Motives for an elliptic curve

Abstract

In this paper we describe the rigid tensor triangulated subcategory of Voevodsky's triangulated category of motives generated by the motive of an elliptic curve as a derived category of dg modules over a commutative differential graded algebra in the category of representations over some reductive group.1 For the CM case, we only consider the complex multiplication is defined over k.Furthermore, the piecewhere V (n,0) ⊗ V ⊗i ⊗V ⊗j are pairwise non-isomorphic irreducible representations over a K-algebra End Res K/Q Gm⊗K ((F K ) ⊗n ) and V i,j are pairwise non-isomorphic irreducible representation over Res K/Q G m ⊗ K = T K . For simplicity, we delete V i,j and one may think that both V andV are endowed with the G m -action. In fact, End Res K/Q Gm⊗K ((F K ) ⊗n ) is a special case defined in the Section 3.9 of [1], which is called B n,K . Ancona's main result -Theorem 4.1 in [1] implies that the decomposition like Lemma 2.1 is holding for the CM elliptic motives:3 Here λ t is the transpose (or conjugate) of λ, which is defined by interchanging rows and columns in the Young diagram associated to λ. 4 We view cn as an idempotent in End((F K ) ⊗n ), which lies in End Res K/Q Gm⊗K ((F K ) ⊗n ).