Polylogarithm for families of commutative group schemes
Authors:
Annette Huber and Guido Kings
Journal:
J. Algebraic Geom. 27 (2018), 449-495
DOI:
https://doi.org/10.1090/jag/717
Published electronically:
April 11, 2018
MathSciNet review:
3803605
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We generalize the definition of the polylogarithm classes to the case of commutative group schemes, both in the sheaf theoretic and the motivic setting. This generalizes and simplifies the existing cases.
References
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- Giuseppe Ancona, Stephen Enright-Ward, and Annette Huber, On the motive of a commutative algebraic group, Doc. Math. 20 (2015), 807–858. MR 3398728
- Giuseppe Ancona, Annette Huber, and Simon Pepin Lehalleur, On the relative motive of a commutative group scheme, Algebr. Geom. 3 (2016), no. 2, 150–178. MR 3477952, DOI https://doi.org/10.14231/AG-2016-008
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- D. Burns and C. Greither, On the equivariant Tamagawa number conjecture for Tate motives, Invent. Math. 153 (2003), no. 2, 303–359. MR 1992015, DOI https://doi.org/10.1007/s00222-003-0291-x
- D.-C. Cisinski and F. Déglise, Triangulated categories of mixed motives, preprint, 2009, http://arxiv.org/abs/0912.2110.
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- Alexandru Dimca, Sheaves in topology, Universitext, Springer-Verlag, Berlin, 2004. MR 2050072
- Brad Drew, Réalisations tannakiennes des motifs mixtes triangulés, PhD thesis, Université Paris 13, June 2013.
- Brad Drew, Motivic Hodge modules, preprint, 2018, arXiv:1801.10129.
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- Matthew Thomas Gealy, On the Tamagawa number conjecture for motives attached to modular forms, ProQuest LLC, Ann Arbor, MI, 2006. Thesis (Ph.D.)–California Institute of Technology. MR 2709207
- Annette Huber and Bruno Kahn, The slice filtration and mixed Tate motives, Compos. Math. 142 (2006), no. 4, 907–936. MR 2249535, DOI https://doi.org/10.1112/S0010437X06002107
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- Annette Huber and Jörg Wildeshaus, Classical motivic polylogarithm according to Beilinson and Deligne, Doc. Math. 3 (1998), 27–133. MR 1643974
- Annette Huber, The comparison theorem for the Soulé-Deligne classes, The Bloch-Kato conjecture for the Riemann zeta function, London Math. Soc. Lecture Note Ser., vol. 418, Cambridge Univ. Press, Cambridge, 2015, pp. 210–238. MR 3497681
- Luc Illusie, Complexe cotangent et déformations. I, Lecture Notes in Mathematics, Vol. 239, Springer-Verlag, Berlin-New York, 1971 (French). MR 0491680
- Florian Ivorra, Perverse, Hodge and motivic realizations of étale motives, Compos. Math. 152 (2016), no. 6, 1237–1285. MR 3518311, DOI https://doi.org/10.1112/S0010437X15007812
- Kazuya Kato, $p$-adic Hodge theory and values of zeta functions of modular forms, Astérisque 295 (2004), ix, 117–290 (English, with English and French summaries). Cohomologies $p$-adiques et applications arithmétiques. III. MR 2104361
- Guido Kings, $K$-theory elements for the polylogarithm of abelian schemes, J. Reine Angew. Math. 517 (1999), 103–116. MR 1728545, DOI https://doi.org/10.1515/crll.1999.088
- Guido Kings, The Tamagawa number conjecture for CM elliptic curves, Invent. Math. 143 (2001), no. 3, 571–627. MR 1817645, DOI https://doi.org/10.1007/s002220000115
- Guido Kings, Eisenstein classes, elliptic Soulé elements and the $\ell $-adic elliptic polylogarithm, The Bloch-Kato conjecture for the Riemann zeta function, London Math. Soc. Lecture Note Ser., vol. 418, Cambridge Univ. Press, Cambridge, 2015, pp. 239–296. MR 3497682
- Guido Kings, On $p$-adic interpolation of motivic Eisenstein classes, Elliptic curves, modular forms and Iwasawa theory, Springer Proc. Math. Stat., vol. 188, Springer, Cham, 2016, pp. 335–371. MR 3629656, DOI https://doi.org/10.1007/978-3-319-45032-2_10
- Marc Levine, 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., vol. 407, Kluwer Acad. Publ., Dordrecht, 1993, pp. 167–188. MR 1367296, DOI https://doi.org/10.1007/978-94-017-0695-7_7
- Masayoshi Nagata, A generalization of the imbedding problem of an abstract variety in a complete variety, J. Math. Kyoto Univ. 3 (1963), 89–102. MR 158892, DOI https://doi.org/10.1215/kjm/1250524859
- S. Pepin Lehalleur, The motivic t-structure for relative 1-motives, preprint, 2015, arXiv:1512.00266.
