A. Venkatesh raised the following question, in the context of torsion automorphic forms: can the mod 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 ) 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 .
"Standard conjecture of Künneth type with torsion coefficients." Algebra Number Theory 11 (7) 1573 - 1596, 2017. https://doi.org/10.2140/ant.2017.11.1573