Abstract
The author gives an overview on recent progress in the study of torsion zero cycles. The new methods that have been developped to obtain finiteness results on the torsion in the codimension two Chow group are described in this paper as well as their connection to other well known conjectures in arithmetic geometry (Tate-Conjecture, Beilinson-Conjecture, Bloch-Kato-Conjecture).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bloch, S., Kato, K. [B-K] L-functions and Tamagawa numbers of motives, in: The Grothendieck Festschrift vol. I, Progress Math. 86, Birkhäuser Boston 1990, 333–400.
Consani, K. [Con] Double Complexes and Euler L-Functions, preprint 1998.
Colliot-Thélène, J.-L., Raskind, W. [CT-R1] K2-cohomology and the second Chow group, Math. Ann. 270 (1985), 169–199.
Colliot-Thélène, J.-L., Raskind, W. [CT-R2] Groupes de Chow de codimension 2 des variétés sur un corps de nombres: un théorème de finitude pour la torsion,Inv. Math. 105 (1991), 221–245.
Dee, J. [Dee] Thesis, University of Cambridge, in preparation.
Flach, M. [Fl1] A finiteness theorem for the symmetric square of an elliptic curve, Inv. Math. 109 (1992), 307–327.
Flach, M. [Fl2] Thesis, University of Cambridge 1990.
Han, S.[Han] Thesis, University of Chicago 1998.
Langer A.[L1] Selmer groups and torsion zero cycles on a selfproduct of a semistable elliptic curve, in: Documenta Math. Bd. II (1997), 47–59.
Langer A.[L2] 0-cycles on the elliptic modular surface of level 4, to appear in Tohoku Math. J., 1998.
Langer A.[L3] Local points of motives in semistable reduction, to appear in Compositio Math.
Langer, A.[L4] Zero cycles on Hilbert-Blumenthal surfaces, preprint 1998.
Langer, A., Saito, S. [L-S] Torsion zero cycles on the selfproduct of a modular elliptic curve, Duke Math. J. vol. 85, no. 2 (1996), 315–357.
Langer, A., Raskind, W.[L-R] 0-cycles on the selfproduct of a CM-elliptic curve over ℚ, to appear in J. f. d. R. u. A. Math.
Mildenhall, S.[Mi] Cycles in a product of elliptic curves and a group analogous to the class group, Duke Math. J. 67 (1992), 387–406.
Nekovář, J.[Ne] On p-adic height-pairings, in: Seminaire de Théorie de Nombres 1990–1991 (1993), 127–202.
Otsubo, N. [O] Selmer groups and zero-cycles on the Fermat quartic surface, preprint, University of Tokyo 1998.
Salberger, P.[Sal] Chow groups of codimension two and 1-adic realization of motivic cohomology, in: Séminaires de théorie de nombres, Birkhäuser 1993.
Schneider, P.[S] P-adic points of motives, in: Motives. Proc. Symp. Pure Math. 55 pt 2, American Math. Society, Providence (1994), 225–249.
Sato, K. [Sa] Finiteness of a certain motivic cohomology of varieties: Application to regular maps, preprint 1998.
Saito, S.[SaS] Class field theory for curves over local fields, J. Number Theory 21 (1985), 44–80.
Wiles, A. [W] Modular elliptic curves and Fermat’s last theorem, Annals of Math. 142 (1995), 443–551.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Langer, A. (2000). Finiteness of Torsion in the Codimension-Two Chow Group: An Axiomatic Approach. In: Gordon, B.B., Lewis, J.D., Müller-Stach, S., Saito, S., Yui, N. (eds) The Arithmetic and Geometry of Algebraic Cycles. NATO Science Series, vol 548. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4098-0_9
Download citation
DOI: https://doi.org/10.1007/978-94-011-4098-0_9
Publisher Name: Springer, Dordrecht
Print ISBN: 978-0-7923-6194-7
Online ISBN: 978-94-011-4098-0
eBook Packages: Springer Book Archive