A Candidate for the abelian category of mixed elliptic motives

TL;DR

This paper proposes a new category of mixed motives generated by elliptic curve motives, computes its cohomology, and constructs nontrivial motives related to symmetric powers, advancing the understanding of mixed elliptic motives.

Contribution

It defines a candidate abelian category for mixed elliptic motives and computes its cohomology, linking it to motivic cohomology under a conjectural assumption.

Findings

Cohomology of the category matches expected motivic cohomology groups under conjectural assumptions.

Constructs families of nontrivial motives with specific weight graded pieces.

Provides a foundational framework for the abelian category of mixed elliptic motives.

Abstract

In this paper we suggest a definition for the category of mixed motives generated by the motive h^1(E) for E an elliptic curve without complex multiplication. We then compute the cohomology of this category. Modulo a strengthening of the Beilinson-Soule conjecture, we show that the cohomology of our category agrees with the expected motivic cohomology groups. Finally for each pure motive (Sym^{n}h^1(E))(-1) we construct families of nontrivial motives whose highest associated weight graded piece is $Sym^{n}h^1(E))(-1). This paper was essentially written in the late 1990's whilst the author was at the University of Chicago. The author apologizes for the tardiness of this posting, and hopes the reader will still find the content interesting.