- Morihiko Saito, Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24 (1988), no. 6, 849–995 (1989) (French). MR 1000123, DOI https://doi.org/10.2977/prims/1195173930
- Morihiko Saito, Mixed Hodge modules, Publ. Res. Inst. Math. Sci. 26 (1990), no. 2, 221–333. MR 1047415, DOI https://doi.org/10.2977/prims/1195171082
- Cohomologie $l$-adique et fonctions $L$, Lecture Notes in Mathematics, Vol. 589, Springer-Verlag, Berlin-New York, 1977 (French). Séminaire de Géometrie Algébrique du Bois-Marie 1965–1966 (SGA 5); Edité par Luc Illusie. MR 0491704
- Vladimir Voevodsky, Triangulated categories of motives over a field, Cycles, transfers, and motivic homology theories, Ann. of Math. Stud., vol. 143, Princeton Univ. Press, Princeton, NJ, 2000, pp. 188–238. MR 1764202
- Jörg Wildeshaus, Realizations of polylogarithms, Lecture Notes in Mathematics, vol. 1650, Springer-Verlag, Berlin, 1997. MR 1482233
- Jörg Wildeshaus, On the interior motive of certain Shimura varieties: the case of Picard surfaces, Manuscripta Math. 148 (2015), no. 3-4, 351–377. MR 3414481, DOI https://doi.org/10.1007/s00229-015-0747-5
References
- Joseph Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. I, Astérisque 314 (2007), x+466 pp. (2008) (French, with English and French summaries). MR 2423375
- Joseph Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. II, Astérisque 315 (2007), vi+364 pp. (2008) (French, with English and French summaries). MR 2438151
- Joseph Ayoub, Note sur les opérations de Grothendieck et la réalisation de Betti, J. Inst. Math. Jussieu 9 (2010), no. 2, 225–263 (French, with English and French summaries). MR 2602027
- Joseph Ayoub, La réalisation étale et les opérations de Grothendieck, Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), no. 1, 1–145 (French, with English and French summaries). MR 3205601
- Paul Balmer and Marco Schlichting, Idempotent completion of triangulated categories, J. Algebra 236 (2001), no. 2, 819–834. MR 1813503
- Giuseppe Ancona, Stephen Enright-Ward, and Annette Huber, On the motive of a commutative algebraic group, Doc. Math. 20 (2015), 807–858. MR 3398728
- Giuseppe Ancona, Annette Huber, and Simon Pepin Lehalleur, On the relative motive of a commutative group scheme, Algebr. Geom. 3 (2016), no. 2, 150–178. MR 3477952
- A. Beĭlinson and A. Levin, The elliptic polylogarithm, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 123–190. MR 1265553
- A. Beilinson, G. Kings, and A. Levin, Topological polylogarithms and $p$-adic interpolation of $L$-values of totally real fields, Math. Ann. (2018), https://doi.org/10.1007/s00208-018-1645-4.
- D. Burns and C. Greither, On the equivariant Tamagawa number conjecture for Tate motives, Invent. Math. 153 (2003), no. 2, 303–359. MR 1992015
- D.-C. Cisinski and F. Déglise, Triangulated categories of mixed motives, preprint, 2009, http://arxiv.org/abs/0912.2110.
- P. Deligne, Le groupe fondamental de la droite projective moins trois points, Galois groups over $\textbf {Q}$ (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 16, Springer, New York, 1989, pp. 79–297 (French). MR 1012168
- Alexandru Dimca, Sheaves in topology, Universitext, Springer-Verlag, Berlin, 2004. MR 2050072
- Brad Drew, Réalisations tannakiennes des motifs mixtes triangulés, PhD thesis, Université Paris 13, June 2013.
- Brad Drew, Motivic Hodge modules, preprint, 2018, arXiv:1801.10129.
