Standard conjecture of Künneth type with torsion coefficients

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

Vol. 11, No. 7, 2017

Junecue Suh

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors