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Algebraic cycles on varieties over finite fields
Post-edited
Authors : Pirutka, Alena (Author of the conference)
CIRM (Publisher )
14C15 - (Equivariant) Chow groups and rings; motives
14C25 - Algebraic cycles
14G15 - Finite ground fields
14H05 - Algebraic functions; function fields
14J70 - Algebraic hypersurfaces
CIRM (Publisher )
Abstract :
Let $X$ be a projective variety over a field $k$. Chow groups are defined as the quotient of a free group generated by irreducible subvarieties (of fixed dimension) by some equivalence relation (called rational equivalence). These groups carry many information on $X$ but are in general very difficult to study. On the other hand, one can associate to $X$ several cohomology groups which are "linear" objects and hence are rather simple to understand. One then construct maps called "cycle class maps" from Chow groups to several cohomological theories.
In this talk, we focus on the case of a variety $X$ over a finite field. In this case, Tate conjecture claims the surjectivity of the cycle class map with rational coefficients; this conjecture is still widely open. In case of integral coefficients, we speak about the integral version of the conjecture and we know several counterexamples for the surjectivity. In this talk, we present a survey of some well-known results on this subject and discuss other properties of algebraic cycles which are either proved or expected to be true. We also discuss several involved methods.
MSC Codes : In this talk, we focus on the case of a variety $X$ over a finite field. In this case, Tate conjecture claims the surjectivity of the cycle class map with rational coefficients; this conjecture is still widely open. In case of integral coefficients, we speak about the integral version of the conjecture and we know several counterexamples for the surjectivity. In this talk, we present a survey of some well-known results on this subject and discuss other properties of algebraic cycles which are either proved or expected to be true. We also discuss several involved methods.
14C15 - (Equivariant) Chow groups and rings; motives
14C25 - Algebraic cycles
14G15 - Finite ground fields
14H05 - Algebraic functions; function fields
14J70 - Algebraic hypersurfaces
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 04/06/2015 Conference Date : 18/05/2015 Subseries : Research talks arXiv category : Algebraic Geometry ; Number Theory Mathematical Area(s) : Algebraic & Complex Geometry Format : QuickTime (.mov) Video Time : 00:48:58 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2015-05-18_Pirutka.mp4 
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Information on the Event
Event Title : AGCT - Arithmetic, Geometry, Cryptography and Coding Theory / AGCT - Arithmétique, géométrie, cryptographie et théorie des codesEvent Organizers : Bassa, Alp ; Couvreur, Alain ; Kohel, David Dates : 18/05/15 - 22/05/15 Event Year : 2015 Event URL : http://conferences.cirm-math.fr/1193.html
Citation Data
DOI : 10.24350/CIRM.V.18764303Cite this video as: Pirutka, Alena (2015). Algebraic cycles on varieties over finite fields. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18764303 URI : http://dx.doi.org/10.24350/CIRM.V.18764303 |
Bibliography
- Atiyah, Mi.F., & Hirzebruch, F. (1962). Analytic cycles on complex manifolds. Topology, 1, 25-45 - http://dx.doi.org/10.1016/0040-9383(62)90094-0
- Charles, F., & Pirutka, A. (2015). La conjecture de Tate entière pour les cubiques de dimension quatre. Compositio Mathematica, 151(2), 253-264 - http://dx.doi.org/10.1112/S0010437X14007386
- Colliot-Thélène, J.-L., Sansuc, J.-J., Soulé, C. (1983). Torsion dans le groupe de Chow de codimension deux. Duke Mathematical Journal, 50, 763-801 - http://dx.doi.org/10.1215/S0012-7094-83-05038-X
- Pirutka, A. (2011). Sur le groupe de Chow de codimension deux des variétés sur les corps finis. Algebra & Number Theory, 5(6), 803-817 - http://dx.doi.org/10.2140/ant.2011.5.803
- Tate, J. (1994). Conjectures on algebraic cycles in $\ell$-adic cohomology. In U. Jannsen, S. Kleiman, & J.-P. Serre (Eds.), Motives. [1] (pp. 71-83). Providence, RI: American Mathematical Society. (Proceedings of Symposia in Pure Mathematics. 55-1) - http://www.ams.org/bookstore-getitem?item=PSPUM-55-1-S