- Torsten Ekedahl, On the adic formalism, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 197–218. MR 1106899
- Matthew Thomas Gealy, On the Tamagawa number conjecture for motives attached to modular forms, ProQuest LLC, Ann Arbor, MI, 2006. Thesis (Ph.D.)–California Institute of Technology. MR 2709207
- Annette Huber and Bruno Kahn, The slice filtration and mixed Tate motives, Compos. Math. 142 (2006), no. 4, 907–936. MR 2249535
- Annette Huber and Guido Kings, Degeneration of $l$-adic Eisenstein classes and of the elliptic polylog, Invent. Math. 135 (1999), no. 3, 545–594. MR 1669288
- Annette Huber and Guido Kings, Bloch-Kato conjecture and Main Conjecture of Iwasawa theory for Dirichlet characters, Duke Math. J. 119 (2003), no. 3, 393–464. MR 2002643
- Annette Huber and Jörg Wildeshaus, Classical motivic polylogarithm according to Beilinson and Deligne, Doc. Math. 3 (1998), 27–133. Correction Doc. Math. 3 (1998), 297–299. MR 1643974, MR 1662477
- Annette Huber, The comparison theorem for the Soulé-Deligne classes, The Bloch-Kato conjecture for the Riemann zeta function, London Math. Soc. Lecture Note Ser., vol. 418, Cambridge Univ. Press, Cambridge, 2015, pp. 210–238. MR 3497681
- Luc Illusie, Complexe cotangent et déformations. I, Lecture Notes in Mathematics, Vol. 239, Springer-Verlag, Berlin-New York, 1971 (French). MR 0491680
- Florian Ivorra, Perverse, Hodge and motivic realizations of étale motives, Compos. Math. 152 (2016), no. 6, 1237–1285. MR 3518311
- Kazuya Kato, $p$-adic Hodge theory and values of zeta functions of modular forms, Astérisque 295 (2004), ix, 117–290 (English, with English and French summaries). Cohomologies $p$-adiques et applications arithmétiques. III. MR 2104361
- Guido Kings, $K$-theory elements for the polylogarithm of abelian schemes, J. Reine Angew. Math. 517 (1999), 103–116. MR 1728545
- Guido Kings, The Tamagawa number conjecture for CM elliptic curves, Invent. Math. 143 (2001), no. 3, 571–627. MR 1817645
- Guido Kings, Eisenstein classes, elliptic Soulé elements and the $\ell$-adic elliptic polylogarithm, The Bloch-Kato conjecture for the Riemann zeta function, London Math. Soc. Lecture Note Ser., vol. 418, Cambridge Univ. Press, Cambridge, 2015, pp. 239–296. MR 3497682
- Guido Kings, On $p$-adic interpolation of motivic Eisenstein classes, Elliptic curves, modular forms and Iwasawa theory, Springer Proc. Math. Stat., vol. 188, Springer, Cham, 2016, pp. 335–371. MR 3629656
- Marc Levine, 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., vol. 407, Kluwer Acad. Publ., Dordrecht, 1993, pp. 167–188. MR 1367296
- Masayoshi Nagata, A generalization of the imbedding problem of an abstract variety in a complete variety, J. Math. Kyoto Univ. 3 (1963), 89–102. MR 0158892
- S. Pepin Lehalleur, The motivic t-structure for relative 1-motives, preprint, 2015, arXiv:1512.00266.
- Morihiko Saito, Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24 (1988), no. 6, 849–995 (1989) (French). MR 1000123
- Morihiko Saito, Mixed Hodge modules, Publ. Res. Inst. Math. Sci. 26 (1990), no. 2, 221–333. MR 1047415
- Cohomologie $l$-adique et fonctions $L$, Lecture Notes in Mathematics, Vol. 589, Springer-Verlag, Berlin-New York, 1977 (French). Séminaire de Géometrie Algébrique du Bois-Marie 1965–1966 (SGA 5); Edité par Luc Illusie. MR 0491704
- Vladimir Voevodsky, Triangulated categories of motives over a field, Cycles, transfers, and motivic homology theories, Ann. of Math. Stud., vol. 143, Princeton Univ. Press, Princeton, NJ, 2000, pp. 188–238. MR 1764202
- Jörg Wildeshaus, Realizations of polylogarithms, Lecture Notes in Mathematics, vol. 1650, Springer-Verlag, Berlin, 1997. MR 1482233
- Jörg Wildeshaus, On the interior motive of certain Shimura varieties: the case of Picard surfaces, Manuscripta Math. 148 (2015), no. 3-4, 351–377. MR 3414481
Additional Information
Annette Huber
Affiliation:
Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, 79104 Freiburg, Germany
MR Author ID:
309501
Email:
annette.huber@math.uni-freiburg.de
Guido Kings
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
MR Author ID:
624982
Email:
guido.kings@mathematik.uni-regensburg.de
Received by editor(s):
October 28, 2015
Received by editor(s) in revised form:
December 23, 2017
Published electronically:
April 11, 2018
Additional Notes:
The research of the second author was supported by the DFG through SFB 1085: Higher invariants
Article copyright:
© Copyright 2018
University Press, Inc.