Vol. 11, No. 7, 2017

Download this article
Download this article For screen
For printing
Recent Issues

Volume 12
Issue 5, 1001–1309
Issue 4, 751–999
Issue 3, 493–750
Issue 2, 227–492
Issue 1, 1–225

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Subscriptions
Editorial Board
Editors' Addresses
Editors' Interests
Scientific Advantages
Submission Guidelines
Submission Form
Editorial Login
Author Index
To Appear
 
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Standard conjecture of Künneth type with torsion coefficients

Junecue Suh

Vol. 11 (2017), No. 7, 1573–1596
Abstract

A. Venkatesh raised the following question, in the context of torsion automorphic forms: can the mod p analogue of Grothendieck’s standard conjecture of Künneth type be true (especially for compact Shimura varieties)? In the first theorem of this article, by using a topological obstruction involving Bockstein, we show that the answer is in the negative and exhibit various counterexamples, including compact Shimura varieties.

It remains an open geometric question whether the conjecture can fail for varieties with torsion-free integral cohomology. Turning to the case of abelian varieties, we give upper bounds (in p) for possible failures, using endomorphisms, the Hodge–Lefschetz operators, and invariant theory.

The Schottky problem enters into consideration, and we find that, for the Jacobians of curves, the question of Venkatesh has an affirmative answer for every prime number p.

Keywords
standard conjectures, Künneth decomposition, algebraic cycles
Mathematical Subject Classification 2010
Primary: 14C25
Secondary: 14H40, 55S05
Milestones
Received: 17 July 2016
Revised: 17 April 2017
Accepted: 5 June 2017
Published: 7 September 2017
Authors
Junecue Suh
Mathematics Department
University of California, Santa Cruz
Santa Cruz, CA
United States