Nondivisible cycles on products of very general Abelian varieties
Author:
Humberto A. Diaz
Journal:
J. Algebraic Geom.
DOI:
https://doi.org/10.1090/jag/775
Published electronically:
December 14, 2020
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Abstract | References | Additional Information
Abstract: In this paper, we give a recipe for producing infinitely many nondivisible codimension 
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 
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.
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-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. 
for the triple product of a general elliptic curve, Asian J. Math. 4 (2000), no. 4, 987-996.
Additional Information
Humberto A. Diaz
Affiliation:
Department of Mathematics, Washington University, St. Louis, Missouri 63130
Email:
humberto@wustl.edu
DOI:
https://doi.org/10.1090/jag/775
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
