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Motives for an elliptic curve

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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.

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Notes

  1. For the CM case, we only consider the complex multiplication is defined over k.

  2. For the definition of Schur functor and notations of partitions, we refer to Section 1.3 and 1.4 of [9].

  3. Here \(\lambda ^t\) is the transpose (or conjugate) of \(\lambda \), which is defined by interchanging rows and columns in the Young diagram associated to \(\lambda \).

  4. We view \(c_n\) as an idempotent in \(End((\mathbf {F}_\mathbb {K})^{\otimes n})\), which lies in \(End_{\mathbf {Res}_{\mathbb {K}/\mathbb {Q}}\mathbb {G}_m \otimes \mathbb {K}}((\mathbf {F}_{\mathbb {K}})^{\otimes n})\).

  5. This could be thought as the “motivic version” of the cycle algebra \(\mathcal {E}_{a, b}^i\).

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Acknowledgements

This paper is part of the authors Ph.D. dissertation written at Universität Duisburg-Essen. I am grateful to my advisor Professor Marc Levine for his constant guidance, encouragement and patience during this work. I would like to thank Giuseppe Ancona, Spencer Bloch, Owen Patashnick, Markus Spitzweck, Rin Sugiyama and Tomohide Terasoma for many helpful discussions. Finally, I’d like to thank the referee for useful comments.

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  1. Yau Mathematical Sciences Center, Tsinghua University, Beijing, China

    Jin Cao

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  1. Jin Cao
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Correspondence to Jin Cao.

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Communicated by Vasudevan Srinivas.

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Cao, J. Motives for an elliptic curve. Math. Ann. 372, 189–227 (2018). https://doi.org/10.1007/s00208-018-1690-z

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