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
.
Keywords
standard conjectures, Künneth decomposition, algebraic
cycles
