Nondivisible cycles on products of very general Abelian varieties
Author:
Humberto A. Diaz
Journal:
J. Algebraic Geom. 30 (2021), 407-432
DOI:
https://doi.org/10.1090/jag/775
Published electronically:
December 14, 2020
MathSciNet review:
4283547
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Abstract |
References |
Additional Information
Abstract: In this paper, we give a recipe for producing infinitely many nondivisible codimension $2$ cycles on a product of two or more very general Abelian varieties. In the process, we introduce the notion of “field of definition” for cycles in the Chow group modulo (a power of) a prime. We show that for a quite general class of codimension $2$ cycles, that we call “primitive cycles”, the field of definition is a ramified extension of the function field of a modular variety. This ramification allows us to use Nori’s isogeny method (modified by Totaro) to produce infinitely many nondivisible cycles. As an application, we prove the Chow group modulo a prime of a product of three or more very general elliptic curves is infinite, generalizing work of Schoen.
References
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References
- S. Bloch, Torsion algebraic cycles and a theorem of Roitman, Compositio Math. 39 (1979), no. 1, 107–127. MR 539002
- Spencer Bloch, Algebraic cycles and values of $L$-functions, J. Reine Angew. Math. 350 (1984), 94–108. MR 743535, DOI 10.1515/crll.1984.350.94
- Spencer Bloch and Hélène Esnault, The coniveau filtration and non-divisibility for algebraic cycles, Math. Ann. 304 (1996), no. 2, 303–314. MR 1371769, DOI 10.1007/BF01446296
- S. Bloch and V. Srinivas, Remarks on correspondences and algebraic cycles, Amer. J. Math. 105 (1983), no. 5, 1235–1253. MR 714776, DOI 10.2307/2374341
- Joe Buhler, Chad Schoen, and Jaap Top, Cycles, $L$-functions and triple products of elliptic curves, J. Reine Angew. Math. 492 (1997), 93–133. MR 1488066, DOI 10.1515/crll.1997.492.93
- Herbert Clemens, Homological equivalence, modulo algebraic equivalence, is not finitely generated, Inst. Hautes Études Sci. Publ. Math. 58 (1983), 19–38 (1984). MR 720930
- Richard M. Hain, Torelli groups and geometry of moduli spaces of curves, Current topics in complex algebraic geometry (Berkeley, CA, 1992/93) Math. Sci. Res. Inst. Publ., vol. 28, Cambridge Univ. Press, Cambridge, 1995, pp. 97–143. MR 1397061
- Uwe Jannsen, Continuous étale cohomology, Math. Ann. 280 (1988), no. 2, 207–245. MR 929536, DOI 10.1007/BF01456052
- A. S. Merkur′ev and A. A. Suslin, $K$-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Dokl. Akad. Nauk SSSR 264 (1982), no. 3, 555–559 (Russian). MR 659762
- Madhav V. Nori, Cycles on the generic abelian threefold, Proc. Indian Acad. Sci. Math. Sci. 99 (1989), no. 3, 191–196. MR 1032704, DOI 10.1007/BF02864390
- Peter Norman and Frans Oort, Moduli of abelian varieties, Ann. of Math. (2) 112 (1980), no. 3, 413–439. MR 595202, DOI 10.2307/1971152
- Andreas Rosenschon and V. Srinivas, The Griffiths group of the generic abelian 3-fold, Cycles, motives and Shimura varieties, Tata Inst. Fund. Res. Stud. Math., vol. 21, Tata Inst. Fund. Res., Mumbai, 2010, pp. 449–467. MR 2906032
- Chad Schoen, On certain exterior product maps of Chow groups, Math. Res. Lett. 7 (2000), no. 2-3, 177–194. MR 1764315, DOI 10.4310/MRL.2000.v7.n2.a4
- Chad Schoen, On the image of the $l$-adic Abel-Jacobi map for a variety over the algebraic closure of a finite field, J. Amer. Math. Soc. 12 (1999), no. 3, 795–838. MR 1672878, DOI 10.1090/S0894-0347-99-00303-3
- Chad Schoen, The Chow group modulo $l$ for the triple product of a general elliptic curve, Asian J. Math. 4 (2000), no. 4, 987–996. MR 1870669, DOI 10.4310/AJM.2000.v4.n4.a15
- Burt Totaro, Complex varieties with infinite Chow groups modulo 2, Ann. of Math. (2) 183 (2016), no. 1, 363–375. MR 3432586, DOI 10.4007/annals.2016.183.1.7
- Claire Voisin, The generalized Hodge and Bloch conjectures are equivalent for general complete intersections, Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), no. 3, 449–475 (2013) (English, with English and French summaries). MR 3099982, DOI 10.24033/asens.2193
- Dave Witte Morris, Introduction to arithmetic groups, Deductive Press, 2015. MR 3307755
Additional Information
Humberto A. Diaz
Affiliation:
Department of Mathematics, Washington University, St. Louis, Missouri 63130
MR Author ID:
1174879
Email:
humberto@wustl.edu
Received by editor(s):
December 6, 2018
Received by editor(s) in revised form:
July 24, 2019, September 10, 2019, December 6, 2019, March 9, 2020, July 13, 2020, August 10, 2020, August 17, 2020, September 30, 2020, and October 5, 2020
Published electronically:
December 14, 2020
Article copyright:
© Copyright 2020
University Press, Inc.
