Skip to main content Accessibility help
×
×
Home

ON THE RESIDUE OF EISENSTEIN CLASSES OF SIEGEL VARIETIES

  • FRANCESCO LEMMA (a1)

Abstract

Eisenstein classes of Siegel varieties are motivic cohomology classes defined as pull-backs by torsion sections of the polylogarithm prosheaf on the universal abelian scheme. By reduction to the Hilbert–Blumenthal case, we prove that the Betti realization of these classes on Siegel varieties of arbitrary genus have non-trivial residue on zero-dimensional strata of the Baily–Borel–Satake compactification. A direct corollary is the non-vanishing of a higher regulator map.

Copyright

References

Hide All
1. Bannai, K., Kings, G., p-adic elliptic polylogarithm, p-adic Eisenstein series and Katz measure, Am. J. Math. 132 (6) (2010), 16091654.
2. Beilinson, A. and Levin, A., The elliptic polylogarithm, in Motives: Proceedings of Symposia in Pure Mathematics (Jannsen, U., Editor), vol. 55, Part 2 (1994), 123190.
3. Beilinson, A. and Levin, A., The elliptic polylogarithm, preprint version of [2].
4. Birkenhake, C. and Lange, H., Complex abelian varieties, 2nd edition, Grundlehren der mathematischen Wissenschaften, vol. 302 (Springer, Berlin, 2004), xii+635.
5. Blottière, D., Réalisation de Hodge du polylogarithme d'un schéma abélien, J. Inst. Math. Jussieu 8 (1) (2009), 138.
6. Blottière, D., Les classes d'Eisenstein des variétés de Hilbert-Blumenthal, IMRN 17 (2009), 32363263.
7. Borel, A., Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem, J. Differ. Geom. 6 (1972), 543560.
8. Burgos, J. I. and Wildeshaus, J., Hodge modules on Shimura varieties and their higher direct images in the Baily-Borel-Satake compactification, Ann. Sci. Ecole Norm. Sup. 37 (3) (2004), 363413.
9. Deninger, C. and Murre, J., Motivic decomposition of abelian schemes and the Fourier transform, J. Reine Angew. Math. 422 (1991), 201219.
10. Kashiwara, M. and Schapira, P., Sheaves on manifolds, A Series of Comprehensive Studies in Mathematics, vol. 292 (Springer-Verlag, Berlin, 1994), x+512.
11. Kings, G., K-theory elements of the polylogarithm of abelian schemes, J. Reine Angew. Math. 517 (1999), 103116.
12. Kings, G., The Tamagawa number conjecture for CM elliptic curves, Invent. Math. 143 (3) (2001), 571627.
13. Laumon, G., Fonctions zêtas des variétés de Siegel de dimension 3, Astérisque 302 (2005), 166.
14. Morel, S., Complexes pondérés sur les compactifications de Baily-Borel-Satake: le cas des variétés de Siegel, J. Am. Math. Soc. 21 (1) (2008), 2361.
15. Pink, R., Arithmetical compactifications of mixed Shimura varieties, PhD Thesis, (Bonn, 1990), available at https://people.math.ethz.ch/~pink/dissertation.html.
16. Van der Geer, G., Hilbert modular surfaces, Ergebnisse der Mathematik und ihrer Grezgebiete 3, vol. 16 (Springer-Verlag, Berlin, 1988), x+291.
17. Wildeshaus, J., Realization of polylogarithms, LNM, vol. 1650 (Springer, Berlin, 1995), xii+343.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed