Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T15:09:14.571Z Has data issue: false hasContentIssue false
  • English
  • Français

A Candidate for the Abelian Category of Mixed Elliptic Motives

Published online by Cambridge University Press:  15 November 2013

Owen Patashnick
Get access

Abstract

In this work, we s uggest a defnition for the category of mixed motives generated by the motive h1 (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 (Symnh1 (E)) (–1) we construct families of nontrivial motives whose highest associated weight graded piece is (Symnh1 (E)) (–1).

Keywords

Keywords arithmetic geometrycohomologyalgebraic cyclesK-theorymotiveLie algebras and their representationselliptic curveselliptic polylogarithmL-function14C2514C3014G9914Q0514F4214C2519D9913D1517B4518E1518F3014H5211G5511G4014G10
Type
Research Article
Information
Journal of K-Theory , Volume 12 , Issue 3 , December 2013 , pp. 569 - 600
Copyright
Copyright © ISOPP 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Beilinson, A., Deligne, P.: Interpretation motivique de la conjecture de Zagier, Proc. Symp. in Pure Math. 55(2) (1994), 97–122.Google Scholar
2.Beilinson, A., Levin, A.: The Elliptic Polylogarithm, Proc. Symp. in Pure Math. 55(2) (1994), 123–192.CrossRefGoogle Scholar
3.Bloch, S.: Remarks on Elliptic Motives, Regulators in Analysis, Geometry and Number Theory, Progr. Math. 171 (2000), 17–27.CrossRefGoogle Scholar
4.Bloch, S.: Algebraic Cycles and the Lie Algebra of Mixed Tate Motives, Journal of the AMS 4(4) (1991), 771–791.Google Scholar
5.Bloch, S.: Algebraic Cycles and Higher K-theory, Adv. Math. 61 (1986), 267–304.CrossRefGoogle Scholar
6.Bloch, S., Kriz, I.: Mixed Tate Motives, Annals Math. 140 (1994), 557–605.CrossRefGoogle Scholar
7.Chen, K-T.: Reduced Bar Construction on de Rham Complexes, Algebra, Topology, and Category Theory, ed. by Heller, A., and Tierney, M., Academic press, 1976.Google Scholar
8.Deninger, C.: Higher regulators and Hecke L-series of imaginary quadratic fields I, Invent. Math. 96 (1989) 1–69.Google Scholar
9.Edidin, D., Graham, W.: Equivariant Intersection Theory, arXiv:alg-geom/9603008v3.Google Scholar
10.Fulton, W.: Young Tableaux, Cambridge University Press, 1997.Google Scholar
11.Fulton, W., and Harris, J.: Representation Theory, a first course, Springer-Verlag, New York, 1991.Google Scholar
12.Goncharov, A.: Mixed Elliptic Motives, in Galois Representations in Arithmetic Algebraic Geometry (Durham, 1996), London Math. Soc. Lecture Note Ser. 254, Cambridge Univ. Press, Cambridge, 1998, 147–221.Google Scholar
13.Goncharov, A.: Volumes of Hyperbolic Manifolds and Mixed Tate Motives, J. Amer. Math Soc. 12 (1999), 569–618.Google Scholar
14.Goncharov, A., Levin, A.: Zagier's conjecture on L(E,2), Invent. Math. 132 (1998), 393–432.Google Scholar
15.Hartshorne, R.: Algebraic Geometry, Springer-Verlag, USA, 2006.Google Scholar
16.Kriz, I, May, J.P.: Operads, Algebras, Modules, and Motives, Astérisque 233, 1995.Google Scholar
17.Levin, A.: Elliptic Polylogarithms in K-theory, J. Math. Sci. 81(3) (1996), 2657–2666.Google Scholar
18.Levine, M.: Tate Motives and the Vanishing Conjectures for Algebraic K-theory, Algebraic K-Theory and Algebraic Topology, Lake Louise, AB, 1991, NATO Adv. Sci. Inst. Ser C Math Phys. Sci. 407, Kluwer Acad. Publ., Dordrecht, (1993), 167–188.Google Scholar
19.Levine, M.: Mixed Motives, Mathematical Surveys and Monographs 57, AMS, Providence RI, 1998.Google Scholar
20.Milnor, J., and Moore, J.: On the structure of Hopf Algebras, Annals Math. 81 (1965), 211–264.Google Scholar
21.Patashnick, Owen.: The Hodge Realization for (E), in preparation.Google Scholar
22.Patashnick, Owen.: Elliptic motivic Massey products, in preparation.Google Scholar
23.Quillen, D.: On the (co-)homology of commutative rings, Proc. Symp. in Pure Math. 17, AMS, 1968, 65–87.Google Scholar
24.Rolshausen, K.: PhD thesis, Université Louis Pasteur, Strasbourg, 1995.Google Scholar
25.Saavedra, R.: Catégories Tannakiennes, LNM 265, Springer-Verlag, Heidelberg, 1972.Google Scholar
26.Scholl, A.J., Motives for Modular Forms, Invent. Math. 100 (1990), 419–130.Google Scholar
27.Serre, J. P.: Motifs, Astérisque 198-199-200 (1991), 333–349.Google Scholar
28.Sullivan, D.: Infinitesimal Computations in Topology, Publ. Math IHES 47 (1978), 269–331.Google Scholar
29.Totaro, B.: Milnor K-theory is the simplest part of algebraic K-theory, K-Theory 6 (1992), 177–189.Google Scholar
30.Voevodsky, V., Suslin, A., Friedlander, E.: Cycles, Transfers, and Motivic Homology Theories, Annals Math. Studies 143, Princeton University Press, 2000.Google Scholar
31.Weibel, C.: An Introduction to Homological Algebra, Cambridge University Press, 1994.CrossRefGoogle Scholar
32.Wildeshaus, J.: On an Elliptic Analogue of Zagier's conjecture, Duke Math J. 87(2) (1997), 355–107.Google Scholar
33.Wildeshaus, J.: On the Generalized Eisenstein Symbol, in Motives, polylogarithms and Hodge theory, Part I, Int. Press, Somerville, MA, 2002, 291–114.Google Scholar
34.Wildeshaus, J.: Realizations of Polylogarithms, LNM 1650, Springer-Verlag, Heidelberg, 1997.CrossRefGoogle Scholar