Remarks on Elliptic Motives

This paper grew out of conversations at the conference with A. Goncharov and A. Levin on their paper [7]. I want to offer an interpretation of their results along lines developed in [4]. The idea, which I learned from A. Beilinson and P. Deligne [5], is to assume one is given some category Μ of pure motives and then to try to construct a Hopf algebra or a co-Lie algebra H in the category Μ such that corepresentations of H in Μ give rise to (and conjecturally are equivalent to) mixed motives whose weight graded pieces lie in Μ. The focus becomes the study of H and its representations just as in number theory one studies the Galois group and its representations. In the case of motives, H is constructed using algebraic cycles, and cycle classes in some Weil cohomology lead to realizations of the mixed motives.