These results give the first examples of L-equivalence for curves (necessarily over non-algebraically closed
Let $E$ be an elliptic curve. When the symmetric group $Sigma_{g+1}$ of order $(g+1)!$ acts on $E^{g+1}$ in the natural way, the subgroup $E_0^{g+1}$, consisting of those $(g+1)$-tuples whose coordinates sum to zero, is stable under the action of $Si