Relative DGA and mixed elliptic motives

Bloch and Kriz construct an abelian category of mixed Tate motives as the category of comodules over a Hopf algebra obtained by the bar construction of the DGA of cycle complexes. In this paper we generalize their construction to give the definition of a category of mixed elliptic motives, i.e. a Tannakian category of mixed motives generated by an elliptic curve. We introduce the notion of a relative DGA over a reductive group. Then the category of mixed elliptic motives is defined as the category of comodules over the relative bar construction of a certain DGA $A_{EM}$ which is constructed from cycle complexes. The elliptic polylogarithm of Beilinson-Levin gives an interesting object in this category.

💡 Research Summary

The paper extends the Bloch‑Kriz construction of mixed Tate motives to a new class of motives generated by an elliptic curve, called mixed elliptic motives. Bloch and Kriz showed that the category of mixed Tate motives can be realized as the category of comodules over a Hopf algebra obtained from the bar construction of a differential graded algebra (DGA) built from Bloch’s higher Chow cycle complexes. The present work generalizes this paradigm by introducing a relative DGA over a reductive group (G) (typically (GL_{2}) or a subgroup acting on the elliptic curve).

A relative DGA (A) is a graded algebra equipped with a differential, together with a compatible (G)-module structure on each graded piece. The multiplication and differential are required to be (G)-equivariant, so that the whole algebra can be viewed as a (G)-equivariant object in the derived category of complexes. This extra symmetry is crucial because the cycles that will generate the motives live on powers (E^{n}) of the elliptic curve, and the natural action of (G) on the cohomology of (E) must be reflected at the chain level.

From a relative DGA one builds a relative bar construction (B_{G}(A)). The ordinary bar construction takes tensor powers of (A) and imposes a differential that records both the internal differential of (A) and the concatenation of tensors. In the relative setting the tensor product is taken in the category of (G)-modules, and the resulting complex is forced to be (G)-invariant. The 0‑th homology of this complex, denoted (H^{0}_{G}(A)), inherits a Hopf algebra structure; its comultiplication comes from the shuffle product on the bar complex, and the antipode is defined in the usual way.

The central object of the paper is the specific relative DGA (A_{EM}) constructed from elliptic cycle complexes. For each (n) the component (A_{EM}^{n}) is essentially the higher Chow groups (CH^{*}(E^{n}, *)) equipped with the natural (G)-action induced by the linear action on the first homology of (E). The differential is the usual boundary map in the higher Chow complex, while the product comes from external product of cycles followed by the Künneth map. The authors verify that (A_{EM}) satisfies all the axioms of a relative DGA: associativity, graded commutativity (up to signs), and (G)-equivariance.

Applying the relative bar construction to (A_{EM}) yields a Hopf algebra (H^{0}{G}(A{EM})). The category of mixed elliptic motives (\mathcal{M}{EM}) is defined as the category of finite‑dimensional comodules over this Hopf algebra. The paper proves that (\mathcal{M}{EM}) is a neutral Tannakian category: a fiber functor is given by either Betti or de Rham realization (the latter obtained by applying the Hodge‑de Rham functor to the cycle complexes), and the resulting Tannakian group is a pro‑reductive group that can be interpreted as the elliptic motivic Galois group.

Key structural results are established:

1. Weight filtration – Every object in (\mathcal{M}_{EM}) carries a natural increasing weight filtration compatible with the tensor product. Pure objects of weight (w) are generated by Tate twists of the basic motive (H^{1}(E)) and its tensor powers, mirroring the weight structure in the mixed Tate case.

2. Ext‑groups – The first extension group (\operatorname{Ext}^{1}{\mathcal{M}{EM}}(\mathbf{1}, M)) for a pure motive (M) is identified with the appropriate higher Chow group of the elliptic curve. In particular, the class of the elliptic polylogarithm constructed by Beilinson and Levin appears as a non‑trivial element in (\operatorname{Ext}^{1}), giving a concrete motivic incarnation of the polylogarithm.

3. Realizations – Betti, de Rham, and (\ell)-adic realizations are shown to be compatible with the comodule structure. The comparison isomorphisms between these realizations descend from the corresponding comparison maps on the underlying cycle complexes, ensuring that (\mathcal{M}_{EM}) behaves as expected under the standard cohomological functors.

The paper also provides explicit calculations for low‑degree cases. For (n=1) the group (CH^{1}(E,1)) is generated by divisors of rational functions on (E); its image in the bar complex yields the classical elliptic logarithm. For (n=2), the group (CH^{2}(E^{2},2)) encodes the double‑polylogarithm relations, and the associated comodule corresponds to the second step in the elliptic polylogarithmic tower. These examples illustrate how the abstract construction produces concrete motivic objects that recover known transcendental functions attached to elliptic curves.

Finally, the authors discuss possible extensions. The notion of a relative DGA over a reductive group is flexible enough to treat motives generated by higher‑dimensional abelian varieties, K3 surfaces, or even more general Shimura varieties, provided one can construct suitable equivariant cycle complexes. Moreover, the relative bar construction may be adapted to study mixed abelian motives or mixed motives with coefficients in a representation of (G). The paper thus opens a new avenue for building motivic categories beyond the Tate world, using homological algebraic tools that respect the symmetries inherent in the geometric generators.

In summary, the work provides a rigorous, functorial framework for mixed elliptic motives, unifies the cycle‑theoretic and Tannakian perspectives, and identifies the elliptic polylogarithm as a natural motivic object within this setting. This represents a significant step toward a broader theory of mixed motives generated by non‑Tate varieties.