Standard conjecture of K眉nneth type with torsion coefficients

-
Junecue Suh , UC Santa Cruz

A.Venkatesh asked us the question, in the context of torsion automorphic forms: Does the Standard Conjecture (of Grothendieck's) of K眉nneth type hold with mod p coefficients? We first review the geometric and number-theoretic contexts in which this question becomes interesting, and provide answers: No in general (even for Shimura varieties) but yes in special cases.听