A Candidate for the Abelian Category of Mixed Elliptic Motives

Abstract

In this work, 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-Soulé 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).

Explicit Motivic Mixed Elliptic Chabauty-Kim

“…As such a Q-linear category is only conjectural in the mixed elliptic case (see [Pat13]), we work with categories of Galois representations. While it is in some ways nicer to work over Q, it is not necessary, as the end result of the Chabauty-Kim method is still p-adic.…”

“…• Adapt the virtual cocycles of [Bro17a, §10] to a more general context • Develop a theory of elliptic motivic polylogarithms and use it as in [CDC20] • Use the "Special Elements" and coproduct formulas of [Pat13] One major obstacle in defining elliptic motivic polylogarithms is the lack of a canonical de Rham path between any two points in an elliptic curve (and more generally, any higher genus curve). This amounts to the lack of a global fiber functor as in the case of P 1 \ {0, 1, ∞}.…”

“…A coproduct formula and a conjectural basis of O(U E ) have already been written down in [Pat13] in terms of the bar construction on cycle complexes. Ongoing work of the author of this paper and Owen Patashnick seeks to use the coproduct formoula of loc.cit.…”

“…) is known as the category of mixed elliptic motives. A candidate for this category was constructed in [Pat13].…”

Motives for an elliptic curve

“…• Patashnick [23] constructs a different cycle algebra for an elliptic curve E without CM and defines one candidate for the abelian category of motives for E. Compared to his work, the advantage of our construction is its identification with a full subcategory of DM gm (k, Q).…”